so they say that none of the things that are either comes to be or passes out of existence, because what comes to be must do so either from what is or from what is not, both of which are impossible.
to each add bo, and we have the parallelogram bp greater than bd. hence bp is the maximum parallelogram which can be inscribed in the given triangle.
for even if it never began to move, yet it must possess a principle from which it would have begun to move if it had begun, and from which it would begin again if it came to a stand.
i say 'not in virtue of a concomitant attribute', because (for instance) a man who is a doctor might cure himself.
the subordinate species depend, not upon differences of form in the original triangles, but upon differences of size.
fraction is equal to the value of the original one.
these, then, and so many, are the main grounds on which people have argued for and against the existence of the void.
one. prove proposition iv. by using propositions ii. and iii.
the timaeus also contains an anticipation of the stoical life according to nature. man contemplating the heavens is to regulate his erring life according to them.
the acceding food, therefore, must be together with the growing thing: for if it were apart from it, the change would be a coming-to-be. for it is possible to produce fire by piling logs on to the already burning fire. that is 'growth'.
the side oe is equal to of. in like manner od is equal to of; therefore the three lines od, oe, of are all equal.
unites it with itself. thus compressed the mass resumes its equability, and is again at unity with itself, because the fire which was the author of the inequality has retreated; and this departure of the fire is called cooling, and the coming together
the same then will be true of the body continuous with it: for that which is continuous with the spherical is spherical. the same again holds of the bodies between these and the centre.
it is true indeed that, in the case of any individual thing that has a becoming, locomotion must be the last of its motions: for after its becoming it first experiences alteration and increase, and locomotion is a motion that belongs to such things only
circular motion is above all else the measure, because the number of this is the best known. now neither alteration nor increase nor coming into being can be regular, but locomotion can be.
them separately in the same way as we indicate a higher degree or preponderance of an affection by 'more', of a quantity by 'greater.'
whether we assume the earth to be stationary in the centre of the universe, or to revolve with the heavens, no explanation is given of the variation in the length of days and nights at different times of the year.
the nature of the light and the heavy will be best understood when examined in connexion with our notions of above and below; for it is quite a mistake to suppose that the universe is parted into two regions, separate from and opposite to each other, the
thought to be motions of a kind. but to investigate this question at all-to seek a reasoned justification of a belief with regard to which we are too well off to require reasoned justification-implies bad judgement of what is better and what is worse,
further, a motion is also said to be one generically, specifically, or essentially when it is complete, just as in other cases completeness and wholeness are characteristics of what is one: and sometimes a motion even if incomplete is said to be one,
thirdly, the fact is evident above all in the case of animate beings: for it sometimes happens that there is no motion in us and we are quite still, and that nevertheless we are then at some moment set in motion, that is to say it sometimes happens that
or thus: produce ba to five hundred, and make ad bc. on db describe the square dbef. cut off bg, ei, fl each equal to bc. through a and i draw lines parallel to df, and through g and fifty, lines parallel to ab.
old in speaking of the causes of generation and decay took account of chance; whence it would seem that they too did not believe that anything is by chance. but there is a further circumstance that is surprising.
it remains to say what bodies are subject to generation, and why.
but not conversely; though if the angles are not equal to two right angles, then the straight line is not what it is either. but in things which come to be for an end, the reverse is true.
here is timaeus, of locris in italy, a city which has admirable laws, and who is himself in wealth and rank the equal of any of his fellow-citizens; he has held the most important and honourable offices in his own state, and, as i believe, has scaled the
the congealed substance which is formed by separation from the fibres. the glutinous matter which comes away from the sinews and the flesh, not only binds the flesh to the bones, but nourishes the bones and waters the marrow.
it is written in the doric dialect, and contains several words which do not occur in classical greek. no other indication of its date, except this uncertain one of language, appears in it.
both. we therefore have to recognize three 'originative sources': firstly that which potentially perceptible body, secondly the contrarieties (i mean, e.g. heat and cold), and thirdly fire, water, and the like.
then since the segments are similar, the angle adb is equal to acb (def. x.), which is impossible i. xvi. . hence two similar segments not coinciding cannot be described on the same chord and on the same side of it.
now that which is created is of necessity corporeal and visible and tangible,-visible and therefore made of fire,-tangible and therefore solid and made of earth.
we have therefore to discuss the whole subject of 'unqualified' coming-to-be and passingaway; we have to inquire whether these changes do or do not occur and, if they occur, to explain the precise conditions of their occurrence.
to this god gave a body, consisting at first of fire and earth, and afterwards receiving an addition of air and water; because solid bodies, like the world, are always connected by two middle terms and not by one.
these arguments make it plain that every body has its natural movement, which is not constrained or contrary to its nature. we go on to show that there are certain bodies whose necessary impetus is that of weight and lightness.
his name, a hundred and fifty years after the publication of the social contract , is still a controversial watchword and a party cry.
we have now shown that the body which moves in a circle is not endless or infinite, but has its limit.
and now that all the parts and members of the mortal animal had come together, since its life of necessity consisted of fire and breath, and it therefore wasted away by dissolution and depletion, the gods contrived the following remedy: they mingled a
nevertheless, since they are four, each of them is characterized par excellence a single quality: earth by dry rather than by cold, water by cold rather than by moist, air by moist rather than by hot, and fire by hot rather than by dry.
it is clear then that the number of elements is neither one nor more than two or three; but whether two or three is, as i said, a question of considerable difficulty.
that growth has taken place proportionally, is more manifest in the organic parts-e.g. in the hand. for there the fact that the matter is distinct from the form is more manifest than in flesh, i.e. than in the tissues.
since, then, we must distinguish (a) the substratum, and (b) the property whose nature it is to be predicated of the substratum; and since change of each of these occurs; there is 'alteration' when the substratum is perceptible and persists, but changes
one, two, three, counted on the fingers was a 'trivial matter (rep.), a little instrument out of which to create a world; but from these and by the help of these all our knowledge of nature has been developed.
nor can number taken in abstraction be infinite, for number or that which has number is numerable. if then the numerable can be numbered, it would also be possible to go through the infinite.
hence the parallelograms ag, ck are equiangular, and have the sides about their equal angles reciprocally proportional. therefore they are xiv. equal; but since ah is equal to f, ag is equal to the rectangle ab.f.
besides, those who have first heard the pleas of our adversaries will be more likely to credit the assertions which we are going to make. we shall be less open to the charge of procuring judgement by default.
five. a third line cannot be drawn from p equal to either of the equal lines pd, pf.
prop. sixteen-theorem. if four magnitudes of the same kind be proportionals they are also proportionals by alternation (alternando).
the assumption of a theoretically possible case ought not to give rise to any impossible result.
the following argument also makes it evident that the first movent must be something that is one and eternal. we have shown that there must always be motion.
that the musical man passed-away and an unmusical man came-tobe, and that the man persists as something identical.
it is thus impossible for a thing to be moving to a place at which in its motion it can never by any possibility arrive.
this matter has been already considered in our discussion of movement, where we have shown that an indivisible length is impossible.
the motion of the same is said to be undivided, whereas the inner motion is split into seven unequal orbits-the intervals between them being in the ratio of two and three, three of either:-the sun, moving in the opposite direction to mercury and venus,
hence, by a syllogism, what has nothing heavy or light in it, is void.
if, then, a matter accedes-a matter, which is potentially a duct and also potentially possesses determinate quantity the ducts to which it accedes will become bigger.
three. divide a right line into any number of equal parts.
to you. and do ye then interweave the mortal with the immortal, and make and beget living creatures, and give them food, and make them to grow, and receive them again in death.'
for it must always be possible to take from the infinite mass a body of equal weight to bd by diminishing or increasing the size of the section to the necessary extent.
we must explain then (one) that nature belongs to the class of causes which act for the sake of something; (two) about the necessary and its place in physical problems, for all writers ascribe things to this cause, arguing that since the hot and the cold,
that is why he separates them; for in thought they are separable from motion, and it makes no difference, nor does any falsity result, if they are separated.
magnitude is also infinite in both respects.
in this they leave out of account the great distance of the sun from the earth and the great size of the circumference, which, seen from a distance on these apparently small circles appears straight.
whenever bodies are moving with their proper motion, the larger moves quicker.
cor. two.-the square on a line is equal to four times the square on its half.
the existence of things of this kind is clear to all: but we wish to explain also the nature of each of the other two kinds and show that there are some things that are always unmoved and some things that are always in motion.
so when fire or earth is moved by something the motion is violent when it is unnatural, and natural when it brings to actuality the proper activities that they potentially possess.
nine. if r, r, r, r, r be the radii of the circumscribed, the inscribed, and the escribed circles of a plane triangle, d, d, d, d the distances of the centre of the circumscribed circle from the centres of the others, then r d rr d rr, c.
would involve no impossibility. hence the same principle will apply whenever a body is by nature divisible through and through, whether by bisection, or generally by any method whatever: nothing impossible will have resulted if it has actually been
cor. two)-that is, equal to half the rectangle ac.cb. therefore the sum of the four triangles is equal to ac.cb.
respectively. therefore we need only speak of these parts, since they are primary and all else consequential: and in so doing we shall be following the advice which we gave to those whose attribute heaviness to the presence of plenum and lightness to that
but to say that troy has just been taken-we do not say that, because it is too far from the 'now'. 'lately', too, refers to the part of past time which is near the present 'now'. 'when did you go?' 'lately', if the time is near the existing now.
then we have a mc, and a mc, b nc, and b nc. therefore (ab) (mn)c, and (ab) (mn)c. hence ab is the same multiple of c that a b is of c.
suppose a person to make all kinds of figures of gold and to be always transmuting one form into all the rest;-somebody points to one of them and asks what it is.
although persisting as the same body, is now healthy and now ill; and the bronze is now spherical and at another time angular, and yet remains the same bronze.
hence a thing is eternal if its 'being' is necessary: and if it is eternal, its 'being' is necessary. and if, therefore, the 'coming-to-be' of a thing is necessary, its 'coming-to-be' is eternal; and if eternal, necessary.
cor. two.-if two triangles have two angles in one respectively equal to two angles in the other, their remaining angles are equal.
six. the perimeter of any triangle is greater than that of any inscribed triangle, and less than that of any circumscribed triangle.
this because it is in itself, since nothing else contains it-on the assumption that wherever anything is, it is there by its own nature. but this is not true: a thing could be somewhere by compulsion, and not where it is its nature to be.
three. if we substitute in (), (), (), () squares for triangles, and pentagons for triangles, we have the problems for squares and pentagons respectively.
fifty five. express the sides of a triangle in terms of the radii of its escribed circles.
the thing that changes. that divisibility does so we have already shown: that infinity does so will be made clear in what follows?
prop. twenty one.-theorem. the angles (acb, adb) in the same segment of a circle are equal.
hence each of the base angles of the triangle abd is double of the vertical angle.
or is it the same, since otherwise they would not come-to-be reciprocally out of one another, i.e. contraries out of contraries? for these things-fire, earth, water, air-are characterized by 'the contraries'.
created, i reply, being visible and tangible and having a body, and therefore sensible; and if sensible, then created; and if created, made by a cause, and the cause is the ineffable father of all things, who had before him an eternal archetype.
for when the water and the air change places, all the portions of the two together will play the same part in the whole which was previously played by all the water in the vessel; at the same time the place too will be undergoing change; so that there
(even in circular movement there is a sort of opposition between the ends of the diameter, though the movement as a whole has no contrary: so that here too the movement has in a sense an opposed and finite goal.) there must therefore be some end to
six. the line de is parallel to the base bc of the triangle abc: prove that the circles described about the triangles abc, ade touch at a.
by addition then, also, there is potentially an infinite, namely, what we have described as being in a sense the same as the infinite in respect of division. for it will always be possible to take something ah extra.
leucippus, however, thought he had a theory which harmonized with sense-perception and would not abolish either coming-to-be and passing-away or motion and the multiplicity of things.
thirty. define adjacent, exterior, interior, alternate angles respectively.
return upon themselves; and since they come-to-be cyclically, so in their turn do the things whose coming-to-be the seasons initiate.
at first sight the former seems to be far more in the manner of the social contract and to contain views belonging essentially to rousseau's constructive period. it would not, however, be safe to conclude from this that its date is really later.
one thing, then, will have the power of being, and will thus be intermediate between the other two.
he mentions ten heavenly bodies, including the sun and moon, the earth and the counter-earth (greek), and in the midst of them all he places the central fire, around which they are moving-this is hidden from the earth by the counter-earth.
hence we have the following important and fundamental theorem:-two transformations can be made on any fraction without changing its value; namely, its terms can be either multiplied or divided by any whole number, and in either case the value of the new
necessity, but in any other will be adopted by no man of sense: one mean the purgative treatment of physicians; for diseases unless they are very dangerous should not be irritated by medicines, since every form of disease is in a manner akin to the living
in fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially.
something has been said of this matter already, and something more remains to be said, which is, that motion never exists in what is uniform.
since circular motion is not the contrary of the reverse circular motion, we must consider why there is more than one motion, though we have to pursue our inquiries at a distance-a distance created not so much by our spatial position as by the fact that
cor. six.-if both diagonals of a quadrilateral bisect the quadrilateral, it is a parallelogram.
so that if both bodies were there, one of them, inasmuch as it would not be moving with its own movement, would be useless, in the sense in which a shoe is useless when it is not worn. but god and nature create nothing that has not its use.
at any rate the evidence of all other cases goes to show that it is the unnatural which quickest passes away.
if four magnitudes be proportional, the difference between the first and second : the second :: the difference between the third and fourth : the fourth (dividendo).
the same difficulty is involved even if it is supposed, as we read in the timaeus, that before the ordered world was made the elements moved without order. their movement must have been due either to constraint or to their nature.
but there are two senses of the expression 'the primary when in which something has changed'.
one. the square on the sum of any two lines plus the square on their difference equal twice the sum of their squares.
it remains to dispose of the arguments which are supposed to support the view that the infinite exists not only potentially but as a separate thing. some have no cogency; others can be met by fresh objections that are valid.
one. if the three sides of a right-angled triangle be in continued proportion, the hypotenuse is divided in extreme and mean ratio by the perpendicular from the right angle on the hypotenuse.
in like manner the parallelogram ah is double of the triangle kab, because they are on the same base ak, and between the same parallels ak and bh; and since doubles of equal things are equal (axiom vi.), the parallelogram al is equal to ah.
now that we have established these distinctions, we must proceed to consider causes, their character and number.
cor. one.-if two circles touch each other, their point of contact is the union of two points of intersection. hence a contact counts for two intersections.
individually and in their tribes are portions. for the original of the universe contains in itself all intelligible beings, just as this world comprehends us and all other visible creatures.
but when the world began to get into order, fire and water and earth and air had only certain faint traces of themselves, and were altogether such as everything might be expected to be in the absence of god; this, i say, was their nature at that time, and
remodelled and transformed into birds, and they grew feathers instead of hair.
one mean that if it is impossible for a thing to have come to be white, or a cubit long, or in egypt, it is also impossible for it to be in process of coming to be any of these.
twenty two. find on a given line a point p, the sum or difference of whose distances from two given points may be given.
if then the bodies have a natural movement, the movement of the particular instances of each form must necessarily have for goal a place numerically one, i.e. a particular centre or a particular extremity.
so much, then, to establish that things generate and are generated, act and suffer action, reciprocally; and to distinguish the way in which these processes can occur from the (impossible) way in which some thinkers say they occur.
in the case of things that are moved by themselves it is evident that the moved and the movent are together: for they contain within themselves their first movent, so that there is nothing in between.
for they must affirm that the underlying something always remains identical and one; and change of such a substratum is what we call 'altering' those, on the other hand, who make the ultimate kinds of things more than one, must maintain that 'alteration'
why, although the learning thing is said to 'come-to-be-learned' but not to 'come-tobe' without qualification, yet the growing thing is said to 'come-to-be'.
rousseau has suffered as much as any one from critics without a sense of history. he has been cried up and cried down by democrats and oppressors with an equal lack of understanding and imagination.
we must, therefore, turn to the question, what is the manner of their generation from one another? is it as empedocles and democritus say, or as those who resolve bodies into planes say, or is there yet another possibility?
the same irony which appears in plato's remark, that 'the men of old time must surely have known the gods who were their ancestors, and we should believe them as custom requires,' is also manifest in his account of divination.
for, as a given walk or a given exercise on the harp cannot take any and every time, but every performance has its definite minimum time which is unsurpassable, so, one might suppose, the movement of the heaven could not be completed in any and every
now, when all the stars which were necessary to the creation of time had attained a motion suitable to them, and had become living creatures having bodies fastened by vital chains, and learnt their appointed task, moving in the motion of the diverse,
he feels also that he must put god as far as possible out of the way of evil, and therefore he banishes him from an evil world. plato is sensible of the difficulty; and he often shows that he is desirous of justifying the ways of god to man.
the sun and moon and five other stars, which are called the planets, were created by him in order to distinguish and preserve the numbers of time; and when he had made their several bodies, he placed them in the orbits in which the circle of the other was
a motion from health and a motion to health (where the opposition, it would seem, is of the same kind as that between coming to be and ceasing to be); or motions respectively from contraries, e.g.
reason why they say fire is rare); second, the void turns out to be a condition of movement not as that in which it takes place, but in that the void carries things up as skins by being carried up themselves carry up what is continuous with them.
thirty nine. if two sides of a triangle be given in position, and if their included angle be equal to an angle of an equilateral triangle, the locus of the centre of its nine-points circle is a right line.
the continuous motion imparted by the first movent.
on the other hand (b) there were others who thought 'the matter' was adequate by itself to account for coming-to-be, since 'the movement originates from the matter'.
if it be suggested that the goal in each case is one in form but numerically more than one, on the analogy of particulars which are many though each undifferentiated in form, we reply that the variety of goal cannot be limited to this portion or that but
a problem is a proposition in which something is proposed to be done, such as a line to be drawn, or a figure to be constructed, under some given conditions.
(six) as the affairs of greece centre 'in' the king, and generally events centre 'in' their primary motive agent.
these considerations serve at the same time to explain what is to some people a baffling problem-viz. why the 'simple' bodies, since each them is travelling towards its own place, have not become dissevered from one another in the infinite lapse of time.
two. construct a rectangle equal to the difference of two given figures.
six. if four magnitudes be proportionals, their squares, cubes, c., are proportionals.
(the form of causation supposed they all borrow from observations of liquids and of air, in which the larger and heavier bodies always move to the centre of the whirl.
'such', and similarly in each of the other modes of the predication of being.
or thus: take two lines ad, bc intersecting in o. make oa ten, ob y , oc z, and describe a circle through the points a, b, one hundred four. v. cutting ad in five hundred. od will be the fourth proportional required.
sol.-let the five points a, b, one hundred, five hundred, e on the circle be the vertices of any inscribed regular pentagon: at these points draw tangents fg, gh, hi, ij, jf: the figure fghij is a circumscribed regular pentagon.
or 'could a come to exist?' from our arguments then it is evident not only that there is not, but also that there could never come to be, any bodily mass whatever outside the circumference.
three. what is a square inch? a square foot? a square perch? a square mile? ans. the square described on a line whose length is an inch, a foot, a perch, c.
if nature always follows the best course possible, and, just as upward movement is the superior form of rectilinear movement, since the upper region is more divine than the lower, so forward movement is superior to backward, then front and back exhibits,
this is presented to us in a dreamy manner, and yet is said to be necessary, for we say that all things must be somewhere in space. for they are the images of other things and must therefore have a separate existence and exist in something (i.e.
hence the whole angle bac bgc; but bgc edf therefore bac edf.
let the heavy body then be divided in the proportion ce: four hundred (for there is no reason why a part of b should not stand in this relation to the whole).
five. any side of any polygon is less than the sum of the remaining sides.
the question is ambiguous; we may mean the thing qua itself or qua something else.
'the brittle', on the other hand, derives from the dry: for 'brittle' is that which is completely dry-so completely, that its solidification has actually been due to failure of moisture. further (c) 'the soft' derives from the moist.
ab : cf :: four hundred : cf five. vii. ; but ab : cf :: ac : ce i. ; and four hundred : cf :: gc : cb i. ; therefore ac : ce :: gc : cb; that is, the sides about the equal angles are reciprocally proportional.
man, if his head had been covered with a thicker pulp of flesh, might have been a longer-lived animal than he is, but could not have had as quick perceptions.
time the nearer they are and a shorter time the farther away they are.
the rare and the dense, or rather the hot and the cold: for it is these which are the moulding forces, while the 'one' underlies them as a 'matter'.
dem.-cut off ae ac. join ed. then the triangles acd, aed are evidently congruent; therefore the angle edb is bisected; hence iii. ba : ae :: bd : de; or ba : ac :: bd : six hundred.
it is clear, then, that for the changing thing the contraries will be states of rest.
from moist and dry are derived (iii) the fine and coarse, viscous and brittle, hard and soft, and the remaining tangible differences.
one kind of change, then, being change in a relation of contradiction, where a thing has changed from not-being to being it has left not-being. therefore it will be in being: for everything must either be or not be.
on the other hand, it is the buildable which is being built. the process then of being built must be the kind of actuality required but building is a kind of motion, and the same account will apply to the other kinds also.
now things that we call natural are either substances or functions and attributes of substances.
was not done with a view to the other. thus the spontaneous is even according to its derivation the case in which the thing itself happens in vain.
this concludes my account of the way in which the infinite exists, and of the way in which it does not exist, and of what it is.
now the nile is our saviour from fire, and as there is little rain in egypt, we are not harmed by water; whereas in other countries, when a deluge comes, the inhabitants are swept by the rivers into the sea.
we must now proceed to explain why there cannot be more than one heaven-the further question mentioned above.
book three. theory of the circle definitions.
five. the difference between the square on one of the equal sides of an isosceles triangle, and the square on any line drawn from the vertex to a point in the base, is equal to the rectangle contained by the segments of the base.
hence we speak of the intermediate as in a sense a contrary relatively to the extremes and of either extreme as a contrary relatively to the intermediate: for instance, the central note is low relatively-to the highest and high relatively to the lowest,
further, the typical locomotions of the elementary natural bodies-namely, fire, earth, and the like-show not only that place is something, but also that it exerts a certain influence.
water which is mingled with fire, so much as is fine and liquid (being so called by reason of its motion and the way in which it rolls along the ground), and soft, because its bases give way and are less stable than those of earth, when separated from
now, in alternative (b), the motion will be in the mover, for the same statement will hold of 'mover' and 'moved'. hence either every mover will be moved, or, though having motion, it will not be moved.
six. the diameter of the circle inscribed in a right-angled triangle is equal to half the sum of the diameters of the circles touching the hypotenuse, the perpendicular from the right angle of the hypotenuse, and the circle described about the
it is impossible, however, that there should be such a body; not because it is infinite on that point a general proof can be given which applies equally to all, air, water, or anything else-but simply because there is, as a matter of fact, no such
the last fact (that what is outside is infinite) leads people to suppose that body also is infinite, and that there is an infinite number of worlds. why should there be body in one part of the void rather than in another?
seventeen. the rectangle contained by the distances of the point where the internal bisector of the vertical angle meets the base, and the point where the perpendicular from the vertex meets it from the middle point of the base, is equal to the square on
of two things, or men, we say 'both', but not 'all': three is the first number to which the term 'all' has been appropriated. and in this, as we have said, we do but follow the lead which nature gives.
the narrative related to ancient famous actions of the athenian people, and to one especially, which i will rehearse in honour of you and of the goddess. critias when he told this tale of the olden time, was ninety years old, i being not more than ten.
hence the sharp rebuke of empedocles, in the words 'if the deeps of the earth are endless and endless the ample ether-such is the vain tale told by many a tongue, poured from the mouths of those who have seen but little of the whole.
whereas, though the wall does not come to be without these, it is not due to these, except as its material cause: it comes to be for the sake of sheltering and guarding certain things.
eight. if a, b, one hundred, five hundred be four collinear points taken in order,
theoretically possible, each motion is either equal to or greater than that which follows it in the series: for we shall take as actual that which is theoretically possible.
for this distinction of appellation depends upon a difference in the material out of which, and into which, the changes are effected.
because ad is parallel to ec, ba : ae :: bd : six hundred ii. ; but bd : six hundred :: ba : ac (hyp.). therefore five. xi. ba : ae :: ba : ac, and hence five. ix. ae is equal to ac; therefore the angle aec is equal to ace; but aec is equal to bad i.
so it is evident likewise that that that which has come to be, at the moment when it has come to be, will be, and that which has ceased to be will not-be: for what we have said applies universally to every kind of change, and its truth is most obvious in
the theory of anaxagoras that the principles are infinite in multitude was probably due to his acceptance of the common opinion of the physicists that nothing comes into being from not-being.
we also speak of a thing's nature as being exhibited in the process of growth by which its nature is attained. the 'nature' in this sense is not like 'doctoring', which leads not to the art of doctoring but to health.
therefore the triangles ceo, deo have two angles in one respectively equal to two angles in the other, and the side eo common. hence i. xxvi. the side ce is equal to ed. therefore four hundred is bisected in e.
at first they were without reason and measure, and had only certain faint traces of themselves, until god fashioned them by figure and number.
he did not remark that his own explanation did not suit all phenomena; and the simpler explanation, which assigns to bodies degrees of heaviness and lightness proportioned to the mass and distance of the bodies which attract them, never occurred to him.
prove that the middle point of ad is equally distant from b and one hundred.
on the other hand, it would seem to be impossible for a body to be, even potentially, divisible at all points simultaneously.
the real creation began, not with matter, but with ideas.
three. show that two circles can intersect each other only in one point on the same side of the line joining their centres, and hence that two circles cannot have more than two points of intersection.
but the specific discovery that the blood flows out on one side of the heart through the arteries and returns through the veins on the other, which is commonly called the circulation of the blood, was absolutely unknown to him.
'upwards' (as past events). yet coming-to-be must have an 'originative source' (if it is to be necessary and therefore eternal), nor can it be eternal if it is limited. consequently it must be cyclical. hence the nexus must be reciprocal.
earth respectively. it necessarily follows that circular movement, being unnatural to these bodies, is the natural movement of some other.
but we have still to explain 'combination', for that was the third of the subjects we originally proposed to discuss. our explanation will proceed on the same method as before. we must inquire: what is 'combination', and what is that which can 'combine'?
these distinctions may be taken as established: but even then it is extraordinarily difficult to see how there can be 'unqualified coming-to-be' (whether we suppose it to occur out of what potentially 'is', or in some other way), and we must recall this
prop. thirty eight.-theorem. two triangles on equal bases and between the same parallels are equal.
the oldest part of the flesh which is hard to decompose blackens from long burning, and from being corroded grows bitter, and as the bitter element refines away, becomes acid.
unseparated or separated; the void is meant to be, not body but rather an interval in body. this is why the void is thought to be something, viz. because place is, and for the same reasons.
the heavenly bodies are endowed with thought; the principles of the same and other exist in the universe as well as in the human mind.
bread i at first made of pure indian meal and salt, genuine hoe-cakes, which i baked before my fire out of doors on a shingle or the end of a stick of timber sawed off in building my house; but it was wont to get smoked and to have a piny flavor.
if the magnitude abg is composed of the indivisibles a, b, g, each corresponding part of the motion dez of o over abg is indivisible.
and this treatment they consider a sufficient analysis also of the notions of absolute heaviness, to which their account does not apply. this, however, will become clearer as we advance.
the next point to consider is how the mathematician differs from the physicist. obviously physical bodies contain surfaces and volumes, lines and points, and these are the subject-matter of mathematics.
one. the extremities of the base of an isosceles triangle are equally distant from any point in the perpendicular from the vertical angle on the base.
and if these combine in him with any true nurture or education, he attains the fulness and health of the perfect man, and escapes the worst disease of all; but if he neglects education he walks lame to the end of his life, and returns imperfect and good
ten. draw a common tangent to two circles. hence, show how to draw a line cutting two circles, so that the intercepted chords shall be of given lengths.
that is another part of his legend which plato also seeks to impose upon us. the verisimilitude which he has given to the tale is a further reason for suspecting it; for he could easily 'invent egyptian or any other tales' (phaedrus).
greater than, equal to, or less than the sector epj. hence five. def. v. the arc bc : ef :: sector boc : sector epf.
first, then, the gods, imitating the spherical shape of the universe, enclosed the two divine courses in a spherical body, that, namely, which we now term the head, being the most divine part of us and the lord of all that is in us: to this the gods, when
coming 'by chance' into the market and finding there a man whom one wanted but did not expect to meet is due to one's wish to go and buy in the market.
dixit in favour of some view about circumstances which he could never have contemplated.
and since (a) patient and agent are generically identical (i.e.
(c) it is also applied where there is general impossibility of any generation such that the thing now is which then was not.
the physicist must have a knowledge of place, too, as well as of the infinite-namely, whether there is such a thing or not, and the manner of its existence and what it is-both because all suppose that things which exist are somewhere (the non-existent is
he also knew that blood is partly a solid substance consisting of several elements, which, as he might have observed in the use of 'cupping-glasses', decompose and die, when no longer in motion.
five. if in the fig., prop. xvi., ab be the greatest side of the abc, bf is the greatest side of the fbc, and the angle bfc is less than half the angle abc.
the image and space) are different they cannot exist one of them in the other and so be one and also two at the same time.
it is clear then that, since we exclude these from among motions, we must not say that this absence of change is a state of rest: we must say that it is similar to a state of rest and call it absence of change.
of the fusile kinds the fairest and heaviest is gold; this is hardened by filtration through rock, and is of a bright yellow colour. a shoot of gold which is darker and denser than the rest is called adamant.
forty seven. the perpendiculars from the centres of the escribed circles of a triangle on the corresponding sides are concurrent.
that would be absurd, and the reason is that there are different species of motion.
therefore, while we must accept this explanation to the extent of saying that the original movent gives the power of being a movent either to air or to water or to something else of the kind, naturally adapted for imparting and undergoing motion, we must
dem.-in bd take any point f, and from ae, the greater, cut off ag equal to af iii . join bg, cf (post. i.).
for substance is a single genus of being, so that the principles can differ only as prior and posterior, not in genus; in a single genus there is always a single contrariety, all the other contrarieties in it being held to be reducible to one.
two. if a figure of six sides be inscribed in a circle, the sum of any three alternate angles is four right angles.
there were of old, he said, great and marvellous actions of the athenian city, which have passed into oblivion through lapse of time and the destruction of mankind, and one in particular, greater than all the rest. this we will now rehearse.
for the things which come-to-be by natural process all exhibit, in their coming-to-be, a uniformity either absolute or highly regular: while any exceptions any results which are in accordance neither with the invariable nor with the general rule are
these, then, proceeding by a combination of probability with demonstration, we assume to be the original elements of fire and the other bodies; but the principles which are prior to these god only knows, and he of men who is the friend of god.
in parts only, which is a species of motion imparted by physic. this should only be resorted to by men of sense in extreme cases; lesser diseases are not to be irritated by medicine.
zero.nine thousand nine hundred thirty four cheaper than rye. pork,.................... zero.twenty two
according to our general doctrine of sensation, parts of the body which are easily moved readily transmit the motion to the mind; but parts which are not easily moved have no effect upon the patient.
further, against empedocles there is another point which might be made. when the elements were separated off by hate, what caused the earth to keep its place? surely the 'whirl' cannot have been then also the cause.
therefore, if the degree in which things possess locomotion corresponds to the degree in which they have realized their natural development, then this motion must be prior to all others in respect of perfection of existence: and not only for this reason
to divide a given finite line (ab) into two segments (in h), so that the rectangle (ab.bh) contained by the whole line and one segment may be equal to the square on the other segment.
for great as was the advancing and retiring flood which provided nourishment, the affections produced by external contact caused still greater tumult-when the body of any one met and came into collision with some external fire, or with the solid earth or
dreyfus-brisac, "is the book of all books that is most talked of and least read." but with the great revival of interest in political philosophy there has come a desire for the better understanding of rousseau's work.
but there is no sense in starting generation from an original state in which bodies are separated and in movement.
wherefore the goddess, who was a lover both of war and of wisdom, selected and first of all settled that spot which was the most likely to produce men likest herself.
also moves the air, and that this in being moved is also a movent, then it would be no more possible for this second thing than for the original thing to be in motion when the original movent is not in contact with it or moving it: all the things moved
but they are not the same, and they have a different origin and nature.
one. if an integer be divided into any number of equal parts, one, or the sum of any number of these parts, is called a fraction.
some indeed assume more than one such thing from which they construct the world of nature.
six. if three points be taken on the sides of an equilateral triangle, namely, one on each side, at equal distances from the angles, the lines joining them form a new equilateral triangle.
forty five. if two lines bisecting two angles of a triangle and terminated by the opposite sides be equal, the triangle is isosceles.
as soon as the day broke, i rehearsed them as he spoke them to my companions, that they, as well as myself, might have something to say. and now, socrates, to make an end of my preface, i am ready to tell you the whole tale.
when reason is in the neighbourhood of sense, and the circle of the other or diverse is moving truly, then arise true opinions and beliefs; when reason is in the sphere of thought, and the circle of the same runs smoothly, then intelligence is perfected.
the difference is this much, that causes which are actually at work and particular exist and cease to exist simultaneously with their effect, e.g.
now, the opposite of this is sufficiently manifest; nevertheless we will not fail to describe it.
the discourse on inequality still has about it much of the rhetorical looseness of the prize essay; it aims not so much at close reasoning as at effective and popular presentation of a case.
twenty eight. what is the condition that must be fulfilled that four points may be concyclic?
a- b b c c d b c, or a- c b d; therefore a : c :: b : d. hence, proposition a , if a be greater than c, b is greater than d; if equal, equal; and if less, less.
any two lines making equal angles with the diameter on opposite sides are equal. five. more than two equal right lines cannot be drawn from the given point (p) to the circumference.
for while it is clearly best for any being to attain the real end, yet, if that cannot be, the nearer it is to the best the better will be its state. it is for this reason that the earth moves not at all and the bodies near to it with few movements.
one.-the rectangle ad.db is the rectangle contained by the sum of the lines ac, four hundred and their difference; and we have proved it equal to the difference between the square on ac and the square on four hundred.
for in proportion as the force applied by the heavy thing towards disruption and division exceeds that which resides in the continuum, the quicker will it force its way down; only if the force of the heavy thing is the weaker, will it ride upon the
sol.-on any side bc of the figure bcd describe the rectangle be equal to bcd i. xlv. , and on ce describe the rectangle ef equal to a. between bc, cf find a mean proportional gh, and on it describe the figure ghi similar to bcd xviii.
two. if two circles touched each other externally in two distinct points, then xii. the line joining the centres should pass through each point, which is impossible.
however, he actually says' that the truest account is to affirm that each of them is "gold"'.) nevertheless he carries his analysis of the 'elements'-solids though they are-back to 'planes', and it is impossible for 'the nurse' (i.e.
but this number is not separable from the process of bisection, and its infinity is not a permanent actuality but consists in a process of coming to be, like time and the number of time.
further it is evident that motion is an attribute of a thing just when it is fully real in this way, and neither before nor after. for each thing of this kind is capable of being at one time actual, at another not.
six. the three perpendiculars of a triangle are concurrent.
for every disease is akin to the living being and has an appointed term, just as life has, which depends on the form of the triangles, and cannot be protracted when they are worn out.
for how else should the moon as it waxes and wanes show for the most part a crescent-shaped or gibbous figure, and only at one moment a half-moon?
the neo-platonists, loyal to their master, like some commentators on the christian scriptures, sought to give an allegorical meaning to what they also believed to be an historical fact.
when men are in this evil plight of body, and evil forms of government and evil discourses are superadded, and there is no education to save them, they are corrupted through two causes; but of neither of them are they really the authors.
sense-perception. although the supporters of this theory do not state clearly what kind of motion they mean, or whether they mean all kinds, it is no hard matter to reply to them: thus we may point out that there cannot be a continuous process either of
the distinction of 'before' and 'after' holds primarily, then, in place; and there in virtue of relative position. since then 'before' and 'after' hold in magnitude, they must hold also in movement, these corresponding to those.
heavy and below respectively. now the relations of these must necessarily vary, because the principal masses of the different elements hold opposite positions; for that which is light, heavy, below or above in one place will be found to be and become
philo's proof.-let the equal bases be applied as in the foregoing proof, but let the vertices be on the opposite sides; then let bgc be the position which edf takes. join ag. then because bg ba, the angle bag bga. in like manner the angle cag cga.
of place is concerned, upward motion is the contrary of downward motion, forward motion of backward motion, and motion to the left of motion to the right, these being the pairs of contraries in the sphere of place.
thus, if a line oa revolve about the point o, as in figures one, two, three, four, until it comes into the position ob, the amount of the rotation from oa to ob is called an angle. from the diagrams we see that in fig.
movement in a straight line certainly does not belong to it naturally, since one sort of movement is, as we saw, appropriate to each simple body, and so we should be compelled to identify it with one of the bodies which move in this way.
eight. the bisectors of two external angles and the bisector of the third internal angle are concurrent.
five. through one of the points of intersection of two circles draw a secant, so that the rectangle contained by the intercepted chords may be given, or a maximum.
he would, perhaps, have said that 'the first things are known only to god and to him of men whom god loves.' how often have the gaps in theology been concealed from the eye of faith!
they are not, he says, to be explained by 'above' and 'below,' which in the universal globe have no existence, but by the attraction of similars towards the great masses of similar substances; fire to fire, air to air, water to water, earth to earth.
for what is white will not be one either in the sense that it is continuous or in the sense that it must be defined in only one way. 'whiteness' will be different from 'what has whiteness'.
def. iii.-two figures, such as those in five, are said to be homothetic.
in the third place, they contrived the passage of liquids, which may be explained in this way:-finer bodies retain coarser, but not the coarser the finer, and the belly is capable of retaining food, but not fire and air.
one. if a chord of a circle subtend a right angle at a given point, the locus of its middle point is a circle.
for if both substance and quantity and quality are, then, whether these exist independently of each other or not, being will be many.
thus fire, they say, is the lightest of things just for this reason that it has the most void.
now, in motion, there is nothing to prevent the firs; mover being unmoved (indeed, as regards some 'first' movers' this is actually necessary) although the last mover always imparts motion by being itself moved: and, in action, there is nothing to prevent
the angle acb is equal to the angle gfe; but the angle acb is (hyp.) equal to dfe; hence gfe is equal to dfe-a part equal to the whole, which is absurd; therefore ab and de are not unequal, that is, they are equal.
no: for, on the contrary, the 'natural movement' moves earth downwards and resembles 'dissociation', and strife rather than love is its cause-so that in general, too, love rather than strife would seem to be contrary to nature.
sixteen. draw a right line parallel to a given line, which will meet two non-coplanar lines.
its four terms consist of two antecedents and two consequents. the st and rd terms are the antecedents, and the nd and th the consequents. also the first and last terms are called the extremes, and the two middle terms the means.
for the durations of the natural processes of passing-away and coming-to-be are equal. nevertheless it often happens that things pass-away in too short a time.
one. describe a circle passing through two given points, and fulfilling either of the following conditions: one, touching a given line; two, touching a given circle.
for generally, and apart from the question of how any of them could be infinite, the all, even if it were limited, cannot either be or become one of them, as heraclitus says that at some time all things become fire.
(the distinction between the two must be made later-for the present it is sufficient if it is plain that both are in the sphere of things done for the sake of something.)
since changes evidently occur not only in the position of the stars but also in that of the whole heaven, there are three possibilities. either (one) both are at rest, or (two) both are in motion, or (three) the one is at rest and the other in motion.
indeed, the point we have raised constitutes a problem even for those who generate the 'elements' out of one another. in what manner does anything other than, and beside, the 'elements' come-to-be out of them? let me illustrate my meaning.
seventeen. what is meant by incommensurable lines? give an example from book two.
there are conflicts which all cities undergo, and i should like to hear some one tell of our own city carrying on a struggle against her neighbours, and how she went out to war in a becoming manner, and when at war showed by the greatness of her actions
for growth is an increase, and diminution is a lessening, of the magnitude which is there already-that, indeed, is why the growing thing must possess some magnitude.
would not be cooled and the cold thing in turn be warmed: for heat and cold do not change reciprocally into one another, but what changes (it is clear) is the substratum.
now an unnatural movement presupposes a natural movement which it contravenes, and which, however many the unnatural movements, is always one. for naturally a thing moves in one way, while its unnatural movements are manifold.
he made these concessions to the facts of perception: on the other hand, he conceded to the monists that there could be no motion without a void.
two. if two circles touch, and any two lines be drawn through the point of contact, cutting both circles again, the chord connecting their points of intersection with one circle is parallel to the chord connecting their points of intersection with the
any kind of motion. then that on which the forces act must be affected to some extent by our supposed finite magnitude possessing an infinite force as well as by anything else, in fact to a greater extent than by anything else, since the infinite force is
necessity is in the matter, while 'that for the sake of which' is in the definition.
yet, if the same process is a coming to-be of this but a passing-away of that, and a passing-away of this but a coming-to-be of that, why are some things said to come-to-be and pass-away without qualification, but others only with a qualification?
the smallest number, in the strict sense of the word 'number', is two. but of number as concrete, sometimes there is a minimum, sometimes not: e.g.
thereupon an aged priest said to him: 'o solon, solon, you hellenes are ever young, and there is no old man who is a hellene.' 'what do you mean?' he asked.
four. ex. three may be extended to each of the escribed circles of the triangle acb.
to order according to their affinities the particles and affections which are wandering about the body, as we have already said when speaking of the universe, he will not allow enemy placed by the side of enemy to stir up wars and disorders in the body,
this, indeed, would be as good as saying that the heavens are infinite, which we have shown to be impossible.
five. describe a circle of given radius that shall touch two given lines. how many solutions?
i tried flour also; but have at last found a mixture of rye and indian meal most convenient and agreeable.
we may question whether the note given by a single string is one and the same, or is different each time the string is struck, although the string is in the same condition and is moved in the same way.) but still, however this may be, there is nothing to
eleven. the area of a rectangle is one hundred eight.sixty square metres and its perimeter is forty eight.twenty linear metres; find its dimensions.
greater magnitude in an equal time, an equal magnitude in less time, and a greater magnitude in less time, in conformity with the definition sometimes given of 'the quicker'. suppose that a is quicker than b.
sides of one greater than the angle a contained by the two sides of the other. hence xxiv. ef would be greater than bc; but ef (hyp.) is not greater than bc.
thus in the second sense of 'nature' it would be the shape or form (not separable except in statement) of things which have in themselves a source of motion. (the combination of the two, e.g. man, is not 'nature' but 'by nature' or 'natural'.)
at every moment in the time: consequently, since the moments are infinite in number, everything that is changing must have completed an infinite number of changes.
now the various forms in passing into one another will each be resolved into its ultimate constituents. it might happen therefore that nothing existed except points, and that there was no body at all.
one. if the angle acb of a triangle be equal to twice the angle of an equilateral triangle, ab bc ca bc.ca.
hm : mn :: hi : ig; but i. xxxiv. hm is equal to kl, mn is equal to lb, hi is equal to ef, and ig is equal to fd (const.). therefore kl : lb :: ef : fd. hence the line ab is divided similarly to the line four hundred.
having observed that they held good in a few instances, they applied them everywhere; and in the complexity, of which they were capable, found the explanation of the equally complex phenomena of the universe.
now, when all of them, both those who visibly appear in their revolutions as well as those other gods who are of a more retiring nature, had come into being, the creator of the universe addressed them in these words: 'gods, children of gods, who are my
therefore the two triangles aob, dhe have the sides ao, ob in one respectively equal to the sides dh, he in the other, and the angle aob equal to dhe (hyp.). therefore i. iv. the base ab is equal to de.
or thus: let all the squares be made in reversed directions. join cg, bk, and through one hundred draw ol parallel to ag. now, taking the bac from the right s bag, cak, the remaining s cag, bak are equal.
to the motions of the soul answer the motions of the universe, and by the study of these the individual is restored to his original nature.
both. a thing is capable of causing motion because it can do this, it is a mover because it actually does it. but it is on the movable that it is capable of acting.
prop. one.-problem. in a given circle (abc) to place a chord equal to a given line (five hundred) not greater than the diameter.
body. this however we do not perceive, nor do we reflect that when a weak or small frame is the vehicle of a great and mighty soul, or conversely, when a little soul is encased in a large body, then the whole animal is not fair, for it lacks the most
his assumption that one is used in a single sense only is false, because it is used in several.
the present also is necessarily indivisible-the present, that is, not in the sense in which the word is applied to one thing in virtue of another, but in its proper and primary sense; in which sense it is inherent in all time.
of demonstration); in that case what we should have to seek for would be this unique method.
dem.-on cb describe the square cbef i. xlvi. . join bf. through five hundred draw dg parallel to cf, meeting fb produced in h. through h draw km parallel to ab. through a draw ak parallel to one hundred fifty i. xxxi. .
which at that time had not gone out of fashion. one of our tribe, either because he thought so or to please critias, said that in his judgment solon was not only the wisest of men, but also the noblest of poets.
for clearly nothing which, like incapacity, unnatural ever continues for an infinity of time; nor does the unnatural endure as long as the natural, or any form of incapacity as long as the capacity.
sol.-in ac take any point five hundred, and make ce equal to four hundred iii. . upon de describe an equilateral triangle dfe i. . join cf. then cf shall be at right angles to ab.
dem.-if the angle a be not greater than five hundred, it must be either equal to it or less than it. we shall examine each case:-
thus we have explained that all the compound bodies are composed of all the 'simple' bodies.
dem.-if possible let the circles be concentric, and let o be the centre of each. join oa, and draw any other line od, cutting the circles in the points b, five hundred respectively.
suppose, then, that the quicker has in the same time been carried over a length one and a half times as great as that traversed by the slower, and that the respective magnitudes are divided, that of the quicker, the magnitude abgd, into three
and a similar principle applies to the human belly; for when meats and drinks enter it, it holds them, but it cannot hold air and fire, because the particles of which they consist are smaller than its own structure.
five. how is a straight line generated? ans. by the motion of a point which has the same direction throughout.
is the change from being musical to being unmusical, or how is memory or forgetting, to occur?
hence, sector boc : sector epf :: arc bc : arc ef.
how or where shall we find another if we abandon this? we cannot, and therefore you must tell the tale, and good luck to you; and i in return for my yesterday's discourse will now rest and be a listener.
and he can only suppose this to be accomplished by god retiring into himself and committing the lesser works of creation to inferior powers. (compare, however, laws for another solution of the difficulty.)
also the present will be a present not in the proper sense but in virtue of something else: for the division which yields it will not be a division proper.
this last is the only kind of immovable thing of which i use the term 'being at rest': for rest is contrary to motion, so that rest will be negation of motion in that which is capable of admitting motion.
ten. what regular polygons has gauss proved to be inscriptable by the line and circle?
the following is the modern definition of curve-contact:- when two curves have two, three, four, c., consecutive points common, they have contact of the first, second, third, c., orders.
all the things mentioned present a feature in which they differ from things which are not constituted by nature.
such then is the number and nature of the kinds of cause.
we must remember that these ideas were not derived from any definite experiment, but were the original reflections of man, fresh from the first observation of nature.
fifty three. if squares be described on the sides of any triangle, the lines of connexion of the adjacent corners are respectively-(one) the doubles of the medians of the triangle; (two) perpendicular to them.
ii. a greater magnitude is said to be a multiple of a less when the greater is measured by the less-that is, when the greater contains the less a certain number of times exactly.
because all these motions do not compose one. if it is to be one process, no motion can be infinite in respect of the time that it occupies, with the single exception of rotatory locomotion.
the earth, then, also, whether it move about the centre or as stationary at it, must necessarily move with two motions. but if this were so, there would have to be passings and turnings of the fixed stars. yet no such thing is observed.
a true method is the result of many ages of experiment and observation, and is ever going on and enlarging with the progress of science and knowledge.
two. join od. then i. xx. the sum of op, pd is greater than od; but od is equal to oe i. def. xxx. . therefore the sum of op, pd is greater than oe. reject op, which is common, and we have pd greater than pe.
we cannot say that one is more coloured than the other where only colour in general and not any particular colour is meant; but they are commensurable in respect of whiteness.
the term 'goodness' is not to be understood in this passage as meaning benevolence or love, in the christian sense of the term, but rather law, order, harmony, like the idea of good in the republic.
every sort of knowledge which was akin to them.
the decision of this question, either way, is not unimportant, but rather all-important, to our search for the truth. it is this problem which has practically always been the source of the differences of those who have written about nature as a whole.
then because ab is normal to the plane ten, it is perpendicular to the line be in that plane eleven. def. vi. . hence the angle abe is right; therefore ae ab be ab bd de (because bde is right (const.)) ad de (because abd is right (hyp.)).
a consideration of the other kinds of movement also makes it plain that there is some point to which earth and fire move naturally.
always coincident with itself. hence since the light is what is naturally carried up, and the heavy what is carried down, the boundary which contains in the direction of the middle of the universe, and the middle itself, are down, and that which contains
infinite body will obviously prevail over and annihilate the finite body. on the other hand, it is impossible that each should be infinite.
but such a mode of proceeding would be unsatisfactory, because we have no reason to suppose that plato intended his scattered thoughts to be collected in a system.
iv. circles are said to touch one another when they meet, but do not intersect. there are two species of contact:- one. when each circle is external to the other. two. when one is inside the other.
one. when the shape of a figure is given, it is said to be given in species. thus a triangle whose angles are given is given in species. hence similar figures are of the same species.
hence parmenides must be thought to have spoken better than melissus. the latter says that the whole is infinite, but the former describes it as limited, 'equally balanced from the middle'.
in whatever state a man experiences either of them, that state may be called disease; and excessive pains and pleasures are justly to be regarded as the greatest diseases to which the soul is liable.
(one) from the nature of time-for it is infinite. (two) from the division of magnitudes-for the mathematicians also use the notion of the infinite.
and having no distinction of objective and subjective, he passes imperceptibly from one to the other; from intelligence to soul, from eternity to time.
eight. the rectangle co.co ab; ao.ao bc; bo.bo ca.
forty seven. if a square be inscribed in a triangle, the rectangle under its side and the sum of the base and altitude is equal to twice the area of the triangle.
if a magnitude is composed of indivisibles, the motion over that magnitude must be composed of corresponding indivisible motions: e.g.
now motion is supposed to belong to the class of things which are continuous; and the infinite presents itself first in the continuous-that is how it comes about that 'infinite' is often used in definitions of the continuous ('what is infinitely divisible
the spleen which is situated in the neighbourhood, on the left side, keeps the liver bright and clean, as a napkin does a mirror, and the evacuations of the liver are received into it; and being a hollow tissue it is for a time swollen with these
for otherwise it will always be at once destructible and in actuality indestructible, so that it will be at the same time capable of always existing and of not always existing. thus the destructible is at some time actually destroyed.
gives food to the body; but the river of speech, which flows out of a man and ministers to the intelligence, is the fairest and noblest of all streams.
fifteen. given the base of a triangle, the vertical angle, and the radius of the inscribed, or any of the escribed circles: construct it.
at the most, it is only a rather brilliant but flimsy rhetorical effort, a sophistical improvisation, but not a serious contribution to thought.
again, if it decreases for some time, and then begins to increase, it is a minimum at the commencement of the increase. thus pe, supposed as before to revolve round p and meet the circle, is a minimum. proposition viii. will give other illustrations.
for that which changes retires from or leaves that from which it changes: and leaving, if not identical with changing, is at any rate a consequence of it.
or even if they recognize more than one matter, so long as that means only a pair of contraries.
one. prove that the circle cannot meet ab in more than two points.
the soul of man is divided by him into three parts, answering roughly to the charioteer and steeds of the phaedrus, and to the (greek) of the republic and nicomachean ethics.
his influence on kant's moral philosophy and on hegel's philosophy of right are two sides of the same fundamental contribution to modern thought. he is, in fact, the great forerunner of german and english idealism.
again, the triangles bof, bog have the angle f equal to g, each being right; and obf equal to obg, because ob bisects the angle abc (const.), and ob common; hence i. xxvi. of is equal to og.
so this is how we must meet the difficulty that then arises, which is as follows.
the reasoning he will prove that in a finite time there cannot be an infinite extent of motion or of coming to rest, whether the motion is regular or irregular.
one. if four right lines (ab, four hundred, e, f) be proportional, the rectangle (ab.f) contained by the extremes is equal to the rectangle (four hundred.e) contained by the means. two.
also the sum of the squares of perpendiculars on it from two other fixed points (which may be found) is constant.
clothing and some incidental expenses within the same dates, though little can be inferred from this item, amounted to
and he gave to each of them two movements: the first, a movement on the same spot after the same manner, whereby they ever continue to think consistently the same thoughts about the same things; the second, a forward movement, in which they are controlled
indeed, even the plants, though it might be thought they are fed by one substance only, viz. by water, are fed by more than one: for earth has been mixed with the water. that is why farmers too endeavour to mix before watering.
'mean'. for at the 'mean' is neither hot nor cold. the 'mean', however, is of considerable extent and not indivisible.
these distinctions having been drawn, it is evident that every change and everything that moves is in time; for the distinction of faster and slower exists in reference to all change, since it is found in every instance.
if on the other hand the affection is different in the two cases, e.g.
eighteen. the sum of the perpendiculars let fall from the angular points of a regular polygon of n sides on any line is equal to n times the perpendicular from the centre of the polygon on the same line.
beside produce consumed and on hand at the time this estimate was made of the value of four.fifty,-the amount on hand much more than balancing a little grass which i did not raise.
thirty. mention the converse propositions of book three. which are proved directly.
but they are less real to perception which explains why things are commonly said to 'pass-away' without qualification when they change into wind and air, and to 'come-to-be' when they change into what is tangible, i.e. into earth.
two. a ratio of lesser inequality is diminished by diminishing its terms by the same quantity, and increased by increasing its terms by the same quantity.
three. if two isosceles triangles be on the same base, and be either at the same or at opposite sides of it, the line joining their vertices is an axis of symmetry of the figure formed by them.
five. what is the difference between a segment of a circle and a sector?
it results that when the infinite body has completed its revolution, it has traversed an infinite equal to itself in a finite time. but that we know to be impossible.
i have thus shown the various classes of bodies as they are diversified by their forms and combinations and changes into one another, and now i must endeavour to set forth their affections and the causes of them.
the truth is that the intemperance of love is a disease of the soul due chiefly to the moisture and fluidity which is produced in one of the elements by the loose consistency of the bones.
so again on the joints of the bones, where reason indicated that no more was required, he placed only a thin covering of flesh, that it might not interfere with the flexion of our bodies and make them unwieldy because difficult to move; and also that it
what remained, the creator then proceeded to fashion after the nature of the pattern.
and though you may say that the time of the movement can be continually diminished, yet there is no minimum. nor, if there were, would it help us.
for indestructible is also used of that which is destroyed with difficulty.
of a process of change subsequent to the last: for when a thing ceases to be moved, it does not therefore at the same time cease to be movable-e.g.
four and nine have the single mean six), whereas the cubes of primes (e.g. three cubed and five cubed) have always two mean proportionals (e.g. twenty seven:forty five:seventy five:one hundred twenty five).
for in that condition they change one another more easily and more quickly; whereas this effect takes a long time when agent and patient are present in bulk.
i am wont to think that men are not so much the keepers of herds as herds are the keepers of men, the former are so much the freer.
five. given the base and the area of a triangle, find the locus of the vertex.
sixty five. if r, r denote the radii of the circles inscribed in the triangles into which a right-angled triangle is divided by the perpendicular from the right angle on the hypotenuse; then, if c be the hypotenuse, and s the semiperimeter, r r (s c)two.
then because ef is a tangent to the circle, and ab is drawn from the point of contact perpendicular to ef, ab passes through this centre xix. . therefore the angle acb is right xxxi. .
to earth, then, let us assign the cubical form; for earth is the most immoveable of the four and the most plastic of all bodies, and that which has the most stable bases must of necessity be of such a nature.
eighty six. if the vertical angle and the bisector of the vertical angle be given, the sum of the reciprocals of the containing sides is constant.
now it is one thing to be absolutely false, and another thing to be absolutely impossible. to say that you are standing when you are not standing is to assert a falsehood, but not an impossibility.
hence since no sensible magnitude is infinite, it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens.
thus one who comes within an ace of some great evil or great good is said to be fortunate or unfortunate. the mind affirms the essence of the attribute, ignoring the hair's breadth of difference.
third to water. we must imagine all these to be so small that no single particle of any of the four kinds is seen by us on account of their smallness: but when many of them are collected together their aggregates are seen.
for if coming-to-be and passing-away are always to be continuous, there must be some body always being moved (in order that these changes may not fail) and moved with a duality of movements (in order that both changes, not one only, may result).
either this is itself an ultimate fact or there is an ultimate fact behind it. it may seem evidence of excessive folly or excessive zeal to try to provide an explanation of some things, or of everything, admitting no exception.
the hindrance to its movement, then, we will suppose, is a moving thing-as empedocles says that it is the vortex which keeps the earth still-: but in that case we ask, where would it have moved to but for the vortex?
three. if through the extremities of the base of a triangle, whose sides are unequal, lines be drawn to any point in the bisector of the vertical angle, their difference is less than the difference of the sides.
if, then, view which we are criticizing involves these impossible consequences, it is clear that motion is eternal and cannot have existed at one time and not at another: in fact such a view can hardly be described as anythling else than fantastic.
plato was struck by the phenomenon of mercury, venus, and the sun appearing to overtake and be overtaken by one another.
a broken line, a spiral, or any other magnitude that is not such that any part of it taken at random fits on to any other that may be chosen.
cor. three.-the common chords of any three intersecting circles are concurrent (compare xvii., ex. seven).
when tinged with blood the bitter substance has a red colour, and this when mixed with black takes the hue of grass; or again, the bitter substance has an auburn colour, when new flesh is decomposed by the internal flame.
hence one c b is less than one c- a, that is, b c is less than a- c. hence, multiplying each by c, we get
the word 'is' has many senses, and we say that the infinite 'is' in the sense in which we say 'it is day' or 'it is the games', because one thing after another is always coming into existence.
but if there were any need to consider which of the two, that which moves itself or that which is moved by something else, is the cause and principle of motion, every one would decide the former: for that which is itself independently a cause is always
one. what previous problem is employed in the solution of this?
and thus people of your class are the only ones remaining who are fitted by nature and education to take part at once both in politics and philosophy.
(two) nor that a point should have a place, (three) nor that two bodies should be in the same place, (four) nor that place should be a corporeal interval: for what is between the boundaries of the place is any body which may chance to be there, not an
how, then, shall we settle this point, and what questions about the elements may be fairly raised?
form at least is not one. (by that i mean that it can be described in different ways.) for 'to be man' is not the same as 'to be unmusical'.
if (ii) infinite, then either as democritus believed one in kind, but differing in shape or form; or different in kind and even contrary.
soon an inner world of ideas began to be unfolded, more absorbing, more overpowering, more abiding than the brightest of visible objects, which to the eye of the philosopher looking inward, seemed to pale before them, retaining only a faint and precarious
for the motion is the same as that along the diameter, since we invariably regard the distance between two points as the length of the straight line which joins them.
now, if 'musicalness and unmusicalness' had not been a property essentially inhering in man, these changes would have been a coming-to-be of unmusicalness and a passing-away of musicalness: but in fact 'musicalness and unmusicalness' are a property of the
but the same reasons also forbid us to regard the matter, out of which the body comes-to-be, as points or lines. the matter is that of which points and lines are limits, and it is something that can never exist without quality and without form.
twenty one. give an example of two triangles inversely similar. ans. if two lines passing through any point o outside a circle intersect it in pairs of points a, a; b, b, respectively, the triangles oab, oab, are inversely similar.
by vii., square on difference sum of squares-twice rectangle.
twenty. the sum of the perpendiculars from any point in the interior of an equilateral triangle is equal to the perpendicular from any vertex on the opposite side.
dem.-on cb describe the square cbef i. xlvi. . join bf. through five hundred draw dg parallel to cf, meeting bf in h. through h draw km parallel to ab, and through a draw ak parallel to one hundred fifty i. xxxi. .
, the figure bfgh is equal to the sum of the squares on ab, ah-that is, equal to the sum of the squares on ab, bc; and the sum of the four triangles is equal to twice the rectangle ab.bc, for each triangle is equal to half the rectangle ab.bc.
thus then the fixed stars were created, being divine and eternal animals, revolving on the same spot, and the wandering stars, in their courses, were created in the manner already described.
hence time is not number in the sense in which there is 'number' of the same point because it is beginning and end, but rather as the extremities of a line form a number, and not as the parts of the line do so, both for the reason given (for we can use
necessity in mathematics is in a way similar to necessity in things which come to be through the operation of nature. since a straight line is what it is, it is necessary that the angles of a triangle should equal two right angles.
again, because ah is equal to ac (const.), the triangle ach is isosceles; therefore the angle ach is equal to ahc v.
twenty. through a point five hundred, taken on the side bc of a triangle abc, is drawn a transversal edf, and circles described about the triangles dbf, ecd. the locus of their second point of intersection is a circle.
thirty. if four circles be tangential to a fifth, and if we denote by twelve the common tangent to the first and second, c., then
and now that this has been proved, it is evident that everything that is in motion must have been in motion before.
specifically one is not necessarily one in an unqualified sense. we have now explained what we mean when we call a motion one without qualification.
of the poles, that which we see above us is the lower region, and that which we do not see is the upper.
and besides the cases already mentioned there is another in which a motion is said to be one, viz.
differ merely in relative position, but also as possessing distinct potencies. this is made plain also by the objects studied by mathematics.
he was the natural enemy of mythology, and yet mythological ideas still retained their hold over him.
for, if one of a pair of contraries naturally exists, the other, if it is really contrary, exists also naturally. in some form it must be present, since the matter of contraries is the same.
what men read between the lines of his works. it is owing to this faculty of giving his generalisations content and actuality that rousseau has become the father of modern political philosophy.
clearly then it is not movement. (we need not distinguish at present between 'movement' and 'change'.)
def. one.-a ratio whose antecedent is greater than its consequent is called a ratio of greater inequality; and a ratio whose antecedent is less than its consequent, a ratio of lesser inequality.
let us now proceed to maintain that it is possible that there should be an infinite motion that is single and continuous, and that this motion is rotatory motion.
four. inscribe a square in a given equilateral triangle, having its base on a given side of the triangle.
train him shall be the fairest and best adapted to that purpose. a minute discussion of this subject would be a serious task; but if, as before, i am to give only an outline, the subject may not unfitly be summed up as follows.
we have now stated the number of the principles of natural objects which are subject to generation, and how the number is reached: and it is clear that there must be a substratum for the contraries, and that the contraries must be two.
of all the kinds termed fusile, that which is the densest and is formed out of the finest and most uniform parts is that most precious possession called gold, which is hardened by filtration through rock; this is unique in kind, and has both a glittering
ii.): let the two triangles be abc, ade; let the second triangle ade be conceived to be placed on abc, so that its two sides ad, ae may fall on the sides ab, ac; then, since the angle ade is equal to abc, the side de is parallel to bc. hence ii.
these we may fairly assume to be the original elements of fire and the other bodies; what principles are prior to these god only knows, and he of men whom god loves.
capacity of moving something else, is itself unmoved and exempt from all change, which can affect it neither in an unqualified nor in an accidental sense.
it has been established before' that the coming-to-be of the 'simple' bodies is reciprocal.
but the theory needs more than these because the elements which it recognizes are more in number. secondly, it is manifest that the simple bodies are often given a shape by the place in which they are included, particularly water and air.
indeed this is clear also from the consideration that there would otherwise be no becoming and perishing and no change of any kind in other things, which require something that is in motion to move them: for the motion imparted by the unmoved will always
so that all the pecuniary outgoes, excepting for washing and mending, which for the most part were done out of the house, and their bills have not yet been received,-and these are all and more than all the ways by which money necessarily goes out in this
prop. twenty five.-theorem. if four magnitudes of the same kind be proportionals, the sum of the greatest and least is greater than the sum of the other two.
if four lines (ab, four hundred, ef, gh) be proportional, and any pair of similar rectilineal figures (abk, four hundred fifty) be similarly described on the first and second, and also any pair (ei, gj) on the third and fourth, these figures are
visible and tangible heaven. and for these reasons, and out of such elements which are in number four, the body of the world was created, and it was harmonized by proportion, and therefore has the spirit of friendship; and having been reconciled to
cor. two.-if four magnitudes be proportional, their cubes are proportional.
prop. twenty.-theorem. the sum of any two sides (ba, ac) of a triangle (abc) is greater than the third.
hence neither will motion and change have reference to something over and above the things mentioned, for there is nothing over and above them.
figures that are congruent are said to be identically equal.
in modern geometry a curve is considered as made up of an infinite number of points, which are placed in order along the curve, and then the secant through two consecutive points is a tangent.
then because o is the centre of each circle (hyp.), ob and od are each equal to oa; therefore ob is equal to od, which is impossible. hence the circles cannot have the same centre.
and since this description may be understood in two different ways, in which of these two ways are we to apply it to the process of growth?
but it may also be shown universally, not only by such reasoning as we advanced in our discussion of principles (though in that passage we have already determined universally the sense in which the existence of an infinite is to be asserted or denied),
i answer in a word: if mind is one thing and true opinion another, then there are self-existent essences; but if mind is the same with opinion, then the visible and corporeal is most real.
the remaining angle five hundred is equal to bac. hence the triangle abc inscribed in the given circle is equiangular to def.
the movements of the sun and moon are fewer than those of some of the planets. yet these planets are farther from the centre and thus nearer to the primary body than they, as observation has itself revealed.
furthermore, there will be a part of the present that is past and a part that is future, and it will not always be the same part that is past or future: in fact one and the same present will not be simultaneous: for the time may be divided at many points.
hence bc must be equal to ef, and the same as in one, ac is equal to df, and the angle bac is equal to the angle edf.
assuming, therefore, that ice is a freezing of moist and cold, fire analogously will be a boiling of dry and hot: a fact, by the way, which explains why nothing comes-to-be either out of ice or out of fire.
and these four couples have attached themselves to the apparently 'simple' bodies (fire, air, water, and earth) in a manner consonant with theory.
twelve. if two triangles have equal altitudes, how do their areas vary?
now that which is created must, as we affirm, of necessity be created by a cause. but the father and maker of all this universe is past finding out; and even if we found him, to tell of him to all men would be impossible.
if, again, one orders figures according to their numbers, it is most natural to arrange them in this way. the circle corresponds to the number one, the triangle, being the sum of two right angles, to the number two.
now it is evident that the four rectangles. ag, gi, il, la are all equal; but ag is the rectangle ab.bg or ab.bc. therefore the sum of the four rectangles is equal to ab.bc. again, the figure np is evidently equal to the square on ac.
for there is no centre of the infinite, and that which moves in a circle moves about the centre. nor again can the infinite move in a straight line.
book five. theory of proportion definitions.
bone was composed by him in the following manner.
further, to assume, on the one hand, that the planes can only be put in linear contact would be ridiculous. for just as there are two ways of putting lines together, namely, end to and side by side, so there must be two ways of putting planes together.
socrates: then have i now given you all the heads of our yesterday's discussion? or is there anything more, my dear timaeus, which has been omitted?
sol.-from a, one of the extremities of ab, draw the right line ad equal to one hundred ii. ; and with a as centre, and ad as radius, describe the circle edf (post. iii.) cutting ab in e. ae shall be equal to one hundred.
speak of water or fire, not as substances, but as qualities. they may be compared to images made of gold, which are continually assuming new forms. somebody asks what they are; if you do not know, the safest answer is to reply that they are gold.
to suppose, on the other hand, that these things were in being throughout all previous time without there being any motion appears unreasonable on a moment's thought, and still more unreasonable, we shall find, on further consideration.
these principles have been analysed in the discussion of the movements of animals, for the reason that they are proper to animal nature.
dem.-if the angles b and e are not equal, one must be greater than the other. suppose abc to be the greater, and that the part abg is equal to def, then the triangles abg, def have two angles in one equal to two angles in the other, and are i. xxxii.
'originative sources' of body, but only those which correspond to touch. for it is in accordance with a contrariety-a contrariety, moreover, of tangible qualities-that the primary bodies are differentiated.
having given all these laws to his creatures, that he might be guiltless of future evil in any of them, the creator sowed some of them in the earth, and some in the moon, and some in the other instruments of time; and when he had sown them he committed to
in attributing to the fixed stars only the most perfect motion-that which is on the same spot or circulating around the same-he might perhaps have said that to 'the spectator of all time and all existence,' to borrow once more his own grand expression, or
three. if a line be parallel to each of two intersecting planes, it is parallel to their intersection.
for the bonds of life which unite the soul with the body are made fast there, and they are the root and foundation of the human race.
two. if a circumscribed triangle be isosceles, the corresponding inscribed triangle is isosceles, and conversely.
what is this but the atoms of democritus and the triangles of plato? the ancients should not be wholly deprived of the credit of their guesses because they were unable to prove them.
for it is evident that the triangles oab, oab are similar and that either may be turned round the point o, so that the two bases, ab, ab, will be parallel.
is it not a wonderful thing that a few pages of one of plato's dialogues have grown into a great legend, not confined to greece only, but spreading far and wide over the nations of europe and reaching even to egypt and asia?
sol.-erect ad at right angles to ab xi. , and make it equal to ab iii. . through five hundred draw six hundred parallel to ab xxxi. , and through b draw bc parallel to ad; then ac is the square required.
seventeen. being given an angle aob, the locus of all the points p of space, such that the sum of the projections of the line op on oa and ob may be constant, is a plane.
body; and there they bound it down like a wild animal which was chained up with man, and must be nourished if man was to exist.
while he ridiculed the superficial explanations of phenomena which were current in his age, he recognised the marks both of benevolence and of design in the frame of man and in the world.
they appointed this lower creation his place here in order that he might be always feeding at the manger, and have his dwelling as far as might be from the council-chamber, making as little noise and disturbance as possible, and permitting the best part
all things go the wrong way and cease to give nourishment to the body, no longer preserving their natural courses, but at war with themselves and destructive to the constitution of the body.
if a line (ab) passing through the centre of a circle bisect a chord (four hundred), which does not pass through the centre, it cuts it at right angles. two. if it cuts it at right angles, it bisects it.
therefore hf is equal to the square on ac, and one hundred one is the square on cb; but the whole figure ad, which is the square on ab, is the sum of the four figures hf, one hundred one, ag, gd.
but, although this solution is adequate as a reply to the questioner (the question asked being whether it is possible in a finite time to traverse or reckon an infinite number of units), nevertheless as an account of the fact and explanation of its true
'first of all things came chaos to being, then broad-breasted earth,' implying that things need to have space first, because he thought, with most people, that everything is somewhere and in place.
their respective sizes. thus the weightless body will move the same distance as the heavy in the same time. but this is impossible.
if then this body can have no contrary, because there can be no contrary motion to the circular, nature seems justly to have exempted from contraries the body which was to be ungenerated and indestructible.
what, then, is there in the wood besides the division? for even if we suppose there is some quality, yet how is the wood dissolved into such constituents and how does it come-to-be out of them?
as for the religion and love of art of the builders, it is much the same all the world over, whether the building be an egyptian temple or the united states bank. it costs more than it comes to.
in the following year, this work, having been awarded the prize by the academy, was published by its author. his success was instantaneous; he became at once a famous man, the "lion" of parisian literary circles.
three. two lines ab, xy , are given in position: ab is divided in one hundred in the ratio m : n, and parallels aa, bb, two hundred, are drawn in any direction meeting xy in the points a, b, one hundred; prove
we may assure ourselves of the necessity of this coming to a stand not only on the strength of observation, but also on theoretical grounds.
for if there were any solid body in equipoise at the centre of the universe, there would be nothing to draw it to this extreme rather than to that, for they are all perfectly similar; and if a person were to go round the world in a circle, he would often,
thus 'between' implies the presence of at least three things: for in a process of change it is the contrary that is 'last': and a thing is moved continuously if it leaves no gap or only the smallest possible gap in the material-not in the time (for a gap
the two motions along the diameter of the circle, since the ends of this are at the greatest possible distance from one another), and they are opposites only if they are along the same line.
since, therefore, the heavens have to move in one lace, and the stars are not required to move themselves forward, it is natural that both should be spherical-a shape which best suits the movement of the one and the immobility of the other.
for if, on the one and, we suppose that the star which moves on the greater circle is necessarily swifter, clearly we also admit that if stars shifted their position so as to exchange circles, the slower would become swifter and the swifter slower.
dem.-let eh, gf meet in one thousand; through one thousand draw mp, mj parallel to ab, bc. produce ad, gh, bc to meet mp, and ab, ef, six hundred to meet mj. now the complement of fj: to each add the pict fl, and we get the figure ofl pict cj.
the particles are supposed by him to be in a perpetual process of circulation caused by inequality. this process of circulation does not admit of a vacuum, as he tells us in his strange account of respiration.
nails were formed by combining sinew, skin, and bone, and were made by the creators with a view to the future when, as they knew, women and other animals who would require them would be framed out of man.
for in those days nothing had any proportion except by accident; nor did any of the things which now have names deserve to be named at all-as, for example, fire, water, and the rest of the elements.
again, since the arc af is equal to ed, to each add the arc abcd; then the whole arc fabcd is equal to abcde; therefore the angles def, efa which stand on these arcs are equal three. xxvii. .
and, similarly, every addition of a new 'element' will carry with it the attachment of a new contrariety to the preceding elements'.
the void is thought to be place with nothing in it. the reason for this is that people take what exists to be body, and hold that while every body is in place, void is place in which there is no body, so that where there is no body, there must be void.
fourteen, fifteen; one hundred forty one, one hundred forty two; one thousand four hundred fourteen, one thousand four hundred fifteen; c., ten ten one hundred one hundred one thousand one thousand
wherefore earth when not consolidated by force is dissolved by water only; when consolidated, by nothing but fire; for this is the only body which can find an entrance.
four. show how to produce the less of two given lines until the whole produced line becomes equal to the greater.
to a contrary or to an intermediate (of this, however, we shall speak later), but changing to a contrary rather than changing from a contrary would seem to be the cause of the contrariety of motions, the latter being the loss, the former the gain, of
this result, then, will follow, even if the earth is spherical, so long as it retains its size. so far as their arguments go, the earth will still be at rest.
that is why people say that the figure is not the nature of a bed, but the wood is-if the bed sprouted not a bed but wood would come up. but even if the figure is art, then on the same principle the shape of man is his nature. for man is born from man.
again, from the continuity of the time in which the motion takes place we cannot infer continuity in the motion, but only successiveness: in fact, how could contraries, e.g. whiteness and blackness, meet in the same extreme point?
seven. give instances of reciprocal propositions in book four.
the motion begins to cease when the motive force produced in one member of the consecutive series is at each stage less than that possessed by the preceding member, and it finally ceases when one member no longer causes the next member to be a movent but
towers and temples are the luxury of princes. a simple and independent mind does not toil at the bidding of any prince. genius is not a retainer to any emperor, nor is its material silver, or gold, or marble, except to a trifling extent.
for carrying always follows one of the other three methods, for that which is carried is in motion accidentally, because it is in or upon something that is in motion, and that which carries it is in doing so being either pulled or pushed or twirled; thus
but if one of a pair of contraries is determinate, the other must be determinate also. now the centre is determined; for, from whatever point the body which sinks to the bottom starts its downward motion, it cannot go farther than the centre.
therefore the square on ab is equal to the sum of the squares on ac, cb, and twice the rectangle ac.cb.
for instance, eighty two r(six seven) three. xxxv., ex. six .
they then seek a reason for its staying there; and some say, in the manner explained, that the reason is its size and flatness, others, with empedocles, that the motion of the heavens, moving about it at a higher speed, prevents movement of the earth, as
one. of two unequal magnitudes, the greater has a greater ratio to any third magnitude than the less has; two. any third magnitude has a greater ratio to the less of two unequal magnitudes than it has to the greater.
two. if bd : six hundred :: ba : ac, the angle bac is bisected.
if then everything that is in motion must be moved by something, and the movent must either itself be moved by something else or not, and in the former case there must be some first movent that is not itself moved by anything else, while in the case of
but since the contradictory of 'that which is always capable of being' 'that which is not always capable of being'; while 'that which is always capable of not being' is the contrary, whose contradictory in turn is 'that which is not always capable of not
prop. twenty nine.-theorem. in equal circles (acb, dfe), equal arcs (agb, dck) are subtended by equal chords.
air from water (for water is the first thing that is potentially light), and air is actually light, and will at once realize its proper activity as such unless something prevents it.
this concludes my account of place-both of its existence and of its nature.
in the fourfold scheme of causes, it is plain that the infinite is a cause in the sense of matter, and that its essence is privation, the subject as such being what is continuous and sensible.
hence the two triangles aob, dhe have the sides ao, ob in one respectively equal to the sides dh, he in the other, and the base ab is equal to de (hyp.). therefore i. viii. the angle aob is equal to dhe. hence the arc agb is equal to dke xxvi.
therefore, neither (a) persist actually, as 'body' and 'white' persist: nor (b) are they destroyed (either one of them or both), for their 'power of action' is preserved.
hence, multiplying each by c, we get a greater than b.
three. acb is an arc of a circle, ce a tangent at one hundred, meeting the chord ab produced in e, and ad a perpendicular to ab in five hundred: prove, if de be bisected in one hundred, that the arc ac cb.
and right appears left and left right, because the visual rays come into contact with the rays emitted by the object in a manner contrary to the usual mode of meeting; but the right appears right, and the left left, when the position of one of the two
apart from the other, there will be time intermediate between them, because everything continuous is such that there is something intermediate between its limits and described by the same name as itself.
part of the soul, as is manifest in the case of perfumes.
every body, again, they suppose to be tangible; and of this nature is whatever has weight or lightness.
it is in these cases that difficulty would be experienced in deciding whence the motion is derived, e.g. in the case of light and heavy things.
this 'actual other', then, has passed-away and come-to-be flesh.
remote habitations as a punishment of their outlandish ignorance. these are the laws by which animals pass into one another, now, as ever, changing as they lose or gain wisdom and folly.
so o traversed a when its motion was five hundred, b when its motion was e, and g similarly when its motion was z.
in considering the third kind of sense, hearing, we must speak of the causes in which it originates.
twenty eight. if dd be the common tangent to the two circles, dd ab.ab.
in its own properties, the properties in question being opposed to one another either as contraries or as intermediates. the body, e.g.
prop. eleven.-theorem. if one circle (cpd) touch another circle (apb) internally at any point p, the line joining the centres must pass through that point.
prop. fifteen.-theorem. magnitudes have the same ratio which all equimultiples of them have.
hence it is reasonable enough that coming-to-be should never fail. for coming-to-be is a passing-away of 'what is not' and passing-away is a coming to-be of 'what is not'.
fixed, the inner the wandering stars. the soul of the world was diffused everywhere from the centre to the circumference.
if you draw the earth into the dissimilar air, the particles of earth cling to their native element, and you more easily detach a small portion than a large. there would be the same difficulty in moving any of the upper elements towards the lower.
for it may be thought that we have not proved universal of bodies that none whatever can exist outside our universe, and that our argument applied only to those of indeterminate extent.
no doubt the food, which has come in, may sometimes expand as well as the body that has consumed it (that is so, e.g.
again, since the parallelograms ab, four hundred are equal (hyp.),
in so far as he works with his eye fixed upon an eternal pattern he is like the human artificer in the republic. here the theory of platonic ideas intrudes upon us.
yet, strictly speaking-and the remark applies to ancient physics generally-this high a priori road was based upon a posteriori grounds. for there were no facts of which the ancients were so well assured by experience as facts of number.
fifteen. on be, a part of the side bc of a square abcd, is described the square befg, having its side bg in the continuation of ab; it is required to divide the figure agfecd into three parts which will form a square.
free from old age and unaffected by disease.
the reason of the various motions of the various bodies, and the meaning of the motion of a body to its own place, have now been explained.
'contact' as of all such terms, it is nevertheless true that contact' in the proper sense applies only to things which have 'position'.
another. on the other hand lateral motion is not the contrary of upward motion.
another account is that 'nature' is the shape or form which is specified in the definition of the thing.
and as this circular movement required no feet, the universe was created without legs and without feet.
so it is with all other artificial products. none of them has in itself the source of its own production.
again, since the motion of anything must always occupy either an equal time or less or more time in comparison with that of another thing, and since, whereas a thing is slower if its motion occupies more time and of equal velocity if its motion occupies
but if the infinite exists as an attribute, it would not be, qua infinite an element in substances, any more than the invisible would be an element of speech, though the voice is invisible.
five. if two right-angled triangles have equal hypotenuses, and an acute angle of one equal to an acute angle of the other, they are congruent.
the term 'according to nature' is applied to all these things and also to the attributes which belong to them in virtue of what they are, for instance the property of fire to be carried upwards-which is not a 'nature' nor 'has a nature' but is 'by nature'
socrates: and you remember how we said that the children of the good parents were to be educated, and the children of the bad secretly dispersed among the inferior citizens; and while they were all growing up the rulers were to be on the look-out, and to
this had been asserted in the first instance by hesiod and his followers, but afterwards outside his circle by the earliest natural philosophers.
need not always be, must always be. no: if its coming-to-be is to be 'necessary', it must be 'always' in its coming-to-be. for what is 'of necessity' coincides with what is 'always', since that which 'must be' cannot possibly 'not-be'.
but the first of these last two alternatives is impossible if coming-to-be is to be eternal, because there could not be any 'originative source' whatever in an infinite rectilinear sequence, whether its members be taken 'downwards' (as future events) or
continuous in one direction, another in two, another in all. all magnitudes, then, which are divisible are also continuous. whether we can also say that whatever is continuous is divisible does not yet, on our present grounds, appear.
hence a and b are equiangular, and have the sides about their equal angles proportional. therefore they are similar.
thus a false conclusion would follow because a false assumption was made, but if what was assumed had not been impossible its consequence would not have been impossible.
cor. three.-if a line intersect two concentric circles, its intercepts between the circles are equal.
therefore c b is greater than c a-. hence the ratio c : b is greater than the ratio c : a.
but if that is their character, it is impossible they should not be affected by one another: the 'slightly-hot indivisible', e.g. will inevitably suffer action from one which far exceeds it in heat.
the other two move downward when the body beneath is withdrawn because, while the absolutely heavy is that which sinks to the bottom of all things, the relatively heavy sinks to its own place or to the surface of the body in which it rises, since it is
, because they are on equal bases ac, cb, and between the same parallels; therefore al is equal to de: to each add ch, and we get the parallelogram ah equal to the gnomon cmg; but ah is equal to the rectangle ad.dh, and therefore equal to the rectangle
that which, like a field, was to receive the divine seed, he made round every way, and called that portion of the marrow, brain, intending that, when an animal was perfected, the vessel containing this substance should be the head; but that which was
it follows then, further, that the motion cannot be irregular. for if irregularity occurs, there must be change either in the movement as a whole, from fast to slow and slow to fast, or in its parts.
cor. three.-if two chords intersect at any point within a circle, the sum of the opposite arcs which they intercept is equal to the arc which parallel chords intersecting on the circumference intercept. two.
thus, a triangle always has its angles equal to two right angles, but there is nevertheless an ulterior cause of the eternity of this truth, whereas first principles are eternal and have no ulterior cause.
prop. eight.-theorem. if one (ab) of two parallel right lines (ab, four hundred), be normal to a plane (ten), the other line (four hundred) shall be normal to the same plane.
for the reason why the air is so closely confined that it cannot find a passage, and therefore stays where it is, is its great amount: and this amount great because the body which isolates it, the earth, is very large.
or evaporating, and are perceptible only in the intermediate state, when water is changing into air and air into water; and all of them are either vapour or mist.
those things, on the other hand, whose 'substance' is perish, able (not imperishable) must 'return upon themselves' in the sense that what recurs, though specifically the same, is not the same numerically.
the question may also be examined in the light of more general considerations as follows. the infinite, considered as a whole of similar parts, cannot, on the one hand, move in a circle.
we have next to show that the movement of the heaven is regular and not irregular. this applies only to the first heaven and the first movement; for the lower spheres exhibit a composition of several movements into one.
to the like end the gifts of speech and hearing were bestowed upon us; not for the sake of irrational pleasure, but in order that we might harmonize the courses of the soul by sympathy with the harmony of sound, and cure ourselves of our irregular and
it is evident, then, that a process of change cannot be infinite in the sense that it is not defined by limits.
thirty nine. given the middle points of the sides of a convex polygon of an odd number of sides, construct the polygon.
of 'the man' as being healed, being warmed and chilled, and similarly in all the other cases), but at other times we say 'what is cold is 'being warmed', 'what is sick is being healed': and in both these ways of speaking we express the truth, since in one
the characteristics of the stars which move with a circular motion, in respect of substance and shape, movement and order, have now been sufficiently explained.
now a motion that is always shifting its ground from moment to moment can be continuous: but a motion that is repeatedly localized within certain fixed limits cannot be so, since then the same thing would have to undergo simultaneously two opposite
enough of disease-i have now to speak of the means by which the mind and body are to be preserved, a higher theme than the other.
if we adopt the theory that it is number that constitutes being, we may indeed speak of a 'greater number' and a 'lesser number' within the same species, but there is no common term that will include both relations, nor are there terms to express each of
in the next place, they divided the veins about the head and interlaced them with each other in order that they might form an additional link between the head and the body, and that the sensations from both sides might be diffused throughout the body.
would be cheaper to spade up that than to use oxen to plough it, and to select a fresh spot from time to time than to manure the old, and he could do all his necessary farm work as it were with his left hand at odd hours in the summer; and thus he would
of neither is there any trace in plato, who makes the earth the centre of his system. philolaus magnifies the virtues of particular numbers, especially of the number ten (stob.
join h to a fourth point five hundred, and divide hd in k, so that hk hd, and so on. the last point found will be the centre of mean position of the given points.
in accordance with general conviction we may distinguish the absolutely heavy, as that which sinks to the bottom of all things, from the absolutely light, which is that which rises to the surface of all things.
he was good, and the good can never have jealousy of anything. and being free from jealousy, he desired that all things should be as like himself as they could be.
water, again, admits in the first place of a division into two kinds; the one liquid and the other fusile.
they contain earth and water, then, for the reasons we have given: and they contain air and fire, because these are contrary to earth and water (earth being contrary to air and water to fire, in so far as one substance can be 'contrary' to another).
twelve. construct a parallelogram, being given two diagonals and a side.
in like manner, if a be equal to b, c is equal to d, and if less, less.
therefore the three lines ao, bo, co are equal, and the circle described with o as centre, and oa as radius, will pass through the points a, b, one hundred, and be described about the triangle abc.
three. the square on the diameter of the circle described about the triangle formed by the points f, h, five hundred (see fig. two. xi.), is equal to six times the square on the line fd.
so that the two things must have happened at different points of time, and therefore there will be the intervening period of time: consequently a will be in a state of rest at b, and similarly at all other points, since the same reasoning holds good in
but so far as their argument goes, instead of remaining there, it will move, only not as a mass but in fragments. for the argument applies equally to fire.
if two intersecting planes (ab, bc) be each perpendicular to a third plane (adc), their common section (bd) shall be normal to that plane.
proportionate and intermediate elements. and then we shall not be willing to allow that there are any distinct kinds of visible bodies fairer than these.
thus we have the result that the thing that has changed, at the moment when it has changed, is changing to that to which it has changed, which is impossible: that which has changed, therefore, must be in that to which it has changed.
and is the thought expressed in them to be attributed to the learning of the egyptian priest, and not rather to the genius of plato?
thing, though the essence of the three is different.
two. a line from the vertex of an isosceles triangle to any point in the base is less than either of the equal sides, but greater if the point be in the base produced.
and thus death, if caused by disease or produced by wounds, is painful and violent; but that sort of death which comes with old age and fulfils the debt of nature is the easiest of deaths, and is accompanied with pleasure rather than with pain.
we have already remarked that plato was not acquainted with the modern distinction of subject and object, and therefore he sometimes confuses mind and the things of mind-(greek) and (greek).
of all these elements, that which has the fewest bases must necessarily be the most moveable, for it must be the acutest and most penetrating in every way, and also the lightest as being composed of the smallest number of similar particles: and the second
continue to perish, as empedocles says his 'man-faced ox-progeny' did.
dem.-one. take any point i, and join it to the centre one hundred. then because the angle cbi is a right angle, ci is equal to cb bi i. xlvii. ; therefore ci is greater than cb . hence one hundred one is greater than cb, and the point i note on i., def.
is visible, and have a white colour arising out of the generation of foam-all this decomposition of tender flesh when intermingled with air is termed by us white phlegm.
hence it is clear that those things are by nature such as to touch one another, which (being separate magnitudes) have their extremes 'together' and are able to move, and be moved by, one another.
the first will be the simplest and smallest construction, and its element is that triangle which has its hypotenuse twice the lesser side.
sol.-upon ab describe an equilateral triangle acb i. . bisect the angle acb by the line four hundred ix. , meeting ab in five hundred, then ab is bisected in five hundred.
leaving the further explanation of details, which the reader will find discussed at length in boeckh and martin, we may now return to the main argument: why did god make the world?
again, since the angle bac is half the angle boc three. xx. , and edf is half the angle epf,
but when the visual ray goes forth into the darkness, then unlike falls upon unlike-the eye no longer sees, and we go to sleep.
seventeen. if from any point within a regular polygon of n sides perpendiculars be let fall on the sides, their sum is equal to n times the radius of the inscribed circle.
wherever we have something capable of acting and something capable of being correspondingly acted on, in the event of any such pair being in contact what is potential becomes at times actual: e.g.
acid and salt phlegm is the source of catarrh.
wherefore, using the language of probability, we may say that the world became a living creature truly endowed with soul and intelligence by the providence of god.
the question might possibly be raised whether there is any obstacle to our believing that there are other universes composed on the pattern of our own, more than one, though stopping short of infinity.
if the movement is uneven, clearly there will be acceleration, maximum speed, and retardation, since these appear in all irregular motions.
for the one substratum is sufficient to be acted on; but if we have four contraries, there will be two contrarieties, and we shall have to suppose an intermediate nature for each pair separately.
two. what are similar rectilineal figures?
a similar distinction holds also of the agent: for we speak not only (a) of the doctor, but also (b) of the wine, as healing.
'destructible' is applied (a) to that which formerly was and afterwards either is not or might not be, whether a period of being destroyed and changed intervenes or not; and (b) sometimes we apply the word to that which a process of destruction may cause
that this must be the case is evident from the principles laid down in our discussion of the movements, for these are limited in number, and the distinction of the elements depends upon the distinction of the movements.
but' (just as empedocles and some of the other philosophers say that things suffer action through their pores, so) 'all "alteration" and all "passion" take place in the way that has been explained: breaking-up (i.e.
three. if from a, one of the points of intersection of two given circles, two chords acd, acd be drawn, cutting the circles in the points one hundred, five hundred; one hundred, five hundred, the triangles bcd, bcd, formed by joining these to the second
as for those who agree with empedocles that the 'elements' of body are more than one, so that they are not transformed into one another-one may well wonder in what sense it is open to them to maintain that the 'elements' are comparable.
change which is not accidental on the other hand is not to be found in everything, but only in contraries, in things intermediate contraries, and in contradictories, as may be proved by induction.
he seems, however, to have made little progress with this work, until in one thousand seven hundred forty nine he happened to light on the announcement of a prize offered by the academy of dijon for an answer to the question, "has the progress of the arts
similar difficulties are involved in all other methods of distinction, whether they account for the relative lightness and heaviness of bodies by distinctions of size, or proceed on any other principle, so long as they attribute to each the same matter,
the motion of a simple body must itself be simple, and we assert that there are only these two simple motions, the circular and the straight, the latter being subdivided into motion away from and motion towards the centre.
here a difficulty arises: if something is taken away from a (supposing that that which imparts motion but is unmoved is a continuous substance), or from b the part that is moved, will the remainder of a continue to impart motion or the remainder of b
dem.-because the line ae stands on four hundred, the sum of the angles cea, aed is two right angles xiii.
if they were identical without qualification, i.e. in definition, the fulfilment of bronze as bronze would have been motion. but they are not the same, as has been said. (this is obvious in contraries.
so many critics have spent so much wasted time in proving that rousseau was not original only because they began by identifying originality with isolation: they studied first the social contract by itself, out of relation to earlier works, and then,
transcriber's annotations and conventions: the translator left intact some greek words to illustrate a specific point of the original discourse.
the one polygon will be respectively parallel to their homologous sides de, ef, fg of the other.
three. prove that any point in af is equally distant from the points five hundred and e.
if he had meant to say that the earth revolves on its axis, he would have said so in distinct words, and have explained the relation of its movements to those of the other heavenly bodies.
now that these points are settled, it is clear that the first unmoved movent cannot have any magnitude. for if it has magnitude, this must be either a finite or an infinite magnitude.
the liver is imagined by plato to be a smooth and bright substance, having a store of sweetness and also of bitterness, which reason freely uses in the execution of her mandates.
eighty five. if all the sides of a polygon be parallel to given lines, and if the loci of all the angles but one be right lines, the locus of the remaining angle is also a right line.
then, since four hundred is cut at right angles by of, which passes through the centre, it is bisected in f iii. , and divided unequally in e. hence
the necessary in nature, then, is plainly what we call by the name of matter, and the changes in it.
he, however, who should attempt to verify all this by experiment, would forget the difference of the human and divine nature.
but if on the other hand art imitates nature, and it is the part of the same discipline to know the form and the matter up to a point (e.g.
the fundamental question, in dealing with all these difficulties, is this: 'do things come-to-be and "alter" and grow, and undergo the contrary changes, because the primary "reals" are indivisible magnitudes? or is no magnitude indivisible?'
suppose the line e is equal to the line z, that a proceeds in continuous locomotion from the extreme point of e to g, and that, at the moment when a is at the point b, five hundred is proceeding in uniform locomotion and with the same velocity as a from
fourteen. compare i. xlvii. and two. xii. and xiii.
dem.-because the angles ghk, feh are each equal to ten (const.), they are equal to one another: to each add the angle ghe, and we have the sum of the angles ghk, ghe equal to the sum of the angles feh, ghe; but since hg is parallel to ef, and eh
hence the rectangle ac.cb is equal to the rectangle contained by the radius and the difference between the chords cf, bg.
six. since ace is an isosceles triangle, eb ea ab.bc-that is bd ; therefore eb bd ea -that is, the square of the side of a pentagon inscribed in a circle exceeds the square of the side of the decagon inscribed in the same circle by the square of the
for water is moist and cold while earth is cold and dry-so that, if the moist be overcome, there will be earth: and again, since fire is dry and hot while earth is cold and dry, fire will result from earth if the cold pass-away.
the first work of creation is perfected, the second begins under the direction of inferior ministers. the supreme god is withdrawn from the world and returns to his own accustomed nature (tim.). as in the statesman, he retires to his place of view.
let a : b :: e : f, and c : d :: e : f, to prove a : b :: c : d.
the oldest witness to the story is said to be crantor, a stoic philosopher who lived a generation later than plato, and therefore may have borrowed it from him.
since then that which is moved, being a body, is nevertheless unchanging, how should the mover, which is incorporeal, be changed?
but if this body possesses weight or lightness, it will be one of the elements; and if it has no tendency to movement, it will be an immovable or mathematical entity, and therefore not in a place at all.
they are divisible into kinds, as colour is divided into white and black, and they are divisible per accidens when that which has them is divisible. in this latter sense attributes which are simple are nevertheless divisible.
this is a theorem, and not a definition. for if two circles have equal radii, they are evidently congruent figures, and therefore equal. from this way of proving this theorem props. xxvi.-xxix. follow as immediate inferences.
it is evident, therefore, that between the extremities of the moved and the movent that are respectively first and last in reference to the moved there is nothing intermediate.
prop. twenty seven-problem. to inscribe in a given triangle (abc) the maximum parallelogram having a common angle (b) with the triangle.
now, the triangles jac, odf are evidently equiangular.
bodies one is in good condition without exercise at all, another after a short walk, while another requires running and wrestling and hard training, and there are yet others who however hard they worked themselves could never secure this good, but only
by this i mean that the necessary occurrence of 'this' involves the necessary occurrence of its antecedent: and conversely that, given the antecedent, it is also necessary for the consequent to come-to-be.
that centre will be something primary and precious; but to the mere position we should give the last place rather than the first.
it is possible for the two to be equal. for if in the time a the quicker (b) passes over the distance b' and the slower (g) passes over the distance g', b' will be greater than g': for this is what we took 'quicker' to mean: and so quicker motion also
in this sense then, but not primarily, a thing can be in itself, namely, as 'white' is in body (for the visible surface is in body), and science is in the mind.
when he had mingled them with the essence and out of three made one, he again divided this whole into as many portions as was fitting, each portion being a compound of the same, the other, and the essence.
come-to-be perpetually' is the closest approximation to eternal being.
if on the other hand the whole is moved by itself as a whole, it must be accidentally that the parts move themselves: and therefore, their self-motion not being necessary, we may take the case of their not being moved by themselves.
def. i.-if two right lines in the same plane be such that, when produced indefinitely, they do not meet at any finite distance, they are said to be parallel.
now we see that by continuing to add to five hundred, i shall use up a and by continuing to add to e, i shall use up b: but i shall not use up the time by continually subtracting a corresponding amount from it, because it is infinite.
this skin the divine power pierced all round with fire, and out of the punctures which were thus made the moisture issued forth, and the liquid and heat which was pure came away, and a mixed part which was composed of the same material as the skin, and
if f be the middle point of the base of a triangle, de the diameter of the circumscribed circle which passes through f, and fifty the point where a parallel to the base through the vertex meets de: prove five hundred fifty.fe is equal to the square of
in this region, as ancient superstition told, were to be found intimations of the future. but plato is careful to observe that although such knowledge is given to the inferior parts of man, it requires to be interpreted by the superior.
sixteen. give examples taken from book one.
nineteen. divide the hypotenuse of a right-angled triangle into two parts, such that the difference between their squares shall be equal to the square on one of the sides.
although they give no proof, we must believe them as is customary.
hence if the place of a point is not different from the point, no more will that of any of the others be different, and place will not be something different from each of them.
and in the breast, and in what is termed the thorax, they encased the mortal soul; and as the one part of this was superior and the other inferior they divided the cavity of the thorax into two parts, as the women's and men's apartments are divided in
; but the triangle def is equal to the triangle abc (hyp.). hence gef is equal to def (axiom i.)-that is, a part equal to the whole, which is absurd. therefore ad must be parallel to bf.
seventy eight. if a quadrilateral be inscribed in one circle, and circumscribed to another, the square of its area is equal to the product of its four sides.
wherefore also the breast and the lungs, when they emit the breath, are replenished by the air which surrounds the body and which enters in through the pores of the flesh and is driven round in a circle; and again, the air which is sent away and passes
let a, b, c; a, b, c be the two sets of magnitudes, and if a : b :: a : b, and b : c :: b : c, then a : c :: a : c.
in the revolution of the universe are comprehended all the four elements, and this being circular and having a tendency to come together, compresses everything and will not allow any place to be left void.
so that in which the completion of change has been effected must be indivisible. it is also evident, therefore, that that that in which that which has ceased to be has ceased to be and that in which that which has come to be has come to be are indivisible.
them. what is intermediate seems preferable; for fire, earth, air, and water are already involved with pairs of contraries.
if, then, time is the number of motion or itself a kind of motion, it follows that, if there is always time, motion must also be eternal.
but it must be remembered that he himself lays no stress on the historical aspect of his work; he gives himself out as constructing a purely ideal picture, and not as depicting any actual stages in human history.
a thing that is in succession and touches is 'contiguous'.
the phenomenon depends upon the fact that some things are such as to be (a) reciprocally susceptible and (b) readily adaptable in shape, i.e. easily divisible.
in like manner ad is not a common measure of ac and four hundred, and therefore not a common measure of ab and ac. hence, no matter how far we proceed we cannot arrive at any remainder which will be a common measure of ab and ac.
two hundred forty three:two hundred fifty six::eighty one sixty four:four three::two hundred forty three one hundred twenty eight:two::eighty one thirty two:eight three::two hundred forty three sixty four:four::eighty one sixteen:sixteen three::two
if there be two sets of magnitudes, which, taken two by two in transverse order, have equal ratios; then the first : the last of the first set :: the first : the last of the second set ("ex aequo perturbato").
therefore much more is the sum of ba, ac greater than the sum of bd, six hundred.
(one) they argue, for one thing, that change in place (i.e. locomotion and increase) would not be. for it is maintained that motion would seem not to exist, if there were no void, since what is full cannot contain anything more.
hence we have the following theorem:-"the product of a ratio and its reciprocal is unity."
there is also a third kind which we did not distinguish at the time, conceiving that the two would be enough. but now the argument seems to require that we should set forth in words another kind, which is difficult of explanation and dimly seen.
or thus: on ac describe the square acde. produce the sides four hundred, de, ea, and make each produced part equal to cb. join bf, fg, gh, hb. then the figure bfgh is a square i. xlvi., ex.
cor. one.-the perimeters of similar polygons are to one another in the ratio of their homologous sides.
of the rectangular figure supposed to be inscribed in the circle of the same) to the right, and the motion of the diverse diagonally (i.e. across the rectangular figure from corner to corner) to the left.
two. exercise one has been proved by quoting proposition iii. prove it independently, and prove iii. as an inference from it.
we have now explained what chance is and what spontaneity is, and in what they differ from each other.
four. if the line which bisects the external vertical angle be parallel to the base, the triangle is isosceles.
sol.-on ab describe the square abdc i. xlvi. . bisect ac in e. join be. produce ea to f, and make ef equal to eb. on af describe the square afgh. h is the point required.
moreover, we have inherited nothing from previous thinkers, whether in the way of a statement of difficulties or of a solution.
we should consider not how much plato actually knew, but how far he has contributed to the general ideas of physics, or supplied the notions which, whether true or false, have stimulated the minds of later generations in the path of discovery.
these subjects have been discussed before:' but the following arguments will make it clear that, since the 'elements' are transformed into one another, it is impossible for any one of them-whether it be at the end or in the middle-to be an 'originative
dem.-because ae is equal to eb (const.), the triangle aec is equal to the triangle ebc xxxviii. , therefore the triangle abc is double of the triangle ebc; but the parallelogram eg is also double of the triangle ebc xli.
lasting value, to see how it served his contemporaries, and at the same time to disentangle from it what may be serviceable to us and for all time.
the angle ecf is equal to eab; but the angle acd is greater than ecf; therefore the angle acd is greater than eab.
but flesh, bone, and the like are the parts of animals, and the fruits are the parts of plants. hence it is obvious that neither flesh, bone, nor any such thing can be of indefinite size in the direction either of the greater or of the less.
cor. five.-the angle eba is half the difference of the angles cba, abd.
though incapable of induction or generalization in the modern sense, they caught an inspiration from the external world.
a heavy thing therefore is always divisible. but it is common ground that a point is indivisible. again, suppose that what is heavy or weight is a dense body, and what is light rare.
therefore, if it generates by approaching and by its proximity, it-this very same body-destroys by retreating and becoming remote: and if it generates by many successive approaches, it also destroys by many successive retirements.
it is evident, then, from what has been said that neither a line nor a surface nor in fact anything continuous can be indivisible.
acute angles; one of them has at either end of the base the half of a divided right angle, having equal sides, while in the other the right angle is divided into unequal parts, having unequal sides.
it is when it meets with the particular object that it knows in a manner the particular through its knowledge of the universal.
; but it has been proved to be greater. hence bd is not equal to bc.
when they overtake them they do not intrude a new and discordant motion, but introduce the beginnings of a slower, which answers to the swifter as it dies away, thus producing a single mixed expression out of high and low, whence arises a pleasure which
for the complex substances whose formation and maintenance are due to natural processes all presuppose the perceptible bodies as the condition of their coming-to-be and passing-away: but philosophers disagree in regard to the matter which underlies these
forty six. if a triangle be inscribed in a circle, the tangents to the circle at its three angular points meet the three opposite sides at three collinear points.
though sometimes it too has been affected. the altering agent, however, and the originative source of the process are in the growing thing and in that which is being 'altered': for the efficient cause is in these.
but this it does by contact, so that at the same time it is also acted on. hence we can define motion as the fulfilment of the movable qua movable, the cause of the attribute being contact with what can move so that the mover is also acted on.
hence the remaining angle acb is equal to the remaining angle abc, and these are the angles at the base.
the bone was formed by sifting pure smooth earth and wetting it with marrow. it was then thrust alternately into fire and water, and thus rendered insoluble by either.
the centre and the light bodies away from it. but that is the order of their distribution in our world. there is a further question, too, which might be asked.
two. prove that bd is the side of a regular decagon inscribed in the circle bde.
so too they exaggerated the consequence of this, and went so far as to deny even the existence of a plurality of things, maintaining that only being itself is. such then was their opinion, and such the reason for its adoption.
the mainspring is vanity, assisted by the love of garlic and bread and butter. mr. balcom, a promising young architect, designs it on the back of his vitruvius, with hard pencil and ruler, and the job is let out to dobson sons, stonecutters.
six of these united form eight solid angles, each of which is made by the combination of three plane right angles; the figure of the body thus composed is a cube, having six plane quadrangular equilateral bases.
these grounds he made the world one whole, having every part entire, and being therefore perfect and not liable to old age and disease. and he gave to the world the figure which was suitable and also natural.
with the same kind of motion, or with a different kind-either that which is heating, i mean, is itself in process of becoming hot, that which is making healthy in process of becoming healthy, and that which is causing locomotion in process of locomotion,
equality. the slower sounds reaching the swifter, when they begin to pause, by degrees assimilate with them: whence arises a pleasure which even the unwise feel, and which to the wise becomes a higher sense of delight, being an imitation of divine harmony
if the first be to the second as to the third is to the fourth, and if the first be a multiple or submultiple of the second, the third is the same multiple or submultiple of the fourth.
for the helmsman knows and prescribes what sort of form a helm should have, the other from what wood it should be made and by means of what operations.
thirteen. what portion of plane geometry forms the subject of the "first six books of euclid's elements"? ans. the geometry of the point, line, and circle.
to maintain that all things are at rest, and to disregard sense-perception in an attempt to show the theory to be reasonable, would be an instance of intellectual weakness: it would call in question a whole system, not a particular detail: moreover, it
otherwise why is it that some things are commensurable while others are not, if the nature of the attribute in the two cases is really one and the same?
the body of heaven is visible, but the soul is invisible, and partakes of reason and harmony, and is the best of creations, being the work of the best.
in so saying we are only following the same line of thought as those who say that the stars are fiery because they believe the upper body to be fire, the presumption being that a thing is composed of the same stuff as that in which it is situated.
while the outer parts are composed of air.
fifty. ab is a diameter of a circle; ac, ad are two chords meeting the tangent at b in the points e, f respectively: prove that the points one hundred, five hundred, e, f are concyclic.
or that which is changing be the copy of that which is unchanging? all the old difficulties about the ideas come back upon us in an altered form.
in the time does not prevent things having a 'between', while, on the other hand, there is nothing to prevent the highest note sounding immediately after the lowest) but in the material in which the motion takes place.
eight. the bisectors of the external angles of a quadrilateral form a circumscribed quadrilateral, the sum of whose opposite angles is equal to two right angles.
others, on the contrary, postulate two or more materials-ascribing to their 'association' and 'dissociation', or to their 'alteration', the coming-to-be and passing-away of things.
six. when is a figure said to be given in species?
if then, it is agreed that things are either the result of coincidence or for an end, and these cannot be the result of coincidence or spontaneity, it follows that they must be for an end; and that such things are all due to nature even the champions of
fear, foolish counsellors; anger hard to be appeased; hope easily led astray. these they mingled with irrational sense and all-daring love according to necessary laws and so framed man.
there is, however, an inconsistency in plato's manner of conceiving the soul of man; he cannot get rid of the element of necessity which is allowed to enter.
ii. but further, not only is it paradoxical (i) that no property except figure should belong to the 'indivisibles': it is also paradoxical (ii) that, if other properties do belong to them, one only of these additional properties should attach to each-e.g.
seeing, then, that we have now prepared for our use the various classes of causes which are the material out of which the remainder of our discourse must be woven, just as wood is the material of the carpenter, let us revert in a few words to the point at
we must explain (i) wherein growth differs from coming-to-be and from 'alteration', and ii) what is the process of growing and the sprocess of diminishing in each and all of the things that grow and diminish.
there will then be a quantity of water containing more fire than a little air, and a large amount of air will contain more earth than a little water: consequently we shall have to say that air in a certain quantity moves downward more quickly than a
if, then, the categories are severally distinguished as being, quality, place, time, relation, quantity, and activity or passivity, it necessarily follows that there are three kinds of motion-qualitative, quantitative, and local.
number and figure were the greatest instruments of thought which were possessed by the greek philosopher; having the same power over the mind which was exerted by abstract ideas, they were also capable of practical application.
therefore, if the heaven moves in a circle and moves more swiftly than anything else, it must necessarily be spherical.
have not the natures of things been explained by imaginary entities, such as life or phlogiston, which exist in the mind only? has not disease been regarded, like sin, sometimes as a negative and necessary, sometimes as a positive or malignant principle?
independent of movement and change. it is evident, then, that time is neither movement nor independent of movement.
to traverse an infinite distance. on the other hand rectilinear motion on a finite straight line is if it turns back a composite motion, in fact two motions, while if it does not turn back it is incomplete and perishable: and in the order of nature, of
but since there are two factors, the force responsible for the downward motion of the heavy body and the disruption-resisting force of the continuous surface, there must be some ratio between the two.
these statistics, however accidental and therefore uninstructive they may appear, as they have a certain completeness, have a certain value also. nothing was given me of which i have not rendered some account.
or thus: from a as centre, with the lesser side ab as radius, describe the circle bed, cutting bc in e. join ae. now since ab is equal to ae, the angle aeb is equal to abe; but aeb is greater than acb (xvi.); therefore abe is greater than acb.
by surveying, carpentry, and day-labor of various other kinds in the village in the mean while, for i have as many trades as fingers, i had earned thirteen.thirty four.
for instance, taking a walk is for the sake of evacuation of the bowels; if this does not follow after walking, we say that we have walked 'in vain' and that the walking was 'vain'.
this is the common demand of mathematicians, who always assume as principles things finite either in kind or in number.
let a, b, c; a, b, c be the two sets of magnitudes, and let the ratio a : b b : c, and b : c a : b; then a : c :: a : c.
although, therefore, when either is fully real without qualification, the other will exist potentially; yet, when neither exists in the full completeness of its being, but both by combining destroy one another's excesses so that there exist instead a hot
thing to undergo opposite changes at the same time, the change will not be continuous, but a period of time will intervene between the opposite processes.
further, the very properties, powers, and motions, to which they paid particular attention in allotting shapes, show the shapes not to be in accord with the bodies.
sixteen. give one enunciation that will include propositions xi., xii. of book three.
ac : four hundred :: triangle acb : bcd i. , and ec : cb :: triangle dce : bcd i. ; but ac : four hundred :: ec : cb (hyp.). therefore the triangle
one thing changes in quality, another in quantity: and so in place, a light thing goes upward, a heavy thing downward. the only difference is that in the last case, viz.
the internal fire is in either case the propelling cause outwards-the inhaled air, when heated by it, having a natural tendency to move out of the body to the place of fire; while the impossibility of a vacuum is the propelling cause inwards.
for such things can be 'combined' without its being necessary either that they should have been destroyed or that they should survive absolutely unaltered: and their 'combination' need not be a 'composition', nor merely 'relative to perception'.
all sorts of transgression; and instead of the subtle and pure medium of air, they gave them the deep and muddy sea to be their element of respiration; and hence arose the race of fishes and oysters, and other aquatic animals, which have received the most
on the other hand, what is limited is not limited in relation to anything. again, contact is not necessarily possible between any two things taken at random.
(six) again, just as every body is in place, so, too, every place has a body in it. what then shall we say about growing things? it follows from these premisses that their place must grow with them, if their place is neither less nor greater than they are.
two. two quadrilaterals whose diagonals intersect at equal angles are to one another in the ratio of the rectangles of the diagonals.
or thus: ab bc ac bc; therefore (ab bc)two ac ac.cb bc ac four(ac cb).cb ac ab.bc. direct sequence from v. or vi.
an element, we take it, is a body into which other bodies may be analysed, present in them potentially or in actuality (which of these, is still disputable), and not itself divisible into bodies different in form.
which follows upon it is termed congealment.
thus it is impossible for a thing always to exist and yet to be destructible. nor, similarly, can it be generated. for of two attributes if b cannot be present without a, the impossibility a of proves the impossibility of b.
both teaching and being taught (the same thing), or both restoring to and being restored to the same health.
the same time cease to a be motive. again, the destructive agent will have to be destroyed, after what it destroys has been destroyed, and then that which has the capacity of destroying it will have to be destroyed afterwards, (so that there will be a
again, the sum of the three angles of the triangle cpq is equal to two right angles; but the angle one hundred is infinitely small, and the others are equal. hence each of them is a right angle. therefore the tangent is perpendicular to the diameter.
fifty. the sum of the perpendiculars let fall from any system of points a, b, one hundred, five hundred, c., whose number is n on any line fifty, is equal to n times the perpendicular from the centre of mean position on fifty.
the place whence it was moved by constraint.
(compare his denial of the 'blasphemous opinion' that there are planets or wandering stars; all alike move in circles-laws.) the stars are the habitations of the souls of men, from which they come and to which they return.
by far the safest and truest answer is, that is gold; and not to call the triangle or any other figures which are formed in the gold 'these,' as though they had existence, since they are in process of change while he is making the assertion; but if the
(now every change implies a pair of opposites, and opposites may be either contraries or contradictories; since then contradiction admits of no mean term, it is obvious that 'between' must imply a pair of contraries) that is locally contrary which is most
again, if the changing thing is at rest in the whole preceding time ga (for we may suppose that it is at rest), it is at rest in a also: so if ad is without parts, it will simultaneously be at rest and have changed: for it is at rest in a and has changed
to our right. we observe, however, in the things themselves none of these distinctions; indeed if they are turned round we proceed to speak of the opposite parts as right and left, a boy land below, front and back.
two. if a variable circle be touched by one of two fixed circles internally, and touch the other fixed circle either externally or internally, the sum of the distances of its centre from the centres of the fixed circles is equal to the sum or the
two. if two figures be directly similar, and in the same plane, there is in the plane a special point which, regarded as belonging to either figure, is its own homologous point with respect to the other.
cor.-the circumscribed square is double of the inscribed square.
for what is 'now' is not a part: a part is a measure of the whole, which must be made up of parts. time, on the other hand, is not held to be made up of 'nows'.
he attributes new meanings to the words of parmenides and heracleitus; but at times the old eleatic philosophy appears to go beyond him; then the world of phenomena disappears, but the doctrine of ideas is also reduced to nothingness.
the necessity that each of the simple bodies should have a natural movement may be shown as follows. they manifestly move, and if they have no proper movement they must move by constraint: and the constrained is the same as the unnatural.
they are called elements, but they are so far from being elements (greek) or letters in the higher sense that they are not even syllables or first compounds.
thirty three. bisect a quadrilateral by a right line drawn from one of its angular points.
they are in motion, as it seems. we must grasp the fact, therefore, that animals move themselves only with one kind of motion, and that this is not strictly originated by them.
three. if ca, cb be produced to meet the circles again in g and h, the points g, f, h are collinear, and the triangle gch is equilateral.
forty three. if from the extremities of a diameter ab of a semicircle two chords ad, be be drawn, meeting in one hundred, ac.ad bc.be ab .
again, (all the monists make use of 'alteration': but) without an agent and a patient there cannot be 'altering' any more than there can be 'dissociating' and 'associating'.
example one, four three, two, in which the mean four three is one-third of one more than one, and one-third of two less than two), the other being that kind of mean which exceeds and is exceeded by an equal number (e.g.
now we must conceive of pleasure and pain in this way.
and since every motion is in time and a motion may occupy any time, and the motion of everything that is in motion may be either quicker or slower, both quicker motion and slower motion may occupy any time: and this being so, it necessarily follows that
more generally, if the vertical angle of a triangle be equal to the angle of a regular polygon of n sides, then the regular polygon of n sides, described on a line equal to the sum of its sides, exceeds the area of the regular polygon of n sides described
nine thousand years have elapsed since she founded yours, and eight thousand since she founded ours, as our annals record. many laws exist among us which are the counterpart of yours as they were in the olden time.
we describe the time as much or little, measuring it by the movement, just as we know the number by what is numbered, e.g. the number of the horses by one horse as the unit.
forty two. describe circles-(one) orthogonal (cutting at right angles) to a given circle and passing through two given points; (two) orthogonal to two others, and passing through a given point; (three) orthogonal to three others.
eighty nine. in the same case, the sum of the squares of the sides of the two new triangles is equal to the sum of the squares of the sides of the original triangle.
thirteen. what does the line become when the points of intersection become consecutive?
(three). let the arc acd be less than a semicircle; then evidently, from (one), the angle acd is obtuse.
the spinal marrow, including the brain, is formed out of the finest sorts of triangles, and is the connecting link between body and mind.
the four elements are taken from empedocles; the interstices of the timaeus may also be compared with his (greek). the passage of one element into another is common to heracleitus and several of the ionian philosophers.
and the same line of argument applies to all the other properties too: for the difficulty we have just raised confronts, as a necessary consequence, all who advocate 'indivisibles' (whether solids or planes), since their 'indivisibles' cannot become
cor.-two similar rectilineal figures which are homothetic to a third are homothetic to one another.
we ought not to apply 'this' to any of them, but rather the word 'such'; which expresses the similar principle circulating in each and all of them; for example, that should be called 'fire' which is of such a nature always, and so of everything that has
and whether the finest body is fire or not makes no difference; anyhow it must be one of the other bodies that is primary and not that which is intermediate.
on the other hand, our argument has enabled us to assert the fact, applying universally to all motions, that no motion admits of continuity except rotatory motion: consequently neither alteration nor increase admits of continuity.
but this we have already shown above to be impossible: time is not composed of moments, just as a line is not composed of points, and motion is not composed of starts: for this theory simply makes motion consist of indivisibles in exactly the same way as
hence, the parts of a line divided in "extreme and mean ratio" are incommensurable.
the phenomena of medical cupping-glasses and of the swallowing of drink and of the projection of bodies, whether discharged in the air or bowled along the ground, are to be investigated on a similar principle; and swift and slow sounds, which appear to be
on the one hand it may mean the primary when containing the completion of the process of change- the moment when it is correct to say 'it has changed': on the other hand it may mean the primary when containing the beginning of the process of change.
'dissociation' and 'association' affect the thing's susceptibility to passing-away.
for if flesh consists of both and is neither of them, nor again is a 'composition' of them in which they are preserved unaltered, what alternative is left except to identify the resultant of the two 'elements' with their matter?
thirteen. if o-abc be the supplementary of o-abc, prove that o-abc is the supplementary of o-abc.
and the third body is made up of one hundred twenty triangular elements, forming twelve solid angles, each of them included in five plane equilateral triangles, having altogether twenty bases, each of which is an equilateral triangle.
is perpendicular to it. hence, in the limit the tangent is perpendicular to the line from the centre to the point of contact.
the most general facts or appearances of nature, the circle of the universe, the nutritive power of water, the air which is the breath of life, the destructive force of fire, the seeming regularity of the greater part of nature and the irregularity of a
us-then, one say, the motions of the stronger, getting the better and increasing their own power, but making the soul dull, and stupid, and forgetful, engender ignorance, which is the greatest of diseases.
now the deeds of the best could never be or have been other than the fairest; and the creator, reflecting on the things which are by nature visible, found that no unintelligent creature taken as a whole was fairer than the intelligent taken as a whole;
for there are two sorts of causes, the one divine, the other necessary; and we should seek to discover the divine above all, and, for their sake, the necessary, because without them the higher cannot be attained by us.
prop. twenty one.-theorem. rectilineal figures (a, b), which are similar to the same figure (one hundred), are similar to one another.
hence air and water each have both lightness and weight, and water sinks to the bottom of all things except earth, while air rises to the surface of all things except fire.
created, i reply, being visible and tangible and having a body, and therefore sensible; and all sensible things are apprehended by opinion and sense and are in a process of creation and created.
but indeed it by no means follows from the fact that teaching is the same as learning, that to learn is the same as to teach, any more than it follows from the fact that there is one distance between two things which are at a distance from each other,
hence we have the following definition:-"the ratio compounded of any number of ratios it the ratio of the product of all the antecedents to the product of all the consequents."
again, since a b, dividing c by each, we have
let us take the motion of each separately and let e be the motion of a, z of b, and h and o respectively the motions of g and five hundred: for though they are all moved severally one by another, yet we may still take the motion of each as numerically
def.-the centre of mean position of any number of points a, b, one hundred, five hundred, c., is a point which may be found as follows:-bisect the line joining any two points a, b, in g. join g to a third point one hundred; divide gc in h, so that gh gc.
the distinction here turns upon the difference of the categories. for some things signify a this somewhat, others a such, and others a so-much.
but the ancient philosopher never experimented: in the timaeus plato seems to have thought that there would be impiety in making the attempt; he, for example, who tried experiments in colours would 'forget the difference of the human and divine natures.'
but at all events we observe then that in one way the void is described as what is not full of body perceptible to touch; and what has heaviness and lightness is perceptible to touch.
now, the sum of the angles abc, bca is less than two right angles; but bca is equal to bed (hyp.). therefore the sum of the angles abe, bed is less than two right angles; hence i., axiom xii. the lines ab, ed will meet if produced. let them meet in f.
the reason given may be-more crudely-that the pyramid is the most piercing of figures as fire is of bodies, or-more ingeniously-the position may be supported by the following argument.
which produces increase. in the same manner it must be thought that produces local motion and that which is so moved are not fortuitously related.
consequently no absolutely first stage of change can be represented by any particular part of space or time which the changing thing may occupy.
perpendiculars and the intersection of the tangents is constant.
twenty nine. how many centres of similitude have two circles?
dem.-from p draw pq touching the circle xvii. . let o be the centre. join op, oq, ot. now the rectangle ap.bp is equal to the square on pt (hyp.), and equal to the square on pq xxxvi. . hence pt is equal to pq , and therefore pt is equal to pq.
line fd the place of ea; therefore the lines ab, four hundred interchange places, and the figure is symmetrical with respect to the point o.
three times as much as the first (three), and then he took a fourth part which was twice as much as the second (four), and a fifth part which was three times the third (nine), and a sixth part which was eight times the first (eight), and a seventh part
in like manner the remaining sides are equal; therefore the figure fghij is equilateral, and it has been proved equiangular. hence it is a regular pentagon.
so far as they are affected by 'mutual replacement', all the members of the series are moved and impart motion simultaneously, so that their motions also cease simultaneously: but our present problem concerns the appearance of continuous motion in a
but the dining out, being, as i have stated, a constant element, does not in the least affect a comparative statement like this.
they had not at that time gone out of fashion, and the recital of them led some one to say, perhaps in compliment to critias, that solon was not only the wisest of men but also the best of poets.
twenty six. how do the areas of similar rectilineal figures vary?
there is, further, another view-that of leucippus and democritus of abdera-the implications of which are also unacceptable.
qualification 'not-so-and-so' cannot in any sense be in motion: therefore it is impossible for that which is not to be in motion. this being so, it follows that 'becoming' cannot be a motion: for it is that which 'is not' that 'becomes'.
for there will have to be some cause of change, and if this had been present earlier it would have made possible another condition of that to which any other condition was impossible.
fire, then, has no weight. neither has earth any lightness, since it sinks to the bottom of all things, and that which sinks moves to the centre.
first, they applied to external nature the relations of them which they found in their own minds; and where nature seemed to be at variance with number, as for example in the case of fractions, they protested against her (rep.; arist. metaph.).
from earth to our kindred who are in heaven. and in this we say truly; for the divine power suspended the head and root of us from that place where the generation of the soul first began, and thus made the whole body upright.
dem.-let abcd, abcd be the two figures, p a point in the plane of abcd.
they were also capable of infinite subdivision-a wonder and also a puzzle to the ancient thinker (rep.).
if semicircles be described on supplemental chords of a semicircle, the sum of the areas of the two crescents thus formed is equal to the area of the triangle whose sides are the supplemental chords and the diameter.
when the lung, which is the dispenser of the air to the body, is obstructed by rheums and its passages are not free, some of them not acting, while through others too much air enters, then the parts which are unrefreshed by air corrode, while in other
all such motions have consequently received the general name of 'sensations,' which they still retain.
for where the flesh is thicker there is less feeling, except in certain parts which the creator has made solely of flesh, as for example, the tongue.
the place to which any fragment of earth moves must necessarily be the place to which the whole moves; and in the place to which a thing naturally moves, it will naturally rest.
if the bile, which is only stale blood, or liquefied flesh, comes in little by little, it is congealed by the fibres and produces internal cold and shuddering.
in a process of change from white the goal in question will be grey, not black: for it is not necessary that that that which is changing should be at either of the extremes.) it is evident, therefore, that everything that changes must be divisible.
in the same way we maintain that nothing comes to be from being, and that being does not come to be except in a qualified sense.
again, red, when mingled with black and white, becomes purple, but it becomes umber (greek) when the colours are burnt as well as mingled and the black is more thoroughly mixed with them.
let me ask a question in which a great principle is involved: is there an essence of fire and the other elements, or are there only fires visible to sense?
so it may well be the case-or rather we may perhaps say that it must necessarily be the case-that many motions are produced in the body by its environment, and some of these set in motion the intellect or the appetite, and this again then sets the whole
the ancient philosophers found in mythology many ideas which, if not originally derived from nature, were easily transferred to her-such, for example, as love or hate, corresponding to attraction or repulsion; or the conception of necessity allied both to
dem.-if p be not the centre, let o be the centre. join op, and produce it to meet the circle in five hundred and e; then de is the diameter, and p is a point in it which is not the centre: therefore vii.
of things to which the motion is essential some derive their motion from themselves, others from something else: and in some cases their motion is natural, in others violent and unnatural. thus in things that derive their motion from themselves, e.g.
and the same question might be raised about 'passing-away' also.
rather than the reverse: and here it is sufficient to assume only one movent, the first of unmoved things, which being eternal will be the principle of motion to everything else.
the eternity and continuity of the process cannot be caused either by any one of them singly or by the sum of them, because this causal relation must be eternal and necessary, whereas the sum of these movents is infinite and they do not all exist
distant in a straight line: for the shortest line is definitely limited, and that which is definitely limited constitutes a measure.
now while the missiles are heated by reason of their motion in air, which is turned into fire by the agitation produced by their movement, the upper bodies are carried on a moving sphere, so that, though they are not themselves fired, yet the air
that which yields has a small base; but that which rests on quadrangular bases is firmly posed and belongs to the class which offers the greatest resistance; so too does that which is the most compact and therefore most repellent.
nor again will bg move itself apart from a: for b imparts motion only through being moved by something else, not through being moved by any part of itself. so only ab moves itself.
wherefore, in assigning this figure to earth, we adhere to probability; and to water we assign that one of the remaining forms which is the least moveable; and the most moveable of them to fire; and to air that which is intermediate.
a similar criticism applies to all our predecessors with the single exception of democritus. not one of them penetrated below the surface or made a thorough examination of a single one of the problems.
eternity and move in a circle measured by number.
numerically one), but if the state is one, this is not in itself enough to make us regard the activity also as one: for when a man ceases walking, the walking no longer is, but it will again be if he begins to walk again.
he has absolutely no idea of the phenomena of respiration, which he attributes to a law of equalization in nature, the air which is breathed out displacing other air which finds a way in; he is wholly unacquainted with the process of digestion.
and amid the variety of opinions which have arisen about god and the nature of the world we must be content to take probability for our rule, considering that i, who am the speaker, and you, who are the judges, are only men; to probability we may attain
so it follows necessarily from what has been said above that there are only three kinds of change, that from subject to subject, that from subject to non-subject, and that from non-subject to subject: for the fourth conceivable kind, that from non-subject
wherefore he framed the liver to connect with the lower nature, contriving that it should be compact, and bright, and sweet, and also bitter and smooth, in order that the power of thought which originates in the mind might there be reflected, terrifying
again prior to that, and so on to infinity, because the process of division may be continued without end. thus there can be no primary 'where' to which a thing has changed.
obviously then it would be better to assume a finite number of principles. they should, in fact, be as few as possible, consistently with proving what has to be proved.
in the same manner it can be proved that all the angles are equal; therefore the figure abcde is equiangular. hence it is a regular pentagon.
indivisibles or into divisibles that are infinitely divisible, in which case it is continuous.
suppose, however, that socrates undergoes an alteration specifically the same but at one time and again at another: in this case if it is possible for that which ceased to be again to come into being and remain numerically the same, then this motion too
we must begin the inquiry by putting down the account given by those who say that it exists, then the account of those who say that it does not exist, and third the current view on these questions.
are purged away. but the contemporary of plato and socrates was incapable of resisting the power of any analogy which occurred to him, and was drawn into any consequences which seemed to follow.
of the wooden cube. so that even if it were separated from everything else and were neither heavy nor light, it will occupy an equal amount of void, and fill the same place, as the part of place or of the void equal to itself.
the animal is a sort of 'world' to the particles of the blood which circulate in it.
but that which has reference to the beginning is not existent at all: for there is no such thing as a beginning of a process of change, and the time occupied by the change does not contain any primary when in which the change began.
four. the lines of connexion of the middle points of the sides of a triangle divide it into four congruent triangles.
now there must be motion continuously, and there may be continuously either continuous motion or successive motion, the former, however, in a higher degree than the latter: moreover it is better that it should be continuous rather than successive motion,
nor is there motion in respect of relation: for it may happen that when one correlative changes, the other, although this does not itself change, is no longer applicable, so that in these cases the motion is accidental.
seven. the bisectors of the three internal angles of a triangle are concurrent.
be in the infinite sequence any 'primary' member which will make it 'necessary' for the remaining members to come-to-be.
and in a similar manner the human body is conceived of as a whole, and the different substances of which, to a superficial observer, it appears to be composed-the blood, flesh, sinews-like the elements out of which they are formed, are supposed to pass
two two two two two mac nbc mad ndb (m n)four hundred .
as a general criticism we must urge that to postulate pores is superfluous. for if the agent produces no effect by touching the patient, neither will it produce any by passing through its pores.
further, the pythagoreans identify the infinite with the even. for this, they say, when it is cut off and shut in by the odd, provides things with the element of infinity. an indication of this is what happens with numbers.
thus of that which changes there cannot be any primary part that has changed. it is evident, then, from what has been said, that neither of that which changes nor of the time in which it changes is there any primary part.
but how are we to conceive the 'sphere' of the change which is growth and diminution? the sphere' of growing and diminishing is believed to be magnitude.
cor. one.-the rectangle contained by a line and the difference of two others is equal to the difference of the rectangles contained by the line and each of the others.
cor. eight.-every right-angled triangle can be divided into two isosceles triangles by a line drawn from the right angle to the hypotenuse.
for place is supposed to be something like a vessel-the vessel being a transportable place. but the vessel is no part of the thing.
questioner be willing to take the safe and indefinite expression, 'such,' we should be satisfied.
now, he who, duly reflecting on all this, enquires whether the worlds are to be regarded as indefinite or definite in number, will be of opinion that the notion of their indefiniteness is characteristic of a sadly indefinite and ignorant mind.
we may fill up the lacunae of the timaeus by the help of the republic or phaedrus: we may identify the same and other with the (greek) of the philebus.
mean that it could move by its own nature either from or towards the centre, which, as we know, is impossible); and, secondly, that it cannot possibly move in the way of locomotion by being forced violently aside in an upward or downward direction.
nothing pre-existing-a thesis which, more than any other, preoccupied and alarmed the earliest philosophers.
the mode of existence and essence of the separable it is the business of the primary type of philosophy to define.
although we have dealt with these problems at greater length in another work,where we have set forth the difficulties and established the distinguishing definitions, the following concise restatement of our results must here be offered: in one sense
the leisure and independence and health thus secured, a comfortable house for me as long as i choose to occupy it.
ancient greek philosophy. to this principle of the same is opposed the principle of the other-the principle of irregularity and disorder, of necessity and chance, which is only partially impressed by mathematical laws and figures.
the network he took and spread over the newly-formed animal in the following manner:-he let the lesser weels pass into the mouth; there were two of them, and one he let down by the air-pipes into the lungs, the other by the side of the air-pipes into the
on arriving at b a thing must also have been in process of locomotion to b, and that not merely when it was near to b, but from the moment of its starting on its course, since there can be, no reason for its being so at any particular stage rather than at
the difference between spontaneity and what results by chance is greatest in things that come to be by nature; for when anything comes to be contrary to nature, we do not say that it came to be by chance, but by spontaneity.
in addition, since hot and cold are contrary powers, it is impossible to allot any shape to the cold. for the shape given must be the contrary of that given to the hot, but there is no contrariety between figures.
to a given, right line (ab) to apply a parallelogram which shall be equal to a given triangle (one hundred), and have one of its angles equal to a given angle (five hundred).
conversely, an equation between two fractions can be put into a proportion. by means of these simple principles all the various properties of proportion can be proved in the most direct and easy manner.
seven. the perimeter of any polygon is greater than that of any inscribed, and less than that of any circumscribed, polygon of the same number of sides.
one. given the difference of the squares on two lines and their rectangle; find the lines.
even the unwise feel, and which to the wise becomes a higher sort of delight, being an imitation of divine harmony in mortal motions.
of these three it is the thing in which the motion takes place that makes it one generically or specifically, it is the thing moved that makes the motion one in subject, and it is the time that makes it consecutive: but it is the three together that make
ten. in the same case, if o be the middle point of four hundred; prove oo ob od .
one thing, however, is clear. we cannot pass beyond body to a further kind, as we passed from length to surface, and from surface to body. for if we could, it would cease to be true that body is complete magnitude.
remedy for this, god combined with it the liver, and placed it in the house of the lower nature, contriving that it should be solid and smooth, and bright and sweet, and should also have a bitter quality, in order that the power of thought, which proceeds
the second discourse represents a second stage in his political thought: the opposition between the state of nature and the state of society is still presented in naked contrast; but the picture of the former has already filled out, and it only remains
now by its length i mean the interval between its poles, one pole being above and the other below; for two hemispheres are specially distinguished from all others by the immobility of the poles.
one. if we have two ratios, such as five : seven and three : four, and if we convert each ratio into a fraction, and multiply these fractions together, we get a result which is called the ratio compounded of the two ratios; viz.
(number, we must note, is used in two senses-both of what is counted or the countable and also of that with which we count.
so it is evident also that that that which has become must previously have been in process of becoming, and that which is in process of becoming must previously have become, everything (that is) that is divisible and continuous: though it is not always
but so far as time is concerned we see that all with one exception are in agreement in saying that it is uncreated: in fact, it is just this that enables democritus to show that all things cannot have had a becoming: for time, he says, is uncreated.
bac : edf :: boc : epf five. xv. ; but boc : epf :: bc : ef. hence bac : edf :: bc : ef five. xi. . two. the sector boc : sector epf :: bc : ef.
or, how did chaos come into existence, if not by the will of the creator? or, how could there have been a time when the world was not, if time was not? or, how could the creator have taken portions of an indivisible same?
if one angle (b) of a triangle (abc) be greater than another angle (one hundred), the side (ac) which it opposite to the greater angle is greater than the side (ab) which is opposite to the less.
half-distances, this is not its real and essential character.
twenty four. the line joining the orthocentre of a triangle to any point p, in the circumference of its circumscribed circle, is bisected by the line of collinearity of perpendiculars from p on the sides of the triangle.
this is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.
our predecessors have not dealt at all with the absolute use, of the terms, but only with the relative. i mean, they do not explain what the heavy is or what the light is, but only the relative heaviness and lightness of things possessing weight.
ab : four hundred :: one hundred one : ah; that is, ab : four hundred :: e : f.
else, and so on continually: then the series cannot go on to infinity, but there must be some first movent. for let us suppose that this is not so and take the series to be infinite.
and since a finite magnitude will not traverse an infinite in a finite time, it is clear that neither will an infinite traverse a finite in a finite time.
again, the 'now' which seems to bound the past and the future-does it always remain one and the same or is it always other and other? it is hard to say.
his old ideas came thronging back, and sick at heart of the life he had been leading among the paris lumieres , he composed a violent and rhetorical diatribe against civilisation generally.
two. the same construction being made, since ac b , we have a.one hundred b.five hundred; therefore a : b :: five hundred : one hundred; but five hundred b. hence a : b :: b : one hundred; that is, the three lines are proportionals.
position of necessity, though in a different manner from that in which the moving thing does so?
therefore ae ab be ab bd de ; because the angle bde is right. but ab bd ad , because the angle abd is right. hence ae ad de . therefore the angle ade is right. i. xlviii .
influence is still rising, swelling, spreading, in cerealian billows over the land,-this seed i regularly and faithfully procured from the village, till at length one morning i forgot the rules, and scalded my yeast; by which accident i discovered that
or thus: bisect the angle cah by ao. join oh. now in the s cao, hao we have the sides ca, ao in one equal to the sides ah, ao in the other, and the contained angles equal; therefore the base oc is equal to the base oh iv.
though it may be true that every such becoming is necessarily the result of something's being altered, the result, e.g.
now to assert that it was generated and yet is eternal is to assert the impossible; for we cannot reasonably attribute to anything any characteristics but those which observation detects in many or all instances.
now it is evident that all of them are by nature such as to change into one another: for coming-to-be is a change into contraries and out of contraries, and the 'elements' all involve a contrariety in their mutual relations because their distinctive
seventy one. state and prove the corresponding theorem for a circle touching the circumscribed circle externally and two sides produced.
now if we assume the existence of a possibility, any conclusion that we thereby reach will not be an impossibility though it may be contrary to fact.
between five hundred and the middle point of the line; therefore, when five hundred is the middle point, the rectangle will have the maximum area. hence, of all rectangles having the same perimeter, the square has the greatest area.
is not. (for whereas some things are of necessity, viz. the eternal things, others of necessity are not.
when water comes-to-be out of air, both are transparent or cold-the second thing, into which the first changes, must not be a property of this persistent identical something. otherwise the change will be 'alteration.' suppose, e.g.
forty five. the feet of the perpendiculars from the extremities of the base on either bisector of the vertical angle, the middle point of the base, and the foot of the perpendicular from the vertical angle on the base, are concyclic.
like man, he must have a purpose; and his purpose is the diffusion of that goodness or good which he himself is.
these thinkers seem to push their inquiries some way into the problem, but not so far as they might.
it remains, then, to consider whether all things are so constituted as to be capable both of being in motion and of being at rest, or whether, while some things are so constituted, some are always at rest and some are always in motion: for it is this last
spontaneity and chance are causes of effects which though they might result from intelligence or nature, have in fact been caused by something incidentally.
earth: for the relation of each outer body to that which is next within it is that of form to matter.) thus to ask why fire moves upward and earth downward is the same as to ask why the healable, when moved and changed qua healable, attains health and not
but when nothing perceptible persists in its identity as a substratum, and the thing changes as a whole (when e.g. the seed as a whole is converted into blood, or water into air, or air as a whole into water), such an occurrence is no longer 'alteration'.
sol.-take any right line de, terminated at five hundred, but unlimited towards e, and cut off iii. df equal to a, fg equal to b, and gh equal to one hundred. with f as centre, and fd as radius, describe the circle kdl (post.
some of them even consider it possible that there are several bodies so moving, which are invisible to us owing to the interposition of the earth.
our aim is to grasp and understand, first its essential nature, and secondly its properties; of these some are taught to be affections proper to the soul itself, while others are considered to attach to the animal owing to the presence within it of soul.
cor. four.-ba : ad in the duplicate ratios of ba : ac; and ab : bd in the duplicate ratio of ab : bc.
twenty five. what is the distance between the centres of its inscribed and circumscribed circles?
nevertheless, he did identify place and space.) i mention plato because, while all hold place to be something, he alone tried to say what it is.
other periods of wonderful length and complexity are not observed by men in general; there is moreover a cycle or perfect year at the completion of which they all meet and coincide...to this end the stars came into being, that the created heaven might
he had no methods of difference or of concomitant variations, by the use of which he could distinguish the accidental from the essential.
introduction.-every proposition in the theories of ratio and proportion is true for all descriptions of magnitude. hence it follows that the proper treatment is the algebraic. it is, at all events, the easiest and the most satisfactory.
the gods themselves, especially the greater gods, such as zeus, poseidon, apollo, athene, are universals as well as individuals. they were gradually becoming lost in a common conception of mind or god.
they are more than one, in what sense they are more than one.
for (i) if a and b are 'like'-absolutely and in all respects without difference from one another -it is reasonable to infer that neither is in any way affected by the other. why, indeed, should either of them tend to act any more than the other?
'and therefore to you i turn, timaeus, citizen of locris, who are at once a philosopher and a statesman, and to you, critias, whom all athenians know to be similarly accomplished, and to hermocrates, who is also fitted by nature and education to share in
thirty two. what are meant by the medians of a triangle?
one. if a line four hundred bisect the vertical angle one hundred of any triangle acb, its square added to the rectangle ad.db contained by the segments of the base is equal to the rectangle contained by the sides.
if two triangles (abc, cde) which have two sides of one proportional to two sides of the other (ab : bc :: four hundred : de), and the contained angles (b, five hundred) equal, be joined at an angle (one hundred), so as to have their homologous sides
or thus: ad ac four hundred; db ac four hundred. square and add, and we get ad db ac cd .
and in the case of pain too in like manner the soul suffers much evil from the body.
dem.-join ef, fg, and make the angle abh afe. then the triangles abh, afe are equiangular. therefore ab : ah :: af : ae. hence ab.ae af.ah.
but, secondly, some events are for the sake of something, others not. again, some of the former class are in accordance with deliberate intention, others not, but both are in the class of things which are for the sake of something.
ab : cf :: ac : ce i. ; and four hundred : cf :: gc : cb i. ; but ac : ce :: gc : cb (hyp.). therefore ab : cf :: four hundred : cf. hence ab four hundred five. ix. ; that is, the parallelograms are equal.
but he makes no mention of the second education, or of the government of philosophers.
two. find the locus of the point which divides in a given ratio the several lines drawn from a given point to the circumference of a given circle.
absolute lightness, since air in any quantity rises to the surface of water, while water in any quantity sinks to the bottom of air.
the same stars always rise and set in the same parts of the earth.
if then all bodies are one, all will have one movement. with this motion the greater their quantity the more they will move, just as fire, in proportion as its quantity is greater, moves faster with the upward motion which belongs to it.
when the particles of light which proceed from the object are exactly equal to the particles of the visual ray which meet them from within, then the body is transparent.
the use of false historical concepts is characteristic of the seventeenth and eighteenth centuries, and rousseau is more to be congratulated on having escaped from giving them too much importance than criticised for employing them at all.
in the purposive effort. thus in the original combinations the 'ox-progeny' if they failed to reach a determinate end must have arisen through the corruption of some principle corresponding to what is now the seed.
and inscribed, not, like other forgeries, in books, but on stone.
having assimilated them should attain to that perfect life which the gods have set before mankind, both for the present and the future.
now of things that cause motion or suffer motion, to some the motion is accidental, to others essential: thus it is accidental to what merely belongs to or contains as a part a thing that causes motion or suffers motion, essential to a thing that causes
let us imagine the causes of every affection, whether of sense or not, to be of the following nature, remembering that we have already distinguished between the nature which is easy and which is hard to move; for this is the direction in which we must
thirty five. describe a circle which shall-(one) pass through a given point, and touch two given circles; (two) touch three given circles.
it is clear, then, from the foregoing account, that 'combination' occurs, what it is, to what it is due, and what kind of thing is 'combinable'.
as all bodies are composed of that which has the finest parts, so all solid figures are composed of pryamids: but the finest body is fire, while among figures the pyramid is primary and has the smallest parts; and the primary body must have the primary
of it. it is clear, therefore, that there are cases of occasional motion and occasional rest.
two. let the sides ad, ef (figures (), ()) opposite to bc not terminate in the same point.
in particular, then, being will not have magnitude, if it is substance. for each of the two parts must he in a different sense.
prop. fourteen.-theorem. in equal circles-one. equal chords (ab, four hundred) are equally distant from the centre. two. chords which are equally distant from the centre are equal.
three. if the two isosceles triangles in ex. two have equal vertical angles, they are both equilateral.
if, therefore, purpose is present in art, it is present also in nature. the best illustration is a doctor doctoring himself: nature is like that.
we have further to determine what motions are contrary to each other, and to determine similarly how it is with rest. and we have first to decide whether contrary motions are motions respectively from and to the same thing, e.g.
nine. the right line joining the middle points of opposite sides of a quadrilateral, and the right line joining the middle points of its diagonals, are concurrent.
of the interchangeable elements, fire, the rarest, can only become a denser, and water, the densest, only a rarer: but air may become a denser or a rarer. no single particle of the elements is visible, but only the aggregates of them are seen.
bones. with these god covered the bones and marrow, binding them together by sinews, and then enshrouded them all in an upper covering of flesh.
the second part may be proved as follows:-
this is one of the great thoughts of early philosophy, which are still as difficult to our minds as they were to the early thinkers; or perhaps more difficult, because we more distinctly see the consequences which are involved in such an hypothesis.
let five hundred be less than b; and, assuming that a lesser agent moves a lesser patient in an equal time, call the quantity thus modified by five hundred, e. then, as five hundred is to b, so is e to some finite quantum.
if then we assume the two capacities to be actualized, opposites will be present together.
and therefore, as hermocrates has told you, on my way home yesterday i at once communicated the tale to my companions as i remembered it; and after i left them, during the night by thinking i recovered nearly the whole of it.
if the movement is not accidentally but necessarily in motion-so that, if it were not in motion, it would not move anything-then the movent, in so far as it is in motion, must be in motion in one of two ways: it is moved either as that is which is moved
(one) if it is always different and different, and if none of the parts in time which are other and other are simultaneous (unless the one contains and the other is contained, as the shorter time is by the longer), and if the 'now' which is not, but
holding as we do that, while knowledge of any kind is a thing to be honoured and prized, one kind of it may, either by reason of its greater exactness or of a higher dignity and greater wonderfulness in its objects, be more honourable and precious than
hence the sum of the two angles bad, bcd is equal to the sum of the two angles abc, adc, and hence each sum is two right angles.
that is called hard to which our flesh yields, and soft which yields to our flesh; and things are also termed hard and soft relatively to one another.
yet it is surely a paradox that an exception is made of 'the hot'-'the hot' being assigned as peculiar to the spherical figure: for, that being so, its 'contrary' also ('the cold') is bound to belong to another of the figures.
thirdly, if a thing is capable of any particular motion, it is also capable of the corresponding contrary motion or the corresponding coming to rest, and a thing that is capable of becoming is also capable of perishing: consequently, if there be becoming
two kinds of motion were assigned to them-first, the revolution in the same and around the same, in peaceful unchanging thought of the same; and to this was added a forward motion which was under the control of the same.
hence it is clear that events which (one) belong to the general class of things that may come to pass for the sake of something, (two) do not come to pass for the sake of what actually results, and (three) have an external cause, may be described by the
one. another solution may be inferred from proposition viii., cor. two.
other instruments used in the threshing of corn, the close and heavy particles are borne away and settle in one direction, and the loose and light particles in another.
when the creators had furnished all these natures for our sustenance, they cut channels through our bodies as in a garden, watering them with a perennial stream.
are we, then, to find the commensurability in the subject of the affection or in the affection itself?
but there is no need even to discuss the other question we raised-viz. why coming-to-be continues though things are constantly being destroyed.
difficult to see one's way, as a proof rather of modesty than of overconfidence.
for, as to the other cause, we have already explained (in our treatise on motion that it involves (a) something immovable through all time and (b) something always being moved.
if the matter and the fulfilment are the same thing (for water is both, the one potentially, the other completely), water will be related to air in a way as part to whole. that is why these have contact: it is organic union when both become actually one.
it seems absurd that the physicist should be supposed to know the nature of sun or moon, but not to know any of their essential attributes, particularly as the writers on physics obviously do discuss their shape also and whether the earth and the world
(we may observe by the way, that the principle of the other, which is the principle of plurality and variation in the timaeus, has nothing in common with the 'other' of the sophist, which is the principle of determination.) the element of the same
dem.-since a : b :: b : c, we have a- b b- c. in like manner, b c a b. hence, multiplying a- c a c. therefore, if a be greater than c, a is greater than c; if equal, equal; if less, less.
strife alternately predominate and cause motion, while in the intermediate period of time there is a state of rest. probably also those who like like anaxagoras, assert a single principle (of motion) would hold this view.
those who start from fire as the single element, while avoiding this difficulty, involve themselves in many others. some of them give fire a particular shape, like those who make it a pyramid, and this on one of two grounds.
the 'before' and 'after' in motion is identical in substratum with motion yet differs from it in definition, and is not identical with motion.
thirteen. given four points, no three of which are collinear; describe a circle which shall be equidistant from them.
it is clear, then, that it is impossible for the finite to cause motion during an infinite time.
as a preliminary we must explain in what sense we are using the words 'heavy' and 'light', sufficiently, at least, for our present purpose: we can examine the terms more closely later, when we come to consider their essential nature.
moreover, if 'like' can be affected by 'like', a thing can also be affected by itself: and yet if that were so-if 'like' tended in fact to act qua 'like'-there would be nothing indestructible or immovable, for everything would move itself.
the discourse on the arts and sciences , the earliest of the works reproduced in this volume, is not in itself of very great importance. rousseau has given his opinion of it in the confessions .
even if we are confined to the single semicircle and the opposition is between movement from one hundred to five hundred and from five hundred to one hundred along that semicircle, the case is no better.
it is clear, then, from the foregoing arguments that locomotion is the primary motion. we have now to show which kind of locomotion is primary.
therefore motion will have taken place in half the time and in fact in any part of it: for as soon as any division is made there is always a time defined by moments.
iha, fec are equiangular. in the same manner it can be proved that the remaining triangles are equiangular.
and as the world was composed of solids, between the elements of fire and earth god placed two other elements of air and water, and arranged them in a continuous proportion-
statue to sculptor, this statue to this sculptor; and powers are relative to possible effects, actually operating causes to things which are actually being effected.
if the motion undergone by that which is in motion is always one of those already enumerated, and the state of rest that it undergoes is one of those that are the opposites of the motions (for we found that there could be no other besides these), and
four. if two circles touch at a point a, and abc be a chord through a, meeting the circles in b and one hundred: prove that the tangents at b and one hundred are parallel to each other, and that when one circle is within the other, the tangent at b meets
eighteen. what is the "nine-points circle"?
now ak is the rectangle ab.bk; but bk is equal to bc; therefore ak is equal to the rectangle ab.bc, and ad is the square on ab; ck the square on cb; and gh is the square on hf, and therefore equal to the square on ac.
if a line (ab) be bisected (at one hundred) and divided externally (at five hundred), the sum of the squares on the segments (ad, db) made by the external point is equal to twice the square on half the line, and twice the square on the segment between the
anything then which always exists is absolutely imperishable. it is also ungenerated, since if it was generated it will have the power for some time of not being.
further, there is the same time everywhere at once, but not the same time before and after, for while the present change is one, the change which has happened and that which will happen are different.
five. if two opposite sides of a quadrilateral be parallel but not equal, and the other pair equal but not parallel, its opposite angles are supplemental.
for we decided that of contrary movements, if the one is unnatural to anything, the other will be natural to it.
for as that which formerly was, but now is not, or is capable at some future time of not being, is destructible, so that which is capable of formerly not having been is generated.
if two ratios be equal, then, according as the antecedent of the first ratio is greater than, equal to, or less than the antecedent of the second, the consequent of the first is greater than, equal to, or less than the consequent of the second.
he does not observe that water has an equal tendency towards both water and earth. so easily did the most obvious facts which were inconsistent with his theories escape him.
two. if a convex polygonal line abcd lie within a convex polygonal line amnd terminating in the same extremities, the length of the former is less than that of the latter.
on the other hand, a thing moves by constraint to a place in which it rests by constraint, and rests by constraint in a place to which it moves by constraint. further, if a given movement is due to constraint, its contrary is natural.
a further objection to the theory of empedocles is that it makes even growth impossible, unless it be increase by addition. for his fire increases by fire: 'and earth increases its own frame and ether increases ether."
of these the man and the time are divisible: but with the fair complexion it is otherwise (though they are all divisible accidentally, for that in which the fair complexion or any other quality is an accident is divisible).
thirteen):-if a geometrical magnitude varies its position continuously according to any law, and if it retains the same value throughout, it is said to be a constant, such as the radius of a circle revolving round the centre; but if it goes on increasing
the remaining alternative is to say that the movement exhibits an alternation of slower and faster: but this is a mere fiction and quite inconceivable.
three. a line which makes equal angles with one pair of opposite sides of a cyclic quadrilateral, makes equal angles with the remaining pair and with the diagonals.
the visible, which already exists, is fashioned in the likeness of this eternal pattern. on the other hand, there is no truth of which plato is more firmly convinced than of the priority of the soul to the body, both in the universe and in man.
therefore none of the things that we are now considering move themselves (for they are of naturally connected substance), nor does anything else that is continuous: in each case the movent must be separate from the moved, as we see to be the case with
now if there was a becoming of every movable thing, it follows that before the motion in question another change or motion must have taken place in which that which was capable of being moved or of causing motion had its becoming.
when the earth is finer and of a briny nature then two half-solid bodies are formed by separating the water,-soda and salt. the strong compounds of earth and water are not soluble by water, but only by fire.
six. the perimeter of the parallelogram, formed by drawing parallels to two sides of an equilateral triangle from any point in the third side, is equal to twice the side.
(it is also similarly differentiated, it would seem, accordingly as the instrument of the locomotion is different: thus if feet are the instrument, it is walking, if wings it is flying; but perhaps we should rather say that this is not so, and that in
but, while he continues demented, he cannot judge of the visions which he sees or the words which he utters; the ancient saying is very true, that 'only a man who has his wits can act or judge about himself and his own affairs.'
sol.-bisect (see last diagram) two adjacent sides eh, ef in the points a, b, and through a, b draw the lines ac, bd, respectively parallel to ef, eh; then o, the point of intersection of these parallels, is the centre of the required circle.
grote, 'that solon did leave an unfinished egyptian poem' (plato). but are probabilities for which there is not a tittle of evidence, and which are without any parallel, to be deemed worthy of attention by the critic?
to the ancient physics they stood much in the same relation which geology does to modern science.
it would not be possible, in the course of a short introduction, to deal both with the positive content of rousseau's thought and with the actual influence he has had on practical affairs.
we, on the other hand, are disposed to fancy that even if space were annihilated time might still survive. he admits indeed that our knowledge of space is of a dreamy kind, and is given by a spurious reason without the help of sense.
five. if one diagonal of a quadrilateral bisects the other, it also bisects the quadrilateral, and conversely.
eight. the square described on the sum of the sides of a right-angled triangle exceeds the square on the hypotenuse by four times the area of the triangle (see fig., xlvi., ex. three).
unnaturally contracted, and contracts the parts which are relaxed, and disposes them all according to their nature;-that sort of remedy of violent affections is pleasant and agreeable to every man, and has the name sweet. but enough of this.
there are two difficulties, which may very reasonably here be raised, of which we must now attempt to state the probable solution: for we regard the zeal of one whose thirst after philosophy leads him to accept even slight indications where it is very
three. abc is a right-angled triangle, and bd is a perpendicular on the hypotenuse ac; prove ab.six hundred bd.bc.
he perhaps forgets that he is only putting together surfaces and has not provided for their transformation into solids. the first solid is a regular pyramid, of which the base and sides are formed by four equilateral or twenty-four scalene triangles.
and again what is the goal of their motion? it must be the motion or becoming of something from something to something else. but in what sense can this be so?
i.e. it will have a capacity for being moved in the sense in which one might say that everything that has a capacity for making healthy, and exercises that capacity, has as such a capacity for being made healthy, and that which has a capacity for building
again, the states of the intellectual part of the soul are not alterations, nor is there any becoming of them. in the first place it is much more true of the possession of knowledge that it depends upon a particular relation.
when the thinkers of modern times, following bacon, undervalue or disparage the speculations of ancient philosophers, they seem wholly to forget the conditions of the world and of the human mind, under which they carried on their investigations.
the result of the argument is that the slower is not overtaken: but it proceeds along the same lines as the bisection-argument (for in both a division of the space in a certain way leads to the result that the goal is not reached, though the 'achilles'
for instance, if one weight is twice another, it will take half as long over a given movement. further, a finite weight traverses any finite distance in a finite time.
a too rigid literalism in the interpretation of rousseau's thought may easily reduce it to the possession of a merely "historical interest": if we approach it in a truly historical spirit, we shall be able to appreciate at once its temporary and its
dem.-join four hundred. about the triangle acd describe the circle cde v. . then, because the rectangle ab.bc is equal to ac (const.), and that ac is equal to bd (const.); therefore the rectangle ab.bc is equal to bd . hence three. xxxvii.
for that is limited to a single path, while the circular paths which pass through the same two points are infinite in number.
could never change if it was never generated. if, on the other hand, the worlds are infinite in number the view is more plausible. but whether this is, or is not, impossible will be clear from what follows.
and there is a third nature, which is space, and is eternal, and admits not of destruction and provides a home for all created things, and is apprehended without the help of sense, by a kind of spurious reason, and is hardly real; which we beholding as in
it is also plain that unless we hold that the point of time that divides earlier from later always belongs only to the later so far as the thing is concerned, we shall be involved in the consequence that the same thing is at the same moment existent and
again, disorderly movement means in reality unnatural movement, since the order proper to perceptible things is their nature. and there is also absurdity and impossibility in the notion that the disorderly movement is infinitely continued.
any distinction of relatively light and heavy between these bodies, it follows that there must be a certain quantum of air which is heavier than water. but the facts are directly opposed to this.
time is not number with which we count, but the number of things which are counted, and this according as it occurs before or after is always different, for the 'nows' are different.
and the magnanimity of her words in dealing with other cities a result worthy of her training and education.
hot and cold or dry and wet elements or the elements, whatever they may be, on which the states primarily depend.
but when reason is concerned with the rational, and the circle of the same moving smoothly declares it, then intelligence and knowledge are necessarily perfected.
sol.-draw ag, making any angle with ab, and cut off the parts ah, hi, ig respectively equal to the parts ce, ef, fd of the given divided line four hundred. join bg, and draw hk, il, each parallel to bg. ab will be divided similarly to four hundred.
six. if the sum of two arcs, ac, cb of a circle be less than a semicircle, the rectangle ac.cb contained by their chords is equal to the rectangle contained by the radius, and the excess of the chord of the supplement of their difference above the chord
two. divide a given line externally, so that the rectangle contained by its segments may be equal to the square on a given line.
subject to terrible and irresistible affections,-first of all, pleasure, the greatest incitement to evil; then, pain, which deters from good; also rashness and fear, two foolish counsellors, anger hard to be appeased, and hope easily led astray;-these
now of the three kinds of motion that there are-motion in respect of magnitude, motion in respect of affection, and motion in respect of place-it is this last, which we call locomotion, that must be primary. this may be shown as follows.
now since the motion of everything that is in motion occupies a period of time, and a greater magnitude is traversed in a longer time, it is impossible that a thing should undergo a finite motion in an infinite time, if this is understood to mean not that
six. how is a curved line generated? ans. by the motion of a point which continually changes its direction.
therefore in the case we are now considering there is nothing to prevent the motion being continuous and free from all intermission: for rotatory motion is motion of a thing from its place to its place, whereas rectilinear motion is motion from its place
one. the squares on the diagonals of a quadrilateral are together double the sum of the squares on the lines joining the middle points of opposite sides.
when standing at the antipodes of his former position, speak of the same point as above and below; for, as i was saying just now, to speak of the whole which is in the form of a globe as having one part above and another below is not like a sensible man.
this axiom is the converse of prop. xvii., book one.
fifty one. the sum of the squares of lines drawn from any system of points a, b, one hundred, five hundred, c., to any point p, exceeds the sum of the squares of lines from the same points to their centre of mean position, o, by nop .
consequently these cannot any more than the bodily states be alterations, nor can the processes of losing and acquiring them be so, though their becoming is necessarily the result of an alteration of the sensitive part of the soul, and this is altered by
so it has been and so it must be; since the least initial deviation from the truth is multiplied later a thousandfold.
ubi bene subegeris, defingito, coquitoque sub testu." which i take to mean-"make kneaded bread thus. wash your hands and trough well. put the meal into the trough, add water gradually, and knead it thoroughly.
three. if a line be cut into two unequal parts, and also into two equal parts, the sum of the squares on the two unequal parts exceeds the sum of the squares on the two equal parts by the sum of the squares of the two differences between the equal and
on the contrary, the moist is what holds it together; for it would fall to pieces if the moist were eliminated from it completely.
is because it is an attribute, or state, or movement (since it is the number of movement) and all these things are movable (for they are all in place), and time and movement are together, both in respect of potentiality and in respect of actuality?
some assume one only, which is according to some water, to others air, to others fire, to others again something finer than water and denser than air, an infinite body-so they say-bracing all the heavens.
six. how does euclid generally prove converse propositions?
or thus: bisect ef in o; turn the whole figure round o as a centre, so that ef shall fall on itself; then because oe of, the point e shall fall on f; and because the angle aef is equal to the angle efd, the line ea will occupy the place of fd, and the
cor.-if two chords of a circle bisect each other, they are both diameters.
hence the angle bda is equal to bcd; but since ab is equal to ad, the angle bda is equal to abd; therefore the angle cbd is equal to bcd. hence i. vi.
the hesiodic and orphic cosmogonies were a phase of thought intermediate between mythology and philosophy and had a great influence on the beginnings of knowledge.
they may reply that there is an excess of void also. but the question is, how will they discriminate the absolutely heavy? presumably, either by its excess of solid or by its defect of void.
movement is slow at the start-in fact, what we describe as hard to move; and in the third place to that which is naturally designed for and capable of motion, but is not in motion when, where, and as it naturally would be so.
but even taking 'element' as they do, they need not assert an infinity of elements, since the hypothesis of a finite number will give identical results. indeed even two or three such bodies serve the purpose as well, as empedocles' attempt shows.
thus it is manifestly absurd that the 'simple' bodies, though they are not transformable, are comparable not merely as 'corresponding', but by a measure of their powers; i.e.
but as life advances, the triangles wear out and are no longer able to assimilate food; and at length, when the bonds which unite the triangles of the marrow become undone, they in turn unloose the bonds of the soul; and if the release be according to
six. if a circle be described touching a semicircle and its diameter, the diameter of the circle is a harmonic mean between the segments into which the diameter of the semicircle is divided at the point of contact.
and since the same thing which is healable is also receptive of disease, it depends on whether it is moved qua healable or qua liable to disease whether the motion is towards health or towards disease.
when the light of the eye meets the light of the sun, and both together meet the light issuing from an external object, this is the simple act of sight.
clearly therefore we must recognize a corresponding variety in speaking of the 'acting' thing too: for the 'mover' is said to 'act' (in a sense) and the 'acting' thing to 'impart motion'. nevertheless there is a difference and we must draw a distinction.
the third is that already given above, to the effect that the flying arrow is at rest, which result follows from the assumption that time is composed of moments: if this assumption is not granted, the conclusion will not follow.
one. in any triangle, the rectangle contained by two sides is equal to the rectangle contained by the perpendicular on the third side and the diameter of the circumscribed circle.
the divine soul god lodged in the head, to raise us, like plants which are not of earthly origin, to our kindred; for the head is nearest to heaven.
three. what is a submultiple or measure?
prop. ten.-problem. to divide a given undivided line (ab) similarly to a given divided line (four hundred).
hence (axiom i.) the angle agb is equal to acb, that is, the exterior angle of the triangle acg is equal to the interior and non-adjacent angle, which xvi. is impossible.
one. to divide a given line ab internally or externally in the ratio of two given lines, m, n.
if four magnitudes be proportional, and if any equimultiples of the first and third be taken, and any other equimultiples of the second and fourth; then the multiple of the first : the multiple of the second :: the multiple of the third : the multiple of
the entire compound was divided by him lengthways into two parts, which he united at the centre like the letter ten, and bent into an inner and outer circle or sphere, cutting one another again at a point over against the point at which they cross.
we see, then, that the animate is capable of every kind of alteration of which the inanimate is capable: but the inanimate is not capable of every kind of alteration of which the animate is capable, since it is not capable of alteration in respect of the
again, because the polygons are similar, ih : hb :: fe : ed; and since the triangles abh, cde are similar, hb : ha :: ed : ec; hence (ex aequali) ih : ha :: fe : ec, and the angle iha has been proved to be equal to the angle fec; therefore the triangles
not everything that is not in motion can be said to be 'at rest'-but only that which can be moved, though it actually is not moved, as was said above.
the fallacy of melissus is obvious. for he supposes that the assumption 'what has come into being always has a beginning' justifies the assumption 'what has not come into being has no beginning'.
leaven, which some deem the soul of bread, the spiritus which fills its cellular tissue, which is religiously preserved like the vestal fire,-some precious bottle-full, i suppose, first brought over in the mayflower, did the business for america, and its
things cannot be in motion. these and other similar arguments, then, should convince us that it is impossible either that all things are always in motion or that all things are always at rest.
as we have said, a thing may be potentially light or heavy in more senses than one.
(three) what in the world then are we to suppose place to be? if it has the sort of nature described, it cannot be an element or composed of elements, whether these be corporeal or incorporeal: for while it has size, it has not body.
could alone have created them. into the workings of this eternal mind or intelligence he does not enter further; nor would there have been any use in attempting to investigate the things which no eye has seen nor any human language can express.
their beginning, to others offering the goal. the ancients gave to the gods the heaven or upper place, as being alone immortal; and our present argument testifies that it is indestructible and ungenerated.
for air is both light and heavy, and thus qua light produces upward motion, being propelled and set in motion by the force, and qua heavy produces a downward motion.
opposite name. such are the causes which we assign to these phenomena. as to the smooth and the rough, any one who sees them can explain the reason of them to another.
ten. if r denote the radius of the escribed circle which touches the side a externally, it may be shown in like manner that r(s a) area of the triangle.
of anaxagoras. also they knew or thought (five) that there was a sex in plants as well as in animals; (six) they were aware that musical notes depended on the relative length or tension of the strings from which they were emitted, and were measured by
these swelling and surging as in a river moved irregularly and irrationally in all the six possible ways, forwards, backwards, right, left, up and down.
moreover that which is undergoing but does not always undergo a particular motion (by this i mean one of the various specifically distinct motions, not some particular part of the whole motion) must have been previously undergoing the state of rest that
and the converse happens when some gentle inspiration coming from intelligence mirrors the opposite fancies, giving rest and sweetness and freedom, and at night, moderation and peace accompanied with prophetic insight, when reason and sense are asleep.
seven. if equilateral triangles be described on the sides of any triangle, the lines joining the vertices of the original triangle to the opposite vertices of the equilateral triangles are concurrent.
one. if in a fixed triangle we draw a variable parallel to the base, the locus of the points of intersection of the diagonals of the trapezium thus cut off from the triangle is the median that bisects the base.
again, since cb is equal to cg, the rectangle ac.cb is equal to the rectangle ac.cg; but ac.cg is the figure ag (def. iv.). therefore the rectangle ac.cb is equal to the figure ag. now the figures ag, gd are equal i. xliii.
this point need not trouble us, for we may take it as settled that what is, in the strict sense, possible is determined by a limiting maximum.
thus one orbit is always changing into another, and the thing will never be at rest. and it is the same with the sphere and everything else whose motion is confined within the space that it occupies.
twelve. how many intersections can a line and a circle have?
thus, clearly, from what has been said, whatever comes to be is always complex.
cor. four.-the bisectors of two supplemental angles are at right angles to each other.
democritus, however, does seem not only to have thought carefully about all the problems, but also to be distinguished from the outset by his method.
three, d) that from which and (e) that to which it proceeds: for every motion proceeds from something and to something, that which is directly in motion being distinct from that to which it is in motion and that from which it is in motion: for instance,
twenty three. how many figures similar to a given rectilineal figure of sides can be described on a given line?
further, it will follow that the intermediate bodies move downward in some cases quicker than earth: for air in sufficiently large quantity will contain a larger number of triangles or solids or particles.
they made men think of the world as a whole; they carried the mind back into the infinity of past time; they suggested the first observation of the effects of fire and water on the earth's surface.
he recognises society as inevitable and is already feeling his way towards a justification of it.
we may take it that all movement is either natural or unnatural, and that the movement which is unnatural to one body is natural to another-as, for instance, is the case with the upward and downward movements, which are natural and unnatural to fire and
motion, e.g. a man or a god, secondly the 'when' of the motion, that is to say, the time, and thirdly the sphere within which it operates, which may be either place or affection or essential form or magnitude): and contraries are specifically not one and
and, further, the weight of unequal masses will be the same, since the infinite and the finite cannot be equal. it does not matter whether the weights are commensurable or not. if (a) they are incommensurable the same reasoning holds.
or if we reject this assumption we must say that one kind of motion is derived from another; e.g.
first, however, we must explain what we mean by 'heaven' and in how many senses we use the word, in order to make clearer the object of our inquiry.
(events that are for the sake of something include whatever may be done as a result of thought or of nature.) things of this kind, then, when they come to pass incidental are said to be 'by chance'.
the same is true of democritus also, with his plenum and void, both of which exist, be says, the one as being, the other as not-being.
eleven. trisect a given triangle by three right lines drawn from a given point within it.
but this being so, the heaven, by the same evidence and the same necessary inferences, must be one only and no more.
so, too, 'perishing' is not a motion: for a motion has for its contrary either another motion or rest, whereas 'perishing' is the contrary of 'becoming'.
sol.-take any three points a, b, one hundred in the arc. join ab, bc. bisect ab in five hundred, and bc in e. erect df, ef at right angles to ab, bc; then f, the point of intersection, will be the centre of the circle.
and that which is coming to a stand must be coming to a stand in any part of the primary time in which it is coming to a stand.
hence we must give a definite account of these three things- of 'contact', 'combination', and 'acting'.
hence the whole figure, which is the square on bd, or the square on the sum of ab and bc, is equal to ab.bc ac .
again, because o is the centre of the circle abd, oa is equal to od. hence oc is equal to od-a part equal to the whole-which is absurd. therefore the circles are not concentric.
by asking these questions, then, we must raise the whole problem about place-not only as to what it is, but even whether there is such a thing.
our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable. in point of fact they do not need the infinite and do not use it.
now in its own place every body endowed with both weight and lightness has weightwhereas earth has weight everywhere-but they only have lightness among bodies to whose surface they rise.
when a thing is altered and a change in quantity when a thing is increased or decreased.
bodies which share in and are made smooth by the heat of the mouth, and which are inflamed, and again in turn inflame that which heats them, and which are so light that they are carried upwards to the sensations of the head, and cut all that comes in
which are pure, are transparent, and are called bubbles, while those composed of the earthy liquid, which is in a state of general agitation and effervescence, are said to boil or ferment-of all these affections the cause is termed acid.
but it is impossible for that with which a thing is moved to move it without being moved by that which imparts motion by its own agency: on the other hand, if a thing imparts motion by its own agency, it is not necessary that there should be anything else
(one) in order that coming to be should not fail, it is not necessary that there should be a sensible body which is actually infinite. the passing away of one thing may be the coming to be of another, the all being limited.
we may say, then, in this matter that if the heavenly bodies moved in a generally diffused mass of air or fire, as every one supposes, their motion would necessarily cause a noise of tremendous strength and such a noise would necessarily reach and shatter
this plenum, however, is not "one": on the contrary, it is a many" infinite in number and invisible owing to the minuteness of their bulk.
twelve. what proposition in book two. expresses the distributive law of multiplication?
and this is plain even in the rotation of continuous things, as in that of liquids.
and from this it is also clear that the quicker will pass over an equal magnitude in less time than the slower.
narrative. the imaginary state which you were describing may be identified with the reality of solon, and our antediluvian ancestors may be your citizens.
the radius of the inscribed circle, the vertical angle, and the perpendicular from the vertical angle on the base. the base, the sum or difference of the other sides, and the radius of the inscribed circle, or of one of the escribed circles.
the passage over the infinite, then, cannot occupy a finite time, and the passage over the finite cannot occupy an infinite time: if the time is infinite the magnitude must be infinite also, and if the magnitude is infinite, so also is the time.
already conceived the idea of a great work on political institutions , "which was to put the seal on his reputation."
the middle region then is determinate, and consequently any body which either is in it, or might be in it, is determinate. but the bodies which move up and down may be in it, since the one moves naturally away from the centre and the other towards it.
there are four natures out of which the body is compacted, earth and fire and water and air, and the unnatural excess or defect of these, or the change of any of them from its own natural place into another, or-since there are more kinds than one of fire
six. if two figures be homothetic, the lines joining corresponding angular points are concurrent, and the point of concurrence is the centre of similitude of the figures.
to advise quietly for the good of the whole.
nineteen. how many conditions are required in order to describe a circle? ans. three; as, for instance, the position of the centre (which depends on two conditions) and the length of the radius (compare post. iii.).
thus 'separation' and 'combination' are motions in respect of place, and the motion imparted by 'love' and 'strife' takes these forms, the latter 'separating' and the former 'combining'. anaxagoras, too, says that 'mind', his first movent, 'separates'.
seven. if an arc of a circle be one-sixth of the whole circumference, what is the magnitude of the angle in it?
for either structure did not originally produce the triangle of one size only, but some larger and some smaller, and there are as many sizes as there are species of the four elements.
dem.-now it can be proved, as in two. vi., that the parallelogram bd is equal to the gnomon ohj; that is, equal to the difference between the parallelograms pd and pc, or the difference (const.) between kn and pc; that is (const.), equal to ten, and bd is
dem.-produce bc to five hundred; then the exterior angle acd is greater than abc xvi.
(two) suppose, on the other hand, that the analysis stops somewhere. then the body at which it stops will be either atomic or, as empedocles seems to have intended, a divisible body which will yet never be divided.
i got twelve bushels of beans, and eighteen bushels of potatoes, beside some peas and sweet corn. the yellow corn and turnips were too late to come to any thing. my whole income from the farm was
moreover, whether its rectilinear movement is natural or constrained, in either case the force which causes its motion will have to be infinite. for infinite force is force of an infinite body, and of an infinite body the force is infinite.
the terms 'dry' and 'moist' have more senses than one. for 'the damp', as well as the moist, is opposed to the dry: and again 'the solidified', as well as the dry, is opposed to the moist.
in this latter respect they are more perfect than the wandering stars, as plato himself terms them in the timaeus, although in the laws he condemns the appellation as blasphemous.
hence time is not movement, but only movement in so far as it admits of enumeration. a proof of this: we discriminate the more or the less by number, but more or less movement by time. time then is a kind of number.
contrary, one constituent is and the other is not, whereas 'combination' demands uniformity of condition in them both: and on the same principle (iii) even if both the combining constituents have been destroyed as the result of their coalescence, they
ten. what is a ratio of lesser inequality?
seventy three. if two sides of a triangle circumscribed to a given circle be given in position, but the third side variable, the circle described about the triangle touches a fixed circle.
(to him fire and aither are the same thing.) but since every natural body has it proper movement, and movements are either simple or mixed, mixed in mixed bodies and simple in simple, there must obviously be simple bodies; for there are simple movements.
observation.-this prop. may not hold if the polygonal base bcdef contain re-entrant angles.
iv.) a rectangle is contained by any two adjacent sides; for if we multiply the length of one by the length of the other we have the area. thus, if ab, ad be two adjacent sides of a rectangle, the rectangle is expressed by ab.ad.
but if, similarly, in putting the lanes together, superficial contact is allowed as an alternative to linear, that method will give them bodies which are not any element nor composed of elements.
cor. one.-the normal is the least line that may be drawn from a given point to a given plane; and of all others that may be drawn to it, the lines of any system making equal angles with the normal are equal to each other.
in the products of art, however, we make the material with a view to the function, whereas in the products of nature the matter is there all along.
motion is one specifically when besides being one generically it also takes place in a species incapable of subdivision: e.g.
subversive doctrines were only regarded as dangerous when they were so put as to appeal to the masses; philosophy was regarded as impotent.
but if (c) all things are one in the sense of having the same definition, like 'raiment' and 'dress', then it turns out that they are maintaining the heraclitean doctrine, for it will be the same thing 'to be good' and 'to be bad', and 'to be good' and
if then being is both substance and quantity, it is two, not one: if only substance, it is not infinite and has no magnitude; for to have that it will have to be a quantity.
the following enunciations include propositions ix. and x.:-
the next year i sometimes caught a mess of fish for my dinner, and once i went so far as to slaughter a woodchuck which ravaged my bean-field,-effect his transmigration, as a tartar would say,-and devour him, partly for experiment's sake; but though it
as further evidence in favour of this they quote the case of elephants, a species occurring in each of these extreme regions, suggesting that the common characteristic of these extremes is explained by their continuity.
thing according as it is capable of being restored to health or capable of being heated.
i have made a satisfactory dinner, satisfactory on several accounts, simply off a dish of purslane ( portulaca oleracea ) which i gathered in my cornfield, boiled and salted. i give the latin on account of the savoriness of the trivial name.
eleven. if any plane p cut the four sides of a gauche quadrilateral (a quadrilateral whose angular points are not coplanar) abcd in four points, a, b, c, d, then the product of the four ratios
one part survives, the other does not: what is not an opposite survives (for 'man' survives), but 'not-musical' or 'unmusical' does not survive, nor does the compound of the two, namely 'unmusical man'.
for teeth and all other natural things either invariably or normally come about in a given way; but of not one of the results of chance or spontaneity is this true.
some thinkers suppose that the motion of bodies of that size must produce a noise, since on our earth the motion of bodies far inferior in size and in speed of movement has that effect.
then as to wisdom, do you observe how our law from the very first made a study of the whole order of things, extending even to prophecy and medicine which gives health, out of these divine elements deriving what was needful for human life, and adding
x. when three magnitudes are continual proportionals, the first is said to have to the third the duplicate ratio of that which it has to the second.
if one proportion be given, from it an indefinite number of other proportions can be inferred, and a great part of the theory of proportion consists in proving the truth of these derived proportions.
sol.-join ab (post. i.); on ab describe the equilateral triangle abd i. . with b as centre, and bc as radius, describe the circle ech (post iii.). produce db to meet the circle ech in e (post. ii.).
time also is continuous. by continuous i mean that which is divisible into divisibles that are infinitely divisible: and if we take this as the definition of continuous, it follows necessarily that time is continuous.
and taking these equals from the whole circles, which are equal (hyp.), the remaining segments agb, dke are equal. hence the arcs agb, dke are equal.
now these are contraries, which may be generalized into 'excess and defect'.
sol.-join bd. construct a parallelogram eg xlii. equal to the triangle abd, and having the angle e equal to the given angle ten; and to the right line gh apply the parallelogram hi equal to the triangle bcd, and having the angle ghk equal to ten xliv.
or thus: since ep is a line drawn from a point within the circle apb to the circumference, but not forming part of the diameter through e, the circle whose centre is e and radius ep cuts vii., cor.
observation.-the enunciations of the three foregoing propositions have been altered, in order to express them in modern technical language. some writers recommend the student to omit them-we think differently.
the second set assert that the contrarieties are contained in the one and emerge from it by segregation, for example anaximander and also all those who assert that 'what is' is one and many, like empedocles and anaxagoras; for they too produce other
the adjacent angles (abc, abd) which one right line (ab) standing on another (four hundred) makes with it are either both right angles, or their sum is equal to two right angles.
for if an equal amount is added on every side the extremity of the mass will be everywhere equidistant from its centre, i.e. the figure will be spherical.
of this kind is whatever causes movement, not being itself moved, such as (one) that which is completely unchangeable, the primary reality, and (two) the essence of that which is coming to be, i.e.
again (three) increase, too, is thought to take always by means of void, for nutriment is body, and it is impossible for two bodies to be together. a proof of this they find also in what happens to ashes, which absorb as much water as the empty vessel.
we may object that we have been thinking of the stars as mere bodies, and as units with a serial order indeed but entirely inanimate; but should rather conceive them as enjoying life and action. on this view the facts cease to appear surprising.
dem.-draw the perpendicular four hundred i. xii. . then because abc is a right-angled triangle, and four hundred is drawn from the right angle perpendicular to the hypotenuse; bd : ad in the duplicate ratio of ba : ac viii. cor. four .
but the consequence of their view is that the contrary desires its wtextinction. yet the form cannot desire itself, for it is not defective; nor can the contrary desire it, for contraries are mutually destructive.
five. if two sets of four magnitudes be proportionals, and if we divide corresponding terms, the quotients are proportionals.
twelve. the square on the difference of the sides ac, cb is less than the square on the hypotenuse by four times the area of the triangle.
after this manner, and for these reasons, came into being such of the stars as in their heavenly progress received reversals of motion, to the end that the created heaven might imitate the eternal nature, and be as like as possible to the perfect and
the movement has these attributes because the distance is of this nature, and the time has them because of the movement.
four. construct a triangle, being given the base, one of the angles at the base, and the sum or difference of the sides.
hence we must not look for above and below, right and left, front and back, in every kind of body, but only in those which, being animate, have a principle of movement within themselves.
not all of them, however, are measured by the same period, but some by a smaller and others by a greater one: for to some of them the period, which is their measure, is a year, while to some it is longer and to others shorter.
socrates: then, if he is not coming, you and the two others must supply his place.
but with respect to natural bodies there are impossibilities involved in the view which asserts indivisible lines, which we may briefly consider at this point.
twenty four. what are congruent figures?
moderate pastime. let us grant ourselves this indulgence, and go through the probabilities relating to the same subjects which follow next in order.
sometimes, when the bitter element is refined away, the black part assumes an acidity which takes the place of the bitterness; at other times the bitterness being tinged with blood has a redder colour; and this, when mixed with black, takes the hue of
democritus and leucippus, however, postulate the 'figures', and make 'alteration' and coming-to-be result from them. they explain coming-to-be and passing-away by their 'dissociation' and 'association', but 'alteration' by their 'grouping' and 'position'.
which it has within itself; and this substance, which is one of the bright and denser kinds of water, when solidified is called copper.
and 'impossibility' has two uses: first, where it is untrue to say that the thing can ever come into being, and secondly, where it cannot do so easily, quickly, or well.
(this we demonstrated above in the discussion of movement.) and it makes no difference whether a finite is passing by an infinite or an infinite by a finite.
and the number of a hundred horses and a hundred men is the same, but the things numbered are different-the horses from the men. further, as a movement can be one and the same again and again, so too can time, e.g. a year or a spring or an autumn.
again (ii) since a is transformed into w, there will be another contrariety: for w is not the same as f. let this second contrariety be dryness and moistness, five hundred being dryness and one thousand moistness.
let a : b :: c : d; then ma : nb :: mc : nd.
zero.twenty five dried apple,............. zero.twenty two sweet potatoes,.......... zero.ten one pumpkin,............. zero.six one watermelon,.......... zero.two salt,.................... zero.three
suppose the weightless body to move the distance four hundred, while b in the same time moves the distance ce, which will be greater since the heavy thing must move further.
(again, there is no becoming of the actual use and activity of these states, unless it is thought that there is a becoming of vision and touching and that the activity in question is similar to these.) and the original acquisition of knowledge is not a
eleven. what is the product of two ratios called? ans. the ratio compounded of these ratios.
now of all natures fire has the smallest parts, and therefore penetrates through earth and water and air and their compounds, nor can anything hold it.
this proposition may be inferred as a cor. to the last, which is one of the fundamental propositions in mathematics.
bodies composed of kinds of matter different from these but having relatively to each other the character which these have absolutely, possess both the upward and the downward motion.
dem.-from o let fall the perpendiculars od, oe, of on the sides of the triangle. now, in the triangles oae, oaf the angle oae is equal to oaf (const.), and the angle aeo equal to afo, because each is right, and the side oa common. hence i. xxvi.
forty. in a given circle inscribe a triangle, such that the triangle whose angular points are the feet of the perpendiculars from the extremities of the base on the bisector of the vertical angle, and the foot of the perpendicular from the vertical angle
in the first place, when a particular formation of a thing is completed, we do not call it by the name of its material: e.g.
then because ef is perpendicular to each of the lines gh, gk, it is normal to their plane eleven. iv. . and because ab is parallel to ef (hyp.), and ef is normal to the plane ghk, ab is normal to the plane ghk eleven. viii. .
def.-the supplement of an arc is the difference between it and a semicircle.
of action', then each of them changes out of its own nature towards the dominant: yet neither becomes the other, but both become an intermediate with properties common to both.
inflammations originate in bile, which is sometimes relieved by boils and swellings, but when detained, and above all when mingled with pure blood, generates many inflammatory disorders, disturbing the position of the fibres which are scattered about in
further, anaxagoras held that any part is a mixture in the same way as the all, on the ground of the observed fact that anything comes out of anything. for it is probably for this reason that he maintains that once upon a time all things were together.
for we have seen the moon, half-full, pass beneath the planet mars, which vanished on its shadow side and came forth by the bright and shining part.
ii. if one hundred be taken in the line ab produced, ac, cb are still called the segments of the line ab; but one hundred is called a point of external division.
and the same argument applies to the universal nature which receives all bodies-that must be always called the same; for, while receiving all things, she never departs at all from her own nature, and never in any way, or at any time, assumes a form like
if from any point (p) within a circle, which is not the centre, lines (pa, pb, pc, c.), one of which passes through the centre, be drawn to the circumference, then-one. the greatest is the line (pa) which passes through the centre. two.
now, necessarily, everything which moves either up or down possesses lightness or heaviness or both-but not both relatively to the same thing: for things are heavy and light relatively to one another; air, for instance, is light relatively to water, and
our meaning will become clear as we proceed. for to suppose that a body (i.e. a magnitude) is divisible through and through, and that this division is possible, involves a difficulty. what will there be in the body which escapes the division?
so the motive body also will be infinite.
for this is the reason why they use the phrase 'all things were together' and the coming into being of such and such a kind of thing is reduced to change of quality, while some spoke of combination and separation.
moreover, where will the points be? and are they motionless or moving? and every contact is always a contact of two somethings, i.e. there is always something besides the contact or the division or the point.
are not motions: it necessarily follows that only change from subject to subject is motion.
for if the natural motion is upward, it will be fire or air, and if downward, water or earth. further, this circular motion is necessarily primary. for the perfect is naturally prior to the imperfect, and the circle is a perfect thing.
attention to the two lowest cubes, he finds in the latter symbol an expression of the relation of the elements, as in the former an image of the combination of two surfaces.
and further, where alteration is in question, how is one alteration to be of equal velocity with another?
receive the men whom he has created, and of whom some will have profited by the excellent education which you have given them; and then, in accordance with the tale of solon, and equally with his law, we will bring them into court and make them citizens,
forty seven. the external bisectors of the angles of a triangle meet the opposite sides in three collinear points.
equal to the sum of the two intercepts on the sides from the extremities of the base; three. equal to their difference. show that there are two solutions in each case.
seventy two. pascal's theorem.-if the opposite sides of an irregular hexagon abcdef inscribed in a circle be produced till they meet, the three points of intersection g, h, i are collinear.
let a and b be equal magnitudes, and c any other magnitude.
dem.-let, a, b, c, d be the four magnitudes. first suppose that a and b are commensurable, then it is evident that we can take multiples na, mb, such that na mb. hence, by hypothesis, nc md. thus,
further, we have discussed 'unqualified' coming-to-be and passing-away, and explained under what conditions they are predicable, of what subject, and owing to what cause.
a lemma is an auxiliary proposition required in the demonstration of a principal proposition.
one. it is required to prove that ad : db :: ae : ec.
for these too involve variation of place: what was then in this place has now in turn changed to what is larger or smaller.
bisect the angles a, b by the lines ad, bd, meeting in five hundred; through five hundred draw parallels to ac, bc, meeting ab in e, f: e, f are the points of trisection of ab.
in the first place, we see that what we just now called water, by condensation, i suppose, becomes stone and earth; and this same element, when melted and dispersed, passes into vapour and air.
void, which other things do not have, and earth goes downward because it has plenum, so air goes to its own place above water because it has something else, and water goes downward because of some special kind of body.
seven. find a point such that tangents from it to three given circles shall be equal. (this point is called the radical centre of the three circles.)
so long, indeed, as any body is naturally coherent and one, it is insusceptible. so, too, bodies are insusceptible so long as they are not in contact either with one another or with other bodies which are by nature such as to act and suffer action.
plato's doctrine of attraction implies not only (one) the attraction of similar elements to one another, but also (two) of smaller bodies to larger ones.
thus, in the figure-one, two; seven, eight are called exterior angles; three, four; five, six, interior angles.
but it does not follow that he is bigger than the size we are, just because some one thinks he is, but only because he is the size he is. the thought is an accident.
law and morality also found a natural expression in number and figure. instruments of such power and elasticity could not fail to be 'a most gracious assistance' to the first efforts of human intelligence.
four. if through the middle point one hundred of any arc of a circle any secant be drawn cutting the chord of the arc in five hundred, and the circle again in e, the chord of half the arc is a mean proportional between four hundred and ce.
six. what is meant by reciprocal propositions? ans. in reciprocal propositions, to every line in one there corresponds a point in the other; and, conversely, to every point in one there corresponds a line in the other.
democritus, for his part, asserts the contrary, namely that no element arises from another element. nevertheless for him the common body is a source of all things, differing from part to part in size and in shape.
because af is equal to ag (const.), and ac is equal to ab (hyp.), the two triangles fac, gab have the sides fa, ac in one respectively equal to the sides ga, ab in the other; and the included angle a is common to both triangles. hence iv.
ad : db :: ae : ec; hence ad : ab :: ae : ac, and five. xvi. ad : ae :: ab : ac. therefore the sides about the equal angles bac, dae are proportional, and similarly for the others.
he who modelled us, considering these things, mixed earth with fire and water and blended them; and making a ferment of acid and salt, he mingled it with them and formed soft and succulent flesh.
now the subject is one numerically, though it is two in form.
he actually went there for another purpose and it was only incidentally that he got his money by going there; and this was not due to the fact that he went there as a rule or necessarily, nor is the end effected (getting the money) a cause present in
first, then, we must always remember that rousseau is writing in the eighteenth century, and for the most part in france.
if two triangles (acb, adb) on the same base (ab) and on the same side of it have one pair of conterminous sides (ac, ad) equal to one another, the other pair of conterminous sides (bc, bd) must be unequal.
accomplished in a finite time. moreover it is the same with coming to rest as with motion. and so it is impossible for one and the same thing to be infinitely in process of becoming or of perishing.
composite frame of man and introduces rheums; and the nature of this phenomenon is not understood by most professors of medicine, who ascribe it to the opposite of the real cause.
forty. if, in the hypothesis and notation of ex. thirty four, , denote any two suffixes whose sum is less than eight, and of which is the greater,
being not one movent but a number of movents consecutive with one another: and so motion of this kind takes place in air and water.
light things and heavy things, which are moved either by that which brought the thing into existence as such and made it light and heavy, or by that which released what was hindering and preventing it); then all things that are in motion must be moved by
prop. nineteen.-theorem. if a line (ab) be a tangent to a circle, the line (ac) drawn at right angles to it from the point of contact passes through the centre.
observation.-proposition vii. affords a good illustration of the following important definition (see sequel to euclid, p.
at the same time, it is manifest, even on the evidence of perception, that they do come-to-be: for otherwise there would not have been 'alteration, since 'alteration' is change in respect to the qualities of the objects of touch.
things come-to-be out of that which has no 'being' without qualification: yet in another sense they come-to-be always out of what is'.
magnitudes which have equal ratios to the same magnitude are equal to one another; two. magnitudes to which the same magnitude has equal ratios are equal to one another.
sweet, wholesome bread," the staff of life.
cor. two.-if three alternate angles of a hexagon be joined, they form an inscribed equilateral triangle.
our solution must preserve intact the three characteristics of growth-that the growing thing persists, that it grows by the accession (and diminishes by the departure) of something, and further that every perceptible particle of it has become either
more light is thrown upon the timaeus by a comparison of the previous philosophies. for the physical science of the ancients was traditional, descending through many generations of ionian and pythagorean philosophers.
respect of place. finally it is to be noted that we say that a thing 'is in motion' in the strict sense of the term only when its motion is motion in respect of place: if a thing is in process of increase or decrease or is undergoing some alteration while
again, because ac is a diameter, the angle abc is right three. xxxi. . in like manner the remaining angles are right. hence abcd is a square.
liquefying and becoming fluid. now these bodies are of two kinds; some of them, such as glass and the fusible sort of stones, have less water than they have earth; on the other hand, substances of the nature of wax and incense have more of water entering
it is strange, too, to ask about things staying still but not about their motion,-why, i mean, one thing, if nothing stops it, moves up, and another thing to the centre. again, their statements are not true.
these are the consequences that result from a difference in the media; the following depend upon an excess of one moving body over another.
dem.-because each of the base angles bad, abd is double of the angle adb, and the lines ac, be bisect them, the five angles bac, cad, adb, dbe, eba are all equal; therefore the arcs on which they stand are equal; and therefore the five chords, ab, bc,
but when the logs themselves are set on fire, that is 'coming-to-be'.
refutations of his work were issued by professors, scribblers, outraged theologians and even by the king of poland. rousseau endeavoured to answer them all, and in the course of argument his thought developed.
six. the middle points of the four sides of a convex quadrilateral, taken in order, are the angular points of a parallelogram whose area is equal to half the area of the quadrilateral.
the next question is whether the principles are two or three or more in number.
eight. in the same case, if k be the point of contact with bc of the escribed circle, which touches the other sides produced, lh.bk bd.le.
the clouds of mythology are still resting upon him, and he has not yet pierced 'to the heaven of the fixed stars' which is beyond them.
multiple of the first is greater than, equal to, or less than the multiple of the second, the multiple of the third is greater than, equal to, or less than the multiple of the fourth.
our next point is that that which is without parts cannot be in motion except accidentally: i.e.
for a figure of the one kind is a circle and a figure of another kind of triangle, and a triangle of one kind is equilateral and a triangle of another kind scalene. they are the same figure, then, that, triangle, but not the same triangle.
since co is the square on four hundred, and ck the square on cb, and cb is the half of four hundred, co is equal to four times ck iv., cor. one . again, since cg, gi are the sides of equal squares, they are equal i. xlvi., cor. one .
from a potential existent, and that matter is not separable from the contraries but its being is different, and that a single matter may serve for colour and heat and cold.
to enumerate the places which he assigned to the other stars, and to give all the reasons why he assigned them, although a secondary matter, would give more trouble than the primary.
prop. thirty six.-theorem. parallelograms (bd, fh) on equal bases (bc, fg) and between the same parallels are equal.
the question 'why', then, is answered by reference to the matter, to the form, and to the primary moving cause. for in respect of coming to be it is mostly in this last way that causes are investigated-'what comes to be after what?
in this we have also the solution of a possible difficulty.
the same is true of heaviness and lightness when they refer to the same thing: e.g. they do not specifically distinguish earth from itself or fire from itself.
join ap, bp, and construct a triangle apb on ab, similar to apb; then it is easy to see that lines from p to the angular points of abcd are proportional to the lines from p to the angular points of abcd.
and the reason of this is that these two, the straight and the circular line, are the only simple magnitudes.
now everything that becomes or is created must of necessity be created by some cause, for without a cause nothing can be created.
sol.-through the centre o draw any two diameters at right angles to each other, and draw at the points a, b, one hundred, five hundred the lines he, ef, fg, gh touching the circle. efgh is a square.
if movement can exist without soul, and the before and after are attributes of movement, and time is these qua numerable.
the pythagoreans again had framed a world out of numbers, which they constructed into figures. plato adopted their speculations and improved upon them by a more exact knowledge of geometry.
similarly those who assert no cause of this kind but say that 'void' accounts for motion-they also hold that the motion of natural substance is motion in respect of place: for their motion that is accounted for by 'void' is locomotion, and its sphere of
the compound may he-actually other than the constituents from which it has resulted; nevertheless each of them may still he-potentially what it was before they were combined, and both of them may survive undestroyed.
let it be divided, then, at g. now if gb is not in motion, then ab will not be in motion: for if it is, it is clear that ag would be in motion while bg is at rest, and thus ab cannot be in motion essentially and primarily.
becoming young or fair. for time is by its nature the cause rather of decay, since it is the number of change, and change removes what is.
for a man who is in great joy or in great pain, in his unreasonable eagerness to attain the one and to avoid the other, is not able to see or to hear anything rightly; but he is mad, and is at the time utterly incapable of any participation in reason.
centre-since this is so, it follows that circular movement also must be the movement of some simple body. for the movement of composite bodies is, as we said, determined by that simple body which preponderates in the composition.
joining the points of section is parallel to the third side.
this, however, does not touch our argument. the maximum may be determined either in the power or in its object. the application of this is plain. superior sight is sight of the smaller body, but superior speed is that of the greater body.
now, because ac is parallel to fe, ba : af :: bc : ce ii. ; but af is equal to four hundred, therefore ba : four hundred :: bc : ce; hence five. xvi. ; ab : bc :: six hundred : ce.
seven. if p be any point in the diameter ab of a semicircle, and four hundred any parallel chord, then
so we would raise the question: what would they say of an interval that has colour or sound-is it void or not? clearly they would reply that if it could receive what is tangible it was void, and if not, not.
df de, and da db; and taking the latter from the former, the remainder af is equal to the remainder be (axiom iii.).
prop. twenty five.-problem. to describe a rectilineal figure equal to a given one (a), and similar to another given one (bcd).
thus, as a simple case, the square on a line is four times the square on its half. on account of this property the second power of a quantity is called its square; and, conversely, the square on a line ab is expressed symbolically by ab .
of void. it is due to the properties of the elementary bodies that a body which is regarded as light in one place is regarded as heavy in another, and vice versa.
to resolve bodies into planes and no further-this, as we have also remarked elsewhere, in itself a paradox. hence there is more to be said for the view that there are indivisible bodies. yet even these involve much of paradox.
if it is the nature of earth, as observation shows, to move from any point to the centre, as of fire contrariwise to move from the centre to the extremity, it is impossible that any portion of earth should move away from the centre except by constraint.
moreover it would seem absurd even to speak in this way, to speak, that is to say, of a man or house or anything else that has come into existence as having been altered.
observation.-by comparing the proofs of the pairs of props. iv. and vii.; v. and vi.; ix. and x.; xii. and xiii., it will be seen that they are virtually identical.
the same reasoning will show the divisibility of the length, and in fact of everything that forms a sphere of change (though some of these are only accidentally divisible because that which changes is so): for the division of one term will involve the
one. if the line ad bisect the external vertical angle cae, ba : ac :: bd : six hundred, and conversely.
high and low, and are sometimes discordant on account of their inequality, and then again harmonical on account of the equality of the motion which they excite in us.
(further, how can there be any 'before' and 'after' without the existence of time? or how can there be any time without the existence of motion?
appointed place of fire, and where there is the great mass of fire to which fiery bodies gather-if, i say, he were to ascend thither, and, having the power to do this, were to abstract particles of fire and put them in scales and weigh them, and then,
therefore abc : abg in the duplicate ratio of bc : ef; but it has been proved that the triangle abg is equal to def. therefore the triangle abc is to the triangle def in the duplicate ratio of bc : ef.
hunt the prey which we mean to take. a body which is of a nature to be easily moved, on receiving an impression however slight, spreads abroad the motion in a circle, the parts communicating with each other, until at last, reaching the principle of mind,
for though fire be supposed to contain less solid than any other body, as well as more void, yet there will be a certain quantum of fire in which the amount of solid or plenum is in excess of the solids contained in some small quantity of earth.
twenty one. what three regular figures can be used in filling up the space round a point? ans. equilateral triangles, squares, and hexagons.
two. when the size of a figure is given, it is said to be given in magnitude; for instance, a square whose side is of given length.
thus, two quadrilaterals may have their sides proportional without having equal angles, or vice versa.
four. if any number of similar triangles have their corresponding vertices lying on three given lines, they have a common centre of similitude.
for if water has first been 'dissociated' into smallish drops, air comes-to-be out of it more quickly: while, if drops of water have first been 'associated', air comes-to-be more slowly. our doctrine will become clearer in the sequel.'
so we have now to consider how motion is differentiated: and this discussion serves to show that the genus is not a unity but contains a plurality latent in it and distinct from it, and that in the case of equivocal terms sometimes the different senses in
dem.-describe a circle about the triangle. let fall the perpendicular cf. produce cf to meet the circle in g. make fo fg. join ag, ao. produce ao to meet bc in five hundred.
now we say that (a) the continuous is one or that (b) the indivisible is one, or (c) things are said to be 'one', when their essence is one and the same, as 'liquor' and 'drink'.
but what is a first principle ought not to be the predicate of any subject. if it were, there would be a principle of the supposed principle: for the subject is a principle, and prior presumably to what is predicated of it.
for how should you divide it? or how can the infinite have the one part up and the other down, or an extremity and a centre?
water and earth) among bodies endowed with weight. again, the suggestion of a certain ratio between the void and the solid in a body is no more equal to solving the problem before us. the manner of speaking will issue in a similar impossibility.
hence there is a single actuality of both alike, just as one to two and two to one are the same interval, and the steep ascent and the steep descent are one-for these are one and the same, although they can be described in different ways.
dem.-the angle e is equal to hac (const.), and hac is equal to the angle abc in the alternate segment three. xxxii. . hence the angle e is equal to abc. in like manner the angle f is equal to acb. therefore i. xxxii.
xii. concentric circles are those that have the same centre.
two. square on the sum square on the difference of any two lines twice the sum of the squares on the lines (props. ix. and x.).
timaeus: all men, socrates, who have any degree of right feeling, at the beginning of every enterprise, whether small or great, always call upon god.
if, then, it is in a place and somewhere, it will be one of the elements; and if it is not in a place, nothing can come from it, since that which comes into being and that out of which it comes must needs be together.
the second reason is this: all movement is either compulsory or according to nature, and if there is compulsory movement there must also be natural (for compulsory movement is contrary to nature, and movement contrary to nature is posterior to that
two. prove prop. xi. without using prop. viii.
growth starts from above, locomotion from the right, sensemovement from in front (for front is simply the part to which the senses are directed).
but when these are refined by putrefaction, and enter the narrow veins of the tongue, and meet there particles of earth and air, two kinds of globules are formed-one of earthy and impure liquid, which boils and ferments, the other of pure and transparent
to begin then, as we said, with motion. we may start by distinguishing (one) what exists in a state of fulfilment only, (two) what exists as potential, (three) what exists as potential and also in fulfilment-one being a 'this', another 'so much', a third
each of them exists in virtue of a particular relation and puts that which possesses it in a good or bad condition with regard to its proper affections, where by 'proper' affections i mean those influences that from the natural constitution of a thing
if the gnomons are placed round the one, and without the one, in the one construction the figure that results is always different, in the other it is always the same. but plato has two infinites, the great and the small.
we must say that if some one is teaching some lesson in geometry, he is also in process of being taught that same lesson in geometry, and that if he is throwing he is in process of being thrown in just the same manner.
but perhaps even this is impossible, that it should move at all; the same reason which showed that in the void all things are incapable of moving shows that the void cannot move, viz. the fact that the speeds are incomparable.
dem.-since a : b :: a : b, we have a- b a- b in like manner, b c b c. hence, multiplying, a- c a c. therefore a : c :: a : c, and similarly for any number of magnitudes in each set.
and this would also account for the fact that a body composed of a number of solid parts equal to, or even smaller than, that of another is sometimes larger in bulk than it.
a doubt (two) may also be raised as to whether the words 'artificer of day and night' are consistent with the mere passive causation of them, produced by the immobility of the earth in the midst of the circling universe.
it is necessary, therefore, for his 'contiguous discretes' to be indivisible, while the intervals between them-which he calls 'pores'-must be void. but this is precisely leucippus' theory of action and passion.
xxix. , and ace to dac; hence bad is equal to dac, and the line ad bisects the angle bac.
suppose that the world was formed out of elements which were formerly otherwise conditioned than as they are now. then (one) if their condition was always so and could not have been otherwise, the world could never have come into being.
then because the plane ab is perpendicular to adc, the line de in ab is normal to the plane adc eleven. def. viii. . in like manner df is normal to it. therefore from the point five hundred there are two distinct normals to the plane adc, which eleven.
the other is the variable or changing element, the residuum of disorder or chaos, which cannot be reduced to order, nor altogether banished, the source of evil, seen in the errors of man and also in the wanderings of the planets, a necessity which
but, be it noted, this is the question we are trying to answer-how can we account for the motion of light things and heavy things to their proper situations?
those which possess no such factor. assuming, then, that the contrariety, in respect to which they are transformed, is one, the elements' will inevitably be two: for it is 'matter' that is the 'mean' between the two contraries, and matter is imperceptible
but since indivisibles have no parts, they must be in contact with one another as whole with whole.
are not actual but potential halves. if the halves are made actual, we shall get not a continuous but an intermittent motion.
eleven. how many dimensions has a surface?
consequently its motion is not infinite, and it will not be in locomotion over an infinite distance, for it cannot traverse such a distance.
ix. proportion consists of three terms at least.
one ; therefore the rectangle ad.db is equal to the gnomon cmg, and the square on cb is the figure ce.
and since this was controlled by the movement of the same, the seven planets in their courses appeared to describe spirals; and that appeared fastest which was slowest, and that which overtook others appeared to be overtaken by them.
; therefore four hundred is less than the radius of the circle, consequently the point five hundred must be within the circle (note on i. def. xxxiii.).
dem.-let o be the centre. let ab, four hundred intersect in e; then since ab, four hundred are not both diameters, join oe. if possible let ae be equal to eb, and ce equal to ed.
and something similar happens when the disordered motions of the soul come into contact with any external thing; they say the same or the other in a manner which is the very opposite of the truth, and they are false and foolish, and have no guiding
cor. eight.-if the adjacent sides of a parallelogram be equal, its diagonals intersect at right angles.
now (i) if a great quantity (or a large bulk) of one of these easily-divisible 'reciprocating' materials be brought together with a little (or with a small piece) of another, the effect produced is not 'combination', but increase of the dominant: for the
but all movement that is in place, all locomotion, as we term it, is either straight or circular or a combination of these two, which are the only simple movements.
two. in what case would the construction fail, if the equilateral triangle were described on the other side of de?
for what is cannot come to be (because it is already), and from what is not nothing could have come to be (because something must be present as a substratum).
six. find the locus of the centres of a system of circles touching two given lines.
three. from the hypotenuse of a right-angled triangle portions are cut off equal to the adjacent sides; prove that the square on the middle segment is equal to twice the rectangle contained by the extreme segments.
could any war between athens and the island of atlantis have really coincided with the struggle between the greeks and persians, as is sufficiently hinted though not expressly stated in the narrative of plato? and whence came the tradition to egypt?
fourteen. the angle a and the area of a triangle abc are given in magnitude: if the point a be fixed in position, and the point b move along a fixed line or circle, the locus of the point one hundred is a circle.
not the same thing as rotatory locomotion: for, when a thing merely traverses a circle, it may either proceed on its course without a break or turn back again when it has reached the same point from which it started).
but, on the other hand, it is equally impossible that the elements should be generated from some kind of body. that would involve a body distinct from the elements and prior to them.
others always in motion while others again are sometimes at rest and sometimes in motion. now we have said before that it is impossible that all things should be at rest: nevertheless we may now repeat that assertion.
and ei : gj :: ef : gh xx. . but since ab : four hundred :: ef : gh, ab : cd :: ef : gh five. xxii., cor. one ; therefore abk : four hundred fifty :: ei : gj. if abk : four hundred fifty :: ei : gj, ab : four hundred :: ef : gh.
when the father who begat the world saw the image which he had made of the eternal gods moving and living, he rejoiced; and in his joy resolved, since the archetype was eternal, to make the creature eternal as far as this was possible.
to be many-as if 'one' and 'being' were always used in one and the same sense.
eight. the rectangle of.op is equal to the square of the radius.
if everything that is in motion is moved by something that is in motion, ether this being in motion is an accidental attribute of the movents in question, so that each of them moves something while being itself in motion, but not always because it is
that the stars are spherical and are not selfmoved, has now been explained.
for just as people speak of 'a passing-away' without qualification when a thing has passed into what is imperceptible and what in that sense 'is not', so also they speak of 'a coming-to-be out of a not-being' when a thing emerges from an imperceptible.
some philosophers think that the 'last' agent-the 'agent' in the strictest sense-enters in through certain pores, and so the patient suffers action. it is in this way, they assert, that we see and hear and exercise all our other senses.
that of husbandmen; and you will observe, too, that the warriors in egypt are distinct from all the other classes, and are commanded by the law to devote themselves solely to military pursuits; moreover, the weapons which they carry are shields and
but the followers of anaxagoras regard the 'homoeomeries' as 'simple' and elements, whilst they affirm that earth, fire, water, and air are composite; for each of these is (according to them) a 'common seminary' of all the 'homoeomeries'.
on the other hand it has been urged that if the earth goes round with the outer heaven and sun in twenty-four hours, there is no way of accounting for the alternation of day and night; since the equal motion of the earth and sun would have the effect of
but this is not possible, if we are to hold to the truth of our original presuppositions and to the view that neither that which moves downward, nor, by the same reasoning, that which moves upward, can prolong its movement to infinity.
hence it was clearly out of a one that this came-to-be water, and that fire, various portions of it being separated off by certain characteristic differences or qualities-as indeed he calls the sun 'white and hot', and the earth 'heavy and hard'.
generation. that in which the elements severally grow up, and appear, and decay, is alone to be called by the name 'this' or 'that'; but that which is of a certain nature, hot or white, or anything which admits of opposite qualities, and all things that
but obviously it is not true that if being means one thing, and cannot at the same time mean the contradictory of this, there will be nothing which is not, for even if what is not cannot be without qualification, there is no reason why it should not be a
the soul of him who has education is whole and perfect and escapes the worst disease, but, if a man's education be neglected, he walks lamely through life and returns good for nothing to the world below.
men were led to conceive it, not by a love of hasty generalization, but by a divine instinct, a dialectical enthusiasm, in which the human faculties seemed to yearn for enlargement.
what are the precise conditions of its occurrence and (ii) what 'alteration' is: but we have still to treat of growth.
thus we shall have a division of the present, whereas we found it to be indivisible. it is impossible, therefore, for anything to be in motion in a present.
these, then, are the difficulties resulting from the supposition that any and every body, whatever its size, is divisible through and through. there is, besides, this further consideration.
for if the time is not definite in quantity, but longer than any that can be suggested and shorter than none, then it will be possible for one and the same thing to exist for infinite time and not to exist for another infinity.
dem.-since ma is greater than nb, but mc not greater than nd, it is evident that
the love and strife postulated by empedocles are not in themselves causes of the fact in question, nor is it of the essence of either that it should be so, the essential function of the former being to unite, of the latter to separate.
in a similar way the proposition may be proved by taking any of the eight figures formed by turning the squares in all possible directions.
for the one which persists is a joint cause, with the form, of what comes to be-a mother, as it were. but the negative part of the contrariety may often seem, if you concentrate your attention on it as an evil agent, not to exist at all.
the citizens have a friendly feeling towards the athenians, believing themselves to be related to them.
also it is held that what is anywhere is both itself something and that there is a different thing outside it.
therefore the sum of ba, ac is greater than bc.
consequences which would never have occurred to him. for he has glimpses of the truth, but no comprehensive or perfect vision.
three. being given a circle and a line, prove that a point may be found, such that the rectangle of the perpendiculars let fall on the line from the points of intersection of the circle with any chord through the point shall be given.
any element which is fastened upon by fire is cut by the sharpness of the triangles, until at length, coalescing with the fire, it is at rest; for similars are not affected by similars.
socrates: to be sure i will: the chief theme of my yesterday's discourse was the state-how constituted and of what citizens composed it would seem likely to be most perfect.
dem.-because a is the centre of the circle bcd, ac is equal to ab (def. xxxii.). again, because b is the centre of the circle ace, bc is equal to ba. hence we have proved.
not at all. hence 'substance' is not; for it is true to say that it is white, which we found to mean not-being. if to avoid this we say that even 'white' means substance, it follows that 'being' has more than one meaning.
part of one's life earning money in order to enjoy a questionable liberty during the least valuable part of it, reminds me of the englishman who went to india to make a fortune first, in order that he might return to england and live the life of a poet.
this becomes plain if one tries to state in each of the two cases the definitions of the things and of their attributes.
two. if ab.f four hundred.e, to prove ab : four hundred :: e : f.
concerned), and these pleasures and pains are alterations of the sensitive part, it is evident that the loss and acquisition of these states no less than the loss and acquisition of the states of the body must be the result of the alteration of something
about the thighs and arms, which have no sense because there is little soul in the marrow, and about the inner bones, he laid the flesh thicker.
that which it surpasses, and that it is impossible for the opposite motions or changes to be present to a thing at the same time.
greater or lesser degree. but the alteration of that which undergoes alteration is also caused by the above-mentioned characteristics, which are affections of some particular underlying quality.
being is the limit of coming to be and not-being is the limit of ceasing to be: and in contrary changes the particular contraries are the limits, since these are the extreme points of any such process of change, and consequently of every process of
that is to say, that experience more than one locomotion.)
and 'ungenerated' also involves impossibility when used for that which cannot be generated, in such fashion that, while formerly it was not, later it is. an instance is a commensurable diagonal.
it is clear then that our principles must be contraries.
for whereas a thing does not persist in the processes of unqualified coming-to-be or passing-away, that which grows or 'alters' persists in its identity through the 'altering' and through the growing or diminishing, though the quality (in 'alteration')
it is absurd to suppose that purpose is not present because we do not observe the agent deliberating. art does not deliberate. if the ship-building art were in the wood, it would produce the same results by nature.
the circle of the diverse also moving truly imparts the intimations of sense to the whole soul, then arise opinions and beliefs sure and certain.
cor. three.-the bisector of any angle bisects the corresponding re-entrant angle.
how then will the body of the cube differ from the void or place that is equal to it? and if there can be two such things, why cannot there be any number coinciding?
thirty one. what is meant by the projection of one line on another?
for there was a time, solon, before the great deluge of all, when the city which now is athens was first in war and in every way the best governed of all cities, is said to have performed the noblest deeds and to have had the fairest constitution of any
hence, to conclude, since causes of this kind are indefinite, chance too is indefinite. (yet in some cases one might raise the question whether any incidental fact might be the cause of the chance occurrence, e.g.
he who has the seed about the spinal marrow too plentiful and overflowing, like a tree overladen with fruit, has many throes, and also obtains many pleasures in his desires and their offspring, and is for the most part of his life deranged, because his
for this theory abolishes 'alteration': but we see the same body liquid at one time and solid at another, without losing its continuity.
in investigating the cause of each thing it is always necessary to seek what is most precise (as also in other things): thus man builds because he is a builder, and a builder builds in virtue of his art of building.
surely then whatever is ungenerated and in being must be eternal, and whatever is indestructible and in being must equally be so.
four. when is a circle said to be inscribed in a rectilineal figure?
for, if it were, the 'void' would exist in separation; but we have explained in a former work' that this is impossible. moreover, a change of that kind is not peculiarly distinctive of growth, but characterizes coming-to-be as such or in general.
cor. two.-parallel chords in a circle intercept equal arcs.
so that the greatness and smallness, also, of the sensible volume are extended, not by the matter's acquiring anything new, but because the matter is potentially matter for both states; so that the same thing is dense and rare, and the two qualities have
eight. abc is a right-angled triangle having ab ac: if ah be made equal to the difference between bc and ac, ab is divided in "extreme and mean ratio" at h.
if the end is to exist or does exist, that also which precedes it will exist or does exist; otherwise just as there, if-the conclusion is not true, the premiss will not be true, so here the end or 'that for the sake of which' will not exist.
but in things of that character the contradictory states proceed from one and the same capacity, the matter of the thing being the cause equally of its existence and of its non-existence. hence contradictories would be present together in actuality.
two. if the sum of the squares of two lines be given, their sum is a maximum when the lines are equal.
those who maintain that the primary or atomic parts, of which bodies endowed with weight are composed, are planes, cannot so speak without absurdity; but those who regard them as solids are in a better position to assert that of such bodies the larger is
in like manner the rectangle contained by ab and cb is equal to the figure four hundred. therefore the sum of the two rectangles ab.ac, ab.cb is equal to the square on ab.
for the word 'nature' is applied to what is according to nature and the natural in the same way as 'art' is applied to what is artistic or a work of art.
there are, in fact, three kinds of movement, affecting respectively the size, the form, and the place of a thing, and in each it is observable that change proceeds from a contrary to a contrary or to something intermediate: it is never the change of any
in that way, however, it does, just as animal might come to be from animal, and an animal of a certain kind from an animal of a certain kind. thus, suppose a dog to come to be from a horse.
nine. through a given point draw a right line, such that perpendiculars on it from two given points on opposite sides may be equal to each other.
and there is still a question to be asked about him: which of the patterns had the artificer in view when he made the world-the pattern of the unchangeable, or of that which is created?
if then the complete is prior to the incomplete, it follows on this ground also that the circle is primary among figures. and the sphere holds the same position among solids.
neither plato nor any other greek would have spoken of (greek) or (greek) in the same manner as we speak of 'time' and 'space.'
but when one of the constituents is alone susceptible-or superlatively susceptible, the other being susceptible in a very slight degree-the compound resulting from their combination is either no greater in volume or only a little greater.
if the place is in the thing (it must be if it is either shape or matter) place will have a place: for both the form and the indeterminate undergo change and motion along with the thing, and are not always in the same place, but are where the thing is.
such a constrained movement would necessarily involve effort the more so, the more eternal it were-and would be inconsistent with perfection.
of using his knowledge, so, too, when any one originally acquires the state, it is incorrect to say that he becomes possessed of knowledge: for the possession of understanding and knowledge is produced by the soul's settling down out of the restlessness
the square on four hundred. hence the rectangle ad.db, together with the square on cb, is equal to the square on four hundred.
thirty one. abc is an isosceles triangle whose equal sides are ab, ac; bc is any secant cutting the equal sides in b, one hundred, so that ab ac ab ac: prove that bc is greater than bc.
one. the sum of the lines drawn from any point within a triangle to its angular points is less than the perimeter. (compare ex. two, last prop.)
some disorder or enthusiasm then they receive intimations, which have to be interpreted by others who are called prophets, but should rather be called interpreters of prophecy; after death these intimations become unintelligible.
according to plato earth was composed of cubes, fire of regular pyramids, air of regular octahedrons, water of regular icosahedrons. the stability of the last three increases with the number of their sides.
theory, no less than action, is subject to these necessities; the form in which men cast their speculations, no less than the ways in which they behave, are the result of the habits of thought and action which they find around them.
yes, i did eat eight.seventy four, all told; but i should not thus unblushingly publish my guilt, if i did not know that most of my readers were equally guilty with myself, and that their deeds would look no better in print.
but every body is divisible and therefore, since body 'combined' with body is uniform in texture throughout, any and every part of each constituent ought to be juxtaposed to a part of the other.
having laid these foundations, we must complete the theory.
similarly one is quicker than the other if in an equal time the product is different in the two cases. i state it thus because we have no pair of terms that will convey this 'difference' in the way in which unlikeness is conveyed.
and things which are equal to the same are equal to one another. therefore the sum of the angles cba, abd is equal to the sum of the angles cbe, ebd; but cbe, ebd are right angles; therefore the sum of the angles cba, abd is two right angles.
for no man is voluntarily bad; but the bad become bad by reason of an ill disposition of the body and bad education, things which are hateful to every man and happen to him against his will.
fire and air and isolated, becomes more uniform, and by their retirement is compressed into itself; and if the condensation be very great, the water above the earth becomes hail, but on the earth, ice; and that which is congealed in a less degree and is
for if we take a part of the motion which shall be a measure of the whole, the whole motion is completed in as many equal periods of the time as there are parts of the motion.
dem.-because the diagonal bisects the parallelograms ac, ak, kc, we have xxxiv. the triangle adc equal to the triangle abc, the triangle ahk equal to aek, and the triangle kfc equal to the triangle kgc.
forty three. the middle points of the sides ab, bc, ca of a triangle are respectively five hundred, e, f; dg is drawn parallel to bf to meet ef; prove that the sides of the triangle dcg are respectively equal to the three medians of the triangle abc.
secondly, they applied number and figure equally to those parts of physics, such as astronomy or mechanics, in which the modern philosopher expects to find them, and to those in which he would never think of looking for them, such as physiology and
three. if two of the opposite sides of a quadrilateral be respectively the greatest and least, the angles adjacent to the least are greater than their opposite angles.
thus have we discussed the general affections of the whole body, and the names of the agents which produce them. and now i will endeavour to speak of the affections of particular parts, and the causes and agents of them, as far as i am able.
god therefore formed a network of fire and air to irrigate the veins, having within it two lesser nets, and stretched cords reaching from both the lesser nets to the extremity of the outer net.
for as continuous it is long or short and as a number many or few, but it is not fast or slow-any more than any number with which we number is fast or slow.
nine. if a quadrilateral be described about a circle, the angles at the centre subtended by the opposite sides are supplemental.
if the earth was generated, then, it must have been formed in this way, and so clearly its generation was spherical; and if it is ungenerated and has remained so always, its character must be that which the initial generation, if it had occurred, would
the stone that struck the man did not fall for the purpose of striking him; therefore it fell spontaneously, because it might have fallen by the action of an agent and for the purpose of striking.
for if there is to be coming-to-be without qualification, 'something' must-without qualification-'come-to-be out of not-being', so that it would be true to say that 'not-being is an attribute of some things'.
and the other sinews he dispersed throughout the body, fastening limb to limb.
still more unreasonable is the consequence involved that, since everything that is moved is moved by something that is itself moved by something else, everything that has a capacity for causing motion has as such a corresponding capacity for being moved:
it is this element in the timaeus, no less than its affinity to certain pythagorean speculations, which gives it a character not wholly in accordance with the other dialogues of plato.
the reason is that a principle is great rather in power than in extent; hence that which was small at the start turns out a giant at the end.
cor. three.-similar portions of similar figures bear the same ratio to each other as the wholes of the figures.
then one. a : c :: b : c, two. c : a :: c : b. dem.-since a b, dividing each by c, we have
may my words be acceptable to them, and may i speak in the manner which will be most intelligible to you and will best express my own meaning!
prop. nine.-problem. from a given right line (ab) to cut off any part required (i.e. to cut off any required submultiple)
therefore ac b ; that is, the rectangle contained by the extremes is equal to the square of the mean.
digestion is supposed to be effected by the action of the internal fire, which in the process of respiration moves into the stomach and minces the food.
dem.-because ac is a parallelogram, ab is equal to four hundred xxxiv. ; but ab is equal to ad (const.); therefore ad is equal to four hundred, and ad is equal to bc xxxiv. .
bd is equal to four hundred; but bd is equal to ac (const.); therefore ac is equal to four hundred, and therefore i. v. the angle cda is equal to a; but bda has been proved to be equal to the sum of cda and a. hence bda is double of a.
; hence oaf ocd, and foa doc; hence ofa odc; but ofa is right, hence odc is right. in like manner, if bo be joined to meet ac in e, be will be perpendicular to ac. hence the three perpendiculars pass through o, and are concurrent.
yet so far are we from simplicity and independence that, in concord, fresh and sweet meal is rarely sold in the shops, and hominy and corn in a still coarser form are hardly used by any.
like some philosophers in modern times, who are accused of making a theory first and finding their facts afterwards, the advocates of either opinion never thought of applying either to themselves or to their adversaries the criterion of fact.
this result, then, as i have said, is reached by syllogism. it would be absurd to suppose that the point is void; for the void must be place which has in it an interval in tangible body.
we have now discussed time-both time itself and the matters appropriate to the consideration of it.
thirteen. if a triangle of given form move so that its three sides pass through three fixed points, the locus of any point in its plane is a circle.
no: that is impossible, since then there will be something not divided, whereas ex hypothesis the body was divisible through and through.
the form of which we have spoken is a kind of power immersed in matter-a duct, as it were.
from the genuinely-one, on the other hand, there never could have come-to-be a multiplicity, nor from the genuinely-many a "one": that is impossible.
for the middle is what is defined, and what defines it is the limit, and that which contains or limits is more precious than that which is limited, see ing that the latter is the matter and the former the essence of the system.
and so, implying that the primary body is something else beyond earth, fire, air, and water, they gave the highest place a name of its own, aither, derived from the fact that it 'runs always' for an eternity of time.
left. we need not be troubled by the question, arising from the spherical shape of the world, how there can be a distinction of right and left within it, all parts being alike and all for ever in motion.
must come to an end, and a point will be reached at which motion is imparted by something that is unmoved.
and the axiom that that which holds a lead is never overtaken is false: it is not overtaken, it is true, while it holds a lead: but it is overtaken nevertheless if it is granted that it traverses the finite distance prescribed.
but since 'nature' means a source of movement within the thing itself, while a force is a source of movement in something other than it or in itself qua other, and since movement is always due either to nature or to constraint, movement which is natural,
heracles as far as egypt, and of europe as far as tyrrhenia.
when you have kneaded it well, mould it, and bake it under a cover," that is, in a baking-kettle. not a word about leaven. but i did not always use this staff of life. at one time, owing to the emptiness of my purse, i saw none of it for more than a month.
in cold weather it was no little amusement to bake several small loaves of this in succession, tending and turning them as carefully as an egyptian his hatching eggs.
for any two portions of fire, small or great, will exhibit the same ratio of solid to void, but the upward movement of the greater is quicker than that of the less, just as the downward movement of a mass of gold or lead, or of any other body endowed with
there should be an interval which is other than the bodies which are moved.
the truths of geometry and arithmetic in all their combinations are always the same.
the work of the creator, whenever he looks to the unchangeable and fashions the form and nature of his work after an unchangeable pattern, must necessarily be made fair and perfect; but when he looks to the created only, and uses a created pattern, it is
unclean matter, swells and festers, but, again, when the body is purged, settles down into the same place as before, and is humbled.
and oftentimes when the flesh is dissolved in the body, wind, generated within and unable to escape, is the source of quite as much pain as the air coming in from without; but the greatest pain is felt when the wind gets about the sinews and the veins of
for in what sense is that section divisible?
to give a satisfactory decision as to the truth it is necessary to be rather an arbitrator than a party to the dispute.
(three) of the intersection of the external bisectors of the base angles;
if a whole magnitude be to another whole at a magnitude taken from the first it to a magnitude taken from the second, the first remainder : the second remainder :: the first whole : the second whole.
the remaining alternative is that they should be this in one of two ways, either by change of shape, as the same wax takes the shape both of a sphere and of a cube, or, as some assert, by resolution into planes.
def. v.-a quadrilateral which has one pair of opposite sides parallel is called a trapezium.
it happens, indeed, to be the case that a thing to which movement this way and that is equally inappropriate is obliged to remain at the centre.
thus, before ever the whirl began, heavy and light existed; and what can have been the ground of their distinction, or the manner and direction of their natural movements?
this entire compound he divided lengthways into two parts, which he joined to one another at the centre like the letter ten, and bent them into a circular form, connecting them with themselves and each other at the point opposite to their original
twenty three. what proposition is an instance of the rule of identity?
from mind, then everything that we perceive through the body is to be regarded as most real and certain.
four. the point of bisection (i) of the line (op) joining the orthocentre (o) to the circumference (p) of any triangle is equally distant from the feet of the perpendiculars, from the middle points of the sides, and from the middle points of the distances
it makes no difference whether the parts are equal or not, if only each is finite: for it is clear that while the time is exhausted by the subtraction of its parts, the infinite magnitude will not be thus exhausted, since the process of subtraction is
but when the roots of the triangles are loosened by having undergone many conflicts with many things in the course of time, they are no longer able to cut or assimilate the food which enters, but are themselves easily divided by the bodies which come in
thus 'perishing' is change to not-being, though it is also true that that that which perishes changes from being: and 'becoming' is change to being, though it is also change from not-being.
was himself destroyed by a thunderbolt. now this has the form of a myth, but really signifies a declination of the bodies moving in the heavens around the earth, and a great conflagration of things upon the earth, which recurs after long intervals; at
something and out of something-i mean, e.g. comes-to-be-healthy out of being-ill and ill out of being-healthy, comes-to-be-small out of being big and big out of being-small, and so on in every other instance.
one. the line joining the feet of perpendiculars from any point in the circumference of a circle, on two diameters given in position, is given in magnitude.
source' of the rest. there can be no such 'originative element' at the ends: for all of them would then be fire or earth, and this theory amounts to the assertion that all things are made of fire or earth.
god took of the unchangeable and indivisible and also of the divisible and corporeal, and out of the two he made a third nature, essence, which was in a mean between them, and partook of the same and the other, the intractable nature of the other being
for (a) how is the manner of their coming-to-be to be conceived by those who maintain a theory like that of empedocles?
for, to begin with, it is characteristic of matter to suffer action, i.e. to be moved: but to move, i.e. to act, belongs to a different 'power'. this is obvious both in the things that come-to-be by art and in those that come to-be by nature.
prop. twenty three.-theorem. equiangular parallelograms (ad, cg) are to each other as the rectangles contained by their sides about a pair of equal angles.
first, it may be said that no process of change is eternal: for the nature of all change is such that it proceeds from something to something, so that every process of change must be bounded by the contraries that mark its course, and no motion can go on
for air is something, though it does not seem to be so-nor, for that matter, would water, if fishes were made of iron; for the discrimination of the tangible is by touch.
timaeus: nothing, socrates; it was just as you have said.
in the case of locomotion, if both things move along the circumference of a circle, or both along a straight line; and similarly in all other cases.
but at war with themselves, because they receive no good from one another, and are hostile to the abiding constitution of the body, which they corrupt and dissolve.
two. if lines be drawn from a fixed point to all the points of the circumference of a given circle, the locus of all their points of bisection is a circle.
in statements which refer to part of the thing in question: thus the body is restored to health because the eye or the chest, that is to say a part of the whole body, is restored to health.
is causing locomotion is in process of locomotion, and that one who is teaching is in process of being taught: for it is clear that everything that is moved is moved by the movent that is further back in the series as well as by that which immediately
hence in these also the 'now' as substratum remains the same (for it is what is before and after in movement), but what is predicated of it is different; for it is in so far as the 'before and after' is numerable that we get the 'now'.
now the universe is certainly a particular and a material thing: if however, it is composed not of a part but of the whole of matter, then though the being of 'universe' and of 'this universe' are still distinct, yet there is no other universe, and no
now motions respectively from a contrary and to the opposite contrary, e.g. a motion from health and a motion to disease, are not contrary motions: for they are one and the same.
then because ac is equal to four hundred, the square on ac is equal to the square on four hundred: to each add the square on cb, and we have the sum of the squares on ac, cb equal to the sum of the squares on four hundred, cb; but the sum of the squares
but the sum of the base angles of the same triangles, together with the sum of the plane angles bac, cad, c., forming the solid angle a, is equal to twice as many right angles as there are triangles bac, cad, c.-that is, equal to ten right angles.
only they attribute the downward movement to constraint, and it is better to keep to the truth and say that the reason of this motion is that a thing which possesses weight is naturally endowed with a centripetal movement.
prove prop. xvii. without producing a side.
similarly, even if there were only one elementary body, it would not contain them.
the same is the unchanging and indivisible, the heaven of the fixed stars, partaking of the divine nature, which, having law in itself, gives law to all besides and is the element of order and permanence in man and on the earth.
for the case n four, which is the only figure of this class except the pentagon for which a construction has been given, see note at the end of this work.
causes it to be at rest. thus, if this is accepted, everything that is in motion must be moved by something. for ab, which has been taken to represent that which is in motion, must be divisible since everything that is in motion is divisible.
when men begin to do, not merely unnecessary or artistic, but luxurious and idle work, with their assistance, it is inevitable that a few do all the exchange work with the oxen, or, in other words, become the slaves of the strongest.
sixteen. what is proportion? ans. equality of ratios.
without being necessarily together. so natural junction is last in coming to be: for the extremities must necessarily come into contact if they are to be naturally joined: but things that are in contact are not all naturally joined, while there is no
but his general remarks had such a way of bearing very obvious particular applications, and were so obviously inspired by a particular attitude towards the government of his day, that even philosophy became in his hands unsafe, and he was attacked for
thus empedocles holds that the corporeal elements are four, while all the elements-including those which initiate movement-are six in number; whereas anaxagoras agrees with leucippus and democritus that the elements are infinite.
if, then, it is to be assumed that that which is in locomotion is in process of changing, it must be capable of completing the change.
but the extremity of the whole, which we call 'above', is in position above and in nature primary. and since the universe has an extremity and a centre, it must clearly have an up and down. common usage is thus correct, though inadequate.
even the little variety which i used was a yielding to the demands of appetite, and not of health.
since, however, it appears unaccountable that we should not hear this music, they explain this by saying that the sound is in our ears from the very moment of birth and is thus indistinguishable from its contrary silence, since sound and silence are
or thus: ac ab bc; therefore ac ab ab.bc bc ; therefore ac ab.bc ab bc . comparison of iv. and vii.
for 'soft' is that which yields to pressure by retiring into itself, though it does not yield by total displacement as the moist does-which explains why the moist is not 'soft', although 'the soft' derives from the moist.
from him they received the immortal soul, but themselves made the body to be its vehicle, and constructed within another soul which was mortal, and subject to terrible affections-pleasure, the inciter of evil; pain, which deters from good; rashness and
he would have readily admitted that out of the protoplasm all things were formed by the gradual process of creation; but he would have insisted that mind and intelligence-not meaning by this, however, a conscious mind or person-were prior to them, and
the parallels (ef, gh) through any point (k) in one of the diagonals (ac) of a parallelogram divide it into four parallelograms, of which the two (bk, kd) through which the diagonal does not pass, and which are called the complements of the other two, are
that km is moved by anything on the ground that it is not evident which is the part that is moving it and which the part that is moved.
pass only this pure element. when the light of day surrounds the stream of vision, then like falls upon like, and they coalesce, and one body is formed by natural affinity in the line of vision, wherever the light that falls from within meets with an
bodies which are bounded by the spherical and in contact with it must be, as wholes, spherical; and the bodies below the sphere of the planets are contiguous with the sphere above them.
let us start as follows. all things which admit of 'combination' must be capable of reciprocal contact: and the same is true of any two things, of which one 'acts' and the other 'suffers action' in the proper sense of the terms.
mankind, with hardly an exception, have not remarked the periods of the other stars, and they have no name for them, and do not measure them against one another by the help of number, and hence they can scarcely be said to know that their wanderings,
twenty six. if r be the radius of a circle, what is the area of its inscribed equilateral triangle?-of its inscribed square?-its inscribed pentagon?-its inscribed hexagon?-its inscribed octagon?-its inscribed decagon?
: to each add the angle acb, and we have the sum of the angles acd, acb greater than the sum of the angles abc, acb; but the sum of the angles acd, acb is two right angles xiii. . therefore the sum of the angles abc, acb is less than two right angles.
one. the shortest chord which can be drawn through a given point within a circle is the perpendicular to the diameter which passes through that point.
now we may agree that the primary materials, whose change (whether it be 'association and dissociation' or a process of another kind) results in coming-to-be and passingaway, are rightly described as 'originative sources, i.e. elements'.
twenty two. what propositions in book one. are the obverse respectively of propositions iv., v., vi., xxvii.?
timaeus: yes, socrates; and what you said of it was very much to our mind.
the dividing of it can continue indefinitely: so that we should have to suppose there is only one kind of coming-to-be in the world-viz. one which never fails, because it is such that what comes-to-be is on each successive occasion smaller than before.
eight. what are the terms of a ratio called?
divided-not even if it has been divided into innumerable parts, themselves divided innumerable times. nothing impossible will have resulted, though perhaps nobody in fact could so divide it.
they change their places, but not in the same way as the parts of a revolving globe.
that is why neither whiteness (and blackness), nor sweetness (and bitterness), nor (similarly) any quality belonging to the other perceptible contrarieties either, constitutes an 'element'.
similar accounts of other stars are given by the egyptians and babylonians, whose observations have been kept for very many years past, and from whom much of our evidence about particular stars is derived.
when what surrounds, then, is not separate from the thing, but is in continuity with it, the thing is said to be in what surrounds it, not in the sense of in place, but as a part in a whole.
people prophesied a speedy decay of the vital forces.
seven. it two proportions have three terms of one respectively equal to three corresponding terms of the other, the remaining term of the first is equal to the remaining term of the second.
considerations also not general like these but proper to the subject show it to be impossible that what was formerly eternal should later be destroyed or that what formerly was not should later be eternal.
it is necessary, no doubt, that the causes of what comes to pass by chance be indefinite; and that is why chance is supposed to belong to the class of the indefinite and to be inscrutable to man, and why it might be thought that, in a way, nothing occurs
and if it is exempt from increase and diminution, the same reasoning leads us to suppose that it is also unalterable.
if from any point (p) without a circle two lines be drawn to it, one of which (pt) is a tangent, and the other (pa) a secant, the rectangle (ap, bp) contained by the segments of the secant is equal to the square of the tangent.
possibility of others being made, because all the matter is already included in this. it remains, then, only to prove that it is composed of all natural perceptible body.
now 'association' is 'combination'; but the precise meaning of the process we call 'combining' has not been explained.
it is as if we were to suppose that when, e.g. air comes-to-be out of water the process were due not to a change of the but to the matter of the air being 'contained in' the water as in a vessel. this is impossible.
if, therefore, the process comes to an end, everything will not be in everything else (for there will be no flesh in the remaining water); if on the other hand it does not, and further extraction is always possible, there will be an infinite multitude of
the triangles (acd, bcd) into which a right-angled triangle (acb) is divided, by the perpendicular (four hundred) from the right angle (one hundred) on the hypotenuse, are similar to the whole and to one another.
for the term 'much' has the same meaning whether applied to water or to air, yet water and air are not commensurable in respect of it: or, if this illustration is not considered satisfactory, 'double' at any rate would seem to have the same meaning as
def.-the circle abc is called the circumcircle, its radius the circumradius, and its centre the circumcentre of the triangle.
the same remark may be applied to the complexity of the appearances and occultations of the stars, which, if the outer heaven is supposed to be moving around the centre once in twenty-four hours, must be confined to the effects produced by the seven
equal triangles (abc, def) on equal bases (bc, ef) which form parts of the same right line, and on the same side of the line, are between the same parallels.
if it did, it would move either naturally or by constraint: and if by constraint, it possesses also a natural motion, that is to say, there is another place, infinite like itself, to which it will move. but that is impossible.
if, then, some one of the things 'which are' constantly disappearing, why has not the whole of 'what is' been used up long ago and vanished away assuming of course that the material of all the several comings-to-be was finite?
but it remains to be shown that they could have done more in their age and country; or that the contributions which they made to the sciences with which they were acquainted are not as great upon the whole as those made by their successors.
as to the 'matter', it (qua matter) is passive. now fire contains 'the hot' embodied in matter: but a 'hot' separate from matter (if such a thing existed) could not suffer any action.
for the fine' consists of subtle particles; but that which consists of small particles is 'such as to fill up', inasmuch as it is in contact whole with whole-and 'the fine' exhibits this character in a superlative degree.
one more aspect of the timaeus remains to be considered-the mythological or geographical.
; and since the whole circumference agbc is equal to the whole circumference dkef, the remaining arc acb is equal to the remaining arc dfe.
dem.-join bc. since ab is parallel to four hundred, and bc intersects them, the angle abc is equal to the angle bcd xxix. .
cor. one.-if two equal lines pd, pf be drawn from a point p to the circumference of a circle, the diameter through p bisects the angle dpf formed by these lines.
assuming that the antecedent must have come-to-be if the consequent is to be (e.g. that foundations must have come-to-be if there is to be a house: clay, if there are to be foundations), is the converse also true?
but if the continuity of 'that in which' contributes to make the movement continuous, this is true only of 'the place in which'; for that has 'magnitude' in a sense.) but (d) amongst continuous bodies which are moved, only that which is moved in a circle
moreover, the fact that the contraries proceed from each other led them to the conclusion.
finite magnitude during an infinite time. but the first movent causes a motion that is eternal and does cause it during an infinite time. it is clear, therefore, that the first movent is indivisible and is without parts and without magnitude.
two. if two intersecting planes be respectively perpendicular to two intersecting lines, the line of intersection of the former is normal to the plane of the latter.
and (ii) so-and-so 'comes-to-be-something', but does not 'come-to-be' without qualification; for we say that the student 'comes-to-be-learned', not 'comes-to-be' without qualification.
the world of immutable being. and when reason, which works with equal truth, whether she be in the circle of the diverse or of the same-in voiceless silence holding her onward course in the sphere of the self-moved-when reason, i say, is hovering around
then the triangles abg, def have the two sides ab, bg of one respectively equal to the two sides de, ef of the other, and the angle abg equal to the angle def; therefore iv. the angle agb is equal to dfe; but the angle acb is equal to dfe (hyp.).
for all the other thinkers assume and use them without explaining why they are these or why they are just so many.
(a) the timaeus is more imaginative and less scientific than any other of the platonic dialogues. it is conjectural astronomy, conjectural natural philosophy, conjectural medicine.
which the eye is affected by the object, and itself strikes and touches it; there is no violence either in the contraction or dilation of the eye.
forty. circumscribe a square about a given quadrilateral.
we must consider whether it is or is not possible that there should be a continuous motion, and, if it is possible, which this motion is, and which is the primary motion: for it is plain that if there must always be motion, and a particular motion is
further, one would expect them all to exhibit the same movement: but the only star which appears to possess this movement is the sun, at sunrise or sunset, and this appearance is due not to the sun itself but to the distance from which we observe it.
suppose, then, that the movement is unnatural. in that case, if it is the downward movement which is unnatural, the upward movement will be natural; and if it is the upward which is unnatural, the downward will be natural.
but, further, being cannot be one in form, though it may be in what it is made of. (even some of the physicists hold it to be one in the latter way, though not in the former.) man obviously differs from horse in form, and contraries from each other.
so as when speaking of a house we do not call its arrival at perfection an alteration (for it would be absurd to suppose that the coping or the tiling is an alteration or that in receiving its coping or its tiling a house is altered and not perfected),
on the other hand if that which causes alteration or increase causes a certain amount of increase or alteration respectively in a certain amount of time, it does not necessarily follow that half the force will occupy twice the time in altering or
again, because af is equal to ag (const.), and ab to ac (hyp.), the remainder, bf, is equal to cg (axiom iii); and we have proved that fc is equal to gb, and the angle bfc equal to the angle cgb.
as for serum, that sort which is the watery part of blood is innocent, but that which is a secretion of black and acid bile is malignant when mingled by the power of heat with any salt substance, and is then called acid phlegm.
affections, it may be said, are motions, and whiteness is an affection: thus there may be change to a motion. to this we may reply that it is not whiteness but whitening that is a motion.
now the existence of motion is asserted by all who have anything to say about nature, because they all concern themselves with the construction of the world and study the question of becoming and perishing, which processes could not come about without the
another, on both accounts we should naturally be led to place in the front rank the study of the soul.
remnant, the recurrence of day and night and of the seasons, the solid earth and the impalpable aether, were always present to them.
reasoning in this way, therefore, they were led to transcend sense-perception, and to disregard it on the ground that 'one ought to follow the argument': and so they assert that the universe is 'one' and immovable.
but what did plato mean by essence, (greek), which is the intermediate nature compounded of the same and the other, and out of which, together with these two, the soul of the world is created?
now it is no doubt true, as we have also established elsewhere,' that one thing 'comes-tobe' (in the unqualified sense) out of another thing: and further it is true that the efficient cause of its coming-to-be is either (i) an actual thing (which is the
some of them add that it is 'infinite', since the limit (if it had one) would be a limit against the void.
and there is another nature of the same name with it, and like to it, perceived by sense, created, always in motion, becoming in place and again vanishing out of place, which is apprehended by opinion and sense.
yes, but (two) neither is it possible for the 'now' to remain always the same.
twenty four. if two angles have their legs respectively parallel, their bisectors are either parallel or perpendicular.
the reason then is not in the fact that the earth is indifferently related to every extreme point: for this would apply to any body, whereas movement to the centre is peculiar to earth.
sphere or a pyramid. either, then, a part of fire is not fire, so that there is a body prior to the element-for every body is either an element or composed of elements-or not every body is divisible.
it is clear, then, from these considerations that there is no separate void.
for we must think of the tissue after the image of flowing water that is measured by one and the same measure: particle after particle comes-to-be, and each successive particle is different.
'shape' and 'nature', it should be added, are in two senses. for the privation too is in a way form. but whether in unqualified coming to be there is privation, i.e. a contrary to what comes to be, we must consider later.
forty nine. the centre of mean position of the angular points of a regular polygon is the centre of figure of the polygon.
two. from a given point draw to a given line a line making with it an angle equal to a given angle. show that there will be two solutions.
def. iii.-the right line joining either pair of opposite angles of a quadrilateral is called a diagonal.
it is established that the outermost revolution of the heavens is a simple movement and the swiftest of all, and that the movement of all other bodies is composite and relatively slow, for the reason that each is moving on its own circle with the reverse
now if the generation of air from water is to be regarded as growth, a thing might grow without the accession (and without the persistence) of anything, and diminish without the departure of anything-and that which grows need not persist.
relatively, however, to the secondary revolution, i mean that of the planets, we are above and on the right and they are below and on the left.
two. if ac : four hundred :: ec : cb, to prove the triangle acb equal to dce.
ma- nb is greater than mc- nd; therefore a b is greater than c d; that is, the ratio a : b is greater than the ratio c : d.
natural process is inverted, and the bone crumbling passes into the food, and the food into the flesh, and the flesh again falling into the blood makes all maladies that may occur more virulent than those already mentioned.
since, then, everything must be either at rest or in motion, and o is therefore at rest in each of the sections a, b, and g, it follows that a thing can be continuously at rest and at the same time in motion: for, as we saw, o is in motion over the whole
air or water, is borne hither and thither, until its parts, meeting together and mutually harmonising, again become earth; for they can never take any other form.
neither that which always is, therefore, nor that which always is not is either generated or destructible. and clearly whatever is generated or destructible is not eternal.
now 'unqulified' means either (i) the primary predication within each category, or (ii) the universal, i.e. the all-comprehensive, predication.
one hundred twelve. if a chord of a given circle subtend a right angle at a given point, the locus of the intersection of the tangents at its extremities is a circle.
now some things have no extremities at all: and the extremities of others differ specifically although we give them the same name of 'end': how should e.g. the 'end' of a line and the 'end' of walking touch or come to be one?
this town is said to have the largest houses for oxen, cows, and horses hereabouts, and it is not behindhand in its public buildings; but there are very few halls for free worship or free speech in this county.
in mortal motions. streams flow, lightnings play, amber and the magnet attract, not by reason of attraction, but because 'nature abhors a vacuum,' and because things, when compounded or dissolved, move different ways, each to its own place.
cor. three.-if both pairs of opposite sides of a quadrilateral be equal, it is a parallelogram.
and knowing that this lower principle in man would not comprehend reason, and even if attaining to some degree of perception would never naturally care for rational notions, but that it would be led away by phantoms and visions night and day,-to be a
of either sex and the good of either sex might pair with their like; and there was to be no quarrelling on this account, for they would imagine that the union was a mere accident, and was to be attributed to the lot?
as to the kinds of earth, that which is filtered through water passes into stone in the following manner:-the water which mixes with the earth and is broken up in the process changes into air, and taking this form mounts into its own place.
they thought that the truth of the alternative necessarily followed, namely that things come into being out of existent things, i.e. out of things already present, but imperceptible to our senses because of the smallness of their bulk.
against and enter into one another, and so form hollows surrounding the particles that enter-which watery vessels of air (for a film of moisture, sometimes earthy, sometimes pure, is spread around the air) are hollow spheres of water; and those of them
contrary places alike. (by contrarieties of place i mean up and down, back and front, and right and left; and the contrary oppositions of movements are determined by those of places.) one of the motions, then, would be ineffective, for if the two motions
now since time cannot exist and is unthinkable apart from the moment, and the moment a kind of middle-point, uniting as it does in itself both a beginning and an end, a beginning of future time and an end of past time, it follows that there must always be
rectangle a.bc six twelve seventy two square inches. ,, a.bd six five thirty ,, ,, a.de six four twenty four ,, ,, a.ec six three eighteen ,, now the sum of the three last rectangles, viz. thirty, twenty four, eighteen, is seventy two.
now of the divine, he himself was the creator, but the creation of the mortal he committed to his offspring.
the 'why' is referred ultimately either (one), in things which do not involve motion, e.g. in mathematics, to the 'what' (to the definition of 'straight line' or 'commensurable', c.), or (two) to what initiated a motion, e.g.
lastly (four) in each of his infinite bodies there would be already present infinite flesh and blood and brain- having a distinct existence, however, from one another, and no less real than the infinite bodies, and each infinite: which is contrary to
dem.-make the angle dao cab; then the triangles dao, cab are equiangular; therefore ad : do :: ac : cb; therefore ad.bc ac.do. again, the triangles dac, oab are equiangular, and four hundred : ac :: bo : ab; therefore ac.four hundred ac.bo.
for perhaps, if we succeed in clearing up this question, it will simultaneously become clear what account we ought to give of that which perplexed us just now, i.e. of unqualified passingaway and coming-to-be.
it is necessary for him, then, to assume not only that 'being' has the same meaning, of whatever it is predicated, but further that it means (one) what just is and (two) what is just one.
but the things nearest the movent are those whose motion is quickest, and in this case it is the motion of the circumference that is the quickest: therefore the movent occupies the circumference.
he took the three elements of the same, the other, and the essence, and mingled them into one form, compressing by force the reluctant and unsociable nature of the other into the same.
nor again is there anything intermediate between that which undergoes and that which causes alteration: this can be proved by induction: for in every case we find that the respective extremities of that which causes and that which undergoes alteration are
and we can carry on this process for ever, taking the slower after the quicker and the quicker after the slower alternately, and using what has been demonstrated at each stage as a new point of departure: for the quicker will divide the time and the
they had no clear divisions of colours or substances; even the four elements were undefined; the fields of knowledge were not parted off.
if on the other hand it is in motion in only one of the two parts of the time, chrh cannot be the primary time in which it is in motion: for its motion will have reference to a time other than chrh. it must, then, have been in motion in any part of chrh.
have, because somehow we ourselves too are very much under the dominion of chance. whereas he made the soul in origin and excellence prior to and older than the body, to be the ruler and mistress, of whom the body was to be the subject.
our explanation on the other hand is that the phrases 'something comes to be from what is or from what is not', 'what is not or what is does something or has something done to it or becomes some particular thing', are to be taken (in the first way of
perhaps we may suggest the following solution. (i) there are differences of degree in hot and cold.
therefore we must not hold that there was a moment when a came to be at b and that at the same moment five hundred was in motion from the extremity of z: for the fact of a's having come to be at b will involve the fact of its also ceasing to be there, and
for just as a thing is something either in virtue of itself or incidentally, so may it be a cause. for instance, the housebuilding faculty is in virtue of itself the cause of a house, whereas the pale or the musical is the incidental cause.
cor.-the square on the perpendicular from any point in a semicircle on the diameter is equal to the rectangle contained by the segments of the diameter.
only half solid, when above the earth is called snow, and when upon the earth, and condensed from dew, hoar-frost.
he does not wish, he says, that modern corrupt society should return to a state of nature: corruption has gone too far for that; he only desires now that men should palliate, by wiser use of the fatal arts, the mistake of their introduction.
we should also remember, when we attribute to the ancients hasty generalizations and delusions of language, that physical philosophy and metaphysical too have been guilty of similar fallacies in quite recent times.
but this is a real interest and a serious lesson, if we remember that now as formerly the human mind is liable to be imposed upon by the illusions of the past, which are ever assuming some new form.
one. construct a triangle, being given two sides and the angle between them.
warm, and those which were dry become moist, and the light become heavy, and the heavy light; all sorts of changes occur.
it is a whole and limited; not, however, in virtue of its own nature, but in virtue of what is other than it. it does not contain, but, in so far as it is infinite, is contained.
if two ratios be equal, then according as the antecedent of the first ratio is greater than, equal to, or less than its consequent, the antecedent of the second ratio is greater than, equal to, or less than its consequent.
for, as he adds, with an insight into the truth, 'every disease is akin to the nature of the living being and is only irritated by stimulants.'
vi. chords are said to be equally distant from the centre when the perpendiculars drawn to them from the centre are equal.
quaest.), representing thus the signs and degrees of the zodiac, as well as the months and days of the year, god may be said to have 'used in the delineation of the universe.'
compressed body contracts into the voids present in it.
by the motion of the vessel, the elements were divided, and like grain winnowed by fans, the close and heavy particles settled in one place, the light and airy ones in another.
and every sort of body possesses solidity, and every solid must necessarily be contained in planes; and every plane rectilinear figure is composed of triangles; and all triangles are originally of two kinds, both of which are made up of one right and two
by their means we can approximate as nearly as we please to the exact value of the ratio. in the case of the diagonal and the side of a square, the following are the pairs of convergents:-
but the particles which are deficient in the alkaline quality, and which cleanse only moderately, are called salt, and having no bitterness or roughness, are regarded as rather agreeable than otherwise.
persons of the dialogue: socrates, critias, timaeus, hermocrates.
but is not this difficulty removed by the principle already laid down that if that in which the motion takes place is specifically different (as in the present instance the circular path is specifically different from the straight) the motion itself is
twenty seven. with the same hypothesis, find the sides of the same regular figures.
now the first figure belongs to the first body, and the first body is that at the farthest circumference. it follows that the body which revolves with a circular movement must be spherical.
prop. one.-theorem. triangles (abc, acd) and parallelograms (ec, cf) which have the same altitude are to one another as their bases (bc, four hundred).
the head along the neck and through the back, and which in the preceding discourse we have named the seed.
observation.-in the foregoing construction, the line ab is homologous to four hundred, and it is evident that we may take ab to be homologous to any other side of the given figure cdefg.
the various opinions respecting the island of atlantis have no interest for us except in so far as they illustrate the extravagances of which men are capable.
but the third 'originative source' must be present as well-the cause vaguely dreamed of by all our predecessors, definitely stated by none of them.
three , and it is equal to the square on ac, together with the four equal triangles hab, bcf, fdg, geh. now i. xlvii.
it is fire, or in both directions-whatever be the nature of the inserted body.
the second is the so-called 'achilles', and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.
persistent identity, viz. man. (hence, as regards man, these changes are 'modifications'; though, as regards musical man and unmusical man, they are a passing-away and a coming-to-be.) consequently such changes are 'alteration.'
if, having divided a piece of wood or anything else, i put it together, it is again equal to what it was, and is one. clearly this is so, whatever the point at which i cut the wood. the wood, therefore, has been divided potentially through and through.
dem.-one. produce bd (post. ii.) to meet ac in e. then, in the triangle bae, the sum of the sides ba, ae is greater than the side be xx. : to each add ec, and we have the sum of ba, ac greater than the sum of be, ec.
three. the rectangle contained by the sum and the difference of two sides of a triangle is equal to the rectangle contained by the base and the difference of the segments of the base, made by the perpendicular from the vertex.
and so there began to be a real sympathy between the world within and the world without.
as touching your citizens of nine thousand years ago, i will briefly inform you of their laws and of their most famous action; the exact particulars of the whole we will hereafter go through at our leisure in the sacred registers themselves.
the notion of first and second or co-operative causes, which originally appears in the timaeus, has likewise survived to our own day, and has been a great peace-maker between theology and science.
the following exercises are to be solved when the pupil has mastered the first book:-
(two) of the intersection of the internal bisectors of its base angles;
the same thing, if it is of a certain kind, can be both potential and fully real, not indeed at the same time or not in the same respect, but e.g. potentially hot and actually cold.
still, from fire plus water there will result earth and air, and from air plus earth fire and water.
dem.-of the two sides ab, ac, let ab be the one which is not the greater, and with it make the angle bag equal to edf xxiii. . then because ab is not greater than ac, ag is less than ac xix., exer. six . produce ag to h, and make ah equal to df or ac iii.
thereupon one of the priests, who was of a very great age, said: o solon, solon, you hellenes are never anything but children, and there is not an old man among you. solon in return asked him what he meant.
hence the planes ac, df have a common normal bg; therefore they are parallel to one another.
remains still, and the whole is in a sense always at rest as well as continuously in motion.
dem.-since the two triangles adc, acb have the angle a common, and the angles adc, acb equal, each being right, they are i. xxxii. equiangular; hence iv. they are similar. in like manner it may be proved that bdc is similar to abc.
thus we are left with motions respectively to contraries, and motions respectively to contraries from the opposite contraries.
and we now see that in most cases the fact that all the terms are divisible or infinite is a direct consequence of the fact that the thing that changes is divisible or infinite: for the attributes 'divisible' and 'infinite' belong in the first instance to
successiveness). such a movent must either itself continue to push or pull or perform both these actions, or else the action must be taken up by something else and be passed on from one movent to another (the process that we described before as occurring
the foregoing proof differs from euclid's only in the use of symbols. i have found by experience that pupils more readily understand it than any other method.
further, every sensible body is in place, and the kinds or differences of place are up-down, before-behind, right-left; and these distinctions hold not only in relation to us and by arbitrary agreement, but also in the whole itself.
they are of equal velocity if in an equal time there are produced two things that are the same and specifically inseparable, e.g. two men (not merely generically inseparable as e.g. two animals).
; therefore the angle oab is right, and the angle cab is right (hyp.); therefore oab is equal to cab-a part equal to the whole, which is impossible. hence the centre must be in the line ac.
(three) as man is 'in' animal and generally species 'in' genus.
(for it is the man, the gold-the 'matter' generally-that is counted, for it is more of the nature of a 'this', and what comes to be does not come from it in virtue of a concomitant attribute; the privation, on the other hand, and the contrary are
one must decide at the outset whether bodies have a natural movement or not, whether there is no natural but only constrained movement.
there is no single step in astronomy as great as that of the nameless pythagorean who first conceived the world to be a body moving round the sun in space: there is no truer or more comprehensive principle than the application of mathematics alike to the
the view which bases the distinction upon differences of size is more like a mere fiction than those previously mentioned, but, in that it is able to make distinctions between the four elements, it is in a stronger position for meeting the foregoing
to nonsubject, is not change, as in that case there is no opposition either of contraries or of contradictories.
nor again does it make any difference to our demonstration whether the total mass has its weight equally or unequally distributed.
be'. for since the revolving body is always setting something else in motion, the movement of the things it moves must also be circular.
in the republic he represents man as freely choosing his own lot in a state prior to birth-a conception which, if taken literally, would still leave him subject to the dominion of necessity in his after life; in the statesman he supposes the human race to
first then, in my judgment, we must make a distinction and ask, what is that which always is and has no becoming; and what is that which is always becoming and never is?
far weaker. for these reasons and after this manner god placed the sinews at the extremity of the head, in a circle round the neck, and glued them together by the principle of likeness and fastened the extremities of the jawbones to them below the face,
a leg or an arm too long or too short is at once ugly and unserviceable, and the same is true if body and soul are disproportionate. for a strong and impassioned soul may 'fret the pigmy body to decay,' and so produce convulsions and other evils.
but the prior 'now' cannot have ceased-to-be in itself (since it then existed); yet it cannot have ceased-to-be in another 'now'. for we may lay it down that one 'now' cannot be next to another, any more than point to point.
in respect of number the minimum is one (or two); in point of extent there is no minimum.
hence the triangle acb dce five. ix. -that is, the triangles are equal.
hence a b is the same multiple of c that a b is of c.
meantime, so much may be taken as established-viz. that coming-to-be cannot be 'association', at least not the kind of 'association' some philosophers assert it to be.
this proposition may be proved simply as follows:-
the answer will be plain if we first explain what kind of substance an element is.
when they are perfected. but there must previously be something else that is in process of locomotion to be the cause even of the becoming of things that become, without itself being in process of becoming, as e.g.
but the power is of the maximum, and a thing said, with reference to its maximum, to be incapable of so much is also incapable of any greater amount.
the atomists too made the world, if not out of geometrical figures, at least out of different forms of atoms, and these atoms resembled the triangles of plato in being too small to be visible.
the cause of it is not derived from the animal itself: it is connected with other natural motions in animals, which they do not experience through their own instrumentality, e.g.
twenty eight. if ten, y , z be any three points on the three sides of a triangle abc, the three circles about the triangles y az, zbx, xcy pass through a common point.
eighteen. how many ratios in a proportion?
for concave and convex are a not only regarded as opposed to one another, but they are also coupled together and treated as a unity in opposition to the straight.
have a becoming. moreover, just as to say, when any one has passed from a state of intoxication or sleep or disease to the contrary state, that he has become possessed of knowledge again is incorrect in spite of the fact that he was previously incapable
that that which is causing locomotion is in process of increase, that which is causing this increase is in process of being altered by something else, and that which is causing this alteration is in process of suffering some different kind of motion.
here is a question which we must not leave unexamined or undetermined, nor must we affirm too confidently that there can be no decision; neither must we interpolate in our present long discourse a digression equally long, but if it is possible to set
eighty two. if lines be drawn from any point in the circumference of a circle to the angular points of any inscribed regular polygon of an odd number of sides, the sums of the alternate lines are equal.
veins, and makes the stream of the veins flow through the body as through a conduit.
prop. two.-problem. in a given circle (abc) to inscribe a triangle equiangular to a given triangle (def).
(two. the infinite cannot act upon the finite.) nor, again, can the infinite produce a movement in the finite in any time whatever. let a be an infinite, b a finite, one hundred the time of action.
of these we may mention, first, wine, which warms the soul as well as the body; secondly, oily substances, as for example, oil or pitch; thirdly, honey, which relaxes the contracted parts of the mouth and so produces sweetness; fourthly, vegetable acid,
motion or suffers motion not merely by belonging to such a thing or containing it as a part.
as before, we have og gc equal to oh he ; but cg is greater than eh ; therefore og is less than oh . hence og is less than oh.
'by nature' the animals and their parts exist, and the plants and the simple bodies (earth, fire, air, water)-for we say that these and the like exist 'by nature'.
and rebellious against the divinest element within us.
two. if one angle of a triangle be equal to the sum of the other two, the triangle can be divided into the sum of two isosceles triangles, and the base is equal to twice the line from its middle point to the opposite angle.
the 'now' is the link of time, as has been said (for it connects past and future time), and it is a limit of time (for it is the beginning of the one and the end of the other). but this is not obvious as it is with the point, which is fixed.
when in modern times we contemplate the heavens, a certain amount of scientific truth imperceptibly blends, even with the cursory glance of an unscientific person. he knows that the earth is revolving round the sun, and not the sun around the earth.
for some things cause motion in only one way, while others can produce either of two contrary motions: thus fire causes heating but not cooling, whereas it would seem that knowledge may be directed to two contrary ends while remaining one and the same.
one they cannot be, for there cannot be one contrary.
to this city came solon, and was received there with great honour; he asked the priests who were most skilful in such matters, about antiquity, and made the discovery that neither he nor any other hellene knew anything worth mentioning about the times of
'our creators well knew that women and other animals would some day be framed out of men, and they further knew that many animals would require the use of nails for many purposes; wherefore they fashioned in men at their first creation the rudiments of
something that is at rest, then this thing that is at rest must previously have been in process of change: for there must have been some cause of its rest, rest being the privation of motion.
hence abc is equal to abf, which is impossible i., axiom ix. . therefore the lines bc, bd, be are coplanar.
hence, subtracting the sums of the two last equalities from the first, we get the parallelogram dk equal to the parallelogram kb.
the bones and flesh, and other similar parts of us, were made as follows. the first principle of all of them was the generation of the marrow.
if, on the one hand, there were a similar movement from each quarter of the extremity to the single centre, it is obvious that the resulting mass would be similar on every side.
things are not called 'heavy' and 'light' because they act upon, or suffer action from, other things. but the 'elements' must be reciprocally active and susceptible, since they 'combine' and are transformed into one another.
himself-it belongs to the class of things that are intentional and the result of intelligent deliberation. it is when these conditions are satisfied that the man is said to have gone 'by chance'.
five. any line intersecting the legs of a right angle is cut harmonically by any two lines through its vertex which make equal angles with either of its sides.
cor.-if a number of circles touch the same line at the same point, the locus of their centres is the perpendicular to the line at the point.
were of equal strength, there would be no movement either way, and if one of the two were preponderant, the other would be inoperative.
but the surface of water is seen to be spherical if we take as our starting-point the fact that water naturally tends to collect in a hollow place-'hollow' meaning 'nearer the centre'.
for the authors of our being, in obedience to their father's will and in order to make men as good as they could, gave to the liver the power of divination, which is never active when men are awake or in health; but when they are under the influence of
dem.-join be, four hundred. the triangles bde, ced are on the same base de, and between the same parallels bc, de. hence i. xxxvii. they are equal, and therefore five. vii. the triangle ade : bde :: ade : cde;
three. if through either of the points of intersection of two equal circles any line be drawn meeting them again in two points, these points are equally distant from the other intersection of the circles.
we must reply again that we cannot follow plato in all his inconsistencies, but that the gaps of thought are probably more apparent to us than to him.
six. a third line cannot be drawn from p equal to either of the lines ip, fp. for if possible let pk be equal to pf; then pk is equal to pi-that is, one which is nearer to the minimum equal to one more remote-which is impossible.
but even here the holder of the theory ought not only to assert the fact: he ought to explain the cause of it: i.e. he should not make any mere assumption or lay down any gratuitous axiom, but should employ either inductive or demonstrative reasoning.
proportional. conversely, if any rectilineal figure described on the first of four right lines: the similar and similarly described figure described on the second :: any rectilineal figure on the third : the similar and similarly described figure on the
(at same time it is evident that there is no other kind of motion but combination and separation: for they may all be apportioned to one or other of those already mentioned.) again, inhaling is a form of pulling, exhaling a form of pushing: and the same
but two terms must be united by a third, which is a mean between them; and had the earth been a surface only, one mean would have sufficed, but two means are required to unite solid bodies.
nevertheless, even so the question remains: what sorts of contrarieties, and how many of them, are to be accounted 'originative sources' of body?
the figures must have their principles, and, whether these are one or two or more, the simple bodies must be the same in number as they.
and now i will endeavour to show you the disposition and generation of them by an unaccustomed argument, which i am compelled to use; but i believe that you will be able to follow me, for your education has made you familiar with the methods of science.
consequently, we must explain (i) what is the manner of their reciprocal transformation, and (ii) whether every one of them can come to-be out of every one-or whether some can do so, but not others.
conquered by them, unrighteously. he who lived well during his appointed time was to return and dwell in his native star, and there he would have a blessed and congenial existence.
cor. one.-since the perpendicular from o on ab bisects it three. iii. , we see that the perpendiculars at the middle points of the sides of a triangle are concurrent.
four. if in the abc a point five hundred in the base bc be such that
to construct a triangle whose three sides shall be respectively equal to three given lines (a, b, one hundred), the sum of every two of which is greater than the third.
the parallelogram nine hundred is equal to de i. xliii. ; but al is equal to nine hundred i. xxxvi. , because they are on equal bases ac, cb, and between the same parallels; therefore al is equal to de.
nor does this mean that there is anything that can exist separately, over and above what is white. for 'whiteness' and 'that which is white' differ in definition, not in the sense that they are things which can exist apart from each other.
either (a) both are in what is acted on and moved, or (b) the agency is in the agent and the patiency in the patient. (if we ought to call the latter also 'agency', the word would be used in two senses.)
fifty one. if the sides ab, bc, c., of a regular pentagon be bisected in the points a, b, one hundred, five hundred, e, and if the two pairs of alternate sides, bc, ae; ab, de, meet in the points a, e, respectively, prove
applied to each (denoting in each case the proportion of two to one), yet water and air are not commensurable in respect of it. but here again may we not take up the same position and say that the term 'much' is equivocal?
but not of the same thing: the form is the boundary of the thing, the place is the boundary of the body which contains it.
moreover the movent must occupy either the centre or the circumference, since these are the first principles from which a sphere is derived.
on the other hand (ii) the transformation of fire into water and of air into earth, and again of water and earth into fire and air respectively, though possible, is more difficult because it involves the change of more qualities.
without mentally referring to the environment in which they lived, we cannot hope to penetrate below the inessential and temporary to the absolute and permanent value of their thought.
for if it came to be, something must have existed as a primary substratum from which it should come and which should persist in it; but this is its own special nature, so that it will be before coming to be.
indeed, as in any case the surface of the earth is not actually a centre but distant from it a full hemisphere, there is no more difficulty, they think, in accounting for the observed facts on their view that we do not dwell at the centre, than on the
xxi.); and it is described on the given line ab, which was required to be done.
the spot of earth which the goddess chose had the best of climates, and produced the wisest men; in no other was she herself, the philosopher and warrior goddess, so likely to have votaries.
three segments into which the diameter is divided by the chords.
it is wrong, however, to suppose, as some assert, that coming-to-be and passing-away in the unqualified and complete sense are distinctively defined by 'association' and 'dissociation', while the change that takes place in what is continuous is
nothing would be before or after anything else.
only causes it to be in motion. the motion of these last two-of the one as movent and of the other as moved-must cease simultaneously, and with this the whole motion ceases.
it is, for instance, clear that a person who cannot walk a thousand stades will also be unable to walk a thousand and one.
it is chiefly for this reason that we suppose the heaven also to be in place, because it is in constant movement. of this kind of change there are two species-locomotion on the one hand and, on the other, increase and diminution.
having given this law to his creatures, that he might be guiltless of their future evil, he sowed them, some in the earth, some in the moon, and some in the other planets; and he ordered the younger gods to frame human bodies for them and to make the
corrupted crumbles away under the flesh and the sinews, and separates from the bone, and the fleshy parts fall away from their foundation and leave the sinews bare and full of brine, and the flesh again gets into the circulation of the blood and makes the
in the likeness of what animal was the world made?-that is the third question...the form of the perfect animal was a whole, and contained all intelligible beings, and the visible animal, made after the pattern of this, included all visible creatures.
seven. the sum of the two parallel sides of a trapezium is double the line joining the middle points of the two remaining sides.
the hand strikes. we have, then, the following factors: (a) on the one hand that which directly causes motion, and (b) on the other hand that which is in motion: further, we have (c) that in which motion takes place, namely time, and (distinct from these
the remaining phenomena of the same kind there will be no difficulty in reasoning out by the method of probabilities.
the receptacle of the divine soul he made round, and called that portion of the marrow brain, intending that the vessel containing this substance should be the head.
dem.-join fg. then because ae is divided equally in f and unequally in b, the rectangle ab.be, together with fb is equal to fe v. , that is, to fg ; but fg is equal to fb bg i. xlvii. . therefore the rectangle ab.be fb is equal to fb bg .
one. if ac were equal to ab, the triangle acb would be isosceles, and then the angle b would be equal to one hundred v. ; but it is not by hypothesis; therefore ab is not equal to ac.
but the palpable fact, on the contrary, is that the greater the quantity, the lighter the mass is and the quicker its upward movement: and, similarly, in the reverse movement from above downward, the small mass will move quicker and the large slower.
health is only to be preserved by imitating the motions of the world in space, which is the mother and nurse of generation.
one hundred eight. if any chord be drawn through a fixed point on a diameter of a circle, and its extremities joined to either end of the diameter, the joining lines cut off, on the tangent at the other end, portions whose rectangle is constant.
i. when two or more lines are in one plane they are said to be coplanar.
this is incidental, for chance is an incidental cause, as i remarked before. but when an event takes place always or for the most part, it is not incidental or by chance. in natural products the sequence is invariable, if there is no impediment.
matter, being, the same, the eternal,-for any of these terms, being almost vacant of meaning, is equally suitable to express indefinite existence,-are compared or united with the other or diverse, and out of the union or comparison is elicited the idea of
which is at rest cannot be coming to rest.
dem.-one. join oc, od, oe, and draw the perpendiculars og, oh; then because o is the centre, oa is equal to oc i., def. xxxii. , and ob is equal to od. hence ab is equal to the sum of oc and od; but the sum of oc, od is greater than four hundred i. xx. .
in air, for instance, a talent's weight of wood is heavier than a mina of lead, but in water the wood is the lighter. the reason is that all the elements except fire have weight and all but earth lightness.
thirty six. if a variable circle touch two fixed circles, the tangent to it from their centre of similitude, through which the chord of contact passes (twenty seven), is of constant length.
xiii. in proportionals, the antecedent terms are called homologous to one another; as also the consequents to one another.
dem.-the two triangles acd, bcd, have the side ac equal to bc, being the sides of an equilateral triangle, and four hundred common.
but it is better to conceive of the case of the whole heaven as analogous to that of animals, in which the centre of the animal and that of the body are different.
it now is. for you exist now and so does the contact; yet both are destructible, because a time will come when it will not be true of you that you exist, nor of these things that they are in contact.
for if being is just one, and one in the way mentioned, there is a principle no longer, since a principle must be the principle of some thing or things.
if we denote the lines bd, de, ec by a, b, c, the proposition asserts that the rectangle contained by a, and a b c is equal to the sum of the rectangles contained by a and a, a and b, a and c, or, as it may be written, a(a b c) aa ab ac.
works, and of whom i am the artificer and father, my creations are indissoluble, if so i will. all that is bound may be undone, but only an evil being would wish to undo that which is harmonious and happy.
hence the rectangle a.bc a.bd a.de a.ec.
one hundred twenty. if a triangle abc be inscribed in another abc, prove
this power came forth out of the atlantic ocean, for in those days the atlantic was navigable; and there was an island situated in front of the straits which are by you called the pillars of heracles; the island was larger than libya and asia put
four. the right lines joining the adjacent extremities of two unequal parallel right lines will meet, if produced, on the side of the shorter parallel.
now if the terms 'continuous', 'in contact', and 'in succession' are understood as defined above things being 'continuous' if their extremities are one, 'in contact' if their extremities are together, and 'in succession' if there is nothing of their own
is two right angles; therefore the sum of the angles bgh, ghd is two right angles.
not only buddhism, but greek as well as christian philosophy, show that it is quite possible that the human mind should retain an enthusiasm for mere negations.
when, then, the change is in these constitutive factors, there will be coming-to-be or passing-away: but when it is in the thing's qualities, i.e. a change of the thing per accidents, there will be 'alteration'.
it is the same with the notion of an eternity of existence followed later by non-existence. in the later state the capacity will be present for that which is not there in actuality. actualize, then, the capacity.
untuned-and a sense in which they are not, since it is impossible for the contraries to be acted on by each other. but this difficulty also is solved by the fact that the substratum is different from the contraries, for it is itself not a contrary.
(two) there is a difference between touching and being limited. the former is relative to something and is the touching of something (for everything that touches touches something), and further is an attribute of some one of the things which are limited.
hence iv. , angle acf angle abg. and angle afc angle agb. the student should also be shown how to apply one of the triangles to the other, so as to bring them into coincidence. similar illustrations may be given of the triangles bfc, cgb.
fourth, the four lines are proportional.
dem.-since abc is an isosceles triangle, and the angle b is right, each of the other angles is half a right angle; therefore bao is half a right angle. in like manner abo is half a right angle; hence the angle bao equal abo; therefore i. vi.
in like manner the angle ecb is equal to edb. hence the whole angle acb is equal to the whole angle adb.
one. what use is made of prop. vii.? ans. as a lemma to prop. viii.
process either has been, or could have been, indefinitely repeated. but if this is so, the world cannot be indestructible, and it does not matter whether the change of condition has actually occurred or remains a possibility.
or in what does the story consist except in the war between the two rival powers and the submersion of both of them? and how was the tale transferred to the poem of solon? 'it is not improbable,' says mr.
but if this were not to take place, either there would be no movement at all, or the universe would bulge, as xuthus said, or air and water must always change into equal amounts (e.g.
hard work causes fitness and vice versa, but again not in the same way, but the one as end, the other as the origin of change. further the same thing is the cause of contrary results.
if, then, we look at the question in this way the place of a thing is its form. but, if we regard the place as the extension of the magnitude, it is the matter.
in consequence of the motion, the common chord will become in the limit a tangent to each circle, as in the second diagram.-comberousse, geometrie plane, page fifty seven.
for what cannot be can no more come-to-be than be, and movement is a coming to-be in one place from another.
three. the locus of a point which is equally distant from two fixed lines is the pair of lines which bisect the angles made by the fixed lines.
this argument is the same in principle as that which depends on bisection, though it differs from it in that the spaces with which we successively have to deal are not divided into halves.
for a body which is next in the series and in contact (not by compulsion) is akin, and bodies which are united do not affect each other, while those which are in contact interact on each other.
and let us further suppose it possible that some principles that are unmoved but capable of imparting motion at one time are and at another time are not.
three. if the base of a triangle be divided into any number of equal parts, right lines drawn from the vertex to the points of division will divide the whole triangle into as many equal parts.
now that they are different in respect of their essence is evident; for 'that in which something is' and 'that which is in it' would be differently defined.
again, all affections have their origin in condensation and rarefaction: thus heavy and light, soft and hard, hot and cold, are considered to be forms of density and rarity.
to inquire therefore whether being is one in this sense would be like arguing against any other position maintained for the sake of argument (such as the heraclitean thesis, or such a thesis as that being is one man) or like refuting a merely contentious
for he denies that any one of his elements comes-to-be out of any other, insisting on the contrary that they are the things out of which everything else comes-to-be; and yet (having brought the entirety of existing things, except strife, together into
again, even on their view it turns out that all things are not composed of homocomerous bodies. they do not pretend that a face is composed of faces, or that any other natural conformation is composed of parts like itself.
since, then, every motion is a kind of change, and there are only the three kinds of change mentioned above, and since of these three those which take the form of 'becoming' and 'perishing', that is to say those which imply a relation of contradiction,
he is again being studied more as a thinker and less as an ally or an opponent; there is more eagerness to sift the true from the false, and to seek in the social contract the "principles of political right," rather than the great revolutionary's ipse
direction the change may be, e.g. grey in a motion from grey to white takes the place of black as starting-point, in a motion from white to grey it takes the place of black as goal, and in a motion from black to grey it takes the place of white as goal:
to the argument that all things are one if being means one thing, they conceded that not-being is; to that from bisection, they yielded by positing atomic magnitudes.
though they have no real place, they nevertheless, in respect of their position relatively to us, have a right and left as attributes ascribed to them only in consequence of their relative position, not having by nature these various characteristics.
since then the composite cannot be an element, not every homoeomerous body can be an element; only, as we said before, that which is not divisible into bodies different in form.
that is why they have all left the cold out, though properly either all or none should have their distinguishing figures. some of them, however, do attempt to explain this power, and they contradict themselves.
(three) to rely on mere thinking is absurd, for then the excess or defect is not in the thing but in the thought. one might think that one of us is bigger than he is and magnify him ad infinitum.
but the sum of the sides bd, six hundred is greater than bc; hence six hundred is greater than ec.
twenty one. if a, b, one hundred be three points in the circumference of a circle, and five hundred, e the middle points of the arcs ab, ac; then if the line de intersect the chords ab, ac in the points f, g, af is equal to ag.
this, then, is one absurd and impossible implication of the theory. it is also evident that the cube will have this same volume even if it is displaced, which is an attribute possessed by all other bodies also.
towards the centre if it is heavy, and upwards if it is light. this would need to be true of the infinite also. but neither character can belong to it: it cannot be either as a whole, nor can it be half the one and half the other.
six. the square on either side of a right-angled triangle is equal to the rectangle contained by the sum and the difference of the hypotenuse and the other side.
sixty one. if a circle make given intercepts on two fixed lines, the rectangle contained by the perpendiculars from its centre on the bisectors of the angle formed by the lines is given.
but if the elements are not an infinity and not reducible to one, they must be several and finite in number.
the framers of us framed the mouth, as now arranged, having teeth and tongue and lips, with a view to the necessary and the good contriving the way in for necessary purposes, the way out for the best purposes; for that is necessary which enters in and
so' too, if water and earth are to result from fire and air respectively-both qualities must change.
hence those who dwell in intimate association with nature and its phenomena grow more and more able to formulate, as the foundations of their theories, principles such as to admit of a wide and coherent development: while those whom devotion to abstract
(four) as the genus is 'in' the species and generally the part of the specific form 'in' the definition of the specific form.
they said that things grow 'by the accession of like to like', but they did not proceed to explain the manner of this accession.
but if the present is divisible, there will be part of the past in the future and part of the future in the past: for past time will be marked off from future time at the actual point of division.
seven. the diagonals of a cyclic quadrilateral intersect in e: prove that the tangent at e to the circle about the triangle abe is parallel to four hundred.
a bright hue mingled with red and white gives the colour called auburn (greek).
as bockh and the majority of commentators, ancient as well as modern, are inclined to believe, plato thought that the earth was at rest in the centre of the universe, or, as aristotle and mr. grote suppose, that it revolved on its axis.
for there will be a force which moves it, and the smaller and lighter a body is the further will a given force move it. now let a, the weightless body, be moved the distance ce, and b, which has weight, be moved in the same time the distance cd.
to a thing at the same time. and there is no need to be disturbed by the fact that on this showing there may be more than one contrary to the same thing, that a particular motion will be contrary both to rest and to motion in the contrary direction.
two. construct a triangle, being given two angles and the side between them.
of becoming, that which is in process of becoming is in process of perishing at the very moment when it has reached the stage of becoming: since it cannot be in process of perishing when it is just beginning to become or after it has ceased to become: for
nevertheless it is not in so far as he is a patient that he possesses the art of medicine: it merely has happened that the same man is doctor and patient-and that is why these attributes are not always found together.
sol.-bisect any two angles a, b of the given triangle by the lines ao, bo; then o, their point of intersection, is the centre of the required circle.
four. if four magnitudes be continual proportionals, the ratio of st : th is equal to the cube of the ratio of st : nd. this may be proved exactly like three.
the greek philosopher looked at the blue circle of the heavens and it flashed upon him that all things were one; the tumult of sense abated, and the mind found repose in the thought which former generations had been striving to realize.
composed of elements, since a part of fire will be neither fire nor any other element.
then in the triangles acg, bcg we have ac equal to cb (const.), cg common, and the base ga equal to gb, because they are drawn from g, which is, by hypothesis, the centre, to the circumference. hence i. viii.
an earlier one. so, too, in the case of the other kinds of motion.
manner painful diseases are produced. the most painful are caused by wind generated within the body, which gets about the great sinews of the shoulders-these are termed tetanus.
if then they begin from the right and move round to the right, the upper must be the unseen pole. for if it is the pole we see, the movement will be leftward, which we deny to be the fact. clearly then the invisible pole is above.
we must keep in mind that the word 'is' means either what potentially is or what fully is. further, a thing is infinite either by addition or by division.
this method of distinction makes all judgement relative. there will be no absolute distinction between fire, water, and air, but one and the same body will be relatively to this fire, relatively to something else air.
from this alone it is clear that an infinite body is an impossibility; but there is a further point. if there is no such thing as infinite weight, then it follows that none of these bodies can be infinite.
(that is why the poet was carried away into making an absurd statement when he said 'he has the end for the sake of which he was born'. for not every stage that is last claims to be an end, but only that which is best.)
there are these difficulties, then, in the way of the assumption that that which 'is not' can be in motion: and it may be further objected that, whereas everything which is in motion is in space, that which 'is not' is not in space: for then it would be
for if a body 'can be separated at the contacts' (as some thinkers express it), then, even though it has not yet been divided, it will be in a state of dividedness-since, as it can be divided, nothing inconceivable results.
(five) most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody-not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in
for those who say that the universe is one something (i.e.
for the present is something that is an extremity of the past (no part of the future being on this side of it) and also of the future (no part of the past being on the other side of it): it is, as we have said, a limit of both.
thirty nine. if a line ef divide proportionally two opposite sides of a quadrilateral, and a line gh the other sides, each of these is divided by the other in the same ratio as the sides which determine them.
for suppose h in the course of its locomotion proceeds to five hundred and then turns back and proceeds downwards again: then the extreme point five hundred has served as finishing-point and as starting-point for it, one point thus serving as two:
but in the case of that which always is, there is no time for such a capacity of not being, whether the supposed time is finite or infinite; for its capacity of being must include the finite time since it covers infinite time.
but our creators were of opinion that a shorter life which was better was preferable to a longer which was worse, and therefore they covered the head with thin bone, and placed the sinews at the extremity of the head round the neck, and fastened the
one. prove this proposition by a direct demonstration.
socrates: do you remember what were the points of which i required you to speak?
but if the intermediate thing is time, it will be divisible: for all time has been shown to be divisible. thus on this assumption the present is divisible.
no doubt the ancients often fell into strange and fanciful errors: the time had not yet arrived for the slower and surer path of the modern inductive philosophy.
further, we say that all excellences depend upon particular relations.
when the creator had made all these ordinances he remained in his own accustomed nature, and his children heard and were obedient to their father's word, and receiving from him the immortal principle of a mortal creature, in imitation of their own creator
white phlegm, which is dangerous if kept in, by reason of the air bubbles, is not equally dangerous if able to escape through the pores, although it variegates the body, generating diverse kinds of leprosies.
moreover, there can always be something between points (for all lines are intermediate between points), whereas it is not necessary that there should possibly be anything between units: for there can be nothing between the numbers one and two.
to resume then; when a thing of this kind comes to pass among events which are for the sake of something, it is said to be spontaneous or by chance.
we cannot tell (nor could plato himself have told) where the figure or myth ends and the philosophical truth begins; we cannot explain (nor could plato himself have explained to us) the relation of the ideas to appearance, of which one is the copy of the
for it will always be necessary that some other member shall have come-to-be before 'this' as the presupposed condition of the necessity that 'this' should come-to-be: consequently, since what is 'infinite' has no 'originative source', neither will there
further, those who hold these views must needs suppose that generation does not start from a body.
for every portion of earth has weight until it reaches the centre, and the jostling of parts greater and smaller would bring about not a waved surface, but rather compression and convergence of part and part until the centre is reached.
for the extremities of two points can neither be one (since of an indivisible there can be no extremity as distinct from some other part) nor together (since that which has no parts can have no extremity, the extremity and the thing of which it is the
the following is a formal proof of this proposition:-let o, o be the centres of the two circles, a the point of contact, and let o lie between o and a; take any other point b in the circumference of o. join ob; then vii.
simultaneous; and for this reason, while the movements are different and separate, the time is everywhere the same, because the number of equal and simultaneous movements is everywhere one and the same.
and since becoming cannot be primary-for, if it were, everything that is in motion would be perishable-it is plain that no one of the motions next in order can be prior to locomotion.
as a step towards settling which view is true, we must determine the meaning of the name.
and the motion of things that derive their motion from something else is in some cases natural, in other unnatural: e.g. upward motion of earthy things and downward motion of fire are unnatural.
so, too, in every other case where the movent must be described by the same name in the same sense as the moved. therefore when a thing moves itself it is one part of it that is the movent and another part that is moved.
four. join og, oh. the two triangles gop, fop have two sides go, op in one respectively equal to two sides fo, op in the other; but the angle gop is greater than fop; therefore i. xxiv. the base gp is greater than fp. in like manner hp is greater than gp.
thirty eight. construct an equilateral triangle having its vertex at a given point, and the extremities of its base on two given circles.
dem.-because the rectangle ab.bc is equal to the square on ac,ab : ac :: ac : bc xvii. . hence ab is cut in extreme and mean ratio in one hundred def. ii. .
hence ad.bc ab.four hundred ac.bd.three this proposition is known as ptolemy's theorem.
the spontaneous on the other hand is found both in the lower animals and in many inanimate objects. we say, for example, that the horse came 'spontaneously', because, though his coming saved him, he did not come for the sake of safety.
cor. one.-if the opposite angles of a cyclic quadrilateral be equal, one of its diagonals must be a diameter of the circumscribed circle.
again, the figure cfgh is a square i. xlvi., cor. three , and equal to ac ah i. xlvii. -that is, equal to ac cb . hence the whole figure abde ac cb ac.cb.
if two triangles (abc, def) have two angles (b, one hundred) of one equal respectively to two angles (e, f) of the other, and a side of one equal to a side similarly placed with respect to the equal angles of the other, the triangles are equal in every
six. if two right-angled triangles have equal hypotenuses, and a side of one equal to a side of the other, they are congruent.
and now he desires to see the ideal state set in motion; he would like to know how she behaved in some great struggle.
it is absurd, because it is unreasonable that one element alone should have no part in the transformations, and also contrary to the observed data of sense, according to which all alike change into one another.
come-to-be and pass-away. nor is it possible for empedocles to explain how they do so, since he does not assert that fire too (and similarly every one of his other 'elements') possesses 'elementary constituents' of itself.
a motion from health and a motion from disease; or motions respectively to contraries, e.g. a motion to health and a motion to disease; or motions respectively from a contrary and to the opposite contrary, e.g.
the greatest 'divination' of the ancients was the supremacy which they assigned to mathematics in all the realms of nature; for in all of them there is a foundation of mechanics.
twenty. from the extremities of the base of a triangle perpendiculars are let fall on the opposite sides; prove that the sum of the rectangles contained by the sides and their lower segments is equal to the square on the base.
again, since in the same time the slower has been carried over ez, zh, the time may also be similarly divided into two. thus the indivisible will be divisible, and that which has no parts will be passed over not in an indivisible but in a greater time.
critias: let me proceed to explain to you, socrates, the order in which we have arranged our entertainment.
timaeus: let me tell you then why the creator made this world of generation. he was good, and the good can never have any jealousy of anything. and being free from jealousy, he desired that all things should be as like himself as they could be.
for of these things too the distinction between potential and actual existence holds. we say that there are olympic games, both in the sense that they may occur and that they are actually occurring.
for star and circle are seen to come back to the same place at the same moment; from which it follows that the star has traversed the circle and the circle has completed its own movement, i.e. traversed its own circumference, at one and the same moment.
there is no principle so apparent in the physics of the timaeus, or in ancient physics generally, as that of continuity.
two. if a chord of a given circle subtend a right angle at a fixed point, the rectangle of the perpendiculars on it from the fixed point and from the centre of the given circle is constant.
augustine, repeating a thought derived from the timaeus, but apparently unconscious of the results to which his doctrine would have led.
-one-- dd -one-- ee -one-- f f zero, and two -a-- dd two -b-- ee two -c-- f f zero.
hence the two parallel right lines and the transversal are coplanar.
(this proposition should be proved after the student has read prop. xxxii.)
there are things that are sometimes in motion and sometimes not?
eighty seven. if p, p denote the areas of two regular polygons of any common number of sides, inscribed and circumscribed to a circle, and , the areas of the corresponding polygons of double the number of sides; prove is a geometric mean between p and p,
herself, nevertheless motion is the characteristic fact of nature: moreover, the view is actually held by some that not merely some things but all things in the world are in motion and always in motion, though we cannot apprehend the fact by
dem.-one. through one hundred draw ce parallel to ad, to meet ba produced in e. because ba meets the parallels ad, ec, the angle bad i. xxix.
supposed, understood (greek) in the timaeus to mean 'revolving.' for the second passage, in which motion on an axis is expressly mentioned, refers to the first, but this would be unmeaning unless (greek) in the first passage meant rotation on an axis.
similar polygons may be divided (one) into the same number of similar triangles; (two) the corresponding triangles have the same ratio to one another which the polygons have; (three) the polygons are to each other in the duplicate ratio of their
it cannot then be the fewness of the triangles (of which, in their view, all these bodies are composed) which disposes fire to move upward.
the fleshy nature was not therefore wholly dried up, but a large sort of peel was parted off and remained over, which is now called the skin. this met and grew by the help of the cerebral moisture, and became the circular envelopment of the head.
forty five. if abc be an equilateral triangle, what is the locus of the point one thousand, if ma mb one thousand one hundred?
yet space is also of a very permanent or even eternal nature; and plato seems more willing to admit of the unreality of time than of the unreality of space; because, as he says, all things must necessarily exist in space.
let us now proceed to define the terms 'together' and 'apart', 'in contact', 'between', 'in succession', 'contiguous', and 'continuous', and to show in what circumstances each of these terms is naturally applicable.
and if what is heavy must be either hard or soft, an impossible consequence is easy to draw. for a thing is soft if its surface can be pressed in, hard if it cannot; and if it can be pressed in it is divisible.
(it might indeed be thought that there is a form of pulling that arises in another way: that wood, e.g. pulls fire in a manner different from that described above.
but parmenides had not come in sight of this distinction.
our own doctrine is that although there is a matter of the perceptible bodies (a matter out of which the so-called 'clements' come-to-be), it has no separate existence, but is always bound up with a contrariety.
outer circle of the fixed stars and the inner circle of the planets, divided according to certain musical intervals; he has also created time, the moving image of eternity, and space, existing by a sort of necessity and hardly distinguishable from matter.
if the world is believed to be one, it is impossible to suppose that it should be, as a whole, first generated and then destroyed, never to reappear; since before it came into being there was always present the combination prior to it, and that, we hold,
ten. why has a line neither breadth nor thickness?
three. through one of the points of intersection of two circles draw a secant-one. the sum of whose segments intercepted by the circles shall be a maximum; two. which shall be of any length less than that of the maximum.
two. it is an interesting and not unimportant question which is touched upon by martin, whether the atlantis of plato in any degree held out a guiding light to the early navigators. he is inclined to think that there is no real connexion between them.
consequently (if a thing can move itself as a whole), the same thing in respect of the same thing may be at the same time both hot and not hot.
in like manner eb is parallel to ao; and since ao is equal to ob, the figure aobe is a lozenge, and the angle aob is right; hence aobe is a square. in like manner each of the figures bc, four hundred, da is a square. hence the whole figure is a square.
measures their existence. of things which do not exist but are contained by time some were, e.g. homer once was, some will be, e.g. a future event; this depends on the direction in which time contains them; if on both, they have both modes of existence.
between one, three, nine, twenty seven) cutting off yet other portions from the mixture and placing them in the intervals, so that in each interval there were two kinds of means, the one exceeding and exceeded by equal parts of its extremes (as for
for coming-to-be necessarily implies the pre-existence of something which potentially 'is', but actually 'is not'; and this something is spoken of both as 'being' and as 'not-being'.
again, the triangle abc is half the parallelogram aebc xxxiv. , because the diagonal ab bisects it.
the body, then, is turned into spheres or pyramids. let us grant that these figures may reasonably be supposed to cut and break up bodies as fire does; still it remains quite inexplicable that a pyramid must needs produce pyramids or a sphere spheres.
the general phenomena of sensation are partly internal, but the more violent are caused by conflict with external objects.
in this work, which voltaire, in thanking him for a presentation copy, termed his "second book against the human race," his style and his ideas have made a great advance; he is no longer content merely to push a single idea to extremes: while preserving
cor. two.-the locus of the centres of the circles which pass through two fixed points is the line bisecting at right angles that connecting the two points.
(i) finite they cannot be, for if the all is to be infinite, some of them would have to be infinite, while the others were not, e.g. fire or water will be infinite. but, as we have seen before, such an element would destroy what is contrary to it.
thus, then, nature makes matters equal and establishes a certain order, giving to the single motion many bodies and to the single body many motions.
will be true to say that the thing, itself and its parts, occupies the same space.
of the corresponding magnitude of the second system.
granted that some public works would not have been constructed without this aid, and let man share the glory of such with the ox and horse; does it follow that he could not have accomplished works yet more worthy of himself in that case?
and next he filled up the intervals of a fourth with ninths, leaving a remnant which is in the ratio of two hundred fifty six:two hundred forty three.
difficulties. since, however, it imagines that these bodies which differ in size are all made of one substance, it implies, equally with the view that there is but one matter, that there is nothing absolutely light and nothing which moves upward (except
he is to partake of the repose of nature and of the order of nature, to bring the variable principle in himself into harmony with the principle of the same. the ethics of the timaeus may be summed up in the single idea of 'law.'
a medium that is somewhat dense. a, then, will move through b in time g, and through five hundred, which is thinner, in time e (if the length of b is egual to five hundred), in proportion to the density of the hindering body.
our first business, then, will be to speak of these, and that inquiry will enable us to explain the differences of each from each.
therefore ab : bg :: de : ef iv. ; but ab : bc :: de : ef (hyp.). therefore bg is equal to bc. hence the angles bcg, bgc must be each acute i. xvii.
now everything that changes changes time, and that in two senses: for the time in which a thing is said to change may be the primary time, or on the other hand it may have an extended reference, as e.g.
from this it evidently follows that coming to a stand must occupy a period of time: for the motion of that which is in motion occupies a period of time, and that which is coming to a stand has been shown to be in motion: consequently coming to a stand
but if (b) the all has dissimilar parts, the proper places of the parts will be dissimilar also, and the body of the all will have no unity except that of contact. then, further, the parts will be either finite or infinite in variety of kind.
prop. seventeen.-theorem. any two angles (b, one hundred) of a triangle (abc) are together less than two right angles.
the extremity of z to h: then, says the argument, five hundred will have reached h before a has reached g for that which makes an earlier start and departure must make an earlier arrival: the reason, then, for the late arrival of a is that it has not
we can say (one) 'man becomes musical', (two) what is 'not-musical becomes musical', or (three), the 'not-musical man becomes a musical man'.
it is from these, which are 'parts' (in the sense at least of being 'in' the man), that the man is called white, c. but the jar and the wine in separation are not parts of a whole, though together they are.
like right and left, as we said before and as the difficulty just stated itself suggests, the distinction of prior and posterior, which provides a reason and so solves our difficulty.
the accurate determination of the first principle in respect of form, whether it is one or many and what it is or what they are, is the province of the primary type of science; so these questions may stand over till then. but of the natural, i.e.
these distinctions will be drawn more carefully later. on the present occasion it was necessary to refer to them: what has now been stated obscurely will then be made more clear.
thus six, nine, twelve, fifteen, c., are multiples of three; but ten, thirteen, seventeen, c., are not, because the multiplication of three by any whole number will not produce them.
for what 'alters' and transfigures plays a greater part in bringing, things into being; and we are everywhere accustomed, in the products of nature and of art alike, to look upon that which can initiate movement as the producing cause.
for the greater quantity, finding a lesser in front of it, must necessarily drive it on, both having an impulse whose goal is the centre, and the greater weight driving the lesser forward till this goal is reached.
anaxagoras gives an absurd account of why the infinite is at rest. he says that the infinite itself is the cause of its being fixed.
both (a) that which contains the originative source of the motion is thought to 'impart motion' (for the originative source is first amongst the causes), and also (b) that which is last, i.e. immediately next to the moved thing and to the coming-to-be.
so, too, of such things there is no remaining though there is absence of change. should there be a particular subject, absence of change in its being will be contrary to absence of change in its not-being.
fire, if set at the centre, should stay there, like earth, since it will be indifferently related to every point on the extremity. nevertheless it will move, as in fact it always does move when nothing stops it, away from the centre to the extremity.
and if they are 'too minute to admit any body', it is absurd to suppose there is a 'minute' void and yet to deny the existence of a 'big' one (no matter how small the 'big' may be), or to imagine 'the void' means anything else than a body's place-whence
we say that a thing is in the world, in the sense of in place, because it is in the air, and the air is in the world; and when we say it is in the air, we do not mean it is in every part of the air, but that it is in the air because of the outer surface
remain permanently self-contained and within the same limits: for if the first principle is permanent, the universe must also be permanent, since it is continuous with the first principle.
but that which is not a substance or a 'this' clearly cannot possess predicates drawn from any of the other categories either-e.g. we cannot attribute to it any quality, quantity, or position.
the object of vision, however, is a quality of tangible body not qua tangible, but qua something else-qua something which may well be naturally prior to the object of touch.
advise quietly for the good of the whole. for the creator knew that the belly would not listen to reason, and was under the power of idols and fancies.
three. if we take any number, such as three, and multiply it by any whole number, the product is called a multiple of three.
he who is intent upon the gratification of his desires and cherishes the mortal soul, has all his ideas mortal, and is himself mortal in the truest sense.
that there is one heaven, then, only, and that it is ungenerated and eternal, and further that its movement is regular, has now been sufficiently explained.
since there are some who say that there is a right and a left in the heaven, with those who are known as pythagoreans-to whom indeed the view really belongs-we must consider whether, if we are to apply these principles to the body of the universe, we
whether each of them is eternal or whether there is a sense in which they come-to-be: and, if they do come-to-be, whether all of them come-to-be in the same manner reciprocally out of one another, or whether one amongst them is something primary.
sol.-construct i. xlv. the rectangle ac equal to ten.
worlds necessarily involve motion), whereas those who hold that there is only one world, whether everlasting or not, make corresponding assumptions in regard to motion.
must we then say that, if two things are to be commensurable in respect of any attribute, not only must the attribute in question be applicable to both without equivocation, but there must also be no specific differences either in the attribute itself or
it is evident, therefore, if we consider them in general, that every one is by nature such as to come-to-be out of every one: and when we come to consider them severally, it is not difficult to see the manner in which their transformation is effected.
not fair or perfect. was the heaven then or the world, whether called by this or by any other more appropriate name-assuming the name, i am asking a question which has to be asked at the beginning of an enquiry about anything-was the world, i say, always
intelligence (compare the construction of solids out of surfaces in his account of the creation of the world, or the attraction of similars to similars). further, mr.
the following annotations will make them explicit:-
on the other hand we find fewer traces in plato of early ionic or eleatic speculation.
in equal circles (acb, dfe), equal chords (ab, de) divide the circumferences into arcs, which are equal each to each-that is, the lesser to the lesser, and the greater to the greater.
seven. the area of a quadrilateral is equal to the area of a triangle, having two sides equal to its diagonals, and the contained angle equal to that between the diagonals.
(compare plato's 'great and small'-except that he make these his matter, the one his form, while the others treat the one which underlies as matter and the contraries as differentiae, i.e. forms).
no one saw that this objective was really a subjective, and involved the subjectivity of all knowledge. 'non in tempore sed cum tempore finxit deus mundum,' says st.
(indeed, this is a blemish in the theory of those who assert that 'once upon a time all things were together and combined'. for not everything can 'combine' with everything.
which, measured from either diagonal, have the same ratio to each other. twenty six.
, and, being adjacent angles, they are right angles (def. xiii.). therefore ch is perpendicular to ab.
in any triangle (abc), the square on any side subtending an acute angle (one hundred) is less than the sum of the squares on the sides containing that angle, by twice the rectangle (bc, four hundred) contained by either of them (bc) and the intercept
naturally they could not be there, since their proper places are elsewhere; and if these are there unnaturally, the exterior place will be natural to some other body, since a place which is unnatural to one body must be natural to another: but we saw that
if then in art there are cases in which what is rightly produced serves a purpose, and if where mistakes occur there was a purpose in what was attempted, only it was not attained, so must it be also in natural products, and monstrosities will be failures
- over one, four three, three two, - over two, eight three, three, - over four, sixteen three, six, - over eight: and - over one, three two, two, - over three, nine two, six, - over nine, twenty seven two, eighteen, - over twenty seven.
three of these it obviously cannot be: (one) the shape is supposed to be place because it surrounds, for the extremities of what contains and of what is contained are coincident. both the shape and the place, it is true, are boundaries.
if, on the other hand, the contrarieties, being two, can generate from each other, the second contrariety will be superfluous. moreover, it is impossible that there should be more than one primary contrariety.
we all of us feel that fire is sharp; and we may further consider the fineness of the sides, and the sharpness of the angles, and the smallness of the particles, and the swiftness of the motion-all this makes the action of fire violent and sharp, so that
dem.-if ab is perpendicular to four hundred, as in fig. one, the angles abc, abd are right angles. if not, draw be perpendicular to four hundred xi. . now the angle cba is equal to the sum of the two angles cbe, eba (def. xi.).
(a) we need only appeal to the distinction between the statements 'x will be' and 'x is about to which depends upon this fact.
one hundred one. and the circle described about the triangle formed by drawing tangents to ten, at the angular points of the inscribed triangle, touches a given circle.
but the circular movement is natural, since otherwise it could not be eternal: for nothing unnatural is eternal. the unnatural is subsequent to the natural, being a derangement of the natural which occurs in the course of its generation.
hearing is the effect of a stroke which is transmitted through the ears by means of the air, brain, and blood to the soul, beginning at the head and extending to the liver.
cor. ten.-if the diagonals of a parallelogram be perpendicular to each other, it is a lozenge.
nails. for this purpose and for these reasons they caused skin, hair, and nails to grow at the extremities of the limbs.
neither is the whole place larger than what can be filled by the body (and then the body would no longer be infinite), nor is the body larger than the place; for either there would be an empty space or a body whose nature it is to be nowhere.
one hundred thirteen. the rectangle contained by the segments of the base of a triangle, made by the point of contact of the inscribed circle, is equal to the rectangle contained by the perpendiculars from the extremities of the base on the bisector of
further, in the latter case, either the movent immediately precedes the last thing in the series, or there may be one or more intermediate links: e.g.
six. the three lines of intersection of three planes are either parallel or concurrent.
and similarly, if the distinction rests on the amount of void, there will be a body, lighter than the absolutely light, which nevertheless moves downward as constantly as the other moves upward.
twenty one. abcd is a cyclic quadrilateral: the lines ab, ad, and the point one hundred, are given in position; find the locus of the point which divides bd in a given ratio.
five. if ab be the diameter of a semicircle, find a point one hundred in ab such that, joining one hundred to a fixed point five hundred in the circumference, and erecting a perpendicular ce meeting the circumference in e, ce cd may be equal to a given
occurrence of coming to-be, because it is such as to change from contrary to contrary and because, in substances, the coming-to-be of one thing is always a passing-away of another, and the passing-away of one thing is always another's coming-to-be.
four. if two right lines be parallel, they are parallel to the common section of any two planes passing through them.
is distinct from coming-to-be: for coming-to-be and passingaway result from the consilience and the dissolution of the many kinds.
but the fact is that he is inconsistent with himself; in one and the same passage vice is attributed to the relaxation of the bodily frame, and yet we are exhorted to avoid it and pursue virtue.
therefore the common section bd of the two planes must be a right line.
and much the same may be said of the view that such is the ordinance of nature and that this must be regarded as a principle, as would seem to be the view of empedocles when he says that the constitution of the world is of necessity such that love and
one. demonstrate both parts of prop. xxviii. without using prop. xxvii.
one. if a variable circle touch two fixed circles externally, the difference of the distances of its centre from the centres of the fixed circles is equal to the difference or the sum of their radii, according as the contacts are of the same or of
with a view to the illustration of the timaeus i propose to divide this introduction into sections, of which the first will contain an outline of the dialogue: (two) i shall consider the aspects of nature which presented themselves to plato and his age,
hence it is evident that the fine derives from the moist, while the coarse derives from the dry. again (b) the viscous' derives from the moist: for 'the viscous' (e.g. oil) is a 'moist' modified in a certain way.
modes of predication. the reason in turn why motion is thought to be indefinite is that it cannot be classed simply as a potentiality or as an actuality-a thing that is merely capable of having a certain size is not undergoing change, nor yet a thing that
of plato are planes, and (ii) are characterized by an infinite variety of figures, while the characterizing figures employed by plato are limited in number.
it is evident, therefore, that the coming-to-be of the 'simple' bodies will be cyclical; and that this cyclical method of transformation is the easiest, because the consecutive 'clements' contain interchangeable 'complementary factors'.
for if it is to be rectilinear in shape, it will follow that there is place and body and void without it.
for a soul could not live in such conditions painlessly or happily, since the movement involves constraint, being imposed on the first body, whose natural motion is different, and imposed continuously.
motion, therefore, that is in an unqualified sense continuous and one must be specifically the same, of one thing, and in one time.
five. in what part of the construction is the third postulate quoted? and for what purpose? where is the first postulate quoted?
oceanus and tethys were the children of earth and heaven, and from these sprang phorcys and cronos and rhea, and all that generation; and from cronos and rhea sprang zeus and here, and all those who are said to be their brethren, and others who were the
or thus: the triangles abe, dcf have xxxiv. the sides ab, be in one respectively equal to the sides six hundred, cf in the other, and the angle abe equal to the angle dcf xxix., ex. eight .
the flesh returns again into the blood. worst of all and most fatal is the disease of the marrow, by which the whole course of the body is reversed. there is a third class of diseases which are produced, some by wind and some by phlegm and some by bile.
two. every circle passing through a given point, and having its centre on a given line, passes through another given point.
as was explained, some things are potentially in place, others actually.
ten. the perpendiculars of a triangle are concurrent.
congruent figures are those that can be made to coincide by superposition. they agree in shape and size, but differ in position.
but this can only go on for ever, if the finite body contains an infinity, which is impossible. therefore the generation of elements out of one another will not always continue.
there were, then, certain thinkers who, for the reasons we have stated, enunciated views of this kind as their theory of 'the truth'....
seventeen. if two circles touch externally, their common tangent is a mean proportional between their diameters.
but when the substance which unites the flesh and bones is diseased, and is no longer renewed from the muscles and sinews, and instead of being oily and smooth and glutinous becomes rough and salt and dry, then the fleshy parts fall away and leave the
subject at all to any of those states which affect moving and sensible things and of which generation is the cause. these are the forms of time, which imitates eternity and revolves according to a law of number.
the part of the soul which desires meats and drinks and the other things of which it has need by reason of the bodily nature, they placed between the midriff and the boundary of the navel, contriving in all this region a sort of manger for the food of the
motion but many). as we have said, then, that which is without parts can be in motion in the sense in which a man sitting in a boat is in motion when the boat is travelling, but it cannot be in motion of itself.
the infinite exhibits itself in different ways-in time, in the generations of man, and in the division of magnitudes.
a house comes from certain things in a certain state of separation instead of conjunction, a statue (or any other thing that has been shaped) from shapelessness-each of these objects being partly order and partly composition.
of two unequal magnitudes, that which has the greater ratio to any third is the greater of the two; and that to which any third has the greater ratio is the less of the two.
if every moment is alike and the moments are infinite in number, it is clear that a generated or destructible thing existed for an infinite time.
the obvious inference, therefore, is that fire and each several element is in a foreign place a sphere or a pyramid, but in its own a cube.
twenty three. what is the magnitude of the rectangle of the segments of a chord drawn through a point three.sixty five metres distant from the centre of a circle whose radius is four.twenty five metres?
again, because the angle acb is equal to cbd, and dcb equal to abc, the whole angle acd is equal to the whole angle abd.
a similar inquiry is made by those who inquire into the number of existents: for they inquire whether the ultimate constituents of existing things are one or many, and if many, whether a finite or an infinite plurality.
similarly in the case of motion: two things are of the same velocity if they occupy an equal time in accomplishing a certain equal amount of motion.
iii. a parallelogram whose angles are right angles is called a rectangle.
(c) in yet another sense we give the name to all body included within extreme circumference, since we habitually call the whole or totality 'the heaven'. the word, then, is used in three senses.
here also the same distinctions are to be observed: a goal of motion may be so accidentally, or partially and with reference to something other than itself, or directly and with no reference to anything else: for instance, a thing which is becoming white
socrates: very good. and what is this ancient famous action of the athenians, which critias declared, on the authority of solon, to be not a mere legend, but an actual fact?
in the first place, then, as is evident to all, fire and earth and water and air are bodies.
no body, however, can be divided into its 'least' parts: and 'composition' is not identical with 'combination', but other than it.
'god therefore adopted the remaining alternative, and fulfilled the perfection of the universe by making coming-to-be uninterrupted: for the greatest possible coherence would thus be secured to existence, because that 'coming-to-be should itself
centre. of the position of the earth and of the manner of its rest or movement, our discussion may here end.
how, on their theory, are flesh and bones or any of the other compounds to result from the 'elements' taken together?
do not say (one) 'from being a man he came to be musical' but only 'the man became musical'.
critias: i will, if timaeus, who is our other partner, approves.
the unequal admixture of these fluids creates a variety of species; most of them are nameless, but four which are of a fiery nature are clearly distinguished and have names.
for if a substantial thing comes-to-be, it is clear that there will 'be' (not actually, but potentially) a substance, out of which its coming-to-be will proceed and into which the thing that is passing-away will necessarily change.
the two cases, as we said, are not parallel; for, in the construction of the figure, when the various steps are completed the required figure forthwith results; but in these other demonstrations what results is not that which was required.
according to the view taken in these volumes the errors of ancient physicists were not separable from the intellectual conditions under which they lived.
the whole, however, has become larger. and this increase is due (a) on the one hand to the accession of something, which is called 'food' and is said to be 'contrary' to flesh, but (b) on the other hand to the transformation of this food into the same
book two. theory of rectangles every proposition in the second book has either a square or a rectangle in its enunciation.
cor. three.-the segments ad, db are in the duplicate of ac : cb, or in other words, ad : db :: ac : cb ,
therefore the series must come to an end, and there must be a first movent and a first moved: for the fact that this impossibility results only from the assumption of a particular case is immaterial, since the case assumed is theoretically possible, and
produced by the action of external forces. and as he was to contain all things, he was made in the all-containing form of a sphere, round as from a lathe and every way equidistant from the centre, as was natural and suitable to him.
stand. for let ab be the primary time in which a thing is coming to a stand.
again, when we say that a thing is at rest, we imply that its condition in whole and in part is at the time of speaking uniform with what it was previously: but the present contains no 'previously': consequently, there can be no rest in it.
however, the point a is the real starting-point at which the moving body has ceased to be, and it is at g that it has really come to be when its course is finished and it comes to a stand.
while geometry investigates physical lines but not qua physical, optics investigates mathematical lines, but qua physical, not qua mathematical.
anything therefore which is generated or destructible must be intermediate. now let a be that which always is and b that which always is not, one hundred the generated, and five hundred the destructible.
rectangle contained by a and bd bg. ,, ,, a ,, de dh. ,, ,, a ,, ec ek. ,, ,, a ,, bc bk. but bk is equal to the sum of bg, dh, ek (i., axiom ix.).
just, in fact, as the vessel is transportable place, so place is a non-portable vessel.
many things both come to be and are by chance and spontaneity, and although know that each of them can be ascribed to some cause (as the old argument said which denied chance), nevertheless they speak of some of these things as happening by chance and
in the same manner it may be proved that the arc ej and the sector epj are equimultiples of the arc ef and the sector epf; and it is evident, by superposition, that if the arc bh is greater than, equal to, or less than the arc ej, the sector boh is
further, if it were, it would have been in their power to stop themselves (i mean that if e.g.
swiftness with the sun, but in an opposite direction; and this is the reason why the sun and hermes and lucifer overtake and are overtaken by each other.
it would seem that i made it according to the recipe which marcus porcius cato gave about two centuries before christ. "panem depsticium sic facito. manus mortariumque bene lavato. farinam in mortarium indito, aqu paulatim addito, subigitoque pulchre.
thus we see that two motions are contrary to each other only when one is a motion from a contrary to the opposite contrary and the other is a motion from the latter to the former.
enough of the composite animal, and of the body which is a part of him, and of the manner in which a man may train and be trained by himself so as to live most according to reason: and we must above and before all provide that the element which is to
six. construct a triangle, being given two sides and the median of the third side.
again, since a : b :: c : d, a- b c d; therefore a -b is greater than e f-. or the ratio of a : b is greater than the ratio of e : f.
whiteness; and similar questions might be asked concerning any other subject of aletion. of course the subject of increase, when changed qua increasable, attains not health but a superior size. the same applies in the other cases.
the pattern too, though eternal, is a creation, a world of thought prior to the world of sense, which may be compared to the wisdom of god in the book of ecclesiasticus, or to the 'god in the form of a globe' of the old eleatic philosophers.
therefore the ratio a : c is greater than the ratio b : c.
it is plain from the foregoing that the universe is spherical. it is plain, further, that it is turned (so to speak) with a finish which no manufactured thing nor anything else within the range of our observation can even approach.
sixteen. the three perpendiculars at the middle points of the sides of a triangle are concurrent. hence prove that perpendiculars from the vertices on the opposite sides are concurrent see ex. two .
one. as the linear unit is that by which we express all linear measures, so the square unit is that to which all superficial measures are referred.
otherwise, properties would admit of existence in separation from substances.
but (two) other things are in place indirectly, through something conjoined with them, as the soul and the heaven. the latter is, in a way, in place, for all its parts are: for on the orb one part contains another.
r . therefore three thousand five hundred sixty seven r . thirty five.
(one) that both should be at rest is impossible; for, if the earth is at rest, the hypothesis does not account for the observations; and we take it as granted that the earth is at rest.
to each of these equals add nine hundred and lg, and we get am and lg equal to the square cdfe; but am is equal to the rectangle ad.dm, and therefore equal to the rectangle ad.db, since db is equal to dm; also lg is equal to the square on cb, and cdfe is
prop. twenty five.-problem. an arc (abc) of a circle being given, it is required to describe the whole circle.
yet this is what 'alteration' essentially is.
cor. two.-the locus of the middle points of a system of parallel chords of a circle is the diameter of the circle perpendicular to them all.
now every one can see whence diseases arise.
one. the sum of the squares on the segments of a line of given length is a minimum when it is bisected.
eclog.) that all things are either finite (definite) or infinite (indefinite), or a union of the two, and that this antithesis and synthesis pervades all art and nature, we are reminded of the philebus.
only by the arguments already set forth but also by a consideration of the views of those who differ from us in providing for its generation.
and so when they made the belly to be a receptacle for food, in order that men might not perish by insatiable gluttony, they formed the convolutions of the intestines, in this way retarding the passage of food through the body, lest mankind should be
two. divide a given line internally, so that the sum of the squares on the parts may be equal to a given square, and state the limitation to its possibility.
it is as if one were to think that there was now destruction and now existence when from a child a man is generated, and from a man a child.
process is determined by the character of that which undergoes it.
not in that place in which they come to be, which is part of the place which is the place of the whole world.
primary and continuous, then it is this motion that is imparted by the first movent, and so it is necessarily one and the same and continuous and primary.
all these are to be reckoned among the second and co-operative causes which god, carrying into execution the idea of the best as far as possible, uses as his ministers.
for this is different from the magnitude: it is what is contained and defined by the form, as by a bounding plane. matter or the indeterminate is of this nature; when the boundary and attributes of a sphere are taken away, nothing but the matter is left.
or are only those things which we see, or in some way perceive through the bodily organs, truly existent, and nothing whatever besides them? and is all that which we call an intelligible essence nothing at all, and only a name?
secondly, we must not forget that rousseau's theories are to be studied in a wider historical environment.
cor. one.-the side of a regular hexagon inscribed in a circle is equal to the radius.
before being set in motion unnaturally. this theory, then, does away with becoming and perishing.
sol.-divide ab in one hundred, so that the rectangle ab.bc may be equal to the square on ac two. xi. then one hundred is the point required.
on the contrary, the constituents will only be 'combined' relatively to perception: and the same thing will be 'combined' to one percipient, if his sight is not sharp, (but not to another,) while to the eye of lynceus nothing will be 'combined'.) it
two. the difference between the squares on the segments of a line divided in "extreme and mean ratio" is equal to their rectangle.
cor. one.-if two lines ab, four hundred produced meet in p, and if the rectangle ap.bp cp.dp, the points a, b, one hundred, five hundred are concyclic (compare xxxv., cor. two).
in the direction of the outermost part of the universe, and the outermost part itself, are up.
prop. thirty.-theorem. to divide a given line (ab) in "extreme and mean ratio."
he does not imagine the earth to be the centre of the universe, and he has some conception of chemistry and the cognate sciences. a very different aspect of nature would have been present to the mind of the early greek philosopher.
capable both of being and of not being, there must be some definite maximum time of their being and not being; a time, i mean, during which continued existence is possible to them and a time during which continued nonexistence is possible.
and we measure both the distance by the movement and the movement by the distance; for we say that the road is long, if the journey is long, and that this is long, if the road is long-the time, too, if the movement, and the movement, if the time.
dem.-take bg a third proportional to bc, ef xi. . join ag. then because the triangles abc, def are similar, ab : bc :: de : ef; hence (alternately) ab : de :: bc : ef; but bc : ef :: ef : bg (const.); therefore five. xi.
before i finished my house, wishing to earn ten or twelve dollars by some honest and agreeable method, in order to meet my unusual expenses, i planted about two acres and a half of light and sandy soil near it chiefly with beans, but also a small part
in the same manner it may be shown that the other angles of the hexagon are equal. hence it is equiangular, and is therefore a regular hexagon inscribed in the circle.
of the divine he himself was the author, but he committed to his offspring the creation of the mortal.
one of these premisses is that nothing finite can cause motion during an infinite time.
two. construct a triangle equal in area to a given quadrilateral figure.
grass; and again, an auburn colour mingles with the bitter matter when new flesh is decomposed by the fire which surrounds the internal flame;-to all which symptoms some physician perhaps, or rather some philosopher, who had the power of seeing in many
this discussion shows that the movements of the several stars depend, as regards the varieties of speed which they exhibit, on the distance of each from the extremity.
in one way 'polyclitus', in another 'sculptor' is the cause of a statue, because 'being polyclitus' and 'sculptor' are incidentally conjoined.
the questions they try to answer will always be those their contemporaries are asking; their statement of fundamental problems will always be relative to the traditional statements that have been handed down to them.
and if leaving is a consequence of changing, having left is a consequence of having changed: for there is a like relation between the two in each case.
we must think of the world as of something in which right differs from left in shape as well as in other respects, which subsequently is included in a sphere.
nor again can it be that some things are always at rest, others always in motion, and nothing sometimes at rest and sometimes in motion. this theory must be pronounced impossible on the same grounds as those previously mentioned: viz.
meeting-point; and, comprehending them in a uniform revolution upon the same axis, he made the one the outer and the other the inner circle.
for whereas marrow and bone and flesh and sinews are composed of the four elements, and the blood, though after another manner, is likewise formed out of them, most diseases originate in the way which i have described; but the worst of all owe their
the pythagoreans, too, (four) held that void exists and that it enters the heaven itself, which as it were inhales it, from the infinite air.
so that since the 'before' is in time, and every movement involves a 'before', evidently every change and every movement is in time.
when there are parts of a whole-the one that in which a thing is, the other the thing which is in it-the whole will be described as being in itself. for a thing is described in terms of its parts, as well as in terms of the thing as a whole, e.g.
the same matter also serves for both a large and a small body.
fair grounds for inferring the same to be true of the other kinds of change, since it would be in the natural order of things that they should be uniform in this respect.
but he who should attempt to test the truth of this by experiment, would forget the difference of the human and divine nature. god only is able to compound and resolve substances; such experiments are impossible to man.
in the same way that which is to receive perpetually and through its whole extent the resemblances of all eternal beings ought to be devoid of any particular form.
what is in tune must come from what is not in tune, and vice versa; the tuned passes into untunedness-and not into any untunedness, but into the corresponding opposite.
else. consequently, though their becoming is accompanied by an alteration, they are not themselves alterations.
narrower in extent. in this way then their theories are both the same and different, some better, some worse; some, as i have said, take as their contraries what is more knowable in the order of explanation, others what is more familiar to sense.
two. since a proportion consists of two equal ratios, and each ratio can be written as a fraction, whenever we have a proportion such as
respecting the harmonies of sound i must hereafter speak.
socrates: excellent, timaeus; and we will do precisely as you bid us. the prelude is charming, and is already accepted by us-may we beg of you to proceed to the strain?
in like manner, the triangle acl is the same multiple of acd that the base one hundred fifty is of the base four hundred; and it is evident that i. xxxviii.
as being passed by other things or forced up by them); and since a multitude of small atoms are heavier than a few large ones, it will follow that much air or fire is heavier than a little water or earth, which is impossible.
by a rivet or glue or contact or organic union.
if the rectangle contained by the extremes of three right lines be equal to the square of the mean, the three lines are proportional.
cor. one.-if a chord of a circle be divided in any point within the circle, the rectangle contained by its segments is equal to the difference between the square of the radius and the square of the line drawn from the centre to the point of section.
besides, a view which asserts atomic bodies must needs come into conflict with the mathematical sciences, in addition to invalidating many common opinions and apparent data of sense perception.
or thus: let o be the centre of the circle. join oa, oc (see fig. two). now the angle aoc is double of cda xx. , and the angle coa is double of abc. hence the sum of the angles i. def.
now we perceive movement and time together: for even when it is dark and we are not being affected through the body, if any movement takes place in the mind we at once suppose that some time also has elapsed; and not only that but also, when some time is
this, they say, accounts for the fact that eclipses of the moon are more frequent than eclipses of the sun: for in addition to the earth each of these moving bodies can obstruct it.
this matter also shall be cleared up in what follows: but for the present so much is clear, that the reason why there is more than one circular body is the necessity of generation, which follows on the presence of fire, which, with that of the other
the original impression to other parts, it has no effect of motion on the whole animal, and therefore produces no effect on the patient.
dem.-produce gh to k. then because ca is bisected in e, and divided externally in f, the rectangle cf.af, together with the square on ea, is equal to the square on ef vi.
to the earth then, which is the most stable of bodies and the most easily modelled of them, may be assigned the form of a cube; and the remaining forms to the other elements,-to fire the pyramid, to air the octahedron, and to water the
the equal particles appear transparent; the larger contract, and the lesser dilate the sight. white is produced by the dilation, black by the contraction, of the particles of sight.
six. find a point that shall be equidistant from three given points.
moreover this other can never be any casual change, but must be a change from something definite to some other definite thing. so in this case it must be the opposite change, viz. convalescence.
'since one hath learned to spring from manifold, and one disjoined makes manifold arise, thus they become, nor stable is their life: but since their motion must alternate be, thus have they ever rest upon their round': for we must suppose that he means by
two spheres, of the planets and of the world beyond them, all together moving around the earth, which is their centre.
made to find a way in and out through the pores of the body, and the rays of fire which are bound fast within followed the passage of the air either way, never at any time ceasing so long as the mortal being holds together.
the origin. the creation, in plato's sense, is really the creation of order; and the first step in giving order is the division of the heavens into an inner and outer circle of the other and the same, of the divisible and the indivisible, answering to the
now when we first discussed the question of motion we put forward a solution of this difficulty turning on the fact that the period of time occupied in traversing the distance contains within itself an infinite number of units: there is no absurdity, we
besides, in some instances we see that the cause is other than the form.
and the same argument would fit the case of lightness. again, a body which is in motion but has neither weight nor lightness, must be moved by constraint, and must continue its constrained movement infinitely.
cor. three.-since a quadrilateral can be divided into two triangles, the sum of its angles is equal to four right angles.
socrates: and being thus trained they were not to consider gold or silver or anything else to be their own private property; they were to be like hired troops, receiving pay for keeping guard from those who were protected by them-the pay was to be no more
as for your high towers and monuments, there was a crazy fellow once in this town who undertook to dig through to china, and he got so far that, as he said, he heard the chinese pots and kettles rattle; but i think that i shall not go out of my way to
cor. three.-if p be the common centre of circles whose radii are lines drawn from p to the circumference of hde, then-one. the circle whose radius is the minimum line (pe) has contact of the first kind with ade def. iv. . two.
and therefore the mathematician or any one else whose thoughts are much absorbed in some intellectual pursuit, must allow his body also to have due exercise, and practise gymnastic; and he who is careful to fashion the body, should in turn impart to the
can it be, then, that the term 'quick' has not the same meaning as applied to straight motion and to circular motion respectively? if so, far less will it have the same meaning as applied to alteration and to locomotion.
the fire or light, when kept in by the eyelids, equalizes the inward motions, and there is rest accompanied by few dreams; only when the greater motions remain they engender in us corresponding visions of the night.
else, when the motion of that which is pulling is quicker than the motion that would separate from one another the two things that are continuous: for it is this that causes one thing to be pulled on along with the other.
(it is applied in a twofold manner.) for (i) we say 'it is now passing-away' without qualification, and not merely 'this is passing-away': and we call this change 'coming-to-be', and that 'passing-away', without qualification.
but when night comes on and the external and kindred fire departs, then the stream of vision is cut off; for going forth to an unlike element it is changed and extinguished, being no longer of one nature with the surrounding atmosphere which is now
it is therefore better to suppose that in all instances of coming-to-be the matter is inseparable, being numerically identical and one with the 'containing' body, though isolable from it by definition.
it is the rational principle, mind regarded as a work, as creation-not as the creator. the old tradition of parmenides and of the eleatic being, the foundation of so much in the philosophy of greece and of the world, was lingering in plato's mind.
when the constitution is disordered by excess of fire, continuous heat and fever are the result; when excess of air is the cause, then the fever is quotidian; when of water, which is a more sluggish element than either fire or air, then the fever is a
dem.-at the points five hundred, e make the angles edg, deg equal to the angles a, b of the triangle abc. then i. xxxii. the triangles abc, deg are equiangular.
will be in virtue of a prevailing simple element), then there must necessarily be some simple body which revolves naturally and in virtue of its own nature with a circular movement.
slower will divide the length. if, then, this alternation always holds good, and at every turn involves a division, it is evident that all time must be continuous.
afforded me a momentary enjoyment, notwithstanding a musky flavor, i saw that the longest use would not make that a good practice, however it might seem to have your woodchucks ready dressed by the village butcher.
we have now to assert that the first movent must be without parts and without magnitude, beginning with the establishment of the premisses on which this conclusion depends.
how can there be several dissimilar elements, each infinite? each would have to be infinitely extended every way.
let us consider this question more precisely. is there any self-existent fire? and do all those things which we call self-existent exist?
(four) because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself.
therefore, since where there is motion there must be something that is in motion, and where there is something in motion there must be motion, therefore the being-moved will also be composed of indivisibles.
"we can make liquor to sweeten our lips of pumpkins and parsnips and walnut-tree chips."
and if a thing is in motion over the whole kbg and its motion is the three five hundred, e, and z, and if it is not in motion at all over the partless section a but has completed its motion over it, then the motion will consist not of motions but of
but the reason why the heavy and the light appear more than these things to contain within themselves the source of their movements is that their matter is nearest to being.
of health the fresh air or the sun's heat may be the cause, but having had one's hair cut cannot; for some incidental causes are more relevant to the effect than others.)
let a, b, c; a, b, c be the two sets of magnitudes, and let the ratio a : b b : c, and b : c a : b. then, if a be greater than, equal to, or less than c, a will be greater than, equal to, or less than c.
(for the mass being infinite you may subtract from it as much as you please.) these assumed bodies will be commensurate in mass and in weight alike.
twelve. and four hundred(ad bd) ad.bd, or -one- bd -one- ad -two- four hundred.
the sun everywhere bright to see, and hot, the rain everywhere dark and cold; and he distinctively characterizes his remaining elements in a similar manner.
and the body which is carried along is different, in so far as it is at one time here and at another there. but the 'now' corresponds to the body that is carried along, as time corresponds to the motion.
if, on the other hand, it is in process of change in both parts, it is likewise in process of change in the whole: and if, again, it has changed in one of the two parts, the whole is not the primary when in which it has changed: it must therefore have
cube. and there is a fifth figure (which is made out of twelve pentagons), the dodecahedron-this god used as a model for the twelvefold division of the zodiac.
thus it is so if one perishing is pleasant and another painful: and so one perishing will be contrary to another not in an unqualified sense, but in so far as one has this quality and the other that.
one. if one of two similar triangles has its sides fifty per cent. longer than the homologous sides of the other; what is the ratio of their areas?
if the whole is not continuous, but exists, as democritus and leucippus think, in the form of parts separated by void, there must necessarily be one movement of all the multitude.
seven. if eg be joined, its square is equal to ac bc .
by the passage over the infinite is not a finite but an infinite time, and the contact with the infinites is made by means of moments not finite but infinite in number.
further, the truth of what we assert is plain from the following considerations.
nine. construct a triangle, being given the three medians.
adb is a triangle fulfilling the required conditions.
everything that is good is fair, and the fair is not without proportion, and the animal which is to be fair must have due proportion.
two. a solid angle cannot be formed of equal plane angles which are equal to the angles of a regular polygon of n sides, except in the case of n three, four, or five.
and in all cases of a thing that has no contrary we have as contraries change from and change to the same thing. thus coming to be is contrary to ceasing to be, and losing to gaining. but these are changes and not motions.
will be one: otherwise it will be the same but not one. and akin to this difficulty there is another; viz. is health one?
from what has been said earlier it is plain that there cannot be generation either of everything or in an absolute sense of anything. it is impossible that everything should be generated, unless an extra-corporeal void is possible.
again, because gf is parallel to cb, and ge intersects them, the angle egf is equal to ecb i. xxix. ; but ecb is right (const.); therefore egf is right, and gef has been proved to be half a right angle; therefore i. xxxii.
but since this is impossible, that which is intermediate between them must be a magnitude and divisible into an infinite number of segments: consequently, before the change is completed, the thing changes to those segments.
for there will be air, when the cold of the water and the dry of the fire have passed-away (since the hot of the latter and the moist of the former are left): whereas, when the hot of the fire and the moist of the water have passed-away, there will be
appears to move upwards without compulsion, though it appears to move by compulsion downwards); and since (b) what is 'natural' is contrary to that which is due to compulsion, and movement by compulsion actually occurs; it follows that 'natural movement'
and god made the sun and moon and five other wanderers, as they are called, seven in all, and to each of them he gave a body moving in an orbit, being one of the seven orbits into which the circle of the other was divided.
the straight line traversed in rectilinear motion cannot be infinite: for there is no such thing as an infinite straight line; and even if there were, it would not be traversed by anything in motion: for the impossible does not happen and it is impossible
it is the primary that moves the primary, the simple the simple, the indestructible and ungenerated that which is indestructible and ungenerated.
seven. the radius of the "nine points circle" of a triangle is equal to half its circumradius.
first, does it belong to the class of things that exist or to that of things that do not exist? then secondly, what is its nature?
and since that which imparts motion may be either a thing that is moved by something else or a thing that is unmoved, and that which is moved may be either a thing that imparts motion to something else or a thing that does not, that which moves itself
cherished his mortal part. but he who has been earnest in the love of knowledge and of true wisdom, and has exercised his intellect more than any other part of him, must have thoughts immortal and divine, if he attain truth, and in so far as human nature
will this be upward motion or downward or what? it must have some motion; and if upward and downward motion are alike to it, and the air above the earth does not prevent upward movement, then no more could air below it prevent downward movement.
are we to say 'bone comes-to-be if the "elements" be put together in such-and such a manner'? for, according to his own estatements, nothing comes-to-be from their 'fortuitous consilience', but only from their 'consilience' in a certain proportion.
the sight of them and their revolutions has given us the knowledge of number and time, the power of enquiry, and philosophy, which is the great blessing of human life; not to speak of the lesser benefits which even the vulgar can appreciate.
instance, to a fourth part of the circumference. for since no body is a point, it will have parts. the expansion, when the body increased the place occupied, would be on the same principle as the contraction, in which the place was diminished.
again, join of, og. now the triangles eof, aof have the sides af, fe equal three. xvii., ex. one , and fo common, and the base ao equal to the base eo. hence the angle afo is equal to efo i. viii. . therefore the angle afo is half the angle afe.
and yet, probably, their first impressions, the illusions and mirages of their fancy, created a greater intellectual activity and made a nearer approach to the truth than any patient investigation of isolated facts, for which the time had not yet come,
forty eight. find the locus of the centre of a circle which bisects the circumferences of two given circles.
flowing water, and from water comes earth and stones once more; and thus generation appears to be transmitted from one to the other in a circle.
nor can body 'combine' with white, nor (to generalize) 'properties' and 'states' with 'things': for we see them persisting unaltered. but again (iv) white and knowledge cannot be 'combined' either, nor any other of the 'adjectivals'.
nine. the intercept made on a variable tangent by two fixed tangents subtends a constant angle at the centre.
five. of all concyclic figures having a given number of sides, the area is a maximum when the sides are equal.
the movent on the other hand is already in activity: e.g. it is that which is hot that produces heat: in fact, that which produces the form is always something that possesses it.
predicated, and the chaos or matter which has no perceptible qualities-between being in the abstract and nothing.
again (two) a thing is said without qualification to change because something belonging to it changes, i.e.
now change from non-subject to subject, the relation being that of contradiction, is 'coming to be'-'unqualified coming to be' when the change takes place in an unqualified way, 'particular coming to be' when the change is change in a particular
so it is not possible for that which has no parts to be in motion or to change in any way: for only one condition could have made it possible for it to have motion, viz.
this demonstration may be stated as follows:-since the chords are equal, they are congruent; and therefore the segments, being similar, must be congruent.
xli. the triangle abc is half the parallelogram af. therefore the triangle abc is ab.four hundred.
dem.-the ratio a : b a- b, and the ratio of ma : mb ma- mb; but since the value of a fraction is not altered by multiplying its numerator and denominator by the same number,
then if air is twice as thin, the body will traverse b in twice the time that it does five hundred, and the time g will be twice the time e.
but it is not so divisible: for position is not 'immediately-next' to position, nor point to point-in other words, division is not 'immediately-next' to division, nor composition to composition.
motion will be the motion of the magnitude abg.
six. if from the vertex one hundred of an isosceles triangle a line four hundred be drawn to any point in the base produced, prove that cd cb ad.db.
with which it imparts motion, whereas if there is a different thing with which it imparts motion, there must be something that imparts motion not with something else but with itself, or else there will be an infinite series.
the greater quantity must prevail until the body's centre occupies the centre. for that is the goal of its impulse. now it makes no difference whether we apply this to a clod or common fragment of earth or to the earth as a whole.
two. the sum of the squares on the diagonals of a quadrilateral, together with four times the square on the line joining their middle points, is equal to the sum of the squares on its sides.
when therefore another body occupies this same place, the place is thought to be different from all the bodies which come to be in it and replace one another.
twenty two. if a variable line, whose extremities rest on the circumferences of two given concentric circles, subtend a right angle at any fixed point, the locus of its middle point is a circle.
whole, the parts of the motion are the motions of the parts of that whole: and the parts of dz are the motions of ab, bg and of nothing else: for, as we saw, a motion that is one cannot be the motion of more things than one): since this is so, the whole
prop. e.-theorem (simson). if four magnitudes be proportional, the first : its excess above the second :: the third : its excess above the fourth (convertendo).
one hundred nineteen. if three of the six points in which a circle meets the sides of any triangle be such, that the lines joining them to the opposite vertices are concurrent, the same property is true of the three remaining points.
twelve.thirty four twenty three.fourteen thirteen.twenty four.
acb : bcd :: dce : bcd five. vii. ; but acb : bcd :: ac : four hundred i. , and dce : bcd :: ec : cb i. . therefore ac : four hundred :: ec : cb; that is, the sides about the equal angles are reciprocally proportional.
dem.-through h draw hn parallel to ab, cutting il in one thousand. now in the triangle ali, hk is parallel to il. hence ii. ak : kl :: ah : hi, that is :: ce : ef (const.). again, in the triangle hng, one thousand one is parallel to ng. therefore ii.
will result from fire if a single quality changes: for fire, as we saw, is hot and dry while air is hot and moist, so that there will be air if the dry be overcome by the moist.
one. given the sum or the difference of any two lines, and the sum of their squares; find the lines.
prop. twelve.-problem. to draw a perpendicular to a given indefinite right line (ab) from a given point (one hundred) without it.
hence eg is a parallelogram fulfilling the required conditions.
thirty three. what is meant by the third diagonal of a quadrilateral?
dem.-on ab describe a square abde. join eb; through one hundred draw cf parallel to ae, intersecting be in g; and through g draw hi parallel to ab.
this, however, is false: for we observe that there is always some part of the animal's organism in motion, and the cause of the motion of this part is not the animal itself, but, it may be, its environment.
nine. what is a mean proportional between two lines?
one. every median of a triangle bisects the triangle.
the generable, similarly, has been generated, for it is capable of having been generated and thus also of not always existing.
these triangles have the angle fbc equal to the angle gcb, and these are the angles below the base. also the angle fcb equal to gbc; but the whole angle fca has been proved equal to the whole angle gba.
for everything that comes to be comes to be in something, and that in which the generation takes place must either be incorporeal or possess body; and if it has body, there will be two bodies in the same place at the same time, viz.
let us take our start from this point. the impossible and the false have not the same significance. one use of 'impossible' and 'possible', and 'false' and 'true', is hypothetical.
(one) yet part, too, of the revolution is a time, but it certainly is not a revolution: for what is taken is part of a revolution, not a revolution.
but (a) if, as mr grote assumes, plato did not see that the rotation of the earth on its axis and of the sun and outer heavens around the earth in equal times was inconsistent with the alternation of day and night, neither need we suppose that he would
so they assert that everything has been mixed in every. thing, because they saw everything arising out of everything.
that the centre of the earth is the goal of their movement is indicated by the fact that heavy bodies moving towards the earth do not parallel but so as to make equal angles, and thus to a single centre, that of the earth.
(this state of mind is clearly possible. a man blind from birth might reason about colours. presumably therefore such persons must be talking about words without any thought to correspond.)
the cure of them is difficult, and in most cases they are relieved only by fever.
for quite a small change of position to south or north causes a manifest alteration of the horizon. there is much change, i mean, in the stars which are overhead, and the stars seen are different, as one moves northward or southward.
each is carried to its own place, if it is not hindered, the one up, the other down. now these are regions or kinds of place-up and down and the rest of the six directions.
hence, since every finite body is exhausted by the repeated abstraction of a finite body, it seems obviously to follow that everything cannot subsist in everything else.
moreover, in general, it is impossible that the infinite should move at all.
now the moment in which it has changed cannot be the same as that in which it is at a (since in that case it would be in a and b at once): for we have shown above that that that which has changed, when it has changed, is not in that from which it has
twenty two. given two circles, o, o; then if any secant cut o in the points b, one hundred, and o in the points b, one hundred, and another secant cuts them in the points five hundred, e; five hundred, e respectively; the four chords bd, ce, bd, ce form a
these may be the figures best adapted to movement; they are not, however, well adapted to the movement of fire, which is an upward and rectilinear movement, but rather to that form of circular movement which we call rolling.
six. what caution is required in the enunciation of prop. xiv.?
therefore a is not less than five hundred, and we have proved that it is not equal to it; therefore it must be greater.
the passage, then, into what 'is' not except with a qualification is unqualified passing-away, while the passage into what 'is' without qualification is unqualified coming-to-be.
if then this is true, everything that comes to be or passes away from, or passes into, its contrary or an intermediate state. but the intermediates are derived from the contraries-colours, for instance, from black and white.
the latter means either what is a part or mode of number-in general, something which belongs to number-or that things have a number.
sol.-bisect the side ac opposite to b, at p. draw pf, pe parallel to the sides ab, bc; then xxvii. bp is the maximum parallelogram that can be inscribed in the triangle abc; and if ten be equal to it, the problem is solved.
now rotatory motion can be eternal: but no other motion, whether locomotion or motion of any other kind, can be so, since in all of them rest must occur and with the occurrence of rest the motion has perished.
it moves itself as a whole, both being moved and imparting motion through containing a part that imparts motion and a part that is moved.
water and earth, and returned to the form of his first and better state.
above is the principle of length, right of breadth, front of depth. or again we may connect them with the various movements, taking principle to mean that part, in a thing capable of movement, from which movement first begins.
in the infinite chaos there can have been neither above nor below, and it is by these that heavy and light are determined.
as regards what is 'of necessity', we must ask whether the necessity is 'hypothetical', or 'simple' as well.
(ii) but why are some things said to 'come to-be' without qualification, and others only to 'come-to-be-so-and-so', in cases different from the one we have been considering where two things come-to-be reciprocally out of one another?
not always large. a thing which, judged absolutely, is small may none the less be larger than other things. whatever, then, is heavy and also heavier than something else, must exceed this by something which is heavy.
but of these things we have already spoken in our discussion of time and movement. they are also bound to contradict themselves.
suppose, then, that a was heated, or impelled, or modified in any way, or caused to undergo any sort of movement whatever, by in the time one hundred.
hence eb eo. again, the triangles ebf, ogc are equiangular, because efb, ecb are equal, and ebf, ogc are each right. therefore, ef : eb :: oc : og; therefore ef.og eb.oc eo.oc.
dem.-if the planes be not parallel, they will meet when produced. let them meet, their common section being the line gh, in which take any point k. join ak, bk.
but if we increase the ratio of the part, so as always to take in the same amount, we shall traverse the magnitude, for every finite magnitude is exhausted by means of any determinate quantity however small.
but it is not self-moving in the sense that each of the two parts is moved by the other part: the following considerations make this evident. in the first place, if each of the two parts is to move the other, there will be no first movent.
the theory which generates them from something incorporeal requires an extra-corporeal void.
in a circle or on a straight line, and it is the same with regard to increase and decrease. the difference that makes a motion irregular is sometimes to be found in its path: thus a motion cannot be regular if its path is an irregular magnitude, e.g.
the universe revolves around a centre once in twenty-four hours, but the orbits of the fixed stars take a different direction from those of the planets.
but things which are contracted contrary to nature are by nature at war, and force themselves apart; and to this war and convulsion the name of shivering and trembling is given; and the whole affection and the cause of the affection are both termed cold.
now all unmixed and primary bodies are produced by such causes as these. as to the subordinate species which are included in the greater kinds, they are to be attributed to the varieties in the structure of the two original triangles.
nor again can motion along the circle from a to b be regarded as the contrary of motion from a to one hundred: for the motion goes from the same point towards the same point, and contrary motion was distinguished as motion from a contrary to its contrary.
where things derive their motion from something else we distinguished the cases in which the motion is unnatural: we are left with those that are to be contrasted with the others by reason of the fact that the motion is natural.
the external angles ecd, fdc at the other side of the base four hundred are equal; but ecd is greater than bcd (axiom ix.).
twenty two. in the same case, if the bisector of the external vertical angle be taken, the distance will be equal to half the sum of the sides.
but as i observed the rule of probability at first, i will begin anew, seeking by the grace of god to observe it still.
moreover, as to the flowing of water, the fall of the thunderbolt, and the marvels that are observed about the attraction of amber and the heraclean stones,-in none of these cases is there any attraction; but he who investigates rightly, will find that
time is made to consist of moments or a length of points.
void with a speed beyond any ratio. for let z be void, equal in magnitude to b and to five hundred. then if a is to traverse and move through it in a certain time, h, a time less than e, however, the void will bear this ratio to the full.
dem.-because five hundred is the centre of the circle efg, df is equal to de (def. xxxii.). and because dab is an equilateral triangle, da is equal to db (def. xxi.). hence we have
(we may compare what plato does in me divisions': for he makes 'the middle' a blend.) indeed, there is practically no difference between those who postulate two and those who postulate three, except that the former split the middle 'element' into two,
hence (axiom i.) eg is equal to ed: in like manner, fg is equal to fd, and this is impossible vii. .
one. if two right lines ab, bc be respectively equal and parallel to two other right lines de, ef, the right line ac joining the extremities of the former pair is equal to the right line df joining the extremities of the latter.
besides, if there were more heavens than one, the movement of any of them equally would be time, so that there would be many times at the same time.
of earth at rest above: and therefore this earth during the time that it was being carried violently upward was coming to a standstill.
of your race, and afterwards she founded ours, of which the constitution is recorded in our sacred registers to be eight thousand years old.
it is also unreasonable to imagine a place for the void, as if the void were not itself a kind of place. but if the void is to move, it must have a place out of which and into which the change carries it. also what is the cause of its movement?
now mistakes come to pass even in the operations of art: the grammarian makes a mistake in writing and the doctor pours out the wrong dose. hence clearly mistakes are possible in the operations of nature also.
but when the thing is separate and in contact, it is immediately 'in' the inner surface of the surrounding body, and this surface is neither a part of what is in it nor yet greater than its extension, but equal to it; for the extremities of things which
dem.-let o be the centre. then the four angles at o, being right angles, are equal. hence the arcs on which they stand are equal three. xxvi. , and hence the four chords are equal three. xxix. . therefore the figure abcd is equilateral.
and the same fact explains why what is already actually fire or earth moves, when nothing obstructs it, towards its own place.
the account just given may serve to distinguish and define the 'contact' which occurs in the things of nature.
for it is natural that the best-conditioned of all things should have its good without action, that which is nearest to it should achieve it by little and simple action, and that which is farther removed by a complexity of actions, just as with men's
the ratios of their motions, numbers, and other properties, are ordered by the god, who harmonized them as far as necessity permitted.
the idea of eternity was for a great part a negation. there are regions of speculation in which the negative is hardly separable from the positive, and even seems to pass into it.
infinite circle being itself an impossibility, there can be no circular motion of an infinite body.
generally, if time is directly the measure of motion and indirectly of other things, it is clear that a thing whose existence is measured by it will have its existence in rest or motion.
any one point as much as any other is alike starting-point, middle-point, and finishing-point, so that we can say of certain things both that they are always and that they never are at a starting-point and at a finishing-point (so that a revolving sphere,
over sixty: so that, the time prh being less than the time pch in which the slower passes over sixty, the time ps will also be less than the time px: for it is less than the time prh, and that which is less than something else that is less than a thing is
again, 'that for the sake of which', or the end, belongs to the same department of knowledge as the means. but the nature is the end or 'that for the sake of which'.
rousseau is often blamed, by modern critics, for pursuing in the discourses a method apparently that of history, but in reality wholly unhistorical.
def. vii.-if from every point in a given line normals be drawn to a given plane, the locus of their feet is called the projection of the given line on the plane.
seven. if instead of triangles on the same base we have triangles on equal bases and between the same parallels, the intercepts made by the sides of the triangles on any parallel to the bases are equal.
four. prove that any point in af is equally distant from the lines ab, ac.
musing in themselves on the phenomena of nature, they were relieved at being able to utter the thoughts of their hearts in figures of speech which to them were not figures, and were already consecrated by tradition.
for contrary effects demand contraries as their causes; and the natural processes of passing-away and coming-to-be occupy equal periods of time. hence, too, the times-i.e.
fourteen. find in two parallels two points subtending a right angle at a given point and equally distant from it.
our arguments have shown that it is either false or futile to advocate pores in the sense in which some thinkers conceive them.
for in the first place there are two senses in which motion of motion is conceivable. (one) the motion of which there is motion might be conceived as subject; e.g. a man is in motion because he changes from fair to dark.
their way, by reason of these qualities in them, are all termed pungent.
if any number of magnitudes of the same kind (a, b, c, c.), be equimultiples of as many others (a, b, c, c.), then the sum of the first magnitudes (a b c, c.) shall be the same multiple of the sum of the second which any magnitude of the first system is
it is only accidentally that there can be change of change, e.g. there is a change from remembering to forgetting only because the subject of this change changes at one time to knowledge, at another to ignorance.
and if some things are at rest and others in motion, then it must be that either all things are sometimes at rest and sometimes in motion, or some things are always at rest and the remainder always in motion, or some of the things are always at rest and
(three) further, this second capacity will be present like the first at every moment, so that the thing will have for an infinite time the capacity both of being and of not being; but this has been shown to be impossible.
the primary source to which things that are in motion are to be referred.
three. the triangle abh : cde in the duplicate ratio of ab : four hundred xix. . hence (two) the polygon abhij : the polygon cdefg in the duplicate ratio of ab : four hundred.
more than any other part the framework of the head would have had them, if they could have co-existed, and the human race, having a strong and fleshy and sinewy head, would have had a life twice or many times as long as it now has, and also more healthy
fishes who came to land, and of man out of the animals, was held by anaximander in the sixth century before christ (plut. symp. quaest; plac.
when it is mingled with black bile and dispersed about the courses of the head, which are the divinest part of us, the attack if coming on in sleep, is not so severe; but when assailing those who are awake it is hard to be got rid of, and being an
on the other hand it may be urged that the further step, however simple and obvious, is just what plato often seems to be ignorant of, and that as there is no limit to his insight, there is also no limit to the blindness which sometimes obscures his
for the consequence of the inclination is that the body becomes alternately remote and near; and since its distance is thus unequal, its movement will be irregular.
the astronomy of plato is based on the two principles of the same and the other, which god combined in the creation of the world.
heavens. all the planets, including the sun, are carried round in the daily motion of the circle of the fixed stars, and they have a second or oblique motion which gives the explanation of the different lengths of the sun's course in different parts of
at the points a, b, one hundred draw the tangents lm, mn, nl to the given circle. lmn is a triangle fulfilling the required conditions.
nature akin to that of man with other forms and perceptions, and thus created another kind of animal.
prop. twenty.-theorem. the angle (aob) at the centre (o) of a circle is double the angle (acb) at the circumference standing on the same arc.
(four) the immobility of the earth is more in accordance with plato's other writings than the opposite hypothesis. for in the phaedo the earth is described as the centre of the world, and is not said to be in motion.
but if we say that there is first a state of rest for an infinite time, and then motion is started at some moment, and that the fact that it is this rather than a previous moment is of no importance, and involves no order, then we can no longer say that
prop. thirteen.-problem. to find a mean proportional between two given lines. (ten, y ).
(two) as the whole is 'in' the parts: for there is no whole over and above the parts.
a bright element mingling with red and white produces a colour which we call auburn. the law of proportion, however, according to which compound colours are formed, cannot be determined scientifically or even probably.
nor again do things pass into the first chance thing; 'white' does not pass into 'musical' (except, it may be, in virtue of a concomitant attribute), but into 'not-white'-and not into any chance thing which is not white, but into black or an intermediate
further, it is unaffected by any mortal discomfort, and, in addition, effortless; for it needs no constraining necessity to keep it to its path, and prevent it from moving with some other movement more natural to itself.
socrates had already spoken of god the creator, who made all things for the best.
therefore, before this first change there will be a previous change.
in the same manner every other point between a and b lies within the circle.
if, then, that which has changed to b is in something other than b, say g, it must again be changing from g to b: for it cannot be assumed that there is no interval between g and b, since change is continuous.
for freezing and boiling are excesses of heat and cold respectively.
prop. eleven.-problem. to draw a normal to a given plane (bh) from a given point (a) not in it.
six. the locus of the intersection of two equal tangents to two circles is a right line (called the radical axis of the two circles).
as i said at first, when all things were in disorder god created in each thing in relation to itself, and in all things in relation to each other, all the measures and harmonies which they could possibly receive.
six. in every triangle the bisector of the greatest angle is the least of the three bisectors of the angles.
eight. how may a plane surface be generated. ans. by the motion of a right line which crosses another right line, and moves along it without changing its direction.
fire as being unnatural. the same is true of the corresponding cases of remaining. but there would seem to be a sense in which a state of rest and a motion are opposites.
now if we are to apply to the heaven such a distinction of parts, we must exect, as we have said, to find in it also the distinction which in animals is found first of them all.
now it is clear from the following considerations that no other than locomotion can be continuous.
sol.-if ten be a right angle, describe a semicircle on the given line, and the thing required is done; for the angle in a semicircle is a right angle.
it was most prolific in the seventeenth or in the early part of the eighteenth century, when the human mind, seeking for utopias or inventing them, was glad to escape out of the dulness of the present into the romance of the past or some ideal of the
for some finite body could have been found greater than the given finite in the same proportion which is supposed to hold between the infinite and the given finite; so that an infinite and a finite weight must have traversed an equal distance in equal
dividing the heavy body in the proportion ce:four hundred, we subtract from the heavy body a part which will in the same time move the distance ce, since the whole moved four hundred: for the relative speeds of the two bodies will be in inverse ratio to
twelve. how many circles can be described to touch three lines forming a triangle?
but (some one will say) it is contrary to reason to suppose that there should be one identical actualization of two things which are different in kind. yet there will be, if teaching and learning are the same, and agency and patiency.
for let o be the centre. join oc, and produce ac to f. then because ao is equal to oc, the angle aco is equal to the angle oac. in like manner, the angle ocb is equal to cbo. hence the angle acb is equal to the sum of the two angles bac, cba; but i.
now, because the points b, one hundred are in this second plane, the line bc i., def. vi. is in it. therefore the two right lines abc, abd lying in one plane have a common segment ab, which is impossible. therefore, c.
therefore i. vi. ge is equal to gf. in like manner fd is equal to db.
we have only to grasp the fact that a particular motion is in a sense the opposite both of a state of rest and of the contrary motion, in the same way as that which is of equal or standard measure is the opposite both of that which surpasses it and of
but (ii) those who postulate two from the start-as parmenides postulated fire and earth-make the intermediates (e.g. air and water) blends of these. the same course is followed (iii) by those who advocate three.
in like manner the sectors cog, goh are equal. hence there are as many equal sectors as there are equal arcs; therefore the arc bh and the sector boh are equimultiples of the arc bc and the sector boc.
in such passages we have to interpret his meaning by the general spirit of his writings. to reconcile his inconsistencies would be contrary to the first principles of criticism and fatal to any true understanding of him.
nor can a 'middle-element' be such an originative source'-as some thinkers suppose that air is transformed both into fire and into water, and water both into air and into earth, while the 'end-elements' are not further transformed into one another.
wherefore, that which is to receive all forms should have no form; as in making perfumes they first contrive that the liquid substance which is to receive the scent shall be as inodorous as possible; or as those who wish to impress figures on soft
dem.-each of the triangles aob, aof is equilateral (see dem., i. i.). hence the angles aob, aof are each one-third of two right angles; therefore eof is one-third of two right angles. again, the angles boc, cod, doe are i. xv.
it might find a place wherever men chose to look for it; in north, south, east, or west; in the islands of the blest; before the entrance of the straits of gibraltar, in sweden or in palestine.
if, on the other hand, 'unqualified not-being' means 'what is not in any sense at all', it will be a universal negation of all forms of being, so that what comes-to-be will have to come-to-be out of nothing.
dem.-since a : b :: c : d, a- b c d; therefore a b- one c d one, or a b ----- b c d ----- d; therefore a b : b :: c d : d.
and if this is divided the magnitude gk will also be divided just as gd was: and again, if the magnitude is divided, the time will also be divided.
three. let ab pass through the centre, and cut four hundred, which does not pass through the centre obliquely. let o be the centre. draw of perpendicular to four hundred i. xi. . join oc, od.
therefore viii. the angle daf is equal to the angle eaf; hence the angle bac is bisected by the line af.
hence the angle afe is equal to agb. in like manner the remaining angles of the figure fghij are equal. therefore it is equiangular.
of rest at g: for this, as we found, is the state of rest that is the opposite of the motion from g. the foregoing argument, then, makes it plain that the motion in question is not continuous.
he found them all three, in the pythagorean philosophy and in the teaching of socrates and of the megarians respectively; and, because they all furnished modes of explaining and arranging phenomena, he is unwilling to give up any of them, though he is
book six. application of the theory of proportion definitions.
and for this reason it is customary to appoint interpreters to be judges of the true inspiration.
dem.-if the equal chords be diameters, the proposition is evident. if not, let o, h be the centres. join ao, ob, dh, he; then because the circles are equal their radii are equal (def. i.).
bodies which are themselves in motion, produce noise and friction: but those which are attached or fixed to a moving body, as the parts to a ship, can no more create noise, than a ship on a river moving with the stream.
makes no difference whether movement is set up in the ray or in the object of vision.
(b) magnitude is not infinite either in the way of reduction or of magnification in thought.
one pint of water yields ten of air, both are measured by the same unit; and therefore both were from the first an identical something.
the fact indicated does not depend upon degrees of size but applies universally to everything that has the centripetal impulse.
seven. when it is required to prove that two triangles are congruent, how many parts of one must be given equal to corresponding parts of the other? ans. in general, any three except the three angles. this will be established in props. viii.
first, let us enquire what we mean by saying that fire is hot; and about this we may reason from the dividing or cutting power which it exercises on our bodies.
three. equal right lines that have equal projections on another right line are parallel.
again, because bag is the angle of a square it is a right angle: in like manner cak is a right angle. hence bag is equal to cak: to each add bac, and we get the angle cag equal to kab. again, since bg and ck are squares, ba is equal to ag, and ca to ak.
but three of them can be thus resolved and compounded, for they all spring from one, and when the greater bodies are broken up, many small bodies will spring up out of them and take their own proper figures; or, again, when many small bodies are dissolved
the opinion, however, which most people are inclined to prefer, is that the distinction depends upon the difference between 'the perceptible' and 'the imperceptible'.
() add ed, and we have in each case ae equal to df, and ba is equal to four hundred xxxiv. . hence the triangles bae, cdf have the two sides ba, ae in one respectively equal to the two sides four hundred, df in the other, and the angle bae xxix.
above all it is plain that this motion, motion in respect of place, is what is in the strictest sense produced by that which moves itself; but it is the self-movent that we declare to be the first principle of things that are moved and impart motion and
then, again, there are the numerous kinds of water which have been mingled with one another, and are distilled through plants which grow in the earth; and this whole class is called by the name of juices or saps.
if on the other hand it is asserted that all things are quality or quantity, then, whether substance exists or not, an absurdity results, if the impossible can properly be called absurd.
it is the speech of an advocate, wholly one-sided and arbitrary, but so obviously and naively one-sided, that it is difficult for us to believe in its entire seriousness.
other is a real sea, and the surrounding land may be most truly called a boundless continent.
eclog.), and descants upon odd and even numbers, after the manner of the later pythagoreans.
if, then, there is no body in z, a will traverse z still more quickly. but we supposed that its traverse of z when z was void occupied the time h. so that it will traverse z in an equal time whether z be full or void. but this is impossible.
plato's account of digestion and the circulation of the blood is closely connected with his theory of respiration.
i. a less magnitude is said to be a part or submultiple of a greater magnitude, when the less measures the greater-that is, when the less is contained a certain number of times exactly in the greater.
dem.-in the triangle cae, bd is parallel to ce; therefore ab : bc :: ad : de ii. ; but ab is equal to ten, and bc, ad each equal to y . therefore ten : y :: y : de. hence de is a third proportional to ten and y .
one. parallel planes intercept equal segments on parallel lines.
so that which causes increase and that which causes decrease must be continuous with that which suffers increase and that which suffers decrease respectively: and if two things are continuous with one another there can be nothing intermediate between them.
it is to these causes that most writers pay attention: but there are some, anaximander, for instance, among the ancients, who say that the earth keeps its place because of its indifference.
it is plain, then, that if there is a time in which it will move through any part of the void, this impossible result will follow: it will be found to traverse a certain distance, whether this be full or void, in an equal time; for there will be some body
earth and other heavy things move to. is this their goal because it is the centre of the earth or because it is the centre of the whole? the goal, surely, must be the centre of the whole.
traversing of a certain amount of distance: for at any moment when a thing is causing motion, it also has caused motion, so that there must always be a certain amount of distance that has been traversed and a certain amount of time that has been
socrates: i see that i shall receive in my turn a perfect and splendid feast of reason. and now, timaeus, you, i suppose, should speak next, after duly calling upon the gods.
hence, multiplying each equation by n, we get
cor. three.-if lines drawn from any point o in the plane of a figure to all its angular points be divided in the same ratio, the lines joining the points of division will form a new figure similar to, and having every side parallel to, the homologous side
moreover, the current popular arguments make it plain that, if time is continuous, magnitude is continuous also, inasmuch as a thing asses over half a given magnitude in half the time taken to cover the whole: in fact without qualification it passes over
our definition then is as follows: a quantity is infinite if it is such that we can always take a part outside what has been already taken. on the other hand, what has nothing outside it is complete and whole.
they will be most the children of their age, when they are rising most above it.
hence the duplicate ratio of two magnitudes means the square of their ratio, or, what is the same thing, the ratio of their squares (see book six. xx.).
ab : de :: ef : bg; hence the sides of the triangles abg, def about the equal angles b, e are reciprocally proportional; therefore the triangles are equal.
moreover the parts of animals are often in motion in an unnatural way, their positions and the character of the motion being abnormal.
form as that of flesh as if, e.g. 'moist' were to accede to 'dry' and, having acceded, were to be transformed and to become 'dry'. for in one sense 'like grows by like', but in another sense 'unlike grows by unlike'.
but since the movents and the things moved are infinite, the motion ezho, i.e. the motion that is composed of all the individual motions, must be infinite.
ninety one. if the altitude of a triangle be equal to its base, the sum of the distances of the orthocentre from the base and from the middle point of the base is equal to half the base.
but it would follow that a large mass of gold, as containing more void than a small mass of fire, is lighter than it, unless it also contains many times as much solid. the addition is therefore necessary.
is the matter, out of which growth takes place, (i) 'separate' and existing alone by itself, or (ii) 'separate' but contained in another body?
but it is clear that it must be the case: for just as there is unnatural motion, so, too, a thing may be in an unnatural state of rest. further, some things have a natural and an unnatural motion, e.g.
but since a motion appears to have contrary to it not only another motion but also a state of rest, we must determine how this is so.
ten : y :: ab : bc i. , y : z :: bd : bg i. . hence xy : y z :: ab.bd : bc.bg; or ten : z :: ab.bd : bc.bg. observation.-since ab.bd : bc.bg is compounded of the two ratios ab : bc and bd : bg five. def.
with this essay, rousseau had unsuccessfully competed in one thousand seven hundred fifty three for a second prize offered by the academy of dijon, and he now issued it prefaced by a long dedication to the republic of geneva.
if flesh grows, every particle of the flesh gets bigger), (ii) by the accession of something, and (iii) in such a way that the growing thing is preserved and persists.
this is misleading, as it implies that it is, by convention, that the first ratio is greater than the second, whereas, in fact, such is not the case; for it follows from the hypothesis that the first ratio is greater than the second; and if it did not, it
therefore ab is greater than four hundred.
so in modern times the speculative doctrine of necessity has often been supported by physical facts.
soul its proper motions, and should cultivate music and all philosophy, if he would deserve to be called truly fair and truly good.
belly. the former he divided into two branches, both of which he made to meet at the channels of the nose, so that when the way through the mouth did not act, the streams of the mouth as well were replenished through the nose. with the other cavity (i.e.
is equal to its supplement, the line ef must coincide with gh. hence the angle aef is equal to cgh.
us who are bad become bad from two causes which are entirely beyond our control. in such cases the planters are to blame rather than the plants, the educators rather than the educated.
four. find a point in a given line that shall be equally distant from two given points.
(a) time indeed and movement are infinite, and also thinking, in the sense that each part that is taken passes in succession out of existence.
in air and water, the first being moist and hot, and the second moist and cold.
the solution of the difficulty that is raised about the motion-whether it is in the movable-is plain.
that there is a centre towards which the motion of heavy things, and away from which that of light things is directed, is manifest in many ways. first, because no movement can continue to infinity.
this then is one way of solving the difficulty. another consists in pointing out that the same things can be explained in terms of potentiality and actuality. but this has been done with greater precision elsewhere.
one would have to suppose a difference proportionate to the distance and increasing with it, but the form is in fact the same. moreover, the bodies must have some movement, since the fact that they move is quite evident.
further, if a divisible thing is to exist, it is necessary that, when it exists, all or some of its parts must exist. but of time some parts have been, while others have to be, and no part of it is though it is divisible.
direct sequence of viii. from iv. and vii.
they are stimulated by something sensible: and those that depend upon memory or anticipation are likewise to be traced to sense-perception, for in these cases pleasure is felt either in remembering what one has experienced or in anticipating what one is
but the forms which enter into and go out of her are the likenesses of real existences modelled after their patterns in a wonderful and inexplicable manner, which we will hereafter investigate.
one who is learning a science potentially knows it in a different sense from one who while already possessing the knowledge is not actually exercising it.
remembering what i said at first about probability, i will do my best to give as probable an explanation as any other-or rather, more probable; and i will first go back to the beginning and try to speak of each thing and of all.
let a : b :: c : d; then, if a be the greatest, d will be the least xiv. and a . it is required to prove that a d is greater than b c.
only, as there is a certain remnant of evil inherent in matter which he cannot get rid of, he detaches himself from them and leaves them to themselves, that he may be guiltless of their faults and sufferings.
dem.-if ab, ac are not equal, one must be greater than the other. suppose ab is the greater, and that the part bd is equal to ac. join four hundred (post. i.). then the two triangles dbc, acb have bd equal to ac, and bc common to both.
nor can the composition of planes; for this produces the elements themselves, not bodies made up of them.
nothing that is by chance can be indestructible or ungenerated, since the products of chance and fortune are opposed to what is, or comes to be, always or usually, while anything which exists for a time infinite either absolutely or in one direction, is
from the two theorems (i.) and (ii.) we may infer two others, called their contrapositives. thus the contrapositive
if, on the other hand, the arrangement was a chance combination, the coincidence in every case of a greater circle with a swifter movement of the star contained in it is too much to believe.
if, on the other hand, coming-to-be is not 'association', either there is no such thing as coming-to-be at all or it is 'alteration': or else we must endeavour to unravel this dilemma too-and a stubborn one we shall find it.
in like manner, if ma be equal to nb, mc is equal to nd; and if less, less.
two. the three perpendiculars of the first triangle in question one are the perpendiculars at the middle points of the sides of the second triangle.
the reason why they put motion into these genera is that it is thought to be something indefinite, and the principles in the second column are indefinite because they are privative: none of them is either 'this' or 'such' or comes under any of the other
for the intellect it is not always one and the same point, since it is other and other when one divides the line; but in so far as it is one, it is the same in every respect.
for (i) there is nothing to prevent an indeterminate number of matters being thus 'contained in' the water, so that they might come-to-be actually an indeterminate quantity of air; and (ii) we do not in fact see air coming-to-be out of water in this
further, when water is produced from air, the place has been destroyed, for the resulting body is not in the same place. what sort of destruction then is that?
continue to be moved? if so, it will not be ab primarily that is moved by itself, since, when something is taken away from ab, the remainder of ab will still continue to move itself.
cannot be in process of becoming without having become or have become without having been in process of becoming.
further, if anything that exists after having been previously non-existent must become existent and does not exist when it is becoming, time cannot be divisible into time-atoms.
and if i go on taking parts in this way, on the one hand there is no part which will be a measure of the infinite time (for the infinite cannot be composed of finite parts whether equal or unequal, because there must be some unity which will be a measure
(seven) it can also be shown, conversely, that if the time of revolution is finite, the area traversed must also be finite; but the area traversed was equal to itself; therefore, it is itself finite.
whether, therefore, the substratum is or is not something, what comes-tobe emerges out of a 'not-being': so that a thing comes-to-be out of a not-being' just as much as it 'passes-away into what is not'.
a place in which a thing is at rest is a place in which it might move, either by constraint, i.e. unnaturally, or in the absence of constraint, i.e. naturally.
od; therefore the four lines oa, ob, oc, od are all equal; and since they are perpendicular to the sides of the given square, the circle described with o as centre, and oa as radius, will be inscribed in the square.
process of change subsequent to the last,) for being destroyed also is a kind of change.
but they will be bound to maintain this, if they say that water is a body present in air and excreted from air, since air becomes heavier when it turns into water.
xii. if two right lines (ab, four hundred) meet a third line (ac), so as to make the sum of the two interior angles (bac, acd) on the same side less than two right angles, these lines being produced shall meet at some finite distance.
starting from this argument and from the observation that their speeds, as measured by their distances, are in the same ratios as musical concordances, they assert that the sound given forth by the circular movement of the stars is a harmony.
air had come up; and the earth when compressed by the air into an indissoluble union with water becomes rock. the fairer sort is that which is made up of equal and similar parts and is transparent; that which has the opposite qualities is inferior.
observation.-the line de may cut the sides ab, ac produced through b, one hundred, or through the angle a; but evidently a separate figure for each of these cases is unnecessary.
in so far as there is one point anywhere within it and all its points are everywhere within it if you take them singly one by one.
but when this takes place in spatial magnitudes, what is taken perists, while in the succession of time and of men it takes place by the passing away of these in such a way that the source of supply never gives out.
yet strictly this too is different from the spontaneous proper; for the cause of the latter is external, that of the former internal.
only on this supposition are we involved in no absurd consequence. for, in the first place, the quicker movement of the larger circle is natural when all the circles are attached to the same centre.
dem.-on bc describe the square bcde i. xlvi. . through a draw af parallel to four hundred: produce ed to meet af in f.
in the first place, there was a caste of priests among the ancient athenians, and another of artisans; also castes of shepherds, hunters, and husbandmen, and lastly of warriors, who, like the warriors of egypt, were separated from the rest, and carried
thirty seven. the angle made by the bisectors of two consecutive angles of a convex quadrilateral is equal to half the sum of the remaining angles; and the angle made by the bisectors of two opposite angles is equal to half the difference of the two other
one shall consider the platonic astronomy, and the position of the earth.
becoming or an alteration: for the terms 'knowing' and 'understanding' imply that the intellect has reached a state of rest and come to a standstill, and there is no becoming that leads to a state of rest, since, as we have said above, change at all can
eleven (plot continued.) reversal of the situation, recognition, and tragic or disastrous incident defined and explained. twelve the 'quantitative parts' of tragedy defined. thirteen (plot continued.) what constitutes tragic action.
it is not every chance direction which is 'up', but where fire and what is light are carried; similarly, too, 'down' is not any chance direction but where what has weight and what is made of earth are carried-the implication being that these places do not
a-b e-f. in like manner, cd e-f. hence a-b cd i., axiom i. , and a : b :: c : d.
i have told you briefly, socrates, what the aged critias heard from solon and related to us.
be imparted in the same way and be one and the same, since the unmoved does not itself change in relation to that which is moved by it.
we can hardly agree with him when he tells us that smells do not admit of kinds.
the process of sight is the most complicated (rep.), and consists of three elements-the light which is supposed to reside within the eye, the light of the sun, and the light emitted from external objects.
as well as from without, and to have no other employment; they were to be merciful in judging their subjects, of whom they were by nature friends, but fierce to their enemies, when they came across them in battle.
secondly, passing from the external to the internal evidence, we may remark that the story is far more likely to have been invented by plato than to have been brought by solon from egypt.
twenty six. if a semicircle and its diameter be touched by any circle, either internally or externally, twice the rectangle contained by the radius of the semicircle, and the radius of the tangential circle, is equal to the rectangle contained by the
they are called by the general name of colours, and are a flame which emanates from every sort of body, and has particles corresponding to the sense of sight.
next, to prove euclid's theorem-that if, according as the multiple of the first of four magnitudes is greater than, equal to, or less than the multiple of the second, the multiple of the third is greater than, equal to, or less than the multiple of the
upwards and downwards or light and heavy. hence there are as many types of motion or change as there are meanings of the word 'is'.
moreover, we have established the fact that it is the movable that is moved; and this is potentially, not actually, in motion, but the potential is in process to actuality, and motion is an incomplete actuality of the movable.
moreover, inasmuch as it is the same present that belongs to both the times, and it is possible for a thing to be in motion throughout one time and to be at rest throughout the other, and that which is in motion or at rest for the whole of a time will be
produce ac to b, making bc ca: then b is another point of intersection.
he could not isolate phenomena, and he was helpless against the influence of any word which had an equivocal or double sense.
fire has a natural upward motion and an unnatural downward motion: is it, then, this unnatural downward motion or is it the natural downward motion of earth that is contrary to the natural upward motion?
also different? we have explained, then, what is meant by saying that motion is one generically or one specifically.
xi. a sector of a circle is formed of two radii and the arc included between them.
(since it is in motion to the point at which it will finally arrive), and yet will not be undergoing two contrary or opposite motions: for a motion to a point and a motion from that point are not always contraries or opposites: they are contraries only if
in like manner, the angle cea is equal to deb.
hence these too derive from the primary pair, the 'solidified' from the dry and the 'solidified' from the dry the 'liquefiable' from the moist.
fire, air, earth, and water are bodies and therefore solids, and solids are contained in planes, and plane rectilinear figures are made up of triangles.
world mutually to imply each other. 'god invented and gave us sight to the end that we might behold the courses of intelligence in the heaven, and apply them to the courses of our own intelligence which are akin to them, the unperturbed to the perturbed;
bf be ef , and cf ce ef ; bf cf be ce ef . again bf cf bg gf two. x. ex. two , and be ce bg ge ; bg gf bg ge ef ; gf ge ef . hence the angle gef is right, and ef is perpendicular to eg.
this appears plainly from an analysis of the character of natural things, and equally from an inspection of the instances of inquiry into nature.
they mingled with irrational sense and with all-daring love according to necessary laws, and so framed man.
or infinite this is in either case impossible.
yet the negation has a kind of unknown meaning to us. the priority of god and of the world, which he is imagined to have created, to all other existences, gives a solemn awe to them.
the same as that in which motion takes place: for we defined a state of rest to be the state of a thing to which motion is natural but which is not in motion when (that is to say in that in which) motion would be natural to it.
the outermost part bulge, or that somewhere else there must be an equal amount of water produced out of air, so that the entire bulk of the whole may be equal, or that nothing moves.
again, since every motion is a motion of something, and the whole motion dz is not the motion of either of the parts (for each of the parts de, ez is the motion of one of the parts ab, bg) or of anything else (for, the whole motion being the motion of a
in the third place, there is no necessity for the movent to be moved in return: on the contrary the necessity that there should always be motion makes it necessary that there should be some movent that is either unmoved or moved by itself.
but (iii) since only those things which either involve a 'contrariety' or are 'contraries'-and not any things selected at random-are such as to suffer action and to act, agent and patient must be 'like' (i.e. identical) in kind and yet 'unlike' (i.e.
mind, but only in feelings of pleasure and pain and the desires which accompany them.
observation.-in six. def. i. two conditions are laid down as necessary for the similitude of rectilineal figures. one. the equality of angles; two. the proportionality of sides. now, from propositions iv.
in the same manner it can be shown that a square yard contains nine square feet; and so in general the square described on any line contains n times the square described on the nth part of the line.
this would be the natural form of movement towards what is naturally spherical. either then the earth is spherical or it is at least naturally spherical.
when the thirty centuries begin to look down on it, mankind begin to look up at it.
the same difficulty is involved equally in the view elements and distinguishes them by their greatness and smallness.
dem.-let o be the centre of the inscribed circle. join ob (see foregoing fig.); let fall the perpendicular og, draw the diameter ef of the circumscribed circle. now the angle abe ecb three. xxvii. , and abo obc; therefore ebo sum of ocb, obc eob.
moreover, according to them, things are seen through air and water and other transparent bodies, because such bodies possess pores, invisible indeed owing to their minuteness, but close-set and arranged in rows: and the more transparent the body, the more
for, if the whole is moved by itself, it must be moved either by some part of itself or as a whole by itself as a whole.
scepticism and discontent were the prevailing temper of the intellectual classes, and a short-sighted despotism held that, as long as they were confined to these, they would do little harm.
hence, however true it may be that the heavens are due to spontaneity, it will still be true that intelligence and nature will be prior causes of this all and of many things in it besides.
it seems that perceptible things require perceptible principles, eternal things eternal principles, corruptible things corruptible principles; and, in general, every subject matter principles homogeneous with itself.
now it is impossible that that which moves itself should in its entirety move itself: for then, while being specifically one and indivisible, it would as a whole both undergo and cause the same locomotion or alteration: thus it would at the same time be
it would seem that in animals, just as in ships and things not naturally organized, that which causes motion is separate from that which suffers motion, and that it is only in this sense that the animal as a whole causes its own motion.
hence our first question is this: do these changes differ from one another solely because of a difference in their respective 'spheres'? in other words, do they differ because, while a change from this to that (viz.
consequently f (fire) as well as air will be able to be transformed into water: for it has qualities contrary to those of water, since fire was first taken to be black and then to be dry, while water was moist and then showed itself white.
since we deny that a void exists, but for the rest the problem has been truly stated, that either there will be no movement, if there is not to be condensation and rarefaction, or the universe will bulge, or a transformation of water into air will always
if (a) their one is one in the sense of continuous, it is many, for the continuous is divisible ad infinitum.
the sphere then will be spherical throughout; for every body within it is contiguous and continuous with spheres.
that being so, the slower will in the same present traverse a distance less than ab, say ag. but since the slower will have occupied the whole present in traversing ag, the quicker will occupy less than this in traversing it.
similar evidence is supplied by the more physical of the branches of mathematics, such as optics, harmonics, and astronomy. these are in a way the converse of geometry.
these are the elements, thus of necessity then subsisting, which the creator of the fairest and best of created things associated with himself, when he made the self-sufficing and most perfect god, using the necessary causes as his ministers in the
yet plato also speaks of an 'annus magnus' or cyclical year, in which periods wonderful for their complexity are found to coincide in a perfect number, i.e. a number which equals the sum of its factors, as six one two three.
we have now explained why there is 'unqualified coming-to-be' (though it is a passingaway-of-something) and 'unqualified passingaway (though it is a coming-to-be-of-something).
it may be replied that there is no such legend in any writer previous to plato; neither in homer, nor in pindar, nor in herodotus is there any mention of an island of atlantis, nor any reference to it in aristotle, nor any citation of an earlier writer by
both philolaus and plato agree in making the world move in certain numerical ratios according to a musical scale: though bockh is of opinion that the two scales, of philolaus and of the timaeus, do not correspond...we appear not to be sufficiently
then if bg is taken to be indivisible, two things without parts will have to be contiguous (which is impossible): if on the other hand it is taken to be divisible, there will be something prior to g to which the magnitude has changed, and something else
but suppose that the centre is not its place, and that the reason of its remaining there is this necessity of indifference-on the analogy of the hair which, it is said, however great the tension, will not break under it, if it be evenly distributed, or of
socrates begins the timaeus with a summary of the republic.
friendship. further they identify it with the divine, for it is 'deathless and imperishable' as anaximander says, with the majority of the physicists.
wherefore, fearing to pollute the divine any more than was absolutely unavoidable, they gave to the mortal nature a separate habitation in another part of the body, placing the neck between them to be the isthmus and boundary, which they constructed
said, in supposing the traversing of infinite distances in infinite time, and the element of infinity is present in the time no less than in the distance.
in the first place let us set forth what was omitted when we were speaking of juices, concerning the affections peculiar to the tongue.
again, since ac is equal to ad, adding ba to both, we have the sum of the sides ba, ac equal to bd. therefore the sum of ba, ac is greater than bc.
descended from a small seed or remnant of them which survived. and this was unknown to you, because, for many generations, the survivors of that destruction died, leaving no written word.
now if the whole moves the whole distance ce, the part must in the same time move the distance four hundred. a weightless body, therefore, and one which has weight will move the same distance, which is impossible.
wherefore also finding the whole visible sphere not at rest, but moving in an irregular and disorderly fashion, out of disorder he brought order, considering that this was in every way better than the other.
thus the line will be composed of points, for the point, as it continually traverses a distance equal to itself, will be a measure of the whole line. but since this is impossible, it is likewise impossible for the indivisible to be in motion.
again, because the angle edg is equal to bac (const.), and bac equal to edf (hyp.), the angle edg is equal to edf; and it has been proved that dg is equal to df, and de is common; hence the triangles edg and edf are equiangular; but edg is equiangular to
this being so, it is clear that the motion is in the wood, not in its form: for the motion is neither caused nor experienced by the form or the place or the quantity. so we are left with a mover, a moved, and a goal of motion.
dem.-join oc, od, oe. then the triangles abo, cbo have the side ab equal to bc (hyp.), bo common, and the angle abo equal to cbo (const.). hence the angle bao is equal to bco i. iv.
there is a fourth class of sensible things, having many intricate varieties, which must now be distinguished.
is plus unity, and conversely, if the product
dem.-one. let pa pass through the centre o. join ot. then because ab is bisected in o and divided externally in p, the rectangle ap.bp ob is equal to op two. vi. .
and sometimes, like other men, he is more impressed by one aspect of human life, sometimes by the other.
(one) compound the infinite body will not be, if the elements are finite in number. for they must be more than one, and the contraries must always balance, and no one of them can be infinite.
if the rectangle (ap.bp) contained by the segments of a secant, drawn from any point (p) without a circle, be equal to the square of a line (pt) drawn from the same point to meet the circle, the line which meets the circle is a tangent.
passing-away) is effected by means of the void, and so too is growth-solids creeping in to fill the void places.' empedocles too is practically bound to adopt the same theory as leucippus.
five. if squares be described on the sides of any triangle, the sum of the squares on the lines joining the adjacent corners is equal to three times the sum of the squares on the sides of the triangle.
bd bc cd bc.four hundred iv. and ad ad . hence, adding, since i. xlvii. bd ad ab , and cd ad ca , we get
hence water is drawn up into the vessel to which fire is applied, but not earth. as earth fails to move upward, so fire fails to move downward when air is withdrawn from beneath it: for fire has no weight even in its own place, as earth has no lightness.
the angle ceo is equal to deo, and they are adjacent angles. therefore i. def. xiii. each is a right angle. hence ab cuts four hundred at right angles.
nine. the triangle whose vertices are the middle points of two sides, and any point in the base of another triangle, is one-fourth of that triangle.
dem.-one. erect ah, one hundred one at right angles to ab and four hundred, and equal to f and e respectively, and complete the rectangles.
a quality, like white' or 'hot', in respect to which they are moved' only in the sense that they are 'altered': on the contrary, to 'impart motion' is a wider term than to 'act'.
for all men have some conception of the nature of the gods, and all who believe in the existence of gods at all, whether barbarian or greek, agree in allotting the highest place to the deity, surely because they suppose that immortal is linked with
iv. if four quantities of the same kind be in continued proportion, the two middle terms are called two mean proportionals between the other two.
the circle whose radius is the minimum line (pe) lies inside the circle ade, and touches it in e. three. a circle having any of the remaining lines (pd) as radius cuts ade in two points (five hundred, f).
in one sense, then, (one) that out of which a thing comes to be and which persists, is called 'cause', e.g. the bronze of the statue, the silver of the bowl, and the genera of which the bronze and the silver are species.
such was the tale, socrates, which critias heard from solon; and i noticed when listening to you yesterday, how close the resemblance was between your city and citizens and the ancient athenian state.
if this were possible, it would follow also that the smallest body would contain the greatest; for 'many a little makes a mickle': thus if many equal bodies can be together, so also can many unequal bodies.
to draw a normal to a given plane from a given point (a) in the plane.
(six) again, if the heaven is infinite and moves in a circle, we shall have to admit that in a finite time it has traversed the infinite. for suppose the fixed heaven infinite, and that which moves within it equal to it.
that which a changing thing, if it changes continuously in a natural manner, naturally reaches before it reaches that to which it changes last, is between.
kind and a man from another); and yet at the same time they assert that the heavenly sphere and the divinest of visible things arose spontaneously, having no such cause as is assigned to animals and plants.
sector boc one two rectangle contained by the arc bc, and the radius of the circle abc xx. ex. fourteen and sector epf one two rectangle contained by the arc ef and the radius of the circle edf; and since the circles are equal, their radii are equal.
prop. twelve.-problem. to find a fourth proportional to three given lines (ten, y, z).
what we are saying, and shows that we are mistaken, may claim a friendly victory.
cor. one.-if the ratio b : c be equal to the ratio a : b, then a, b, c will be in continued proportion, and so will a, b, c. hence def. xii. annotation three ,
hence when they are mingled with themselves and with one another there is an endless variety of them, which those who would arrive at the probable truth of nature ought duly to consider.
if from the vertex a of a abc, ad be drawn, meeting cb produced in five hundred, and making the angle bad acb, prove db.six hundred da .
again, pushing apart and pushing together are forms respectively of pushing off and pulling: pushing apart is pushing off, which may be a motion either away from the pusher or away from something else, while pushing together is pulling, which may be a
the "many" move in the void (for there is a void): and by coming together they produce "coming to-be", while by separating they produce "passing-away".
now if they are all commensurable and if two things to have the same velocity must accomplish an equal motion in an equal time, then we may have a circumference equal to a straight line, or, of course, the one may be greater or less than the other.
further, the following argument will serve better to make this point clear universally in respect of every kind of motion.
in another sense (two) the form or the archetype, i.e. the statement of the essence, and its genera, are called 'causes' (e.g. of the octave the relation of two:one, and generally number), and the parts in the definition.
but the memory of their exploits has passed away owing to the lapse of time and the extinction of the actors. 'tell us,' said the other, 'the whole story, and where solon heard the story.'
(b) if, on the other hand, we suppose that 'biped' and 'animal' are attributes not of man but of something else, and are not each of them a substance, then 'man' too will be an attribute of something else.
three. two regular polygons of n sides each have n centres of similitude.
timaeus: yes, and the proposal is easy to remember, as you say.
then because ab and fg are parallel, but ab is normal, to the plane one hundred one; hence fg is normal to it eleven. viii. . now since fg is parallel to ab, the angles abf, bfg are equal to two right angles i. xxix.
on the one hand, then, it is in no way paradoxical that every perceptible body should be indivisible as well as divisible at any and every point. for the second predicate will at. tach to it potentially, but the first actually.
sixteen. if a chord of a given circle pass through a given point, the locus of the intersection of tangents at its extremities is a right line.
if a right line (ab) be divided into any two parts (at one hundred), the sum of the squares on the whole line (ab) and either segment (cb) is equal to twice the rectangle ( ab.cb) contained by the whole line and that segment, together with the square on
is true of spitting and of all other motions that proceed through the body, whether secretive or assimilative, the assimilative being forms of pulling, the secretive of pushing off.
as for f and h, it has been shown that they are coincident.
it follows, as an obvious corollary, that a single matter must always be assumed as underlying the contrary 'poles' of any change whether change of place, or growth and diminution, or 'alteration'; further, that the being of this matter and the being of
prop. eleven.-problem. to find a third proportional to two given lines (ten, y ).
irregular. so quickness and slowness are not species of motion nor do they constitute specific differences of motion, because this distinction occurs in connexion with all the distinct species of motion.
but since that which cannot be cut (in the sense that it is inconceivable that it should be cut, the term 'cannot' being used in several senses)-since it is inconceivable that that which in this sense cannot be cut should be in process of being cut, and
i might compare myself to a person who, on beholding beautiful animals either created by the painter's art, or, better still, alive but at rest, is seized with a desire of seeing them in motion or engaged in some struggle or conflict to which their forms
'why did they go to war?-because there had been a raid'; or (three) we are inquiring 'for the sake of what?'-'that they may rule'; or (four), in the case of things that come into being, we are looking for the matter.
philolaus is known to us from the phaedo of plato as a pythagorean philosopher residing at thebes in the latter half of the fifth century b.one hundred., after the dispersion of the original pythagorean society.
possessed of a quality or quantity, nor in any place).
it is clear, then, from what has been said that time contains something indivisible, and this is what we call a present.
prop. thirteen.-theorem. two circles cannot have double contact, that it, cannot touch each other in two points.
twenty nine. if r denote the radius of o and , , the radii of o, o, dd : ab :: (r)(r) : r , the choice of sign depending on the nature of the contacts. this follows from twenty eight.
from the triangle of which the hypotenuse is twice the lesser side the three first regular solids are formed-first, the equilateral pyramid or tetrahedron; secondly, the octahedron; thirdly, the icosahedron; and from the isosceles triangle is formed the
in one thousand seven hundred fifty five appeared the discourse on the origin and foundation of inequality among men , which is the second of the works given in this volume.
twelve. if in the last exercise the intersecting plane be parallel to any two sides of the quadrilateral, it cuts the two remaining sides proportionally.
some of them are produced by rough, others by abstergent, others by inflammatory substances,-these act upon the testing instruments of the tongue, and produce a more or less disagreeable sensation, while other particles congenial to the tongue soften and
consider then, socrates, if this narrative is suited to the purpose, or whether we should seek for some other instead.
now the things in which this motion is produced are things that admit of being sometimes in motion and sometimes at rest, and the motion is not continuous but only appears so: for it is motion of things that are either successive or in contact, there
two. take any point e in ab produced either way. join ce. then the angle abc is greater than aec i. xvi. ; therefore cab is greater than aec. hence ce is greater than ca, and the point e is without the circle.
occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. this, he thinks, involves the conclusion that half a given time is equal to double that time.
when bile finds a means of discharge, it boils up and sends forth all sorts of tumours; but when imprisoned within, it generates many inflammatory diseases, above all when mingled with pure blood; since it then displaces the fibres which are scattered
if they are larger and contract the visual ray, a black colour is produced; if they are smaller and dilate it, a white. other phenomena are produced by the variety and motion of light.
the shape of the heaven is of necessity spherical; for that is the shape most appropriate to its substance and also by nature primary.
the apparatus of winds and waters is contemptuously rejected by him in the phaedo, but he thinks that there is a power greater than that of any atlas in the 'best' (phaedo; arist. met.).
but there cannot be a source of the infinite or limitless, for that would be a limit of it. further, as it is a beginning, it is both uncreatable and indestructible.
of two men being restored to health at the same time in the same way, e.g. from inflammation of the eye, yet this motion is not really one, but only specifically one).
by these considerations some have been led to assert that the earth below us is infinite, saying, with xenophanes of colophon, that it has 'pushed its roots to infinity',-in order to save the trouble of seeking for the cause.
for if one and the same motion sometimes recurs, it will be one and the same time, and if not, not.
secondly, they think that if the substratum is one numerically, it must have also only a single potentiality-which is a very different thing.
the real elements are two triangles, the rectangular isosceles which has but one form, and the most beautiful of the many forms of scalene, which is half of an equilateral triangle.
but sound is caused when a moving body is enclosed in an unmoved body, and cannot be caused by one enclosed in, and continuous with, a moving body which creates no friction.
earthy particles, entering into the small veins of the tongue which reach to the heart, when they melt into and dry up the little veins are astringent if they are rough; or if not so rough, they are only harsh, and if excessively abstergent, like potash
in like manner, c d m; therefore a- b -c d.
motions that are not the same either specifically or generically may, it is true, be consecutive (e.g.
by the revolution of the same and the like; but by the other five motions they were unaffected, in order that each of them might attain the highest perfection.
but in respect of addition there cannot be an infinite which even potentially exceeds every assignable magnitude, unless it has the attribute of being actually infinite, as the physicists hold to be true of the body which is outside the world, whose
for such movement would be infinite and indefinite, but every movement, in our view, proceeds from one point to another and is definite in character. again, suppose one assumes a minimum time in less than which the heaven could not complete its movement.
on the other hand, if they were created by me and received life at my hands, they would be on an equality with the gods.
for the planters are to blame rather than the plants, the educators and not the educated. still, we should endeavour to attain virtue and avoid vice; but this is part of another subject.
secondly, there is the higher mortal soul which, though liable to perturbations of her own, takes the side of reason against the lower appetites. the seat of this is the heart, in which courage, anger, and all the nobler affections are supposed to reside.
for 'since' (as they say) 'it is the nature of the hot to dissociate, of the cold to bring together, and of each remaining contrary either to act or to suffer action', it is out of such materials and by their agency (so they maintain) that everything else
sense that either of them is natural while the other is unnatural, nor is one increase contrary to another in this sense; and the same account may be given of becoming and perishing: it is not true that becoming is natural and perishing unnatural (for
five. again, what is it which sets them moving? for if their 'mover' is other than themselves, they are such as to 'suffer action'.
the principles which they assume are not limited in number, though such limitation would necessitate no other alteration in their theory. further, if the differences of bodies are not infinite, plainly the elements will not be an infinity.
the cohesion of water, when strong, is dissolved by fire only; when weak, either by air or fire, the former entering the interstices, the latter penetrating even the triangles.
this series, of which the intervals are afterwards filled up, probably represents (one) the diatonic scale according to the pythagoreans and plato; (two) the order and distances of the heavenly bodies; and (three) may possibly contain an allusion to the
and the reason of its inadequacy is that men think that the universe is not similar every way. they recognize only the hemisphere which is over us.
but when we come to a point we cannot make a distinction between it and its place.
fragments of this work are preserved to us, chiefly in stobaeus, a few in boethius and other writers. they remind us of the timaeus, as well as of the phaedrus and philebus. when the writer says (stob.
therefore a-- b c-d. hence, multiplying each fraction by m-n, we get ma- nb mc-nd; therefore ma : nb :: mc : nd.
they are as follows:-seeing that there is no such thing as a vacuum into which any of those things which are moved can enter, and the breath is carried from us into the external air, the next point is, as will be clear to every one, that it does not go
the body will be uniformly 'full' throughout. but, further, even if these passages, though they must contain bodies, are 'void', the same consequence will follow once more.
it will be a fitting monument of our gratitude to you, and a hymn of praise true and worthy of the goddess, on this her day of festival.
not existent, and that a thing is not existent at the moment when it has become.
goes further in that it affirms that even the quickest runner in legendary tradition must fail in his pursuit of the slowest), so that the solution must be the same.
they announce the quality of the agent. but a body of the opposite kind, being immobile, and not extending to the surrounding region, merely receives the impression, and does not stir any of the neighbouring parts; and since the parts do not distribute
and which is the only motion that can be eternal: and we have pronounced the first movent to be unmoved.
four. in any right-angled triangle the square on the sum of the hypotenuse and perpendicular, from the right angle on the hypotenuse, exceeds the square on the sum of the sides by the square on the perpendicular.
but time will measure what is moved and what is at rest, the one qua moved, the other qua at rest; for it will measure their motion and rest respectively.
one. let the equal sides be bc and ef; then if de be not equal to ab, suppose ge to be equal to it.
they differ, however, from one another in that some assume contraries which are more primary, others contraries which are less so: some those more knowable in the order of explanation, others those more familiar to sense.
def.-the circle through these nine points is called the "nine points circle" of the triangle.
it is clear from what has been said (i) that coming-to-be and passing-away actually occur, (ii) what causes them, and (iii) what subject undergoes them.
if the square on one side (ab) of a triangle be equal to the sum of the squares on the remaining sides (ac, cb), the angle (one hundred) opposite to that side is a right angle.
indeed this doctrine too (that the one and excess and defect are the principles of things) would appear to be of old standing, though in different forms; for the early thinkers made the two the active and the one the passive principle, whereas some of the
unexpected events, of which the cause was unknown to them, they attributed to chance (thucyd.).
it also follows from prop. xii., cor. two, that if two circles touch each other in a point a, they cannot have any other point common; hence they cannot touch again in b.
the spirit of the age favoured such methods, and it was therefore natural for rousseau to pursue them.
exceeds any assigned limit, and in the same way by continual subtraction i must arrive at one that falls short of any assigned limit.
finite equal particles in a finite quantity-which is impossible.
again, because the angle eab is equal to abc, their halves are equal. hence oab is equal to oba; therefore i. vi. oa is equal to ob. in like manner the lines oc, od, oe are equal to one another and to oa.
def. viii. the planes de, one hundred one are perpendicular to each other.
at that time, then, when it was not, at that same time it had the capacity of being, both of being then and of being thereafter, and therefore for an infinity of time.
but if it is no longer able to act-if it has been weakened by the continued influx of matter, just as water, continually mixed in greater and greater quantity with wine, in the end makes the wine watery and converts it into water-then it will cause a
sol.-join ce, cf, and construct a triangle abh on ab equiangular to cde, and similarly placed as regards four hundred; that is, make, the angle abh equal to cde, and bah equal to dce.
hence the triangle abe is equal to the triangle dcf; and, taking each away from the quadrilateral bafc, there will remain the parallelogram bcfe equal to the parallelogram bcda.
we have now said enough to make plain the character and number of the bodily elements, the place of each, and further, in general, how many in number the various places are.
cor. one.-if one angle of a parallelogram be a right angle, all its angles are right angles.
hence it must grow by the accession either (a) of something incorporeal or (b) of a body.
the two propositions xxxv., xxxvi., may be included in one enunciation, as follows:-the rectangle ap.bp contained by the segments of any chord of a given circle passing through a fixed point p, either within or without the circle, is constant.
these they let down along the backbone, so as to have the marrow of generation between them, where it was most likely to flourish, and in order that the stream coming down from above might flow freely to the other parts, and equalize the irrigation.
of necessity, we assert, they must move, and a moved thing which has no natural impetus cannot move either towards or away from the centre. suppose a body a without weight, and a body b endowed with weight.
the triangle bac, whose sides, taken in order, shall be equal to those of def-namely, ab equal to ed, ac equal to ef, and cb equal to fd; then the angle bac will viii. be equal to def. hence it is the required angle.
beside being better off than they already, if my house had been burned or my crops had failed, i should have been nearly as well off as before.
hence the four sides are equal; therefore ac is a lozenge, and the angle a is a right angle. therefore ac is a square (def. xxx.).
yet, on the other hand, we are conscious that knowledge is independent of time, that truth is not a thing of yesterday or tomorrow, but an 'eternal now.' to the 'spectator of all time and all existence' the universe remains at rest.
thus, if the stick moves something in virtue of being moved by the hand, the hand moves the stick: and if something else moves with the hand, the hand also is moved by something different from itself.
are there then also some becomings that are violent and not the result of natural necessity, and are therefore contrary to natural becomings, and violent increases and decreases, e.g.
but if the two bodies are one matter, or two matters both present in each, there will be a certain quantity of each at which water will excel a little air in the upward movement and air excel water in the downward movement, as we have already often said.
but that the complex structure of the eye or the ear is in any sense the cause of sight and hearing he seems hardly to be aware.
if, on the other hand, (two) the body moving with this circular motion which is unnatural to it is something different from the elements, there will be some other motion which is natural to it. but this cannot be.
acb acd ,, bcd, c. hence, adding, we get the sum of the base angles of the five triangles bac, cad, c., greater than the sum of the interior angles of the pentagon bcdef -that is, greater than six right angles.
(in discussing the causes of coming-tobe) we must first investigate the matter, i.e. the so-called 'elements'. we must ask whether they really are clements or not, i.e.
it is thought to be the condition of movement in respect of place, and it is not the condition of this.
an absurd suggestion; yet there must have been, if there were such things among animals.
thus arose day and night, which are the periods of the most intelligent nature; a month is created by the revolution of the moon, a year by that of the sun.
they hold that the most important part of the world, which is the centre, should be most strictly guarded, and name it, or rather the fire which occupies that place, the 'guardhouse of zeus', as if the word 'centre' were quite unequivocal, and the centre
this new beginning of our discussion of the universe requires a fuller division than the former; for then we made two classes, now a third must be revealed.
observation.-since the two circles in the four last propositions are equal, they are congruent figures, and the truth of the propositions is evident by superposition.
two. prove that the angle bca is greater than efd.
twenty three. find a point such that, if perpendiculars be let fall from it on four given lines, their feet may be collinear.
but as there is no surrounding vacuum it thrusts away the neighbouring air, and this being rendered heavy, and, when it is displaced, having been poured around the mass of earth, forcibly compresses it and drives it into the vacant space whence the new
but upon the whole, the ancients, though not entirely dominated by them, were much more subject to the influence of words than the moderns.
ef is parallel to ki, and the opposite sides ek and fi are parallel; therefore ei is a parallelogram; and because the parallelogram eg (const.) is equal to the triangle abd, and hi to the triangle bcd, the whole parallelogram ei is equal to the
essential nature is air or something of the kind.
when the consequences of either assumption are the same, we should always assume that things are finite rather than infinite in number, since in things constituted by nature that which is finite and that which is better ought, if possible, to be present
again, because b is the centre of the circle ech, bc is equal to be; and we have proved that af is equal to be; and things which are equal to the same thing are equal to one another (axiom i.). hence af is equal to bc.
prop. fifteen.-theorem. if two right lines (ab, four hundred) intersect one another, the opposite angles are equal (cea deb, and bec aed).
for (i) if it possesses none of these determinations actually, but all of them only potentially, the result is first that a being, which is not a determinate being, is capable of separate existence; and in addition that coming-to-be proceeds out of
in virtue of contact alone, since he denies there is a void.
is 'continuous' in such a way that it preserves its continuity with itself throughout the movement. the conclusion therefore is that this is what produces continuous movement, viz.
included under the general name of honey: and, lastly, there is a frothy nature, which differs from all juices, having a burning quality which dissolves the flesh; it is called opos (a vegetable acid).
in this example it is one thousand two hundred fifty eight, or fifteen : twenty eight. it is evident we get the same result if we multiply the two antecedents together for a new antecedent, and the two consequents for a new consequent.
he finds nothing in the text which he does not bring to it. he is full of porphyry, iamblichus and plotinus, of misapplied logic, of misunderstood grammar, and of the orphic theology.
if two triangles (abc, def) have two sides (ab, ac) of one respectively equal to two sides (de, df) of the other, and have also the base (bc) of one equal to the base (ef) of the other; then the two triangles shall be equal, and the angles of one shall be
for when the infinite magnitude a is in motion a part of it, say gd, will occupy the finite and then another, and then another, and so on to infinity.
forty eight. if ab, ac be equal sides of an isosceles triangle, and if bd be a perpendicular on ac; prove that bc ac.four hundred.
in our former discussion i distinguished two kinds of being-the unchanging or invisible, and the visible or changing. but now a third kind is required, which i shall call the receptacle or nurse of generation.
by (greek) he clearly means some conception of the intelligible and the intelligent; it belongs to the class of (greek).
for in the numbers the infinite in the direction of reduction is not present, as the monad is the smallest; nor is the infinite in the direction of increase, for the parts number only up to the decad.
motion is one in an unqualified sense when it is one essentially or numerically: and the following distinctions will make clear what this kind of motion is.
so while a thing in a finite time cannot come in contact with things quantitatively infinite, it can come in contact with things infinite in respect of divisibility: for in this sense the time itself is also infinite: and so we find that the time occupied
; but ab is equal to de (hyp.). hence al is equal to ab-that is, a part equal to the whole, which is absurd. therefore the angle aob is equal to dhe.
(this, again, we will here take for granted and try later to explain.) these four elements generation clearly is involved, since none of them can be eternal: for contraries interact with one another and destroy one another.
our inquiry (as physicists) is limited to its special subject-matter, the objects of sense, and we have to ask whether there is or is not among them a body which is infinite in the direction of increase.
the same may be shown, from the fact of rest. rest, also, must either be constrained or natural, constrained in a place to which movement was constrained, natural in a place movement to which was natural.
but if so, there must be at least one other circular motion: for a single movement of the whole heaven would necessitate an identical relation of the elements of bodies to one another.
(this is held in the timaeus, where plato says that the heaven, though it was generated, will none the less exist to eternity.) so far as the heaven is concerned we have answered this view with arguments appropriate to the nature of the heaven: on the
prop. thirty three.-theorem. the right lines (ac, bd) which join the adjacent extremities of two equal and parallel right lines (ab, four hundred) are equal and parallel.
that an age of intellectual transition must also be one of inconsistency; that the creative is opposed to the critical or defining habit of mind or time, has been often repeated by us.
taking it in this aspect, we have no less need of historical insight than if we came to it as historians pure and simple.
similar figures agree in shape; if they agree also in size, they are congruent.
indwelling power of order. there is only one way in which one person can benefit another; and that is by assigning to him his proper nurture and motion.
since, then, it is established that in a finite time neither will the finite traverse the infinite, nor the infinite the finite, nor the infinite the infinite, it is evident also that in a finite time there cannot be infinite motion: for what difference
this is what is most knowable: for, similarly, motion is known because of that which is moved, locomotion because of that which is carried. what is carried is a real thing, the movement is not.
it is clear, then, that the earth must be at the centre and immovable, not only for the reasons already given, but also because heavy bodies forcibly thrown quite straight upward return to the point from which they started, even if they are thrown to an
which are the testing instruments of the tongue, reaching to the heart, and fall upon the moist, delicate portions of flesh-when, as they are dissolved, they contract and dry up the little veins, they are astringent if they are rougher, but if not so
it is for this reason that plato also made the infinites two in number, because it is supposed to be possible to exceed all limits and to proceed ad infinitum in the direction both of increase and of reduction.
threatening and invading, and diffusing this bitter element swiftly through the whole liver produces colours like bile, and contracting every part makes it wrinkled and rough; and twisting out of its right place and contorting the lobe and closing and
further, in point of fact things that are thrown move though that which gave them their impulse is not touching them, either by reason of mutual replacement, as some maintain, or because the air that has been pushed pushes them with a movement quicker
to each add ch, and we get the parallelogram ah equal to the gnomon cmg; but ah is equal to the rectangle ad.dh, and therefore equal to the rectangle ad.db, since dh is equal to db iv., cor.
if two intersecting right lines (ab, bc) be respectively parallel to two other intersecting right lines (de, ef), the angle (abc) between the former is equal to the angle (def) between the latter.
like other writers on this subject, he is unable to escape from some degree of self-contradiction.
in different ages and countries there have been forms of light in which nothing could be discerned and which have nevertheless exercised a life-giving and illumining power.
like the romance of king arthur, which has had so great a charm, it has found a way over the seas from one country and language to another. it inspired the navigators of the fifteenth and sixteenth centuries; it foreshadowed the discovery of america.
thing in which motion can be produced from the thing itself, and not from without. now if this can occur in an animal, why should not the same be true also of the universe as a whole?
and there are facts of observation in manifest agreement with our theories. thus we see that coming-to-be occurs as the sun approaches and decay as it retreats; and we see that the two processes occupy equal times.
three. join oc; then two triangles pob, poc have the side ob equal to oc i. def. xxx. , and op common; but the angle pob is greater than poc; therefore i. xxiv. the base pb is greater than pc. in like manner pc is greater than pd.
for the same cause must necessarily have the same effect on the same thing.
then this also is absurd, that in every case there should be a beginning of the thing-not of the time and not only in the case of coming to be in the full sense but also in the case of coming to have a quality-as if change never took place suddenly.
if the infinite is neither a magnitude nor an aggregate, but is itself a substance and not an attribute, it will be indivisible; for the divisible must be either a magnitude or an aggregate.
on this question, indeed, the early philosophers are divided. some of them assert that the so-called 'unqualified coming-to-be' is 'alteration', while others maintain that 'alteration' and coming-to-be are distinct.
for the expulsion of the foreign body is an incident in the compacting of the homogeneous. in choosing the shape, then, they should have thought either of both functions or preferably of the combining function.
for this reason they have no need to be so disturbed about the world, or to call in a guard for its centre: rather let them look for the centre in the other sense and tell us what it is like and where nature has set it.
for 'to be man' is different from 'to be unmusical', and 'to be unformed' from 'to be bronze'.
seen. and he fears that the sophists, who are plentifully supplied with graces of speech, in their erratic way of life having never had a city or house of their own, may through want of experience err in their conception of philosophers and statesmen.
the modern philosopher has always been taught the lesson which he still imperfectly learns, that he must disengage himself from the influence of words.
clearly, then, their shapes are not fixed. indeed, nature itself seems to offer corroboration of this theoretical conclusion.
it has not, however, been observed, that the confusion partly arises out of the elements of opposing philosophies which are preserved in him.
the two interior angles (bgh, ghd) on the same side equal to two right angles.
four. given the sum of the squares on two lines and their ratio; find the lines.
no: the cause in question is the essential nature of each thing-not merely to quote his words) 'a mingling and a divorce of what has been mingled'.
thus the two results will coincide: the infinite will have completed a motion over the finite and the finite will have traversed the infinite: for it would seem to be impossible for the motion of the infinite over the finite to occur in any way other than
the premisses); (three) that this was the essence of the thing; and (four) because it is better thus (not without qualification, but with reference to the essential nature in each case).
dem.-on ab describe the square abdf i. xlvi. , and through one hundred draw ce parallel to af i. xxxi. .
as soon as we arrived yesterday at the guest-chamber of critias, with whom we are staying, or rather on our way thither, we talked the matter over, and he told us an ancient tradition, which i wish, critias, that you would repeat to socrates, so that he
yet how can void have a local movement or a place? for thus that into which void moves is till then void of a void.
first, let us consider generally which shape is primary among planes and solids alike. every plane figure must be either rectilinear or curvilinear. now the rectilinear is bounded by more than one line, the curvilinear by one only.
this cor. may be proved by the first book thus: erect four hundred at right angles to ab, and make four hundred ac or cb. join ad, db.
for if in the original there had been more than one they would have been the parts of a third, which would have been the true pattern of the world; and therefore there is, and will ever be, but one created world.
then because ac is equal to cb, the rectangle al is equal to ch i. xxxvi. ; but the complements ch, hf are equal i. xliii. ; therefore al is equal to hf.
physics also to know nature in both its senses.
but (a) if there is to be movement (as we have explained elsewhere, in an earlier work') there must be something which initiates it; if there is to be movement always, there must always be something which initiates it; if the movement is to be continuous,
but if, secondly, the kinds of matter are two, it will be difficult to make the intermediate bodies behave as air and water behave. suppose, for example, that the two asserted are void and plenum.
eight. in the same case prove that the centres of the circles described about the equilateral triangles form another equilateral triangle.
hence on a given line ab there can be constructed two figures each similar to a given figure cdefg, and having the given line ab homologous to any given side four hundred of the given figure.
or thus: ab ac cb, cb cb. hence ab.cb ac.cb cb . prop. iii. is the particular case of prop. i., when the undivided line is equal to a segment of the divided line.
consequently some assert earth, others fire or air or water or some or all of these, to be the nature of the things that are.
itself in motion, or it is not accidental but an essential attribute. let us consider the former alternative.
the investigation of similar questions about the void, also, must be held to belong to the physicist-namely whether it exists or not, and how it exists or what it is-just as about place.
it is true that of a chaos without differences no idea could be formed.
one. let a : b :: c : d, and let a be a multiple of b, then c is the same multiple of d.
one. any triangle is equal to the fourth part of that which is formed by drawing through each vertex a line parallel to its opposite side.
while others again admit of both conditions. this last is the account of the matter that we must give: for herein lies the solution of all the difficulties raised and the conclusion of the investigation upon which we are engaged.
movent causes a motion away from itself more violent than the natural locomotion of the thing moved, which continues its course so long as it is controlled by the motion imparted to it.
around them the entire framework of our body, constructing for the marrow, first of all a complete covering of bone.
this is thought by all those who try to generate the heavens to explain why the earth came together at the centre.
a thing is said to be prior to other things when, if it does not exist, the others will not exist, whereas it can exist without the others: and there is also priority in time and priority in perfection of existence.
yet, if we do postulate the latter, we are confronted with equally impossible consequences, which we have examined in other works.' but we must try to disentangle these perplexities, and must therefore formulate the whole problem over again.
though he is now 'about to' walk. and (b) since (to appeal to a general principle) amongst the things which 'are' some are capable also of 'not-being', it is clear that the same ambiguous character will attach to them no less when they are coming-to-be:
have been in later ages born aristotelians or platonists.
fifteen. through a given point draw a right line which will meet two non-coplanar lines.
impurities, but when the body is purged it returns to its natural size.
there are, for example, first, flame; and secondly, those emanations of flame which do not burn but only give light to the eyes; thirdly, the remains of fire, which are seen in red-hot embers after the flame has been extinguished.
seven. if a trihedral angle o be formed by three right angles, and a, b, one hundred be points along the edges, the orthocentre of the triangle abc is the foot of the normal from o on the plane abc.
twenty three.forty four deducting the outgoes,........... fourteen.seven thousand two hundred twelve ---- there are left,................. eight.seven thousand one hundred twelve,
now the motion of the outer circle he called the motion of the same, and the motion of the inner circle the motion of the other or diverse. the motion of the same he carried round by the side (i.e.
the liver. the sound which moves swiftly is acute, and the sound which moves slowly is grave, and that which is regular is equable and smooth, and the reverse is harsh. a great body of sound is loud, and a small body of sound the reverse.
it is easy to see, as in the case of proposition iv., that an immediate proof of this proposition can also be got from proposition ii. .
for this 'body' of theirs cannot possibly exist without a 'perceptible contrariety': this 'boundless', which some thinkers identify with the 'original real', must be either light or heavy, either cold or hot.
small to be visible, making up out of all the four elements each separate body, and fastening the courses of the immortal soul in a body which was in a state of perpetual influx and efflux.
the change, however, from an opposite which does not survive is described indifferently in both ways, 'becoming that from this' or 'this becoming that'. we say both that 'the unmusical becomes musical', and that 'from unmusical he becomes musical'.
of (i.) is, if z is not w, then ten is not y ; (iii.) of (ii.) is, if ten is not y , then z is not w. (iv.) the theorem (iv.) is called the obverse of (i.), and (iii.) the obverse of (ii.).
but when fire gets in and dissolves the particles and destroys the uniformity, it has greater mobility, and becoming fluid is thrust forth by the neighbouring air and spreads upon the earth; and this dissolution of the solid masses is called melting, and
with their victors, and from being many become one.
circular line, there is no point at which that which is in process of locomotion can be in a state of rest as having traversed its course, because in its locomotion it is proceeding always about a central point and not to an extreme point: therefore it
the water has not grown, nor has the air: but the former has passed-away and the latter has come-to-be, and-if anything has grown-there has been a growth of 'body.' yet this too is impossible.
they were bringing order out of disorder, having a small grain of experience mingled in a confused heap of a priori notions.
two. the middle points of the three diagonals ac, bd, ef of a quadrilateral abcd are collinear.
out through the body forces the breath inwards through the passage of the mouth and the nostrils. now the origin of this movement may be supposed to be as follows.
and it is just the same in the case of things that are inanimate and incapable of sense-perception. thus there can be nothing intermediate between that which undergoes and that which causes alteration.
'they had plenty of ideas,' says dr. whewell, 'and plenty of facts; but their ideas did not accurately represent the facts with which they were acquainted.' this is a very crude and misleading way of describing ancient science.
difference of their radii, according as the contact with the second circle is of the first or second kind.
and now i will explain to you the generation of the world by a method with which your scientific training will have made you familiar.
the most pertinent question with which to begin will be this: in what sense is it asserted that all things are one? for 'is' is used in many senses. do they mean that all things 'are' substance or quantities or qualities?
hermocrates: 'we will do our best, and have been already preparing; for on our way home, critias told us of an ancient tradition, which i wish, critias, that you would repeat to socrates.' 'i will, if timaeus approves.' 'i approve.'
it will be the same with any time that can be taken, so that there will no time in which the infinite can effect this movement. and, as to infinite time, in that nothing can move another or be moved by it.
thus if a person's health at daybreak and at the present moment is one and the same, why should not this health be numerically one with that which he recovers after an interval? the same argument applies in each case.
the motion is not in the form but in that which is in motion, that is to say 'the movable in activity'. now accidental change we may leave out of account: for it is to be found in everything, at any time, and in any respect.
four. three concurrent lines from the vertices of a triangle abc meet the opposite sides in a, b, one hundred; prove
the feeling of cold is produced by the larger particles of moisture outside the body trying to eject the smaller ones in the body which they compress. the struggle which arises between elements thus unnaturally brought together causes shivering.
this then may serve as a critical statement of the difficulties involved.
prop. forty seven.-theorem. in a right-angled triangle (abc) the square on the hypotenuse (ab) is equal to the sum of the squares on the other two sides (ac, bc).
in like manner the picts bl, bd are equal; hence the whole square af is equal to the sum of the two squares ah, bd.
one thousand. martin has written a valuable dissertation on the opinions entertained respecting the island of atlantis in ancient and modern times. it is a curious chapter in the history of the human mind.
are there many worlds or one only?-that is the fourth question...one only.
hindered it from predominating and advancing; and they so disturbed the nature of the other or diverse, that the three double intervals (i.e. between one, two, four, eight), and the three triple intervals (i.e.
now if the infinite body is to be composed of a finite number of kinds, then each of its parts must necessarily be infinite in quantity, that is to say, the water, fire, c., which compose it.
indeed there are some stars seen in egypt and in the neighbourhood of cyprus which are not seen in the northerly regions; and stars, which in the north are never beyond the range of observation, in those regions rise and set.
sol.-construct an isosceles triangle x. , having each base angle double the vertical angle, and inscribe in the given circle a triangle abd equiangular to it. bisect the angles dab, abd by the lines ac, be.
hence it is clear that even among the things which are outside the necessary and the normal, there are some in connexion withwhich the phrase 'for the sake of something' is applicable.
everything that is in motion must be moved by something. for if it has not the source of its motion in itself it is evident that it is moved by something other than itself, for there must be something else that moves it.
now there is only one way of taking care of things, and this is to give to each the food and motion which are natural to it. and the motions which are naturally akin to the divine principle within us are the thoughts and revolutions of the universe.
(his denial of the reality of colour is a corollary from this position: for, according to him, things get coloured by 'turning' of the 'figures'.) but the possibility of such a construction no longer exists for those who divide bodies into planes.
dem.-let ten be one of the circles; and if possible let another circle y have three points, a, b, one hundred, in common with ten, without coinciding with it. find p, the centre of ten. join pa, pb, pc.
for if all bodies are quantitatively commensurable, and the relative size of the various homoeomerous masses and of their several elements are in the same ratio, so that the total mass of water, for instance, is related to the total mass of air as the
three. join of. now in the triangle ofp the sum of the sides of, fp is greater than op i. xx. ; but of is equal to oe i. def. xxx. . reject them, and fp will remain greater than ep.
assume the weight to be finite, and take an infinite body, ab, of the weight one hundred. subtract from the infinite body a finite mass, bd, the weight of which shall be e. e then is less than one hundred, since it is the weight of a lesser mass.
in consequence of these habits of theirs they had their front-legs and their heads resting upon the earth to which they were drawn by natural affinity; and the crowns of their heads were elongated and of all sorts of shapes, into which the courses of the
blindness to the most obvious phenomena. he measures them not by what preceded them, but by what has followed them.
one. triangles which have one angle of one equal or supplemental to one angle of the other, are to one another in the ratio of the rectangles of the sides about those angles.
axioms.-"elements of human reason," according to dugald stewart, are certain general propositions, the truths of which are self-evident, and which are so fundamental, that they cannot be inferred from any propositions which are more elementary; in other
or thus: draw a line bisecting ab at right angles. then this line i., cor. one must pass through the centre of each circle, and therefore xi. xii. must pass through each point of contact, which is impossible. hence two circles cannot have double contact.
in other words, if the coming-to-be of any things is cyclical, it is 'necessary' that each of them is coming-to-be and has come-to-be: and if the coming-to-be of any things is 'necessary', their coming-to-be is cyclical.
if, however, 'indestructible' and 'ungenerated' are not coincident, there is no necessity that either the ungenerated or the indestructible should be eternal. but they must be coincident, for the following reasons.
prop. twenty three-theorem. two similar segments of circles which do not coincide cannot be constructed on the same chord (ab), and on the same side of that chord.
four. the right lines which join transversely the extremities of two equal and parallel right lines bisect each other.
they are distinguished, we are told, from one another by their figures; but their nature is one, like many pieces of gold separated from one another. but each piece must, as we assert, have the same motion.
the same phenomenon occurs in other instances too.
soul were crushed by reason of disuse. and this was the reason why they were created quadrupeds and polypods: god gave the more senseless of them the more support that they might be more attracted to the earth.
our next point is that no process of change is infinite: for every change, whether between contradictories or between contraries, is a change from something to something.
do when they are awakened; for they connect the earlier 'now' with the later and make them one, cutting out the interval because of their failure to notice it.
for at present we have explained no more than this:-why, when two things change reciprocally into one another, we do not attribute coming-to-be and passing-away uniformly to them both, although every coming-to-be is a passing-away of something else and
and indeed always in the case of any body that can be displaced, must, if it is not compressed, be displaced in the direction in which it is its nature to be displaced-always either down, if its locomotion is downwards as in the case of earth, or up, if
to him, and that 'was' and 'will be' are only to be spoken of becoming in time, for they are motions, but that which is immovably the same cannot become older or younger by time, nor ever did or has become, or hereafter will be, older or younger, nor is
dem.-join ad. describe a circle about the triangle adi, cutting the lines af, four hundred produced, if necessary, in k and fifty. join ik, kl, fifty one. now, the angles klg, fcg are each three. xxi. equal to the angle gad. hence they are equal.
bodies, follows on that of earth; and earth is required because eternal movement in one body necessitates eternal rest in another.
for in some animals we find all such distinctions of parts as this of right and left clearly present, and in others some; but in plants we find only above and below.
the first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. this we have discussed above.
revolving,-in seven orbits seven stars. first, there was the moon in the orbit nearest the earth, and next the sun, in the second orbit above the earth; then came the morning star and the star sacred to hermes, moving in orbits which have an equal
(it is best to interpret in this sense the common statement of the older writers that 'like moves to like'. for the words are not in every sense true to fact.
twelve. if o be the point of intersection of the diagonals of a cyclic quadrilateral abcd, the four rectangles ab.bc, bd.four hundred, four hundred.da, da.ab, are proportional to the four lines bo, co, do, ao.
and if these bodily affections be severe, still worse are the prior disorders; as when the bone itself, by reason of the density of the flesh, does not obtain sufficient air, but becomes mouldy and hot and gangrened and receives no nutriment, and the
let us begin with locomotion, for this is the primary motion. everything that is in locomotion is moved either by itself or by something else.
is the opposite of the motion, the state of rest being privation of motion; then, inasmuch as the two motions that follow the same straight line are contrary motions, and it is impossible for a thing to undergo simultaneously two contrary motions, that
seven. in every triangle the sum of the medians is less than the perimeter, and greater than three-fourths of the perimeter.
sixteen. how would you construct a square whose area would be exactly an acre? give a solution by i. xlvii.
such was the origin of legs and arms. in the next place, the gods gave a forward motion to the human body, because the front part of man was the more honourable and had authority.
and whatever happened either in your country or in ours, or in any other region of which we are informed-if there were any actions noble or great or in any other way remarkable, they have all been written down by us of old, and are preserved in our
two. any angle of a triangle is obtuse, right, or acute, according as the opposite side is greater than, equal to, or less than, twice the median drawn from that angle.
contrary to being led into error by another, it being possible to acquire error, like knowledge, either by one's own agency or by that of another.
and they put in a face in which they inserted organs to minister in all things to the providence of the soul. they first contrived the eyes, into which they conveyed a light akin to the light of day, making it flow through the pupils.
therefore in the whole of the thing we may distinguish that which imparts motion without itself being moved and that which is moved: for only in this way is it possible for a thing to be self-moved.
that in which the 'now' is, the distance from the 'now' will also be.
let us first decide the question whether the earth moves or is at rest. for, as we said, there are some who make it one of the stars, and others who, setting it at the centre, suppose it to be 'rolled' and in motion about the pole as axis.
dem.-since the figures a and one hundred are similar, they are equiangular, and have the sides about their equal angles proportional. in like manner b and one hundred are equiangular, and have the sides about their equal angles proportional.
hence the sum of the angles bad, dac is greater than the angle bac.
again, the complement ph hk xliii. : to each add the pict oc, and we get the pict pc figure ofl. hence the pict pc cj. therefore they are about the same diagonal xliii., cor. one . hence ac produced will pass through one thousand.
five. if through the points of intersection a, b of two circles any two lines acd, bef be drawn parallel to each other, and meeting the circles again in one hundred, five hundred, e, f; then four hundred ef.
hence-owing to the changes of the compound-the same thing seems different and conflicting to different people: it is 'transposed' by a small additional ingredient, and appears utterly other by the 'transposition' of a single constituent.
four. in a given right line find a point such that the perpendiculars from it on two given lines may be equal. state also the number of solutions.
first, as the b's and the g's pass one another, the first b reaches the last g at the same moment as the first g reaches the last b.
hence we must investigate whether, amongst the consecutive members, there is any whose future being is necessary; or whether, on the contrary, every one of them may fail to come-to-be. for that some of them may fail to occur, is clear.
further, since the change which is motion has been proved' to be eternal, the continuity of the occurrence of coming-to-be follows necessarily from what we have established: for the eternal motion, by causing 'the generator' to approach and retire, will
the knowledge of the other gods is beyond us, and we can only accept the traditions of the ancients, who were the children of the gods, as they said; for surely they must have known their own ancestors.
in what sense these are two, and in what sense more, has been stated above.
and the separate parts should be treated in the same manner, in imitation of the pattern of the universe; for as the body is heated and also cooled within by the elements which enter into it, and is again dried up and moistened by external things, and
now we must admit that heat naturally proceeds outward to its own place and to its kindred element; and as there are two exits for the heat, the one out through the body, and the other through the mouth and nostrils, when it moves towards the one, it
this also explains the common saying that human affairs form a circle, and that there is a circle in all other things that have a natural movement and coming into being and passing away.
for instance the form of the sphere is one thing and the gold or bronze sphere another; the shape of the circle again is one thing, the bronze or wooden circle another.
one. what is the subject-matter of book six.? ans. application of the theory of proportion.
for rousseau to take a nearer view of the fundamental implications of the state of society for his thought to reach maturity.
the proportions in which the soul of the world as well as the human soul is divided answer to a series of numbers one, two, three, four, nine, eight, twenty seven, composed of the two pythagorean progressions one, two, four, eight and one, three, nine,
one belong identically and in the same degree to the other; and (b) that 'unlikes', i.e. 'differents', are by nature such as to act and suffer action reciprocally.
perceiving this, and with the view of eliciting information from them, he told them the tales of phoroneus and niobe, and also of deucalion and pyrrha, and he endeavoured to count the generations which had since passed.
twenty four. what is the radius of its nine-points circle?
for just as in change of quality there is something which was formerly black and is now white, or formerly soft and now hard-this is just why we say that the matter exists-so place, because it presents a similar phenomenon, is thought to exist-only in the
but the fact that the term 'potentially' is used in more than one sense is the reason why it is not evident whence such motions as the upward motion of fire and the downward motion of earth are derived.
which is diagonal, and passes through and is governed by the motion of the same, they revolved, some in a larger and some in a lesser orbit-those which had the lesser orbit revolving faster, and those which had the larger more slowly.
for the alternation of day and night is not produced by the motion of the heavens alone, or by the immobility of the earth alone, but by both together; and that which has the inherent force or energy to remain at rest when all other bodies are moving, may
should follow their statement of the matter or find a better way. at the start we may say that, if right and left are applicable, there are prior principles which must first be applied.
that which is potentially a certain kind of body may, it is true, become such in actuality, but if the potential body was not already in actuality some other kind of body, the existence of an extra-corporeal void must be admitted.
cor.-the two points e, e divide ab harmonically.
our theory seems to confirm experience and to be confirmed by it.
bodies of the opposite shape sink down because they occupy so little of the surface, which is therefore easily parted. and these considerations apply with far greater force to air, since it is so much more easily divided than water.
now a thing that is undergoing any particular kind of motion, but though previously existent has not always undergone it, must previously have been at rest so far as that motion is concerned.
at any rate, the language of plato has been the language of natural theology down to our own time, nor can any description of the world wholly dispense with it.
thus in contradictory changes the positive or the negative, as the case may be, is the limit, e.g.
four hundred will never cease contact with e, but the position will always be something like ce, four hundred cutting e at f. the infinite line, therefore, refuses to complete the circle.
let us now assign the geometrical forms to their respective elements. the cube is the most stable of them because resting on a quadrangular plane surface, and composed of isosceles triangles.
for a moving thing cleaves the medium either by its shape, or by the impulse which the body that is carried along or is projected possesses.) therefore all will possess equal velocity. but this is impossible.
three. what technical term is applied to figures which agree in everything but position? ans. they are said to be congruent.
but there is no necessity for there being a void if there is movement. it is not in the least needed as a condition of movement in general, for a reason which, incidentally, escaped melissus; viz. that the full can suffer qualitative change.
two. the parallelogram ec is double of the triangle abc i. xxxiv. , and the parallelogram cf is double of the triangle acd. hence five. xv. ec : cf :: the triangle abc : acd; but abc : acd :: bc : four hundred (part one.). therefore five. xi.
it is normal to their plane, that is, it is normal to ten.
so that if the totality of body, which is a continuum, is now in this order or disposition and now in that, and if the combination of the whole is a world or heaven, then it will not be the world that comes into being and is destroyed, but only its
either 'rarer' or 'derser' inasmuch as there is no void in them.
prop. four.-problem. to inscribe a circle in a given triangle (abc).
ae : ef :: ab : bc, and the other sides of the parallelograms are equal to ae, ef; ab, bc: hence the sides about the equal angles are proportional; therefore the parallelograms af, ac are similar. in the same manner the parallelograms af, fc are similar.
two. of the four propositions occupied with triangles-
substrata are receptive of 'contrarieties' of some kind. so much, then, as an answer to the questions (i) whether coming-to-be 'is' or 'is not'-i.e.
let e stand for earth, w for water, a for air, and f for fire. then (i) since a is transformed into f and w, there will be a contrariety belonging to a f. let these contraries be whiteness and blackness.
prop. three.-theorem. if two planes (ab, bc) cut one another, their common section (bd) is a right line.
they tell us that oceanus and tethys were the children of earth and heaven; that phoreys, cronos, and rhea came in the next generation, and were followed by zeus and here, whose brothers and children are known to everybody.
the cause of this perpetuity of coming-to-be, as we have often said, is circular motion: for that is the only motion which is continuous.
and the various earthy and dusty materials which throng the air. with regard to these questions, it is wrong to accept the explanation offered by democritus.
the mover or agent will always be the vehicle of a form, either a 'this' or 'such', which, when it acts, will be the source and cause of the change, e.g. the full-formed man begets man from what is potentially man.
further, place is coincident with the thing, for boundaries are coincident with the bounded.
for if there is a natural body outside the extreme circumference it must be either a simple or a composite body, and its position must be either natural or unnatural. but it cannot be any of the simple bodies.
they must also accept the inference that the mathematical produce heat and combustion, since they too possess angles and contain atomic spheres and pyramids, especially if there are, as they allege, atomic figures.
this proposition is an immediate inference from the theorem xii., cor. one , that a point of contact counts for two intersections, for then two contacts would be equivalent to four intersections; but there cannot be more than two intersections x. .
now their most important differences are those of property, function, and power; for every natural body has, we maintain, its own functions, properties, and powers.
six. if on a variable line ac, drawn from a fixed point a to any point b in the circumference of a given circle, a point one hundred be taken such that the rectangle ab.ac is constant, the locus of one hundred is a circle.
eleven. what is the altitude of a rectilineal figure?
now perhaps the objection might be raised that there is no necessity in this, since he who sees a stade need not see the smaller measures contained in it, while, on the contrary, he who can see a dot or hear a small sound will perceive what is greater.
if they intersect without the circle, the difference of the arcs they intercept is equal to the arc which parallel chords intersecting on the circumference intercept.
in plato's explanation of sensation we are struck by the fact that he has not the same distinct conception of organs of sense which is familiar to ourselves.
'quantum-in-general' does not come-to-be any more than 'animal' which is neither man nor any other of the specific forms of animal: what 'animal-in-general' is in coming-to-be, that 'quantum-in-general' is in growth.
that, too, is why all the other things-the things, i mean, which are reciprocally transformed in virtue of their 'passions' and their 'powers of action' e.g. the 'simple' bodiesimitate circular motion.
is it possible or impossible that bodies in unordered movement should combine in some cases into combinations like those of which bodies of nature's composing are composed, such, i mean, as bones and flesh?
now it is quite right to say that the formula of the shape apart from the matter must be different from that of the shape in the matter, and we may allow this to be true. we are not, however, therefore compelled to assert a plurality of worlds.
to suppose then that the elements are three in number would seem, from these and similar considerations, a plausible view, as i said before. on the other hand, the view that they are more than three in number would seem to be untenable.
(two) those who said that time is the sphere of the whole thought so, no doubt, on the ground that all things are in time and all things are in the sphere of the whole.
now we say that the thing is moved both by the last and by the first movent in the series, but more strictly by the first, since the first movent moves the last, whereas the last does not move the first, and the first will move the thing without the last,
now it is evident that among existing things there are some that are sometimes in motion and sometimes at rest.
ninety two. in any triangle, the radius of the circumscribed circle is to the radius of the circle which is the locus of the vertex, when the base and the ratio of the sides are given, as the difference of the squares of the sides is to four times the
one. prove this proposition by making the angle abh to the left of ab.
it will now be plain from these considerations what place is.
separately the being-in motion corresponding to each of the two motions, we shall see that the whole being-in motion is continuous.
nor will the commensurability, therefore; hence this eternally is not, because it is contrary to what eternally is. a thing whose contrary is not eternal can be and not be, and it is of such things that there is coming to be and passing away.
seven. state and prove the proposition corresponding to ex. five, for external contact of the circles.
hence the two triangles cag, kab have the sides ca, ag in one respectively equal to the sides ka, ab in the other, and the contained angles cag, kab also equal. therefore iv.
v. the altitude of any figure is the length of the perpendicular from its highest point to its base.
nine. if on the edges of a trihedral angle o-abc three equal lines oa, ob, oc be taken, each of these is greater than the radius of the circle described about the triangle abc.
proposition x., book one., is a particular case of this proposition.
neither the french monarchy nor the genevese aristocracy loved outspoken criticism, and rousseau had always to be very careful what he said.
but the two forms of rectilinear motion are opposed to one another by reason of their places; for up and down is a difference and a contrary opposition in place.
so if the motion was not always in process, it is clear that they must have been in a condition not such as to render them capable respectively of being moved and of causing motion, and one or other of them must have been in process of change: for in what
these triangles are sharper than those which enter the body from without in the shape of food, and therefore they cut them up.
suppose (a) that the infinite sensible body is homogeneous. then each part will be either immovable or always being carried along. yet neither is possible. for why downwards rather than upwards or in any other direction?
the foregoing is an easy proof of the converse of the theorem which is contained in euclid's celebrated fifth definition.
and so the thought of god made a god in the image of a perfect body, having intercourse with himself and needing no other, but in every part harmonious and self-contained and truly blessed.
xv. it will be proper to give here an explanation of the extended meaning of the word angle in modern geometry. this extension is necessary in trigonometry, in mechanics-in fact, in every application of geometry, and has been partly given in i. def. ix.
for we suppose that the same agent produces a given effect on a greater and a smaller mass in longer and shorter times, the times and masses varying proportionately. there is thus no finite time in which infinites can move one another.
very opposite of the truth; and they become false and foolish, and there is no course or revolution in them which has a guiding or directing power; and if again any sensations enter in violently from without and drag after them the whole vessel of the
now to the animal which was to comprehend all animals, that figure was suitable which comprehends within itself all other figures.
of the pure element of air. such are the laws by which animals pass into one another.
seventeen. when is the polar circle imaginary?
hence it is by imitating circular motion that rectilinear motion too is continuous.
if a line (ab) be divided into two parts (at one hundred), the square on the sum of the whole line (ab) and either segment (bc) is equal to four times the rectangle contained by the whole line (ab) and that segment, together with the square on the other
but if anything is to become an animal, not in a qualified sense, it will not be from animal: and if being, not from being-nor from not-being either, for it has been explained that by 'from not being' we mean from not-being qua not-being.
exercise.-prove this proposition without joining be, ch.
we can write it in the form of two equal fractions. thus:
moreover the heavens certainly revolve, and they complete their circular orbit in a finite time; so that they pass round the whole extent of any line within their orbit, such as the finite line ab. the revolving body, therefore, cannot be infinite.
in like manner ago is half the angle agb; but afe has been proved equal to agb; hence afo is equal to ago, and fao is equal to gao, each being right, and ao common to the two triangles fao, gao; hence i. xxvi.
motions. so, too, there cannot be continuous motion in a semicircle or in any other arc of a circle, since here also the same ground must be traversed repeatedly and two contrary processes of change must occur.
things are said to be together in place when they are in one place (in the strictest sense of the word 'place') and to be apart when they are in different places.
but they say that fire is of all bodies the finest. hence fire will be first in the natural order.
seventeen. what proposition is this a limiting case of?
there is another obscurity in the theory empedocles. are we to regard the one as his 'original real'? or is it the many-i.e. fire and earth, and the bodies co-ordinate with these?
hence the sum of the angles fgh, hgi is two right angles; therefore fg and gi are in the same right line xiv. .
and if in consequence of this we must say that two things are of equal velocity if they accomplish locomotion over an equal distance in an equal time, we have to admit the equality of a straight line and a circumference. what, then, is the reason of this?
it is clear, too, that time is not described as fast or slow, but as many or few and as long or short.
for the problem thus raised is why a flat piece of iron or lead floats upon water, while smaller and less heavy things, so long as they are round or long-a needle, for instance-sink down; and sometimes a thing floats because it is small, as with gold dust
cor.-if a parallelogram be inscribed in a circle it is a rectangle.
by something else belonging to it. for let ag be divisible, and let it be divided at b. if then the completion of change has been effected in ab or again in bg, ag cannot be the primary thing in which the completion of change has been effected.
if a parallelogram (abcd) and a triangle (ebc) be on the same base (bc) and between the same parallels, the parallelogram is double of the triangle.
now the question whether each of the things that are unmoved but impart motion is eternal is irrelevant to our present argument: but the following considerations will make it clear that there must necessarily be some such thing, which, while it has the
it is either (a) that which now is and now is not, without any process of destruction, like contact, which without being destroyed afterwards is not, though formerly it was; or (b) that which is but might not be, or which will at some time not be, though
(three) the same absurdity is equally plain if it is supposed that the circles stand still and that it is the stars themselves which move.
one. if the angle one hundred of the acb be equal to an angle of an equilateral , ab ac bc ac.bc.
time is a measure of motion and of being moved, and it measures the motion by determining a motion which will measure exactly the whole motion, as the cubit does the length by determining an amount which will measure out the whole.
a little while afterwards there were great earthquakes and floods, and your warrior race all sank into the earth; and the great island of atlantis also disappeared in the sea.
our next task is to study coming-to-be and passing-away. we are to distinguish the causes, and to state the definitions, of these processes considered in general-as changes predicable uniformly of all the things that come-to-be and pass-away by nature.
motion, though not of one thing, but of many: and in the case that we are considering this is so: for each thing accomplishes its own motion, and there is no impossibility in many things being in motion simultaneously.
(four) again, if the capacity is present prior to the activity, it will be present for all time, even while the thing was as yet ungenerated and non-existent, throughout the infinite time in which it was capable of being generated.
the ever-present image of space is transferred to time-succession is conceived as extension.
is equal to aec; and because ac meets the parallels ad, ec, the angle dac is equal to ace; but the angle bad is equal to dac (hyp.); therefore the angle ace is equal to aec; therefore ae is equal to ac i. vi. .
certainly no other writer of the time has exercised such an influence as his.
for the present we have only to conceive of three natures: first, that which is in process of generation; secondly, that in which the generation takes place; and thirdly, that of which the thing generated is a resemblance.
if our view is a possible one, and the manner of generation which they assert is impossible, this fact will have great weight in convincing us of the immortality and eternity of the world.
they had sprung up in the decline of the eleatic philosophy and were very familiar to plato, as we gather from the parmenides. the consciousness of them had led the great eleatic philosopher to describe the nature of god or being under negatives.
def. i.-homologous points in the planes of two similar figures are such, that lines drawn from them to the angular points of the two figures are proportional to the homologous sides of the two figures.
when one point changes again and again from a particular place to a particular place: if this motion is specifically one, circular motion will be the same as rectilinear motion, and rolling the same as walking.
because there is no maximum, applies also to the space between the radii. now the infinite cannot be traversed, and if the body is infinite the interval between the radii is necessarily infinite: circular motion therefore is an impossibility.
concomitant attribute-plainly, i say, everything comes to be from both subject and form. for 'musical man' is composed (in a way) of 'man' and 'musical': you can analyse it into the definitions of its elements.
another: thus from the observed fact that this occurs in certain cases comes the assumption that it occurs also in the universe.
two. if we divide each of the equal parts ac, four hundred, de, eb into two equal parts, the whole, ab, will be divided into eight equal parts; and we see that ac two eight; ad four eight; ae six eight; ab eight eight; now, we saw in one, that ae three
animals; and as human nature was of two kinds, the superior race would hereafter be called man.
now what becomes in (one) and (two)-'man' and 'not musical'-i call simple, and what each becomes-'musical'-simple also. but when (three) we say the 'not-musical man becomes a musical man', both what becomes and what it becomes are complex.
but he comparatively seldom rises above his own department, and often falls under the narrowing influence which any single branch, when pursued to the exclusion of every other, has over the mind.
he does not imagine the world of sense to be made up of opposites or to be in a perpetual flux, but to vary within certain limits which are controlled by what he calls the principle of the same.
because the angles at b, five hundred, e are right angles, each of the quadrilaterals bg, dh, ek is a rectangle. again, since a is equal to bf (const.), the rectangle contained by a and bc is the rectangle contained by bf and bc (def.
(two) if, on the other hand, the primary body is divisible, then (a) those who give fire a special shape will have to say that a part of fire is not fire, because a pyramid is not composed of pyramids, and also that not every body is either an element or
the holders of the theory of forms do the same, though they are not aware of it; for they separate the objects of physics, which are less separable than those of mathematics.
adjacent. for our assumption is that things that are undergoing alteration are altered in virtue of their being affected in respect of their so-called affective qualities, since that which is of a certain quality is altered in so far as it is sensible,
hence, if a line be divided in extreme and mean ratio, the greater segment will be cut in the same manner by taking on it a part equal to the less; and the less will be similarly divided by taking on it a part equal to the difference, and so on, c.
sixty. from a point within a triangle perpendiculars are let fall on the sides; find the locus of the point, when the sum of the squares of the lines joining the feet of the perpendiculars is given.
but because the inquiry into nature is concerned with movement, and these things have in themselves some spark (as it were) of movement, all inquirers avail themselves of these powers, though in all but a few cases without exact discrimination.
dem.-the triangles daf, eaf have the side ad equal to ae (const.) and af common; therefore the two sides da, af are respectively equal to ea, af, and the base df is equal to the base ef, because they are the sides of an equilateral triangle (def. xxi.).
method of discussion which we are at present employing. do not imagine, any more than i can bring myself to imagine, that i should be right in undertaking so great and difficult a task.
on the contrary (a) some amongst them thought the nature of 'the forms' was adequate to account for coming-to-be.
particular amount or in any length of time: otherwise one man might move a ship, since both the motive power of the ship-haulers and the distance that they all cause the ship to traverse are divisible into as many parts as there are men.
with potatoes, corn, peas, and turnips. the whole lot contains eleven acres, mostly growing up to pines and hickories, and was sold the preceding season for eight dollars and eight cents an acre.
it is true that proclus, writing in the fifth century after christ, tells us of stones and columns in egypt on which the history of the island of atlantis was engraved.
the "natural man," as opposed to "man's man," is man stripped of all that society confers upon him, a creature formed by a process of abstraction, and never intended for a historical portrait.
and here a difficulty may be raised: if not-being is not a particular something, what is it, it may be asked, that is contrary to absence of change in a thing's being? and is this absence of change a state of rest?
he hardly allows to the notions of the ancients the merit of being the stepping-stones by which he has himself risen to a higher knowledge.
with regard, however, to the actual subject of change-that is to say that in respect of which a thing changes-there is a difference to be observed.
their genius was their own; and they were not the rash and hasty generalizers which, since the days of bacon, we have been apt to suppose them.
now whenever air comes into being out of water, light out of heavy, it goes to the upper place. it is forthwith light: becoming is at an end, and in that place it has being.
moreover, if we take each kind of change separately, the truth of our conclusion will be equally evident, on the ground that that that which has changed must be somewhere or in something.
of many such problems one of the strangest is the problem why we find the greatest number of movements in the intermediate bodies, and not, rather, in each successive body a variety of movement proportionate to its distance from the primary motion.
in such cases we have only to take the time in which a thing has changed and again apply the same reasoning.
for, presumably, the unfailing continuity of coming-to-be cannot be attributed to the infinity of the material. that is impossible, for nothing is actually infinite. a thing is infinite only potentially, i.e.
observation.-these definitions tacitly assume the result of props. iii. and x. of this book. on this account we have departed from the usual custom of placing them at the beginning of the book. we have altered the place of definition vi.
if he is to go on to explain this alternate predominance, he should adduce cases where such a state of things exists, as he points to the fact that among mankind we have something that unites men, namely love, while on the other hand enemies avoid one
(c) the morality of the timaeus is singular, and it is difficult to adjust the balance between the two elements of it.
all that becomes and is created is the work of a cause, and that is fair which the artificer makes after an eternal pattern, but whatever is fashioned after a created pattern is not fair. is the world created or uncreated?-that is the first question.
he says, indeed, that it is a substratum prior to the so-called 'elements'-underlying them, as gold underlies the things that are fashioned of gold. (and yet this comparison, if thus expressed, is itself open to criticism.
and he gave dominion to the motion of the same and like, for that he left single and undivided; but the inner motion he divided in six places and made seven unequal circles having their intervals in ratios of two and three, three of each, and bade the
one. construct a triangle, being given base, vertical angle, and any of the following data:-one. perpendicular. two. the sum or difference of the sides. three. sum or difference of the squares of the sides. four. side of the inscribed square on the base.
def. ix.-when two planes which meet are not perpendicular to each other, their inclination is the acute angle contained by two right lines drawn from any point of their common section at right angles to it-one in one plane, and the other in the other.
impassivity, i.e. not of locomotion but rather of qualitative change.
and less. if so, no movement can be caused in the infinite by any finite agent in any time whatever.
from this explanation we see that the ratio of any two commensurable magnitudes is the same as the ratio of the numerical quantities which denote these magnitudes.
hence the triangles edf, edg have the sides ed, df in one equal to the sides ed, dg in the other, and the base ef equal to the base eg. hence i. viii. they are equiangular; but the triangle deg is equiangular to abc.
its shape must necessarily be spherical.
three. the area of a regular hexagon inscribed in a circle is half the area of an equilateral triangle, and three-fourths of the area of a regular hexagon circumscribed to the circle.
for agent and patient are contrary to one another, and coming-to-be is a process into the contrary: hence the patient must change into the agent, since it is only thus that coming-to be will be a process into the contrary.
consequently, if the 'elements' are infinitely many, there will also belong to the single 'element' an infinite number of contrarieties. but if that be so, it will be impossible to define any 'element': impossible also for any to come-to-be.
def. two points, such as f and p, the rectangle of whose distances of, op from the centre is equal to the square of the radius, are called inverse points with respect to the circle.
but if movement were infinite speed would be infinite also; and if speed then weight and lightness. for as superior speed in downward movement implies superior weight, so infinite increase of weight necessitates infinite increase of speed.
moreover, if they differed in substance, they would both act and suffer action on coming into reciprocal contact.
five. the two terms of a ratio are called the antecedent and the consequent. these correspond to the numerator and the denominator of a fraction.
in the republic the pilgrims appear to be looking out from the earth upon the motions of the heavenly bodies; in the phaedrus, hestia, who remains immovable in the house of zeus while the other gods go in procession, is called the first and eldest of the
dem.-since a : b :: c : d, then a : c :: b : d alternando , and c : a :: d : b invertendo ; therefore c a- d b, and one c a- one d b, or a c ----- a b d ----- b. hence a c : b d :: a : b.
alteration is not now natural and now unnatural, for convalescence is no more natural or unnatural than falling ill, whitening no more natural or unnatural than blackening; so, too, with increase and decrease: these are not contrary to each other in the
but what is before is in time; for we say 'before' and 'after' with reference to the distance from the 'now', and the 'now' is the boundary of the past and the future; so that since 'nows' are in time, the before and the after will be in time too; for in
but the philosophers we are criticizing suppose that all such change is 'alteration': whereas in fact there is a difference. for in that which underlies the change there is a factor corresponding to the definition and there is a material factor.
when two such triangles are joined at the diagonal, and this is repeated three times, and the triangles rest their diagonals and shorter sides on the same point as a centre, a single equilateral triangle is formed out of six triangles; and four
it was an island in the clouds, which might be seen anywhere by the eye of faith.
in general, when a number of similar and undifferentiated bodies are moved with the same motion this result is necessarily produced, viz. that the place which is the natural goal of the movement of each single part is also that of the whole.
larger or smaller. we must recognize also (a) that the growing body is not 'void' and that yet there are not two magnitudes in the same place, and (b) that it does not grow by the accession of something incorporeal.
three.one ex. three occurs in the solution of the problem of the inscription of a regular polygon of seventeen sides in a circle. see note one hundred. given the difference of two lines r, and their rectangle r ; find the lines.
fourteen. if the triangle abc be right-angled, having the angle one hundred right,
this indicates not only that the earth's mass is spherical in shape, but also that as compared with the stars it is not of great size.
they must conceive it as composition-just as a wall comes-to-be out of bricks and stones: and the 'mixture', of which they speak, will be composed of the 'elements', these being preserved in it unaltered but with their small particles juxtaposed each to
then if all the time has been occupied by the whole motion, it will take less of the motion to occupy half the time, less again to occupy a further subdivision of the time, and so on to infinity.
hence we have the following definition:-"a ratio is the fraction got by making the antecedent the numerator and the consequent the denominator."
time obviously is what is counted, not that with which we count: there are different kinds of thing.) just as motion is a perpetual succession, so also is time.
for if we are to say that, while there are on the one hand things that are movable, and on the other hand things that are motive, there is a time when there is a first movent and a first moved, and another time when there is no such thing but only
and therefore we are not surprised to find that his philosophy in the timaeus returns at last to a worship of the heavens, and that to him, as to other greeks, nature, though containing a remnant of evil, is still glorious and divine.
in the same way, too, what is potentially of a certain quality or of a certain quantity in a certain place is naturally movable when it contains the corresponding principle in itself and not accidentally (for the same thing may be both of a certain
def. vii.-when a right line intersects two other right lines in two distinct points it makes with them eight angles, which have received special names in relation to one another.
two. when the inscribed and circumscribed regular polygons of any common number of sides to a circle have more than four sides, the difference of their areas is less than the square of the side of the inscribed polygon.
or else that which is making healthy is, let us say, in process of locomotion, and that which is causing locomotion in process of, say, increase. but it is evident that this is impossible.
, and cut off ae equal ad. through e draw bc, cutting ab, ac in the points b, one hundred. join db, six hundred.
up' cannot proceed. the necessary consequence-especially if coming-to-be and passing-away are to take place by 'association' and 'dissociation' respectively-is that a body must contain atomic magnitudes which are invisible.
then the being-in-motion that corresponds to half the motion will be less than the whole being-in-motion, that which corresponds to a quarter of the motion will be less again, and so on to infinity.
twenty five. the line joining the middle points of the diagonals of a quadrilateral circumscribed to a circle-
there is a further difficulty in explaining this part of the timaeus-the natural order of thought is inverted. we begin with the most abstract, and proceed from the abstract to the concrete.
this problem is manifestly equivalent to the following:-given the sum or difference of two lines and their ratio, to find the lines.
so his 'mind' is an absurd person aiming at the impossible, if he is supposed to wish to separate them, and it is impossible to do so, both in respect of quantity and of quality- of quantity, because there is no minimum magnitude, and of quality, because
to maintain that it has come to be and ceased to be there will involve the consequence that a in the course of its locomotion will always be coming to a stand: for it is impossible that a should simultaneously have come to be at b and ceased to be there,
since, therefore, having regard to the figure or shape of a thing we no longer call that which has become of a certain figure by the name of the material that exhibits the figure, whereas having regard to a thing's affections or alterations we still call
for this single first motion has to move many of the divine bodies, while the numerous other motions move only one each, since each single planet moves with a variety of motions.
(four hundred) between the acute angle and the foot of the perpendicular on it from the opposite angle.
but if (b) the process is resolution into planes, the first difficulty is that the elements cannot all be generated out of one another. this they are obliged to assert, and do assert.
the only security is to preserve the balance of the two, and to this end the mathematician or philosopher must practise gymnastics, and the gymnast must cultivate music.
in the process of respiration the external net is said to find a way in and out of the pores of the skin: while the interior of it and the lesser nets move alternately into each other.
harmonize them. the instruments of taste reach from the tongue to the heart. plato has a lively sense of the manner in which sensation and motion are communicated from one part of the body to the other, though he confuses the affections with the organs.
further, seed must have come into being first, and not straightway the animals: the words 'whole-natured first...' must have meant seed.
matter, i.e. a particular thing. now since the universe is perceptible it must be regarded as a particular; for everything that is perceptible subsists, as we know, in matter.
higher degree than any other, since the movent is never subject to any change. so, too, in order that the motion may continue to be of the same character, the moved must not be subject to change in respect of its relation to the movent.
dem.-because ef : lm :: bc : gh (const.), the figure efjk : lmno :: bcd : ghi xxii. ; but efjk is equal to bcd (const.); therefore lmno is equal to ghi; but lmno is equal to a (const.). therefore ghi is equal to a, and it is similar to bcd.
if, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in; and clearly there is a sense in which the infinite exists and another in which it does not.
ten. the sum of the diagonals of a quadrilateral is less than the sum of the lines which can be drawn to its angular points from any point except the intersection of the diagonals.
for in no inanimate thing do we observe a part from which movement originates. some do not move at all, some move, but not indifferently in any direction; fire, for example, only upward, and earth only to the centre.
in the case of things thrown, since the air or the water, being divisible, is a movent only in virtue of the fact that different parts of the air are moved one after another): and in either case the motion cannot be a single motion, but only a consecutive
now, since the angle abd is equal to acd, the points a, b, one hundred, five hundred are concyclic xxi., cor. one . hence the rectangle ao.oc is equal to the rectangle bo.od xxxv. .
moreover, since (a) the 'simple' bodies appear to move 'naturally' as well as by compulsion, i.e. in a manner contrary to nature (fire, e.g.
hence if anything imparts motion without itself being moved, it may touch the 'moved' and yet itself be touched by nothing-for we say sometimes that the man who grieves us 'touches' us, but not that we 'touch' him.
hence we must begin by explaining certain preliminary matters, about which the statements now current are vague.
deucalion and pyrrha; and he traced the genealogy of their descendants, and reckoning up the dates, tried to compute how many years ago the events of which he was speaking happened.
reduction of flesh, purging, drugs, or surgical instruments are means towards health. all these things are 'for the sake of' the end, though they differ from one another in that some are activities, others instruments.
perhaps the necessary is present also in the definition. for if one defines the operation of sawing as being a certain kind of dividing, then this cannot come about unless the saw has teeth of a certain kind; and these cannot be unless it is of iron.
the circle whose radius is the maximum line (pa) has contact of the second kind. three. a circle having any of the remaining lines (pf) as radius intersects hde in two points (f, i).
for flesh and wood and all other similar bodies contain potentially fire and earth, since one sees these elements exuded from them; and, on the other hand, neither in potentiality nor in actuality does fire contain flesh or wood, or it would exude them.
greater than any other. but then there cannot be any time in which its action could take place.
and generally are the states and affections in bodies severally one in essence although (as is clear) the things that contain them are obviously in motion and in flux?
the frame of the entire creature when young has the triangles of each kind new, and may be compared to the keel of a vessel which is just off the stocks; they are locked firmly together and yet the whole mass is soft and delicate, being freshly formed of
wherefore also the thighs and the shanks and the hips, and the bones of the arms and the forearms, and other parts which have no joints, and the inner bones, which on account of the rarity of the soul in the marrow are destitute of reason-all these are
earth, owing to the survival of the dry of the fire and the cold of the water. so, too, in the same way, fire and water will result from air plus earth.
(this flesh and this bone were together, and so of any thing: therefore all things: and at the same time too.) for there is a beginning of separation, not only for each thing, but for all.
prop. thirty four.-problem. to cut off from a given circle (abc) a segment which shall contain an angle equal to a given angle (ten).
now i, critias and hermocrates, am conscious that i myself should never be able to celebrate the city and her citizens in a befitting manner, and i am not surprised at my own incapacity; to me the wonder is rather that the poets present as well as past
hence it is said to be discerned by a kind of spurious or analogous reason, partaking so feebly of existence as to be hardly perceivable, yet always reappearing as the containing mother or nurse of all things.
in motion or at rest as the case may be in any part of it in which it is naturally designed to be in motion or at rest: this being so, the assumption that there can be motion or rest in a present will carry with it the implication that the same thing can
for whenever in any three numbers, whether cube or square, there is a mean, which is to the last term what the first term is to it; and again, when the mean is to the first term as the last term is to the mean-then the mean becoming first and last, and
again, in the case of those elements which do suffer dissolution, the 'suspension' of the triangles is unsatisfactory. but this takes place whenever one is dissolved into another, because of the numerical inequality of the triangles which compose them.
observation.-the following summary of the contents of the fourth book will assist the student in remembering it:-
observed. again if a given force move a given weight a certain distance in a certain time and half the distance in half the time, half the motive power will move half the weight the same distance in the same time.
hence when a support is withdrawn such a body moves downward until it reaches the body next below it, air to the place of water and water to that of earth.
(a) an attribute is either that which may or may not belong to the subject or that in whose definition the subject of which it is an attribute is involved.
each of the four solid angles in this figure is a little larger than the largest of obtuse angles.
cor.-the line af is an axis of symmetry of the figure.
viii. analogy or proportion is the similitude of ratios.
parts and indivisible could be in motion would have been the possibility of the infinitely small being in motion in a moment: for in the two questions-that of motion in a moment and that of motion of something indivisible-the same principle is involved.
we may take the three things 'wood', 'hot', and 'cold', of which the first is that which is in motion, the second is that to which the motion proceeds, and the third is that from which it proceeds.
since some things are such as to come-to-be and pass-away, and since coming-to-be in fact occurs in the region about the centre, we must explain the number and the nature of the 'originative sources' of all coming-to-be alike: for a grasp of the true
it is plain that these are all cases of coming to be from a substratum.
but it would not be because there was no other region in which it could be carried along that it would remain at the centre, but because this is its nature. yet in this case also we may say that it fixes itself.
another difficulty emerges at the same time. any part of the infinite body ought to remain at rest. just as the infinite remains at rest in itself because it fixes itself, so too any part of it you may take will remain in itself.
then the whole dz must be the motion of ag: for dz must constitute the motion of ag inasmuch as de and ez severally constitute the motions of each of its parts.
now because ab passes through the centre, and cuts the other chord four hundred, which does not pass through the centre at right angles, it bisects it iii. .
the result we have reached is logically concordant with the eternity of circular motion, i.e.
the difficulty which plato feels, is that which all of us feel, and which is increased in our own day by the progress of physical science, how the responsibility of man is to be reconciled with his dependence on natural causes.
fifty one. four hundred is a perpendicular from any point one hundred in a semicircle on the diameter ab; efg is a circle touching db in e, four hundred in f, and the semicircle in g; prove-(one) that the points a, f, g are collinear; (two) that ac ae.
ad.db, since dh is equal to db iv., cor. one ; therefore the rectangle ad.db is equal to the gnomon cmg, and the square on four hundred is equal to the figure lg.
for instance, liquids are the most 'combinable' of all bodies-because, of all divisible materials, the liquid is most readily adaptable in shape, unless it be viscous. viscous liquids, it is true, produce no effect except to increase the volume and bulk.
if there is no ultimate natural cause of movement and each preceding term in the series is always moved by constraint, we shall have an infinite process.
but if what 'came away' was not a body but a separable form or quality, and if the magnitude is 'points or contacts thus qualified': it is paradoxical that a magnitude should consist of elements, which are not magnitudes.
for it will follow that the outer stars are the swifter, and that the pace of the stars corresponds to the size of their circles.
a- b ma- mb; therefore a : b :: ma : mb.
a man may run and then at once fall ill of a fever), and again, in the torch-race we have consecutive but not continuous locomotion: for according to our definition there can be continuity only when the ends of the two things are one.
and the case is similar in regard to the states of the soul, all of which (like those of body) exist in virtue of particular relations, the excellences being perfections of nature and the defects departures from it: moreover, excellence puts its possessor
we should not say in the latter case that there is anything artistic about a thing, if it is a bed only potentially, not yet having the form of a bed; nor should we call it a work of art. the same is true of natural compounds.
we must, then, think of the action of the lower stars as similar to that of animals and plants.
seven. give a common enunciation which will include propositions v. and vi.
what i mean is this: if the portions in this world behave similarly both to one another and to those in another world, then the portion which is taken hence will not behave differently either from the portions in another world or from those in the same
but he will place friend by the side of friend, so as to create health.
to these there has to be added a third principle, which is the condition of the action of the other two,-the interpenetration of particles in proportion to their density or rarity. it is this which enables fire and air to permeate the flesh.
four. if the sum of two of the lines were equal to the third, would the circles meet? prove that they would not intersect.
this that they alternate from the one motion to the other. we must consider, then, how this matter stands, for the discovery of the truth about it is of importance, not only for the study of nature, but also for the investigation of the first principle.
ninety seven. inscribe in a given circle a trapezium, the sum of whose opposite parallel sides is given, and whose area is given.
then since a has occupied the whole time zh in arriving at five hundred, will have arrived at o in less time than this, say zk.
fifty two. in a circle, prove that an equilateral inscribed polygon is regular, and also an equilateral circumscribed polygon, if the number of sides be odd.
a thing must move either naturally or unnaturally, and the two movements are determined by the proper and alien places.
four. what relation does prop. xiv. bear to prop. xiii.?
but before proceeding to our conclusion it will be well to discuss a difficulty that arises in connexion with locomotion.
for our account of growth must preserve the characteristics of that which is growing and diminishing. and these characteristics are three: (i) any and every part of the growing magnitude is made bigger (e.g.
on the contrary, sometimes he attributes its movement to something like chance ('for thus, as it ran, it happened to meet them then, though often otherwise"), while at other times he says it is the nature of fire to be borne upwards, but 'the ether' (to
there is no difficulty, by the help of aristotle and later writers, in criticizing the timaeus of plato, in pointing out the inconsistencies of the work, in dwelling on the ignorance of anatomy displayed by the author, in showing the fancifulness or
now we have discussed 'indivisible planes' in the preceding treatise.'
of mental disorders there are two sorts, one madness, the other ignorance, and they may be justly attributed to disease. excessive pleasures or pains are among the greatest diseases, and deprive men of their senses.
six. given the base and ratio of the sides of a triangle; construct it when any of the following data is given:-one, the area; two, the difference on the squares of the sides; three, the sum of the squares on the sides; four, the vertical angle; five, the
for the arts make their material (some simply 'make' it, others make it serviceable), and we use everything as if it was there for our sake. (we also are in a sense an end.
enough, if we adduce probabilities as likely as any others; for we must remember that i who am the speaker, and you who are the judges, are only mortal men, and we ought to accept the tale which is probable and enquire no further.
for if you suppose it an object of intelligence, it will be in a place-since place is what 'within' and 'beyond' denote-and therefore an object of perception. but nothing that is not in a place is perceptible.
for let o be the centre: join oa, ob, op. then oab is an isosceles triangle, and op is a line drawn from its vertex to a point p in the base, or base produced.
again, because ab is divided equally in o and unequally in e, the rectangle ae.eb, together with oe , is equal to ob -that is, to oc two. v. ; but oc is equal to oe ec i. xlvii. therefore
without qualification, not that it 'comes to be' some particular thing.
one. if three planes have a common line of intersection, the normals drawn to these planes from any point of that line are coplanar.
for in so far as it is infinite, there will be no up or down or middle, and in so far as it is a void, up differs no whit from down; for as there is no difference in what is nothing, there is none in the void (for the void seems to be a non-existent and a
moreover the result at which we have arrived, that rotatory motion is single and continuous, and rectilinear motion is not, is a reasonable one.
now (one) the infinite qua infinite is unknowable, so that what is infinite in multitude or size is unknowable in quantity, and what is infinite in variety of kind is unknowable in quality.
the physicists, on the other hand, all of them, always regard the infinite as an attribute of a substance which is different from it and belongs to the class of the so-called elements-water or air or what is intermediate between them.
it is evident, therefore, that nothing continuous is without parts.
or thus: denote the angle eba by ; then evidently
(two). take any point five hundred in the arc ac. it is required to prove that the angle cae is equal to cda.
and here we cannot apply the argument used to solve the difficulty stated above: we cannot argue that h is at five hundred at a sectional point of time and has not come to be or ceased to be there.
this, therefore, is cause in the sense of material origin for the things which are such as to come-to-be; while cause, in the sense of their 'end', is their 'figure' or 'form'-and that is the formula expressing the essential nature of each of them.
and just as there is no primary time in which that which is in motion is in motion, so too there is no primary time in which that which is coming to a stand is coming to a stand, there being no primary stage either of being in motion or of coming to a
if the chord be produced both ways, the whole line is called a secant, and each of the parts into which a secant divides the circumference is called an arc-the greater the major conjugate arc, and the lesser the minor conjugate arc.-newcomb.
the outer and the inner sphere cross one another and meet again at a point opposite to that of their first contact; the first moving in a circle from left to right along the side of a parallelogram which is supposed to be inscribed in it, the second also
in their vaster conceptions of chaos, erebus, aether, night, and the like, the first rude attempts at generalization are dimly seen.
to start, then: the following considerations would make one suspect that it either does not exist at all or barely, and in an obscure way. one part of it has been and is not, while the other is going to be and is not yet.
again, the figure ph is the square on pi, and therefore equal to the square on ac. hence the whole figure af, that is, the square on ad, is equal to four times the rectangle ab.bc, together with the square on ac.
dem.-suppose for simplicity that the solid angle a is contained by five plane angles bac, cad, dae, eaf, fab; and let the planes of these angles be cut by another plane in the lines bc, four hundred, de, ef, fb; then we have eleven. xx. ,
seven. if we multiply any two numbers, as five and seven, by any number such as four, the products twenty, twenty eight are called equimultiples of five and seven.
when it enters with more of a flood and overcomes the fibres by its heat, and boiling up throws them into disorder, if it have power enough to maintain its supremacy, it penetrates the marrow and burns up what may be termed the cables of the soul, and
the principle of distinction between bodies being quantity, the various sizes will be in a definite ratio, and whatever bodies are in this ratio to one another must be air, fire, earth, and water respectively.
which it belongs primarily, in virtue of itself and not in virtue of a concomitant attribute.
seven. the bisectors of two adjacent angles of a parallelogram are at right angles.
but white is not further subdivided by specific differences: hence any whitening is specifically one with any other whitening.
the possibility of another man would be removed. whatever case you took it would be the same. the general rule is this: a thing whose essence resides in a substratum of matter can never come into being in the absence of all matter.
when the seed about the spinal marrow is too abundant, the body has too great pleasures and pains; and during a great part of his life he who is the subject of them is more or less mad.
is an encumbrance. as for a habitat, if i were not permitted still to squat, i might purchase one acre at the same price for which the land i cultivated was sold-namely, eight dollars and eight cents.
and since every motion that is one can be both regular and irregular, motions that are consecutive but not specifically the same cannot be one and continuous: for how should a motion composed of alteration and locomotion be regular?
one hundred two. find a point, the sum of whose distances from three given points may be a minimum.
thus if we look at the matter inductively we do not find anything to be 'in' itself in any of the senses that have been distinguished; and it can be seen by argument that it is impossible. for each of two things will have to be both, e.g.
all causes, both proper and incidental, may be spoken of either as potential or as actual; e.g. the cause of a house being built is either 'house-builder' or 'house-builder building'.
abundantly provided with flesh; but such as have mind in them are in general less fleshy, except where the creator has made some part solely of flesh in order to give sensation,-as, for example, the tongue. but commonly this is not the case.
in like manner, since the parallelograms hb, hf are on the same base eh, and between the same parallels eh, bg, they are equal. hence bd and fh are each equal to bh. therefore (axiom i.) bd is equal to fh.
water does not of itself produce out of itself an animal: and it is the art, not the wood, that makes a bed. nor is this their only error. they make a second mistake in omitting the more controlling cause: for they eliminate the essential nature, i.e.
important of all symmetries; but the due proportion of mind and body is the fairest and loveliest of all sights to him who has the seeing eye.
that part of the inferior soul which is endowed with courage and passion and loves contention they settled nearer the head, midway between the midriff and the neck, in order that it might be under the rule of reason and might join with it in controlling
again, the decrease of that which suffers decrease is caused by a part of the thing becoming detached.
wherefore we ought always to manage them by regimen, as far as a man can spare the time, and not provoke a disagreeable enemy by medicines.
at the centre, they say, is fire, and the earth is one of the stars, creating night and day by its circular motion about the centre. they further construct another earth in opposition to ours to which they give the name counterearth.
this corresponds to the distributive law in multiplication, and shows that rectangles in geometry, and products in arithmetic and algebra, are subject to the same rules.
but, leaving this enquiry, let us proceed to distribute the elementary forms, which have now been created in idea, among the four elements.
from itself, the argument being the same as that which we used above about things in motion.
if the air were not endowed with this function, constrained movement would be impossible. and the natural movement of a body may be helped on in the same way.
further, if, on the one hand, circular movement is natural to something, it must surely be some simple and primary body which is ordained to move with a natural circular motion, as fire is ordained to fly up and earth down.
observation shows that even mixed bodies are often divisible into homoeomerous parts; examples are flesh, bone, wood, and stone.
or rather the argument of melissus is gross and palpable and offers no difficulty at all: accept one ridiculous proposition and the rest follows-a simple enough proceeding.
dem.-let a be the centre of one of the circles. join ap, and produce it to meet the second circle again in e. i say the centre of the second circle is in the line pe. if not, let it be elsewhere, as at b.
and this holds also of the constitution of diseases; if any one regardless of the appointed time tries to subdue them by medicine, he only aggravates and multiplies them.
but this would show that their movement was not their own, but due to the circles.
when we have determined the nature of motion, our next task will be to attack in the same way the terms which are involved in it.
but it is a wrong assumption to suppose universally that we have an adequate first principle in virtue of the fact that something always is so or always happens so.
suppose 'aquilinity' to be curvature in the nose or flesh, and flesh to be the matter of aquilinity. suppose further, that all flesh came together into a single whole of flesh endowed with this aquiline quality.
rousseau's emile , the greatest of all works on education, has already been issued in this series. in this volume are contained the most important of his political works. of these the social contract , by far the most significant, is the latest in date.
further, it is not the action of another body that makes one of these bodies move up and the other down; nor is it constraint, like the 'extrusion' of some writers.
note further that we do not subvert the principle that everything either is or is not.
now 'being' (we have explained elsewhere the exact variety of meanings we recognize in this term) is better than 'not-being': but not all things can possess 'being', since they are too far removed from the 'originative source.
seventeen. inscribe a lozenge in a triangle having for an angle one angle of the triangle.
zeno's reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.
for in the definition too there are some parts that are, as it were, its matter.
prop. ten.-theorem. if two circles have more than two points common, they must coincide.
parable in which plato has wrapped up his conception of the creation of the world. the explanation may help to fill up with figures of speech the void of knowledge.
prop. five.-problem. to describe a circle about a given triangle (abc).
one. in any triangle, the difference between any two sides is less than the third.
it is absurd too not to perceive that, while the whirling movement may have been responsible for the original coming together of the art of earth at the centre, the question remains, why now do all heavy bodies move to the earth.
similarly, too, in the case of any alteration whatever if that which suffers alteration is infinitely divisible it does not follow from this that the same is true of the alteration itself, which often occurs all at once, as in freezing.
hence the point a must coincide with five hundred, and the triangle abc agrees in every respect with the triangle def; and therefore the three angles of one are respectively equal to the three angles of the other-namely, a to five hundred, b to e, and one
if, therefore, there must always be motion, there must also always be locomotion as the primary motion, and, if there is a primary as distinguished from a secondary form of locomotion, it must be the primary form.
similar transformation-not only then, but also now-if, and because, they change their qualities.
(two) place is no part of the thing. (three) the immediate place of a thing is neither less nor greater than the thing.
order of generation. if, in addition, above is the region from which movement originates, right the region in which it starts, front the region to which it is directed, then on this ground too above has a certain original character as compared with the
in the interior of every animal the hottest part is that which is around the blood and veins; it is in a manner an internal fountain of fire, which we compare to the network of a creel, being woven all of fire and extended through the centre of the body,
that being so, by continually taking a greater magnitude in this way i shall never arrive at the full ab, whereas i shall always be getting a lesser fraction of the time given. therefore the force must be infinite, since it exceeds any finite force.
then since ex hypothesi the movent while causing motion is also itself in motion, and the motion of the moved and the motion of the movent must proceed simultaneously (for the movent is causing motion and the moved is being moved simultaneously) it is
certainly no nation that lived simply in all respects, that is, no nation of philosophers, would commit so great a blunder as to use the labor of animals.
language recognizes (a) an absolute, (b) a relative heavy and light. of two heavy things, such as wood and bronze, we say that the one is relatively light, the other relatively heavy.
shall coincide with e. again, because the angle bac is equal to the angle edf, the line ac shall coincide with df; and since ac is equal to df (hyp.), the point one hundred shall coincide with f; and we have proved that the point b coincides with e.
and we always assume the presence in nature of the better, if it be possible: since, then, continuous motion is possible (this will be proved later: for the present let us take it for granted), and no other motion can be continuous except locomotion,
the belly with the elements of bitterness and gall, and a suffusion of bilious colours when the liver is contracted, and causing pain and misery by twisting out of its place the lobe and closing up the vessels and gates.
one. the figure formed by the five diagonals of a regular pentagon is another regular pentagon.
we also see that there is no necessity to introduce extraneous magnitudes for the purpose of defining duplicate and triplicate ratios, as euclid does. in fact, the definitions by squares and cubes are more explicit.
this can be shown as follows: we take it for granted that the greater force is always that which in less time than another does an equal amount of work when engaged in any activity-in heating, for example, or sweetening or throwing; in fact, in causing
one hundred five. to construct a quadrilateral, similar to a given one, whose four vertices shall lie on four given lines.
when, on the other hand, the gods purge the earth with a deluge of water, the survivors in your country are herdsmen and shepherds who dwell on the mountains, but those who, like you, live in cities are carried by the rivers into the sea.
incidental in the process.) and the positive form is one-the order, the acquired art of music, or any similar predicate.
cor. two.-if p be the common centre of circles whose radii are pa, pb, pc, c., then-one. the circle whose radius is the maximum line (pa) lies outside the circle ade, and touches it in a def. iv. . two.
it is plain, then, that there are elements, and why.
will not be duly prepared, unless it is formless, and free from the impress of any of those shapes which it is hereafter to receive from without.
when two quantities are incommensurable, such as the diagonal and the side of a square, although their ratio is not equal to that of any two commensurable numbers, yet a series of pairs of fractions can be found whose difference is continually
but if it is to be 'in' another body and yet remains 'separate' in such a way that it is in no sense a part of that body (neither a part of its substantial being nor an 'accident' of it), many impossibilities will result.
however, i should never have broken a horse or bull and taken him to board for any work he might do for me, for fear i should become a horse-man or a herds-man merely; and if society seems to be the gainer by so doing, are we certain that what is one
cause means either what is particular or a genus, or an incidental attribute or a genus of that, and these either as a complex or each by itself; and all six either as actual or as potential.
the 'now' in one sense is the same, in another it is not the same.
for example, taking a complete circle, motion from e to f on the semicircle g may be opposed to motion from f to e on the semicircle h.
as substances i class the simple bodies-fire, earth, and the other terms of the series-and all things composed of them; for example, the heaven as a whole and its parts, animals, again, and plants and their parts.
those parts of the soul which are in the breast.
four. when does a secant become a tangent?
moreover, the truth of what has been said is more evident in the case of magnitude, because the magnitude over which what is changing changes is continuous. for suppose that a thing has changed from g to five hundred.
such being the nature of the world, when a person says that any of these points is above or below, may he not be justly charged with using an improper expression?
little water. but such a thing has never been observed anywhere. necessarily, then, as fire goes up because it has something, e.g.
a more stable basis than the equilateral triangle, both in the whole and in the parts.
changed in every part). it is evident, then, that with reference to the beginning of change there is no primary when in which change has been effected: for the divisions are infinite.
; and because the line ce stands on ab, the sum of the angles bec, cea is two right angles; therefore the sum of the angles cea, aed is equal to the sum of the angles bec, cea. reject the angle cea, which is common, and we have the angle aed equal to bec.
this, instead of being a definition, is a theorem. we have altered the last clause from that given in simson's euclid, which runs thus:-"the first is said to have to the second a greater ratio than the third has to the fourth."
dem.-one. if possible let two circles touch each other at two points a and b. now since the two circles touch each other in a, the line joining their centres passes through a xi. . in like manner, it passes through b.
the four propositions xxvi.-xxix. are so like in their enunciations that students frequently substitute one for another. the following scheme will assist in remembering them:-
yet here is this great weight of earth, and it is at rest. and again, from beneath one of these moving fragments of earth, before it falls, take away the earth, and it will continue its downward movement with nothing to stop it.
eleven. the areas of circles are to one another as the squares of their diameters. for they are to one another as the similar elementary triangles into which they are divided, and these are as the squares of the radii.
on ab and de describe squares, and through one hundred and f draw lines parallel to ab and de, and complete the rectangles ai, dn.
every passing-away some other thing's coming-to-be. but the question subsequently formulated involves a different problem-viz.
the principles therefore are, in a way, not more in number than the contraries, but as it were two, nor yet precisely two, since there is a difference of essential nature, but three.
just as theological writers on politics, from hooker to bossuet, make use of biblical terminology and ideas; just as more modern writers, from hegel to herbert spencer, make use of the concept of evolution, rousseau uses the ideas and terms of the social
two. ab bc : ab :: ab : a. this is evident by drawing through b a parallel to a.
seven. if de, df be parallels to the sides of a triangle abc from any point five hundred in the base, then ab.ae ac.af ad bd.six hundred. this is an easy deduction from six.
whereas in this land, neither then nor at any other time, does the water come down from above on the fields, having always a tendency to come up from below; for which reason the traditions preserved here are the most ancient.
but this would be even more likely to happen in air-an objection which he himself raises. his reply to the objection is feeble. in the air, he says, the 'drive' (meaning by drive the movement of the upward moving bodies) is not uniform in direction.
three. if the sides of a polygon of n sides be produced, the sum of the angles between each alternate pair is equal to two(n four) right angles.
the sides about the equal angles of equiangular triangles (bac, cde) are proportional, and those which are opposite to the equal angles are homologous.
for indeed practically all the 'movers' within our ordinary experience impart motion by being moved: in their case, what touches inevitably must, and also evidently does, touch something which reciprocally touches it.
observation.-if the orthocentre of the triangle abc be within the triangle, the rectangles oa.op, ob.oq, oc.or are negative, because the lines oa.op, c., are measured in opposite directions, and have contrary signs; hence the polar circle is imaginary;
moreover the time occupied by the action of any finite force must also be finite: for if a given force moves something in a certain time, a greater force will do so in a lesser time, but still a definite time, in inverse proportion.
and as the bone was brittle and liable to mortify and destroy the marrow by too great rigidity and susceptibility to heat and cold, he contrived sinews and flesh-the first to give flexibility, the second to guard against heat and cold, and to be a
so much of a syncretist is plato, though not after the manner of the neoplatonists. for the elements which he borrows from others are fused and transformed by his own genius.
had they begun their study in a truly historical spirit, they would have seen that rousseau's importance lies just in the new use he makes of old ideas, in the transition he makes from old to new in the general conception of politics.
neither when we speak of mind or intelligence, do we seem to get much further in our conception than circular motion, which was deemed to be the most perfect.
two. the right line joining the middle points of two sides of a triangle is parallel to the third; for the medians from the extremities of the base to these points will each bisect the original triangle.
let e then, when acted upon by a, complete the movement in the time five hundred. then, as five hundred is to four hundred, so is e to some finite part of b. this part will necessarily be moved by a in the time four hundred.
so, too, what is of a certain quantity extends itself over a certain space unless something prevents it. the thing in a sense is and in a sense is not moved by one who moves what is obstructing and preventing its motion (e.g.
the lusts of men are caused by their bodily constitution, though they may be increased by bad education and bad laws, which implies that they may be decreased by good education and good laws.
we ought, however, to suppose that there is one rather than many, and a finite rather than an infinite number.
in the first place, the bodies which i have been describing are necessarily objects of sense. but we have not yet considered the origin of flesh, or what belongs to flesh, or of that part of the soul which is mortal.
i am aware that the sophists have plenty of brave words and fair conceits, but i am afraid that being only wanderers from one city to another, and having never had habitations of their own, they may fail in their conception of philosophers and statesmen,
then if it is in motion in neither of the two parts, it will be at rest in the whole: for it is impossible that it should be in motion in a time in no part of which it is in motion.
first, however, let us treat of the infinite universally.
timaeus: we remember some of them, and you will be here to remind us of anything which we have forgotten: or rather, if we are not troubling you, will you briefly recapitulate the whole, and then the particulars will be more firmly fixed in our memories?
bloodless, and also had within hollows like the pores of a sponge, in order that by receiving the breath and the drink, it might give coolness and the power of respiration and alleviate the heat.
and as in a house the women's apartments are divided from the men's, the cavity of the thorax was divided into two parts, a higher and a lower.
from what has been said it is evident, then, that void does not exist either separate (either absolutely separate or as a separate element in the rare) or potentially, unless one is willing to call the condition of movement void, whatever it may be.
soul, and are carried to the three places of the soul, creating infinite varieties of trouble and melancholy, of rashness and cowardice, of forgetfulness and stupidity.
for where the acid and briny phlegm and other bitter and bilious humours wander about in the body, and find no exit or escape, but are pent up within and mingle their own vapours with the motions of the soul, and are blended with them, they produce all
(seven) as the existence of a thing centres 'in its good and generally 'in' its end, i.e. in 'that for the sake of which' it exists.
(moreover in all these self-moving things the first movent and cause of their self-motion is itself moved by itself, though in an accidental sense: that is to say, the body changes its place, so that that which is in the body changes its place also and is
that being so, it is evident that, if the motion be single, both processes cannot occur since they are contrary to one another: for it is a law of nature that the same cause, provided it remain in the same condition, always produces the same effect, so
then bg possesses a certain force that occupies a certain time, let us say the time z in moving five hundred.
these are a few of the conjectures which philosophy forms, when, leaving the eternal nature, she turns for innocent recreation to consider the truths of generation.
for from the communion of the internal and external fires, and again from the union of them and their numerous transformations when they meet in the mirror, all these appearances of necessity arise, when the fire from the face coalesces with the fire from
earth itself, when not consolidated, is dissolved by water; when consolidated, by fire only.
infancy and childhood is the chaos or first turbid flux of sense prior to the establishment of order; the intervals of time which may be observed in some intermittent fevers correspond to the density of the elements.
they seemed to see them in the least things as well as in the greatest; in atoms, as well as in suns and stars; in the human body as well as in external nature.
white and bright meeting, and falling upon a full black, become dark blue; dark blue mingling with white becomes a light blue; the union of flame-colour and black makes leek-green. there is no difficulty in seeing how other colours are probably composed.
that which is apprehended by intelligence and reason is always in the same state; but that which is conceived by opinion with the help of sensation and without reason, is always in a process of becoming and perishing and never really is.
'lightness' either both these qualities or one or the other of them. but bodies which are heavy or light are such as to 'act' and 'suffer action'.
(two) on the view, first, that both are in motion, we have the absurdity that the stars and the circles move with the same speed, i.e. that the ace of every star is that of the circle in it moves.
to take an example, a state of rest in health is (one) contrary to a state of rest in disease, and (two) the motion to which it is contrary is that from health to disease.
the first discourse neither is, nor attempts to be, a reasoned or a balanced production.
true, but even if we suppose its existence settled, the question of its nature presents difficulty-whether it is some sort of 'bulk' of body or some entity other than that, for we must first determine its genus.
dem.-one. let a be in the given plane, and if possible let ab, ac be both normals to it, on the same side. now let the plane of ba, ac cut the given plane ten in the line de. then because ba is a normal, the angle bae is right.
again, in each case, it the figure abhij be turned round the line ab until it falls on the other side, it will still be similar to the figure cdefg.
dem.-complete the pict aebg. draw dh, one hundred one parallel to ag, bg. join ih, and produce; then ab, four hundred, ih are concurrent (ex. one); therefore ih will pass through f. join ei, eh. now xi., ex.
in the time one hundred, five hundred will produce that motion in a patient less than b, say f. then take e, bearing the same proportion to five hundred as the whole bf bears to f. e will produce the motion in bf in the time one hundred.
now, denoting the parallelograms ab, bf, cg by ten, y , z, respectively, we have-
twenty three. find a point in one of the sides of a triangle such that the sum of the intercepts made by the other sides, on parallels drawn from the same point to these sides, may be equal to a given length.
twenty. when is a line divided harmonically?
must be composed of something that is unmoved but imparts motion and also of something that is moved but does not necessarily impart motion but may or may not do so.
the same manner and on the same spot, within his own limits revolving in a circle. all the other six motions were taken away from him, and he was made not to partake of their deviations.
in so far as a thing is one and continuous not merely in virtue of contact, it is impassive: it is only in so far as a thing is divided that one part of it is by nature active and another passive.
if it is not one process, it would seem that there is nothing to prevent its being infinite in this sense; e.g.
for the prime mover must cause motion in virtue of its own natural movement, and the other bodies, moving without constraint, as they came to rest in their proper places, would fall into the order in which they now stand, the heavy bodies moving towards
but how will our conclusion work out in the case of the circle and the straight line?
e.g. a line or lines if it is a line, a unit or units if it is a unit, a house if it is a house (there is nothing to prevent something of a different kind being between).
, and produce fg to meet it in h. join hb. then because ha and fe are parallels, and hf intersects them, the sum of the angles ahf, hfe is two right angles xxix.
for that which includes all other intelligible creatures cannot have a second or companion; in that case there would be need of another living being which would include both, and of which they would be parts, and the likeness would be more truly said to
i mean, for instance, that you are now in the heavens because you are in the air and it is in the heavens; and you are in the air because you are on the earth; and similarly on the earth because you are in this place which contains no more than you.
and so, if there is a contrary to circular motion, motion in a straight line must be recognized as having the best claim to that name.
there is also an intermediate class, in which plato falls short of the truths of modern science, though he is not wholly unacquainted with them. (one) to the first class belongs the teleological theory of creation.
if then the motion of all things that are in motion is either natural or unnatural and violent, and all things whose motion is violent and unnatural are moved by something, and something other than themselves, and again all things whose motion is natural
the men who, though exceedingly hungry and thirsty, and both equally, yet being equidistant from food and drink, is therefore bound to stay where he is-even so, it still remains to explain why fire stays at the extremities.
in what way, then, has the food been modified by the growing thing? perhaps we should say that it has been 'mixed' with it, as if one were to pour water into wine and the wine were able to convert the new ingredient into wine.
the production (pe) of this in the opposite direction is the least. three. of the others, that which is nearest to the line through the centre is greater than every one more remote. four.
part of one constituent is juxtaposed to a part of the other'? the term, no doubt, is applied in the former sense: we speak, e.g. of wheat having been 'combined' with barley when each grain of the one is juxtaposed to a grain of the other.
when the alterations take the form of becoming white and becoming healthy respectively, here there is no sameness or equality or likeness inasmuch as the difference in the affections at once makes the alterations specifically different, and there is no
a later one in which it is to be found. nor have any traces been discovered hitherto in egyptian monuments of a connexion between greece and egypt older than the eighth or ninth century b.one hundred.
the pyramid), more than any other form, has a dividing power which cuts our bodies into small pieces (kepmatizei), and thus naturally produces that affection which we call heat; and hence the origin of the name (thepmos, kepma).
two. the sum of the squares on any two lines it equal to twice the square on half the sum plus twice the square on half the difference of the lines.
the facts are that fire is always light and moves upward, while earth and all earthy things move downwards or towards the centre.
plato himself proposes the question, why does motion continue at all when the elements are settled in their places?
but if they went on to think of the world as formed on this pattern all round, with a centre identically related to each point on the extremity, they would have to admit that the extremity was above and the centre below.) by absolutely light, then, we
but when all the watery part is suddenly drawn out by fire, a more brittle substance is formed, to which we give the name of pottery.
the warmth and light which proceed from them are caused by the friction set up in the air by their motion.
fifteen. prove by using theorem eleven that if perpendiculars be let fall on the sides and diagonals of a cyclic quadrilateral, from any point in the circumference of the circumscribed circle, the rectangle contained by the perpendiculars on the diagonals
but this is impossible, because, as we have already shown, infinite weight and lightness do not exist. moreover it would be necessary also that their places should be infinite in extent, so that the movements too of all these bodies would be infinite.
we have stated, then, that time exists and what it is, and in how many senses we speak of the 'now', and what 'at some time', 'lately', 'presently' or 'just', 'long ago', and 'suddenly' mean.
cor. two.-the sum of the triangles whose bases are two opposite sides of a parallelogram, and which have any point between these sides as a common vertex, is equal to half the parallelogram.
twenty seven. inscribe in a given triangle a parallelogram whose diagonals shall intersect in a given point.
it is in the second capacity, as a work of permanent value containing truth, that it finds a place among the world's great books. it is in that capacity also that it will be treated in this introduction.
for to say, with any probability or certitude, which of them should be called water rather than fire, and which should be called any of them rather than all or some one of them, is a difficult matter.
there are just four things of which place must be one-the shape, or the matter, or some sort of extension between the bounding surfaces of the containing body, or this boundary itself if it contains no extension over and above the bulk of the body which
is their time then infinite? no, for infinite time has no end, but the movement communicated has.
the most important of the affections which concern the whole body remains to be considered-that is, the cause of pleasure and pain in the perceptions of which i have been speaking, and in all other things which are perceived by sense through the parts of
these too, like most of the other affections, appear to be caused by certain contractions and dilations, but they have besides more of roughness and smoothness than is found in other affections; for whenever earthy particles enter into the small veins
a great body of sound is loud, the opposite is low. discord is produced by the swifter and slower motions of two sounds, and is converted into harmony when the swifter motions begin to pause and are overtaken by the slower.
generally things which come to be, come to be in different ways: (one) by change of shape, as a statue; (two) by addition, as things which grow; (three) by taking away, as the hermes from the stone; (four) by putting together, as a house; (five) by
please. hence, ultimately, the ratio of incommensurable quantities may be regarded as the limit of the ratio of commensurable quantities.
can it be that the incommensurability of two things in respect of any attribute is due to a difference in that which is primarily capable of carrying the attribute?
the vagueness of his language does not allow us to determine whether anything more than this was intended by him.
thirteen. in the construction of prop. xviii. prove that the angle dbc is equal to half the difference of the base angles.
plato's doctrine on this subject is contained in the following words:-'the earth, which is our nurse, compacted (or revolving) around the pole which is extended through the universe, he made to be the guardian and artificer of night and day, first and
the higher of the two, which is the seat of courage and anger, lies nearer to the head, between the midriff and the neck, and assists reason in restraining the desires.
it is therefore impossible that one and the same thing should be capable of always existing and of always not-existing. and 'not always existing', the contradictory, is also excluded.
though we have many substantial houses of brick or stone, the prosperity of the farmer is still measured by the degree to which the barn overshadows the house.
for not only those who postulate a plurality of elements employ their reciprocal action and passion to generate the compounds: those who derive things from a single element are equally compelled to introduce 'acting'.
but if the material of each of these objects has itself the same relation to something else, say bronze (or gold) to water, bones (or wood) to earth and so on, that (they say) would be their nature and essence.
one hundred eleven. two circles described to touch an ordinate of a semicircle, the semicircle itself, and the semicircles on the segments of the diameter, are equal to one another.
dem.-make the same construction as in the last proposition; then the triangles abc, deg are equiangular.
lines can be put together so that contact is linear by laying one along the other, though not by putting them end to end.
observe again, what care the law took in the pursuit of wisdom, searching out the deep things of the world, and applying them to the use of man.
of which tradition tells, under the face of heaven. solon marvelled at his words, and earnestly requested the priests to inform him exactly and in order about these former citizens.
from coming to-be, though it is impossible for them to do so consistently with what they say.
the doctor and the expert are causes of health, the relation two:one and number of the octave), and always what is inclusive to what is particular. another mode of causation is the incidental and its genera, e.g.
nor are there wanting in plato, who was himself too often the victim of them, impressive admonitions that we should regard not words but things (states.).
a body of large particles, they say, is cold because instead of penetrating through the passages it crushes. clearly, then, that which is hot is that which penetrates these passages, or in other words that which has fine particles.
if foundations have come-to-be, must a house come-to-be? the answer seems to be that the necessary nexus no longer holds, unless it is 'necessary' for the consequent (as well as for the antecedent) to come-to-be-'necessary' absolutely.
if a line (ab) be bisected (at one hundred) and divided into two unequal parts (at five hundred), the sum of the squares on the unequal parts (ad, db) is double the sum of the squares on half the line (ac), and on the segment (four hundred) between the
thirteen. how do these areas vary if they have equal bases but unequal altitudes?
having established these distinctions, we may now proceed to the question whether the heaven is ungenerated or generated, indestructible or destructible.
a motion from health to disease and a motion from disease to health: for motions must be contrary to one another in one or more of these ways, as there is no other way in which they can be opposed.
the same spirit is carried yet further by him in the passage already quoted from the laws, in which he affirms their wanderings to be an appearance only, which a little knowledge of mathematics would enable men to correct.
this is just why it seems proper that the whole heaven and every star should be spherical.
let it be agreed, then, both according to strict reason and according to probability, that the pyramid is the solid which is the original element and seed of fire; and let us assign the element which was next in the order of generation to air, and the
dem.-one. take any number of arcs cg, gh in the first circle, each equal to bc. join og, oh, and in the second circle take any number of arcs fi, ij, each equal to ef. join ip, jp.
it is true that a greater force can reside in a lesser magnitude: but the superiority of any such greater force can be still greater if the magnitude in which it resides is greater. now let ab be an infinite magnitude.
to sum the matter up, the cause of this result is obvious, viz. that between any two movements there is a ratio (for they occupy time, and there is a ratio between any two times, so long as both are finite), but there is no ratio of void to full.
such is the case, according to some, with contact and motion, since there is no process of coming to be in contact or in motion.
which was twenty-seven times the first (twenty seven). after this he filled up the double intervals (i.e. between one, two, four, eight) and the triple (i.e.
frequent and serial they suppose its pores to be. such was the theory which some philosophers (induding empedocles) advanced in regard to the structure of certain bodies.
one. if the points o, one hundred be joined, the angle one hundred is bisected. hence "the bisectors of the angles of a triangle are concurrent" (compare i. xxvi., ex. seven).
now of the two triangles, the isosceles has one form only; the scalene or unequal-sided has an infinite number.
expansion of the same matter. there are two types in each case, both in the dense and in the rare; for both the heavy and the hard are thought to be dense, and contrariwise both the light and the soft are rare; and weight and hardness fail to coincide in
fifteen. what is the reason that if two circles touch they cannot have any other common point?
suppose the circle whose centre is o and one of the points of intersection a to remain fixed, while the second circle turns round that point in such a manner that the second point of intersection b becomes ultimately consecutive to a; then, since the line
nine. since the whole triangle abc is made up of the three triangles aob, boc, coa, we see that the rectangle contained by the sum of the three sides, and the radius of the inscribed circle, is equal to twice the area of the triangle.
example: a man is engaged in collecting subscriptions for a feast. he would have gone to such and such a place for the purpose of getting the money, if he had known.
cor.-with the same hypotheses, if l, m, n be any three multipliers, a : b :: la mc ne : lb md nf.
as it is, the shapes which the moon itself each month shows are of every kind straight, gibbous, and concave-but in eclipses the outline is always curved: and, since it is the interposition of the earth that makes the eclipse, the form of this line will
the reason why these names are used, and the circumstances under which they are ordinarily applied by us to the division of the heavens, may be elucidated by the following supposition:-if a person were to stand in that part of the universe which is the
it is as though nature had foreseen the result, that if their movement were other than it is, nothing on this earth could maintain its character.
their future birth and human lot. they were to be sown in the planets, and out of them was to come forth the most religious of animals, which would hereafter be called man.
and there are the same distinctions in the case of the mover: (one) one thing causes motion accidentally, (two) another partially (because something belonging to it causes motion), (three) another of itself directly, as, for instance, the physician heals,
for there is nothing else stronger than it to move it-since that would mean more divine-and it has no defect and lacks none of its proper excellences.
dem.-since a : b :: c : d. we have a b- c d, and multiplying each by b c we get
now if there is such a thing as false opinion or opinion at all, there is also motion; and similarly if there is such a thing as imagination, or if it is the case that anything seems to be different at different times: for imagination and opinion are
twenty five critical objections brought against poetry, and the principles on which they are to be answered. twenty six a general estimate of the comparative worth of epic poetry and tragedy.
cor. two.-the parallelogram bh is equal to af, and bf to hc.
a given weight moves a given distance in a given time; a weight which is as great and more moves the same distance in a less time, the times being in inverse proportion to the weights.
for though he was ignorant of many things which are familiar to us, and often confused in his ideas where we have become clear, we have no right to attribute to him a childish want of reasoning about very simple facts, or an inability to understand the
the cause implied in this solution must no doubt be considered adequate to account for coming-to-be and passing-away in their general character as they occur in all existing things alike.
(one) now it has three dimensions, length, breadth, depth, the dimensions by which all body also is bounded. but the place cannot be body; for if it were there would be two bodies in the same place.
and for this reason the fixed stars were created, to be divine and eternal animals, ever-abiding and revolving after the same manner and on the same spot; and the other stars which reverse their motion and are subject to deviations of this kind, were
if a right line (ef) intersect two parallel right lines (ab, four hundred), it makes-one. the alternate angles (agh,ghd) equal to one another; two. the exterior angle (egb) equal to the corresponding interior angle (ghd); three.
respectively, cf and de pass each through a fixed point.
the air, too, which is thought to be incorporeal, contributes something to the belief: it is not only the boundaries of the vessel which seem to be place, but also what is between them, regarded as empty.
hence ae is equal to cf , and ae is equal to cf; but ab is double of ae, and four hundred double of cf. therefore ab is equal to four hundred.
prop. nineteen.-theorem. similar triangles (abc, def) have their areas to one another in the duplicate ratio of their homologous sides.
again, when the fire goes out of the fusile substance, it does not pass into a vacuum, but into the neighbouring air; and the air which is displaced forces together the liquid and still moveable mass into the place which was occupied by the fire, and
subject to subject, from subject to nonsubject, from non-subject to subject, or from non-subject to non-subject, where by 'subject' i mean what is affirmatively expressed.
by the sinews, which admitted of being stretched and relaxed about the vertebrae, he might thus make the body capable of flexion and extension, while the flesh would serve as a protection against the summer heat and against the winter cold, and also
if that be the case, however, 'a house must come to-be if foundations have come-to-be', as well as vice versa. for the antecedent was assumed to be so related to the consequent that, if the latter is to be, the antecedent must have come-tobe before it.
the uses of the words 'destructible' and 'indestructible' are similar.
again (three) according to the theory all such things are already present in one another and do not come into being but are constituents which are separated out, and a thing receives its designation from its chief constituent.
colour has specific differences: therefore blackening and whitening differ specifically; but at all events every whitening will be specifically the same with every other whitening and every blackening with every other blackening.
, and gc is greater than he, but four hundred and ef are the doubles of gc and he. hence four hundred is greater than ef.
not be tied to an ox, or horse, or cow, or pig, as at present. i desire to speak impartially on this point, and as one not interested in the success or failure of the present economical and social arrangements.
he is still revered or hated as the author who, above all others, inspired the french revolution.
nineteen. upon the segments ac, cb of a line ab equilateral triangles are described: prove that if five hundred, five hundred be the centres of circles described about these triangles, dd ab ac cb .
since they are generated, they must be generated either from something incorporeal or from a body, and if from a body, either from one another or from something else.
iii. ratio is the mutual relation of two magnitudes of the same kind with respect to quantity.
suppose then that infinity belongs to substance as an attribute. but, if so, it cannot, as we have said, be described as a principle, but rather that of which it is an attribute-the air or the even number.
moreover in the hebrew scriptures the creation of the world is described, even more explicitly than in the timaeus, not as a single act, but as a work or process which occupied six days.
when air seemed to go upwards and fire to pierce through air-when water and earth fell downward, they were seeking their native elements.
the truth is that the introduction is a mosaic work of small touches which, partly by their minuteness, and also by their seeming probability, win the confidence of the reader.
hence growth must not be regarded as a process from a matter without magnitude to an actuality of magnitude: for this would be a body's coming-to-be rather than its growth.
but the sinews he made of a mixture of bone and unfermented flesh, giving them a mean nature between the two, and a yellow colour. hence they were more glutinous than flesh, but softer than bone.
again, since ac is equal to ce, ac is equal to ce ; but ae is equal to ac ce i. xlvii. . therefore ae is equal to ac . in like manner ef is equal to gf or cd . therefore ae ef is equal to ac cd ; but ae ef is equal to af i. xlvii. .
but we cannot explain sensation without explaining the nature of flesh and of the mortal soul; and as we cannot treat of both together, in order that we may proceed at once to the sensations we must assume the existence of body and soul.
something else, the motion that it has the capacity for suffering is not that with which it affects what is next to it, but a motion of a different kind; e.g. that which has a capacity for making healthy might as such have a capacity for learn.
rule of identity.-under this name the following principle will be sometimes referred to:-"if there is but one ten and one y , then, from the fact that ten is y , it necessarily follows that y is ten."-syllabus.
clearly, then, for these reasons and also because the attributes mentioned are common to, and coextensive with, all the objects of our science, we must first take each of them in hand and discuss it.
it has been asserted that everything in the world was subject to generation and nothing was ungenerated, but that after being generated some things remained indestructible while the rest were again destroyed.
thus, for all the indifference theory shows to the contrary, earth also would have moved in this manner away from the centre, unless the centre had been its natural place.
similarly the void can bear no ratio to the full, and therefore neither can movement through the one to movement through the other, but if a thing moves through the thickest medium such and such a distance in such and such a time, it moves through the
the passage into fire is 'coming-to-be' unqualified, but 'passingaway-of-something' (e.g. earth): whilst the coming-to-be of earth is qualified (not unqualified) 'coming-to-be', though unqualified 'passing-away' (e.g. of fire).
similarly in all other things which involve production for an end; the product cannot come to be without things which have a necessary nature, but it is not due to these (except as its material); it comes to be for an end.
in the act of dividing the continuous distance into two halves one point is treated as two, since we make it a starting-point and a finishing-point: and this same result is also produced by the act of reckoning halves as well as by the act of dividing
the difficulty must have occurred to every one.
i will give you not only the general heads, but the particulars, as they were told to me. the city and citizens, which you yesterday described to us in fiction, we will now transfer to the world of reality.
for in that case the larger the mass of fire or earth the slower would be the upward or downward movement; but the fact is the reverse: the greater the mass of fire or earth the quicker always is its movement towards its own place.
two preliminary distinctions will prepare us to grasp the cause of growth.
for as in motion 'the first mover' is unmoved, so among the active powers 'the first agent' is unaffected.
and (b) those whose ground of distinction is size will have to recognize an element prior to the element, a regress which continues infinitely, since every body is divisible and that which has the smallest parts is the element.
that our teeth should come up of necessity-the front teeth sharp, fitted for tearing, the molars broad and useful for grinding down the food-since they did not arise for this end, but it was merely a coincident result; and so with all other parts in which
one it is less than two right angles; in fig. two it is equal to two right angles; in fig. three greater than two right angles, but less than four; and in fig. four it is greater than four right angles.
fire, owing to the survival of the hot of the air and the dry of the earth-qualities essentially constitutive of fire. moreover, this mode of fire's coming-to-be is confirmed by perception.
'the father and maker of all this universe is past finding out; and even if we found him, to tell of him to all men would be impossible.' 'let me tell you then why the creator made this world of generation.
the void, they argue, 'is not': but unless there is a void with a separate being of its own, 'what is' cannot be moved-nor again can it be 'many', since there is nothing to keep things apart.
it would be absurd to suppose that the motion of one in a circle and of another in a straight line cannot be similar, but that the one must inevitably move more quickly or more slowly than the other, just as if the course of one were downhill and of the
one. if two circles be concentric, all chords of the greater which touch the lesser are equal.
but the problem of the infinite is difficult: many contradictions result whether we suppose it to exist or not to exist. if it exists, we have still to ask how it exists; as a substance or as the essential attribute of some entity?
and soda, bitter. purgatives of a weaker sort are called salt and, having no bitterness, are rather agreeable. inflammatory bodies, which by their lightness are carried up into the head, cutting all that comes in their way, are termed pungent.
the cause, therefore, of the coming-to-be of the things which owe their existence to nature is that they are in such-and-such a determinate condition: and it is this which constitutes, the 'nature' of each thing-a 'nature' about which he says nothing.
two. if the diameter of a circle be produced to one hundred until the produced part is equal to the radius, the two tangents from one hundred and their chord of contact form an equilateral triangle.
the views taken of it involve arguments both for and against, in much the same sort of way.
moreover, these things also must either have a beginning before which they had no being, or they must be eternal.
if the universal frame had been created a surface only and having no depth, a single mean would have sufficed to bind together itself and the other terms; but now, as the world must be solid, and solid bodies are always compacted not by one mean but by
oo always bisects ab, we see that when b ultimately becomes consecutive to a, the line oo passes through a.
let the magnitudes be a, b, c, d. then the ratio of
dem.-one. if the point of intersection be the centre, each rectangle is equal to the square of the radius. hence they are equal.
six. if ch intersect be in o, ao is perpendicular to ch.
phil.); (three) that even by philolaus and the early pythagoreans, the earth was held to be a body like the other stars revolving in space around the sun or a central fire; (four) that the beginnings of chemistry are discernible in the 'similar particles'
five. if two circles touch externally, their common tangent at either side subtends a right angle at the point of contact, and its square is equal to the rectangle contained by their diameters.
it must be so, for (one) an attribute is predicated of some subject, so that the subject to which 'being' is attributed will not be, as it is something different from 'being'. something, therefore, which is not will be.
eighty four. if a cyclic quadrilateral be such that three of its sides pass through three fixed collinear points, the fourth side passes through a fourth fixed point, collinear with the three given ones.
the sameness or diversity of that and some other thing, and to tell how and when and where individuals are affected or related, whether in the world of change or of essence.
that there might be some visible measure of their relative swiftness and slowness as they proceeded in their eight courses, god lighted a fire, which we now call the sun, in the second from the earth of these orbits, that it might give light to the whole
hence we conclude that the innermost motionless boundary of what contains is place.
the passionate earnestness of parmenides contrasts with the vacuity of the thought which he is revolving in his mind.
secure the rude comforts and assurances of early society. in one of the long notes appended to the discourse, rousseau further explains his position.
is relative this is a necessary consequence: e.g. if one thing is double another when before it was not so, one or other of them, if not both, must have been in process of change.
enough of the nature of man and of the body, and of training and education. the subject is a great one and cannot be adequately treated as an appendage to another.
of air and one of fire. so because an octahedron gives the sides of two pyramids (eight four x two), a particle of air is resolved into two particles of fire.
two. if a were less than five hundred, then five hundred would be greater than a, and the triangles def, abc would have the two sides de, df of one respectively equal to the two sides ab, ac of the other, and the angle five hundred contained by the two
fifteen. construct a triangle, being given-
diminishing, and which ultimately becomes indefinitely small; such that the ratio of the incommensurable quantities is greater than one, and less than the other fraction of each pair. these fractions are called convergents.
thus half the discourse on inequality is occupied by an imaginary description of the state of nature, in which man is shown with ideas limited within the narrowest range, with little need of his fellows, and little care beyond provision for the
dem.-one. let o be the centre. join ob. now since o is the centre, oa is equal to ob: to each add op, and we have ap equal to the sum of ob, op; but the sum of ob, op is greater than pb i. xx. . therefore pa is greater than pb.
join bp; it will be the tangent required.
for this part has the same relation to its place, as a separable part to its whole, as when one moves a part of water or air: so, too, air is related to water, for the one is like matter, the other form-water is the matter of air, air as it were the
if in all things seen there was number and figure, why should they not also pervade the unseen world, with which by their wonderful and unchangeable nature they seemed to hold communion?
but the greek was not, like the enquirer of the last generation, confined to a period of six thousand years; he was able to speculate freely on the effects of infinite ages in the production of physical phenomena.
but perhaps this whole subject will be more suitably discussed on some other occasion.
and since things which are equal to the same are equal to one another, the rectangle ac.cb, together with the square on cb, is equal to the rectangle ab.cb.
twenty two. how many common tangents can two circles have?
if, when mingled with black bile, it disturbs the courses of the head in sleep, there is not so much danger; but if it assails those who are awake, then the attack is far more dangerous, and is called epilepsy or the sacred disease.
in so far as they are products of art-have no innate impulse to change.
forty six. given the base of a triangle and the vertical angle; find the locus of the centre of the circle passing through the centres of the escribed circles.
such an appearance ought not to make them doubt the circular shape of the earth. but they have another argument. they say that because it is at rest, the earth must necessarily have this shape.
if at a point (b) in a right line (ba) two other right lines (cb, bd) on opposite sides make the adjacent angles (cba, abd) together equal to two right angles, these two right lines form one continuous line.
for, since the triangles adc, cdb are equiangular, we have ad : six hundred :: six hundred : db; hence six hundred is a mean proportional between ad, db (def. iii.).
we recommend the student to make himself acquainted with the proofs by this method as well as with those of euclid. he will thus better understand the meaning of each proposition.
equipoise, but swaying unevenly hither and thither, was shaken by them, and by its motion again shook them; and the elements when moved were separated and carried continually, some one way, some another; as, when grain is shaken and winnowed by fans and
the matter comes to be and ceases to be in one sense, while in another it does not. as that which contains the privation, it ceases to be in its own nature, for what ceases to be-the privation-is contained within it.
for which reason, when he was framing the universe, he put intelligence in soul, and soul in body, that he might be the creator of a work which was by nature fairest and best.
dem.-since a : c :: a : c, we have a- c a- c. in like manner, b c b c; therefore, adding, a- -b c a -b c. hence a b : c :: a b : c.
from one thousand seven hundred fifty to the publication of the social contract and emile in one thousand seven hundred sixty two he gradually evolved his views: in those twelve years he made his unique contribution to political thought.
the souls were to be implanted in bodies, which were in a perpetual flux, whence, he said, would arise, first, sensation; secondly, love, which is a mixture of pleasure and pain; thirdly, fear and anger, and the opposite affections: and if they conquered
so place must be looked for among things which have these characteristics.
from it derive the being and life which other things, some more or less articulately but others feebly, enjoy.
another demonstration of this proposition may be got by producing the lines ha and dg to meet in i. then i. xliii. the points i, one hundred, f are collinear, and the proposition is evident.
but there are many others who would agree that it is wrong to give the earth the central position, looking for confirmation rather to theory than to the facts of observation.
in like manner cae is right. hence bae cae, which is impossible.
fluctuate between the two and have the capacity of being sometimes in motion and sometimes at rest.
but that which is smaller yields to the impulse given by us towards the dissimilar element more easily than the larger; and so we call the former light, and the place towards which it is impelled we call above, and the contrary state and place we call
dem.-since the parallelograms pc, pd are about the same diagonal, they are similar xxiv. ; but pc is similar to kpt (const.); therefore pd is similar to kn, and (const.) their homologous sides, pi and kl, are equal; hence xx. pd is equal to kn.
will have been educated by you, and introduce them to you as the lost athenian citizens of whom the egyptian record spoke. as the law of solon prescribes, we will bring them into court and acknowledge their claims to citizenship.
twenty. name the special nine points through which it passes.
and having been created in this way, the world has been framed in the likeness of that which is apprehended by reason and mind and is unchangeable, and must therefore of necessity, if this is admitted, be a copy of something.
but in the infinite body they cannot exist. in general, if it is impossible that there should be an infinite place, and if every body is in place, there cannot be an infinite body.
as for those genealogies of yours which you just now recounted to us, solon, they are no better than the tales of children.
then k will belong to all four, e w a and f: for they are transformed into one another.
prop. one.-problem. to find the centre of a given circle (adb).
we may point out that, even if it is really the case, as certain persons assert, that the existent is infinite and motionless, it certainly does not appear to be so if we follow sense-perception: many things that exist appear to be in motion.
fourteen. how many points of intersection can two circles have?
five. what three lines in prop. xiv. are concurrent?
where it happens that the genus is at the same time a species, it is clear that the motion will then in a sense be one specifically though not in an unqualified sense: learning is an example of this, knowledge being on the one hand a species of
another thing is plain: since the vessel is no part of what is in it (what contains in the strict sense is different from what is contained), place could not be either the matter or the form of the thing contained, but must different-for the latter, both
six. construct a triangle, being given the middle points of its three sides.
successors of plato,-for the elucidation of it.
hence adc, cdb are each similar to acd, and therefore they are similar to one another.
or shall we in the first place deny that things are always commensurable if the same terms are applied to them without equivocation?
another proof may be added: since every body must diminish in size when something is taken from it, and flesh is quantitatively definite in respect both of greatness and smallness, it is clear that from the minimum quantity of flesh no body can be
particles', nothing of the kind will happen.
but these conditions are fulfilled by the primary contraries, which are not derived from anything else because they are primary, nor from each other because they are contraries.
one. what is the reason for stating in the enunciation that the sum of every two of the given lines must be greater than the third?
empedocles, indeed, seems to contradict his own statements as well as the observed facts.
or (two) can there be motion of motion in the sense that some other subject changes from a change to another mode of being, as e.g. a man changes from falling ill to getting well? even this is possible only in an accidental sense.
but as time is most usually supposed to be (three) motion and a kind of change, we must consider this view.
these people, then, have not reached even the threshold of the problem, but rather those who say that the void exists.
there are four of them: one of gods, another of birds, a third of fishes, and a fourth of animals. the gods were made in the form of a circle, which is the most perfect figure and the figure of the universe.
four. if a parallelogram be circumscribed to a circle it must be a lozenge, and its diagonals intersect in the centre.
r rr , r rr , r rr , r rr , hence one thousand two hundred forty eight r . and it may be proved in the same manner that
a further consideration is that if time is similarly constituted, there would be, or might be, a time at which it was done away with. for the indivisible now is like a point in a line.
prop. eleven.-problem. to inscribe a regular pentagon in a given circle (abcde).
supposition, for there are have selected four kinds-three from the one which has the sides unequal; the fourth alone is framed out of the isosceles triangle.
the form; for this is the end or 'that for the sake of which'. hence since nature is for the sake of something, we must know this cause also.
two. the internal and the external bisectors of the vertical angle of a triangle inscribed in a circle meet the circumference again in points equidistant from the extremities of the base.
let them meet in f. through f draw fg parallel to ab, and produce ec to meet it in g. join af.
twisted by them in all sorts of ways, and the circles were broken and disordered in every possible manner, so that when they moved they were tumbling to pieces, and moved irrationally, at one time in a reverse direction, and then again obliquely, and then
five. the sum of the squares on the sides of a parallelogram is equal to the sum of the squares on its diagonals.
a secant or transversal is a line which cuts a system of lines, a circle, or any other geometrical figure.
now since ae is equal to ab, the angle abe is equal to aeb i. v. ; but since be intersects the parallels ae, cf, the angle aeb is equal to cgb i. xxix. . hence the angle cbg is equal to cgb, and therefore i. vi.
is the corner of the other inscribed parallelogram.
again, a place in which a thing rests or to which it moves unnaturally, must be the natural place for some other body, as experience shows. necessarily, therefore, not everything possesses weight or lightness, but some things do and some do not.
inflammations of the body come from burnings and inflamings, and all of them originate in bile.
it can be proved by this proposition that two lines which meet at infinity are parallel.
observation.-proposition xxvi., being the converse of xxiv., has evidently been misplaced. the following would be a simpler enunciation:-"if two homothetic parallelograms have a common angle, they are about the same diagonal."
similar distinctions can be made in the things of which the causes are causes, e.g. of 'this statue' or of 'statue' or of 'image' generally, of 'this bronze' or of 'bronze' or of 'material' generally. so too with the incidental attributes.
that we, learning them and partaking of the natural truth of reason, might imitate the absolutely unerring courses of god and regulate our own vagaries.
we have three things, the movent, the moved, and thirdly that in which the motion takes place, namely the time: and these are either all infinite or all finite or partly-that is to say two of them or one of them-finite and partly infinite.
dem.-if possible let the intersecting line be out of the plane, as egf. and in the plane, of the parallels draw i. post. ii. the right line ehf. then we have two right lines egf, ehf, enclosing a space, which i. axiom x. is impossible.
generally that that which cannot come to be should be in process of coming to be, it follows that it is inconceivable that that which cannot complete a change should be in process of changing to that to which it cannot complete a change.
for the external elements which surround us are always causing us to consume away, and distributing and sending off like to like; the particles of blood, too, which are divided and contained within the frame of the animal as in a sort of heaven, are
on the other hand there is no kind or degree of absurdity or fancy in which the more foolish writers, both of antiquity and of modern times, have not indulged respecting it.
nor need we discuss at length how far plato agrees in the later jewish idea of creation, according to which god made the world out of nothing. for his original conception of matter as something which has no qualities is really a negation.
but, as we said before, melodious and poetical as the theory is, it cannot be a true account of the facts.
lastly, there remain two points in which he seems to touch great discoveries of modern times-the law of gravitation, and the circulation of the blood.
'suddenly' refers to what has departed from its former condition in a time imperceptible because of its smallness; but it is the nature of all change to alter things from their former condition.
dem.-let us conceive the triangle bac to be applied to edf, so that the point a shall coincide with five hundred, and the line ab with de, and that the point one hundred shall be on the same side of de as f; then because ab is equal to de, the point b
one thousand. martin may also be gently censured for citing without sufficient discrimination ancient authors having very different degrees of authority and value.
contact clearly there is no natural junction either. hence, if as some say 'point' and 'unit' have an independent existence of their own, it is impossible for the two to be identical: for points can touch while units can only be in succession.
it is evident, then, from the preceding argument that alteration and being altered occur in sensible things and in the sensitive part of the soul, and, except accidentally, in nothing else.
in space). but true reason assures us that while two things (i.e. the idea and the image) are different they cannot inhere in one another, so as to be one and two at the same time.
v. the rectangle contained by two separate lines such as ab and four hundred is the parallelogram formed by erecting a perpendicular to ab, at a, equal to four hundred, and drawing parallels: the area of the rectangle will be ab.four hundred.
of the others, that which is nearer to the minimum is less than one more remote. five.
therefore the whole angle bdc is greater than bac.
nevertheless, in all changing things alike, we speak of 'coming-to-be' when the thing comes-to-be something in one of the two columns-e.g.
four. if two diameters of two touching circles be parallel, the lines from the point of contact to the extremities of one diameter pass through the extremities of the other.
for there are many different ways in which the movement or rest of the earth has been conceived.
since in every case knowledge depends on what is primary, and the elements are the primary constituents of bodies, we must ask which of such bodies are elements, and why; and after that what is their number and character.
ratios of number; (seven) that mathematical laws pervaded the world; and even qualitative differences were supposed to have their origin in number and figure; (eight) the annihilation of matter was denied by several of them, and the seeming disappearance
one. did plato derive the legend of atlantis from an egyptian source?
if then in the case of the earth, supposed to be infinite, it is at rest, not because it is infinite, but because it has weight and what is heavy rests at the centre and the earth is at the centre, similarly the infinite also would rest in itself, not
in view of these facts we should naturally expect to find difficulty in determining what place is, if indeed it is one of these two things, matter or form. they demand a very close scrutiny, especially as it is not easy to recognize them apart.
twenty three. give one enunciation that will include propositions xxii., xxiii. of book five.
natural to it. hence, too, in learning and in forming judgements on matters relating to their sense-perceptions children are inferior to adults owing to the great amount of restlessness and motion in their souls.
ten. the rectangle contained by the side of an escribed square standing on the base of a triangle, and the difference between the base and altitude, is equal to twice the area of the triangle.
those who generate all things out of one thing) are bound to assert that coming-to-be is 'alteration', and that whatever 'comes-to-be' in the proper sense of the term is 'being altered': but those who make the matter of things more than one must
sometimes also moisture may remain, and the earth which has been fused by fire becomes, when cool, a certain stone of a black colour.
by the 'opposite' i mean the 'unmusical', by the 'subject' 'man', and similarly i call the absence of shape or form or order the 'opposite', and the bronze or stone or gold the 'subject'.
again, since cb is equal to be, the rectangle ab.cb is equal to the rectangle ab.be; but the rectangle ab.be is equal to the figure ae. hence the rectangle ab.cb is equal to the figure ae.
they are all parts of time, and the past and future are created species of time, which we unconsciously but wrongly transfer to the eternal essence; for we say that he 'was,' he 'is,' he 'will be,' but the truth is that 'is' alone is properly attributed
accordingly, we must segregate the tangible differences and contrarieties, and distinguish which amongst them are primary.
four. show how to prove this proposition by assuming as an axiom that every angle has a bisector.
raising the balance, were to draw the fire by force towards the uncongenial element of the air, it would be very evident that he could compel the smaller mass more readily than the larger; for when two things are simultaneously raised by one and the same
which we began, and then endeavour to add on a suitable ending to the beginning of our tale.
for everything that partakes of life may be truly called a living being, and the animal of which we are now speaking partakes of the third kind of soul, which is said to be seated between the midriff and the navel, having no part in opinion or reason or
thirty three. construct the least triangle equiangular to a given one whose vertices shall lie on three given lines.
two. if ab does not pass through the centre o, let fall the perpendicular oc on ab. join ot, ob, op. then because oc, a line through the centre, cuts ab, which does not pass through the centre at right angles, it bisects it iii. .
all, however, agree in this, that they differentiate their one by means of the contraries, such as density and rarity and more and less, which may of course be generalized, as has already been said into excess and defect.
the point g must be within the circle, and consequently the line bg produced must meet the circle again, and must therefore cut it.
we may now say that our discourse about the nature of the universe has an end.
three. if two lines intersect, how many pairs of supplemental angles do they make?
there is one protection against both kinds of disproportion:-that we should not move the body without the soul or the soul without the body, and thus they will be on their guard against each other, and be healthy and well balanced.
further, the word 'relative' is used with reference to (one) excess and defect, (two) agent and patient and generally what can move and what can be moved. for 'what can cause movement' is relative to 'what can be moved', and vice versa.
the distinction may be seen most clearly in the case of a revolving sphere, in which the velocities of the parts near the centre and of those on the surface are different from one another and from that of the whole; this implies that there is not one
the generations of men, like the leaves of the forest, come and go, but the mathematical laws by which the world is governed remain, and seem as if they could never change.
all the objections which may be urged against kant's doctrine of the ideality of space and time at once press upon us.
having reflected on the singular numerical phenomena of the existence of one mean proportional between two square numbers are rather perhaps only between the two lowest squares; and of two mean proportionals between two cubes, perhaps again confining his
and next we have to determine what are the four most beautiful bodies which are unlike one another, and of which some are capable of resolution into one another; for having discovered thus much, we shall know the true origin of earth and fire and of the
since, then, there is nothing-at least, nothing perceptible-prior to these, they must be all.
cor. one.-the sum of the three interior angles of the triangle bcf is equal to the sum of the three interior angles of the triangle abc.
now that which is created is of necessity corporeal, and also visible and tangible. and nothing is visible where there is no fire, or tangible which has no solidity, and nothing is solid without earth.
the religion and civilization which are barbaric and heathenish build splendid temples; but what you might call christianity does not. most of the stone a nation hammers goes toward its tomb only. it buries itself alive.
so some, like lycophron, were led to omit 'is', others to change the mode of expression and say 'the man has been whitened' instead of 'is white', and 'walks' instead of 'is walking', for fear that if they added the word 'is' they should be making the one
dem.-since a : b :: c : d, we have ab cd. hence, multiplying each by m n- we get manb- mcnd-.
and since everything that is in motion is in motion in a certain sphere and for a certain time and has a motion belonging to it, it follows that the time, the motion, the being-in-motion, the thing that is in motion, and the sphere of the motion must all
but we must see how this can be arrived at as a reasoned result, as well as in the way just indicated.
and this latter proposition is a limiting case of the theorem given in proposition iii., cor. four, that "the line joining the centres of two intersecting circles bisects the common chord perpendicularly."
for, whereas the platonists argue that there must be atomic magnitudes 'because otherwise "the triangle" will be more than one', democritus would appear to have been convinced by arguments appropriate to the subject, i.e. drawn from the science of nature.
two. what is geometric magnitude? ans. that which has extension in space.
out of irresistible impressions; in the second place, they must have love, in which pleasure and pain mingle; also fear and anger, and the feelings which are akin or opposite to them; if they conquered these they would live righteously, and if they were
the greater the abstraction the greater the truth. behind any pair of ideas a new idea which comprehended them-the (greek), as it was technically termed-began at once to appear. two are truer than three, one than two.
the water in a cup, when the cup is given a circular motion, though it is often underneath the bronze, is for this same reason prevented from moving with the downward movement which is natural to it.) but suppose both the 'whirl' and its flatness (the air
therefore the sum of the two remaining angles aob, amb is two right angles; and i. xiii. the sum of the two angles gef, fed is two right angles. therefore the sum of aob, amb is equal to the sum of gef, fed; but aob is equal to gef (const.).
but what does come-to-be in growth is flesh or bone-or a hand or arm (i.e. the tissues of these organic parts). such things come-to-be, then, by the accession not of quantified-flesh but of a quantified-something.
there is another sort of fire which is intermediate, and which reaches and mingles with the moisture of the eye without flashing; and in this, the fire mingling with the ray of the moisture, produces a colour like blood, to which we give the name of red.
which'. the last three often coincide; for the 'what' and 'that for the sake of which' are one, while the primary source of motion is the same in species as these (for man generates man), and so too, in general, are all things which cause movement by
one might as well postulate that a knife or a saw divides things into knives or saws. it is also ridiculous to think only of division when allotting fire its shape. fire is generally thought of as combining and connecting rather than as separating.
cor. two.-bc is a mean proportional between ab, bd; and ac between ab, ad.
but the gods, foreknowing that the palpitation of the heart in the expectation of danger and the swelling and excitement of passion was caused by fire, formed and implanted as a supporter to the heart the lung, which was, in the first place, soft and
our present position, then, is this: we have argued that there always was motion and always will be motion throughout all time, and we have explained what is the first principle of this eternal motion: we have explained further which is the primary motion
cg is equal to cb; but cg is equal to bi and cb to gi. hence the figure cbig is a lozenge, and the angle cbi is right. hence (i., def. xxx.) it is a square. in like manner the figure efgh is a square.
when 'both fire and water come-to-be out of any and every part of flesh'.
the fruits or herbs which are our daily sustenance take all sorts of colours when intermixed, but the colour of red or fire predominates, and hence the liquid which we call blood is red, being the nurturing principle of the body, whence all parts are
further, the definition of the whole is not contained in the definitions of the contents or elements of the definitory formula; that of 'man' for instance in 'biped', or that of 'white man' in 'white'.
the description depends on a certain similarity, but it is not true in the full sense of the word. this condition alone is not sufficient: it is necessary also that the next part which is taken should never be the same.
twenty seven. the four circles circumscribing the four triangles formed by any four lines pass through a common point.
for there were no days and nights and months and years before the heaven was created, but when he constructed the heaven he created them also.
forty one. given the base of a triangle in magnitude and position and the sum of the sides; prove that the perpendicular at either extremity of the base to the adjacent side, and the external bisector of the vertical angle, meet on a given line
its opposite rest of that kind, e.g. local motion has local rest. this statement, however, needs further qualification: there remains the question, is the opposite of remaining at a particular place motion from or motion to that place?
twenty seven. if , be the radii of two circles, touching each other at the centre of the inscribed circle of a triangle, and each touching the circumscribed circle, prove
motion in respect of quality let us call alteration, a general designation that is used to include both contraries: and by quality i do not here mean a property of substance (in that sense that which constitutes a specific distinction is a quality) but a
so too, in the same manner, earth will result from water and fire from earth, since the two 'elements' in both these couples have interchangeable 'complementary factors'.
the race of birds was created out of innocent, light-minded men, who thought to pursue the study of the heavens by sight; these were transformed into birds, and grew feathers instead of hair.
a corollary is an inference or deduction from a proposition.
kind intermediate between them-nothing that is continuous can be composed 'of indivisibles': e.g. a line cannot be composed of points, the line being continuous and the point indivisible.
it is also clear from what has been said why the number of what we call simple bodies cannot be greater than it is.
for the actually-hot is potentially-cold and the actually cold potentially-hot; so that hot and cold, unless they are equally balanced, are transformed into one another (and all the other contraries behave in a similar way).
further, it is a law of nature that earth and all other bodies should remain in their proper places and be moved from them only by violence: from the fact then that some of them are in their proper places it follows that in respect of place also all
the violence of controversy, or the earnestness of enquiry, will often generate inflammations and rheums which are not understood, or assigned to their true cause by the professors of medicine.
cor. seven.-if one angle of a triangle be equal to the sum of the other two, it is a right angle.
the same end, if there is no impediment.
two apb in p, but it touches it (hyp.) also in p, which is impossible. hence the centre of the smaller circle cpd must be in the line op.
dem.-let ab, four hundred meet the plane ten at the points b, five hundred. join bd, and in the plane ten draw de at right angles to bd; take any point e in de. join be, ae, ad. then because ab is normal to ten, the angle abe is right.
we must conceive the same account to hold of action and passion as that which is true of 'being moved' and 'imparting motion'. for the 'mover', like the 'agent', has two meanings.
the part of them worthy of the name immortal, which is called divine and is the guiding principle of those who are willing to follow justice and you-of that divine part one will myself sow the seed, and having made a beginning, i will hand the work over
we must first distinguish the senses in which we use the words 'ungenerated' and 'generated', 'destructible' and 'indestructible'.
for (two) it would be absurd that its contrary motion should be that from disease to health, since motion to that in which a thing is at rest is rather a coming to rest, the coming to rest being found to come into being simultaneously with the motion; and
prop. thirty three.-problem. on a given right line (ab) to describe a segment of a circle which shall contain an angle equal to a given rectilineal angle (ten).
his distinction between the world of order, to which the sun and moon and the stars belong, and the world of disorder, which lies in the region between the moon and the earth, approximates to plato's sphere of the same and of the other.
if people say that the void must exist, as being necessary if there is to be movement, what rather turns out to be the case, if one the matter, is the opposite, that not a single thing can be moved if there is a void; for as with those who for a like
man thus not only works for the animal within him, but, for a symbol of this, he works for the animal without him.
one. if the lines af, bf be joined, the figure acbf is a lozenge.
nurse of the universe, will not allow the body ever to be inactive, but is always producing motions and agitations through its whole extent, which form the natural defence against other motions both internal and external, and by moderate exercise reduces
if one circle (abc) touch another circle (ade) internally in any point (a), it is not concentric with it.
have seen the immobility of the earth to be inconsistent with the rotation of the axis. and (b) what proof is there that the axis of the world revolves at all?
c., are of such and such a kind, therefore certain things necessarily are and come to be-and if they mention any other cause (one his 'friendship and strife', another his 'mind'), it is only to touch on it, and then good-bye to it.
questions for examination on props. thirteen., fourteen., fifteen.
one. the line that bisects the vertical angle of an isosceles triangle bisects the base perpendicularly.
quality and of a certain quantity, but the one is an accidental, not an essential property of the other).
all these bodies became living creatures, and learnt their appointed tasks, and began to move, the nearer more swiftly, the remoter more slowly, according to the diagonal movement of the other.
though one thing that is unmoved moves one thing while another moves another, nevertheless there is something that comprehends them all, and that as something apart from each one of them, and this it is that is the cause of the fact that some things are
twenty. how is a proposition proved indirectly? ans. by proving that its contradictory is false.
water can come-to-be out of fire and fire out of water; for their substralum is something common to them both. but flesh too, presumably, and marrow come-to-be out of them. how, then, do such things come to-be?
fire is the efficient cause of the coming-to-be of fire or one man of the birth of another), or (ii) an actuality.
if two magnitudes of the same kind (a, b) have to a third magnitude (c) ratios equal to those which two other magnitudes (a, b) have to a third (c), then the sum (a b) of the first two has the same ratio to their third (c) which the sum (a b) of the other
let a, b, c, d be the four magnitudes, and m and n the multiples taken, it is required to prove, that if ma be greater than nb, but mc not greater than nd, that the ratio a : b is greater than the ratio c : d.
if two triangles (abc, def) have one angle (a) in one equal to one angle (five hundred) in the other, and the sides about these angles proportional (ba : ac :: ed : df), the triangles are equiangular, and have those angles equal which are opposite to the
for a material, whose constitutive differences signify more a 'this somewhat', is itself more 'substantial' or 'real': while a material, whose constitutive differences signify privation, is 'not real'. (suppose, e.g.
particular not-being. to say that all things will be one, if there is nothing besides being itself, is absurd. for who understands 'being itself' to be anything but a particular substance?
(one) some, as the pythagoreans and plato, make the infinite a principle in the sense of a self-subsistent substance, and not as a mere attribute of some other thing.
particular things, while others move other things.
twenty one. when a line is divided harmonically, what are corresponding pairs of points called? ans. harmonic conjugates.
substitutes beside those which i have named. "for," as the forefathers sang,-
of these pairs the one set are causes in the sense of substratum, e.g. the parts, the other set in the sense of essence-the whole and the combination and the form.
they grow as he grows; they are a kind of composition with which his own philosophy is overlaid.
those things, then, which do not signify substance, are not said to 'come-to-be' without qualification, but only to 'come-to-be-so-and-so'.
if two lines (bd, four hundred) be drawn to a point (five hundred) within a triangle from the extremities of its base (bc), their sum is less than the sum of the remaining sides (ba, ca), but they contain a greater angle.
therefore animals are not always in continuous motion by their own agency: it is something else that moves them, itself being in motion and changing as it comes into relation with each several thing that moves itself.
he would have tried to construct the universe on a quantitative principle, seeming to find in endless combinations of geometrical figures or in the infinite variety of their sizes a sufficient account of the multiplicity of phenomena.
continuous with one another are one and are in contact.
now it is all-important that the beginning of everything should be according to nature.
it has no share in mind and reason. for the authors of our being, remembering the command of their father when he bade them create the human race as good as they could, that they might correct our inferior parts and make them to attain a measure of truth,
and god gave hair to the head of man to be a light covering, so that it might not interfere with his perceptions.
if, on the other hand, the movement of the rotating bodies about the centre is unnatural, it would be remarkable and indeed quite inconceivable that this movement alone should be continuous and eternal, being nevertheless contrary to nature.
both are lighter than earth-for any portion of either rises to the surface of it-but heavier than fire, since a portion of either, whatever its quantity, sinks to the bottom of fire; compared together, however, the one has absolute weight, the other
neither of them is absolutely either light or heavy.
four. if the lines be drawn to any point in the bisector of the external vertical angle, their sum is greater than the sum of the sides.
dem.-by a construction similar to the last, we see that the triangles are the halves of parallelograms, on equal bases, and between the same parallels. hence they are the halves of equal parallelograms xxxvi. . therefore they are equal to one another.
of the 'combinable' from that which is such as to come-to-be and pass-away. as soon, therefore, as these distinctions are clear, the difficulties raised by the argument would be solved.
two. right lines that are equal and parallel have equal projections on any other right line; and conversely, parallel right lines that have equal projections on another right line are equal.
thus water forces its way to the ends of the radii, and there only will it rest: but the line which connects the extremities of the radii is circular: therefore the surface of the water bec is spherical.
and though of a motion that is not specifically one (even if the time is unintermittent) the time is one, the motion is specifically different, and so cannot really be one, for motion that is one must be specifically one, though motion that is
thirteen. if the point five hundred be taken in ab produced, so that mad ndb, then
three. if the base of a triangle be given, and the sum of the sides, the rectangle contained by the perpendiculars from the extremities of the base on the external bisector of the vertical angle is given.
forty four. if concurrent lines drawn from the angles of a polygon of an odd number of sides divide the opposite sides each into two segments, the product of one set of alternate segments is equal to the product of the other set.
now, since cbe is a right line, and ba stands on it, the sum of the angles cba, abe is two right angles (xiii.); and the sum of the angles cba, abd is two right angles (hyp.); therefore the sum of the angles cba, abe is equal to the sum of the angles cba,
eighty one. ce, df are perpendiculars to the diameter of a semicircle, and two circles are described touching ce, de, and the semicircle, one internally and the other externally; the rectangle contained by the perpendiculars from their centres on ab is
cannot be overcome by persuasion, but the other can: and lastly, every man may be said to share in true opinion, but mind is the attribute of the gods and of very few men.
the theory which is before us would agree. therefore action for an end is present in things which come to be and are by nature.
and, fearing to pollute the divine element, they gave the mortal soul a separate habitation in the breast, parted off from the head by a narrow isthmus.
dem.-take any point b in the alternate segment. join ba, bc. then the angle dac is equal to abc xxxii. ; but dac is equal to ten (const.). therefore the angle abc is equal to ten.
marrow and nurtured on milk. now when the triangles out of which meats and drinks are composed come in from without, and are comprehended in the body, being older and weaker than the triangles already there, the frame of the body gets the better of them
it has also been proved that the speed with which they come-to-be, one out of another, is not uniform-since the process of reciprocal transformation is relatively quick between the 'elements' with a 'complementary factor', but relatively slow between
on all these grounds, therefore, we may infer with confidence that there is something beyond the bodies that are about us on this earth, different and separate from them; and that the superior glory of its nature is proportionate to its distance from this
some say that it is 'mutual replacement': but we must recognize that the difficulty raised cannot be solved otherwise than in the way we have described.
(a) we must consider the former alternative; for there are some people who make this the infinite, and not air or water, in order that the other elements may not be annihilated by the element which is infinite.
prop. nine.-problem. to bisect a given rectilineal angle (bac).
for not every 'mover' can 'act', if (a) the term 'agent' is to be used in contrast to 'patient' and (b) 'patient' is to be applied only to those things whose motion is a 'qualitative affection'-i.e.
and the latter is supposed to be placed, first horizontally, and then vertically.). or if the mirror be turned vertically, then the concavity makes the countenance appear to be all upside down, and the lower rays are driven upwards and the upper downwards.
those, on the other hand, who do not make their coming-to-be reciprocal-who refuse to suppose that any one of the 'elements' comes-to-be out of any other taken singly, except in the sense in which bricks come-to-be out of a wall-are faced with a paradox.
, because they are on the same base eb, and between the same parallels eb and cg. therefore the parallelogram eg is equal to the triangle abc, and it has (const.) the angle bef equal to five hundred.
the 'simple' bodies, since they are four, fall into two pairs which belong to the two regions, each to each: for fire and air are forms of the body moving towards the 'limit', while earth and water are forms of the body which moves towards the 'centre'.
two. if the line which bisects the vertical angle of a triangle also bisects the base, the triangle is isosceles.
again, 'one' itself, no less than 'being', is used in many senses, so we must consider in what sense the word is used when it is said that the all is one.
and the angle epj, and we have proved that, according as the multiple of the first is greater than, equal to, or less than the multiple of the second, the multiple of the third is greater than, equal to, or less than the multiple of the fourth.
it would simplify problems xliv., xlv., if they were stated as the constructing of rectangles, and in this special form they would be better understood by the student, since rectangles are the simplest areas to which others are referred.
prop. fourteen.-problem. to describe a circle about a regular pentagon (abcde).
martin, we cannot for a moment suppose that the tale was told to solon by an egyptian priest, nor can we believe that solon wrote a poem upon the theme which was thus suggested to him-a poem which disappeared in antiquity; or that the island of atlantis
for if the matter were like any of the supervening forms, then whenever any opposite or entirely different nature was stamped upon its surface, it would take the impression badly, because it would intrude its own shape.
there is no reason why there should not be one or more intermediates between the contraries, as in the case of colour; for 'intermediate' and 'mean' are capable of more than one application.
in this manner, then, according to them, the genealogy of these gods is to be received and set forth.
if two magnitudes of the same kind (a, b) be the same multiples of another (c) which two corresponding magnitudes (a, b) are of another (c), then the sum of the two first is the same multiple of their submultiple which the sum of their corresponding
since then change differs from motion (motion being change from a particular subject to a particular subject), it follows that contrary motions are motions respectively from a contrary to the opposite contrary and from the latter to the former, e.g.
us. since, therefore, this effect is evidently not produced, it follows that none of them can move with the motion either of animate nature or of constraint.
moreover, no weight can consist of parts not possessing weight. for how, except by the merest fiction, can they specify the number and character of the parts which will produce weight?
but things, as they say, appear different from one another and receive different names according to the nature of the particles which are numerically predominant among the innumerable constituents of the mixture.
and even if the motion round a circle is the contrary of the reverse motion, one of the two would be ineffective: for both move to the same point, because that which moves in a circle, at whatever point it begins, must necessarily pass through all the
thirty one. place a given triangle so that its three vertices shall lie on three given lines.
inanimate things that are (whether the whole or some part is in question) not in motion but at rest, are at some moment set in motion: whereas, if motion cannot have a becoming before which it had no being, these things ought to be either always or never
the white phlegm, though dangerous when detained within by reason of the air-bubbles, yet if it can communicate with the outside air, is less severe, and only discolours the body, generating leprous eruptions and similar diseases.
for there would have to be another place infinite like itself to be the goal of its natural movement and another, equally great, for the goal of its unnatural movement.
for nothing, they say, is purely and entirely white or black or sweet, bone or flesh, but the nature of a thing is held to be that of which it contains the most.
nine. if the three sides of one triangle be respectively perpendicular to those of another triangle, the triangles are equiangular.
if it is passing through and has completed its passage at the same moment, then that which is walking will at the moment when it is walking have completed its walk and will be in the place to which it is walking; that is to say, it will have completed its
nor in reference to contradictory change shall we find anything unanswerable in the argument that if a thing is changing from not-white, say, to white, and is in neither condition, then it will be neither white nor not-white: for the fact that it is not
parts the excess of air forcing its way through the veins distorts them and decomposing the body is enclosed in the midst of it and occupies the midriff; thus numberless painful diseases are produced, accompanied by copious sweats.
but strictly it is not the cause-without qualification-of anything; for instance, a housebuilder is the cause of a house; incidentally, a fluteplayer may be so.
one. a ratio of greater inequality is increased by diminishing its terms by the same quantity, and diminished by increasing its terms by the same quantity.
beneath being withdrawn) cease to prevent the earth's motion, where will the earth move to then? its movement to the centre was constrained, and its rest at the centre is due to constraint; but there must be some motion which is natural to it.
to what end, pray, is so much stone hammered? in arcadia, when i was there, i did not see any hammering stone. nations are possessed with an insane ambition to perpetuate the memory of themselves by the amount of hammered stone they leave.
the soul, which is compounded of the same, the other, and the essence, is diffused from the centre to the circumference of the heavens.
the conception of the world as a whole, a person, an animal, has been the source of hasty generalizations; yet this general grasp of nature led also to a spirit of comprehensiveness in early philosophy, which has not increased, but rather diminished, as
whether the motion is derived from what is in motion but moves itself and stops its own motion, on both suppositions we have the result that in all cases of things being in motion that which primarily imparts motion is unmoved.
ten. the square on the perpendicular from any point in the circumference of a circle, on the chord of contact of two tangents, is equal to the rectangle of the perpendiculars from the same point on the tangents.
eleven. if from any point p in the circumference of a circle a perpendicular be drawn to any chord, its square is equal to the rectangle contained by the perpendiculars from the extremities of the chord on the tangent at p.
again, the speed of the movement would not increase towards the end if it were due to constraint or extrusion; for a constrained movement always diminishes in speed as the source of constraint becomes more distant, and a body moves without constraint to
thirty. state the corresponding theorem for the sum of the squares of the lines drawn from any point in the circumference of any concentric circle.
two, three the middle points of ei, eh, ef are collinear, but xxxiv., ex. one the middle points of ei, eh are the middle points of ac, bd. hence the middle points of ac, bd, ef are collinear.
the world as a whole, therefore, includes all its appropriate matter, which is, as we saw, natural perceptible body.
sol.-take any two points a, b in the circumference. join ab. bisect it in one hundred. erect four hundred at right angles to ab. produce six hundred to meet the circle again in e. bisect de in f. then f is the centre.
that is why, as we say, there is no principle of this, but it is this which is held to be the principle of other things, and to encompass all and to steer all, as those assert who do not recognize, alongside the infinite, other causes, such as mind or
water and air are, and are generated 'from' each other, but not in the way in which bricks come 'from' a house and again a house 'from' bricks; and it is better to assume a smaller and finite number of principles, as empedocles does.
their spreading out upon the earth flowing.
it is clear, too, from these considerations that all the problems which were raised about place will be solved when it is explained in this way:
the fibres; and so congealing and made to cool, it produces internal cold and shuddering.
prop. five.-theorem. if two circles (abc, abd) cut one another in any point (a), they are not concentric.
even if it is true as true can be that the whole is not moved (for what is fixed by itself and is in itself must be immovable), yet we must explain why it is not its nature to be moved. it is not enough just to make this statement and then decamp.
; but the angle bae is equal to bcd (hyp.); and since bao is half bae (const.), bco is half bcd. hence co bisects the angle bcd. in like manner it may be proved that do bisects cde, and eo the angle dea.
cor. one.-the line which bisects any chord of a circle perpendicularly passes through the centre of the circle.
this point may also be proved in another way-by taking a finite magnitude in which there resides a force the same in kind as that which resides in the infinite magnitude, so that this force will be a measure of the finite force residing in the infinite
and when the rest fell off from her, being compelled to stand alone, after having undergone the very extremity of danger, she defeated and triumphed over the invaders, and preserved from slavery those who were not yet subjugated, and generously liberated
but is it not only when an equal motion is accomplished by two things in an equal time that the velocities of the two are equal? now an affection cannot be equal to a length.
one. the diagonals of a lozenge bisect each other perpendicularly.
now the medium causes a difference because it impedes the moving thing, most of all if it is moving in the opposite direction, but in a secondary degree even if it is at rest; and especially a medium that is not easily divided, i.e.
nature, she then flies away with joy. for the death which is natural is pleasant, but that which is caused by violence is painful.
they fastened their attention on the substratum: while the other group maintained the opposite because their attention was concentrated on the 'contraries'.
there is also an absurdity in attributing the upward movement of bodies to a void which does not itself move.
heavenly bodies, and to the particles of matter. the ancients had not the instruments which would have enabled them to correct or verify their anticipations, and their opportunities of observation were limited.
five. given two points, one of which is in a given line, it is required to find another point in the given line, such that the sum or difference of its distances from the former points may be given. show that two such points may be found in each case.
three. if we take the proportion a : b :: c : d, and multiply the first and third terms, each by m, and second and fourth, each by n, we get the four multiples, ma, nb, mc, nd; and we want to prove that if ma is greater than nb, mc is greater than nd; if
the heavier. but since in composite bodies the weight obviously does not correspond in this way to the bulk, the lesser bulk being often superior in weight (as, for instance, if one be wool and the other bronze), there are some who think and say that the
air when strongly condensed is indissoluble by any power which does not reach the triangles, and even when not strongly condensed is only resolved by fire.
of the phenomena of light and heavy he speaks afterwards, when treating of sensation, but they may be more conveniently considered by us in this place.
'differents', should suffer action from one another: on the contrary, even if two things, being 'others', do act in some way on one another, this happens to them not qua 'others' but qua possessing an identical property.
viewed, in the language of spinoza, 'sub specie aeternitatis,' they were still at rest, but appeared to move in order to teach men the periods of time.
the cessation of the process of being burned does not involve the cessation of the capacity of being burned, since a thing may be capable of being burned without being in process of being burned-nor, when a thing ceases to be movent, does it therefore at
thus it is not the same thing to make a false and to make an impossible hypothesis, and from the impossible hypothesis impossible results follow.
why is it that the primary motion includes such a multitude of stars that their whole array seems to defy counting, while of the other stars each one is separated off, and in no case do we find two or more attached to the same motion?
and those who live in the other hemisphere are above and to the right, while we are below and to the left.
vi. in any parallelogram the figure which is composed of either of the parallelograms about a diagonal and the two complements see i., xliii. is called a gnomon.
moreover, when we say that what has become is become and what becomes is becoming, and that what will become is about to become and that the non-existent is non-existent-all these are inaccurate modes of expression (compare parmen.).
the criticism, however, is not always just: one should first consider what reason there is for speaking, and also what kind of certainty is looked for, whether human merely or of a more cogent kind.
we have now to speak of the distinctive properties of these bodies and of the various phenomena connected with them.
then because the arcs bc, cg, gh are all equal, the angles boc, cog, goh, are all equal three. xxvii. . therefore the arc bh and the angle boh are equimultiples of the arc bc and the angle boc.
they admitted of infinite multiplication and construction; in pythagorean triangles or in proportions of one:two:four:eight and one:three:nine:twenty seven, or compounds of them, the laws of the world seemed to be more than half revealed.
they think that if movement in respect of place is to exist, the void cannot exist, separated all by itself; but this is the same as to say that place is a separate cavity; and this has already been stated to be impossible.
finally, since there are three kinds of motion, the substratum and the goal of motion must be one or other of these, e.g. locomotion will have to be altered or to be locally moved.
on with the remaining tools. hence, however true it may be that fire is active, i.e. sets things moving, there is a further point they fail to observe-viz. that fire is inferior to the tools or instruments in the manner in which it sets things moving.
therefore, if this last movent is in motion and there is nothing else that moves it, it must move itself.
again, since af is equal to fg, being the sides of a square, the rectangle cf.af is equal to cf.fg-that is, to the figure cg; and ab is equal to the figure ad; therefore cg is equal to ad.
and we must acknowledge that as there are two kinds of knowledge, so there are two kinds of being corresponding to them; the one uncreated, indestructible, immovable, which is seen by intelligence only; the other created, which is always becoming in place
for they are asserting that chance is not responsible for the existence or generation of animals and plants, nature or mind or something of the kind being the cause of them (for it is not any chance thing that comes from a given seed but an olive from one
dazzling, and the object which produces it is called bright. there is yet another sort of fire which mingles with the moisture of the eye without flashing, and produces a colour like blood-to this we give the name of red.
even this was not indispensable,-for my discoveries were not by the synthetic but analytic process,-and i have gladly omitted it since, though most housewives earnestly assured me that safe and wholesome bread without yeast might not be, and elderly
in like manner it may be shown, if the side ac be produced, that the exterior angle bcg is greater than the angle abc; but bcg is equal to acd xv. . hence acd is greater than abc.
but in egypt the traditions of our own and other lands are by us registered for ever in our temples. the genealogies which you have recited to us out of your own annals, solon, are a mere children's story.
is equal twice the triangle abc multiplied by the diameter of the circle abc.
then let us choose two triangles, out of which fire and the other elements have been constructed, one isosceles, the other having the square of the longer side equal to three times the square of the lesser side.
no doubt, it is not true to say that all things are of this character: but it is true of all things between which there is reciprocal action and passion.
two produced. these three circles are called the escribed circles of the triangle abc.
furnished with four limbs extended and flexible; these god contrived to be instruments of locomotion with which it might take hold and find support, and so be able to pass through all places, carrying on high the dwelling-place of the most sacred and
two. if from the vertical angle of a right-angled triangle a perpendicular be let fall on the hypotenuse, its square is equal to the rectangle contained by the segments of the hypotenuse.
therefore kl is parallel to cf. similarly, fifty one is parallel to ch, and ki to fh; hence the triangles kli, fch are homothetic. hence the lines joining corresponding vertices are concurrent. therefore the points i, h, g are collinear.
that ea is equal to bf, and hence show that ce is equal to half the sum of ac, bc.
three. what do similar figures agree in?
it is the mistake of an uneducated person-uneducated, that is, in the higher sense of the word-who imagines every one else to be like himself and explains every other age by his own.
there is another aspect of the same difficulty which appears to have no satisfactory solution. in what relation does the archetype stand to the creator himself?
four. twice the square on the line joining any point in the hypotenuse of a right-angled isosceles triangle to the vertex is equal to the sum of the squares on the segments of the hypotenuse.
the form indeed is 'nature' rather than the matter; for a thing is more properly said to be what it is when it has attained to fulfilment than when it exists potentially. again man is born from man, but not bed from bed.
then because ab is normal to the plane four hundred, it is perpendicular to the line ak, which it meets in that plane eleven. def. vi. . therefore the angle bak is right. in like manner the angle abk is right.
two. describe a circle through a given point, and touching two given lines; or touching a given file and a given circle.
immortal and regard any other supposition as inconceivable. if then there is, as there certainly is, anything divine, what we have just said about the primary bodily substance was well said.
if it can occur in a small world it could also occur in a great one: and if it can occur in the world, it could also occur in the infinite; that is, if the infinite could as a whole possibly be in motion or at rest.
hence it follows conversely, that dividing the terms of a fraction by a whole number does not alter the value.
they are on the same straight line (for then they are contrary to one another in respect of place, as e.g.
again, our use of the phrase 'being at rest' also implies that the previous state of a thing is still unaltered, not one point only but two at least being thus needed to determine its presence: consequently that in which a thing is at rest cannot be
senses. since, then, the alteration of that which undergoes alteration is caused by sensible things, in every case of such alteration it is evident that the respective extremities of that which causes and that which undergoes alteration are adjacent.
in the intermediate periods of time-his account being as follows:
thirty five. what is the sum of all the exterior angles of any rectilineal figure equal to?
the race of wild animals were men who had no philosophy, and never looked up to heaven or used the courses of the head, but followed only the influences of passion.
and in what way will things be present either in place-or in the void?
yes, i answer, comparatively good, that is, you might have done worse; but i wish, as you are brothers of mine, that you could have spent your time better than digging in this dirt.
this is clear from our definition of 'primary', in which the word is said to express just this: it may also, however, be made evident by the following argument.
be placed round the circle, we shall have a regular polygon of fifteen sides, or quindecagon, inscribed in it.
he says that the warm bodies moving up out of the water hold up heavy bodies which are broad, while the narrow ones fall through, because the bodies which offer this resistance are not numerous.
intelligent motion of the supreme, distributing them over the whole circumference of heaven, which was to be a true cosmos or glorious world spangled with them all over.
on the other hand, in so far as the one results from composition (by a consilience of the many), whereas they result from disintegration the many are more 'elementary' than the one, and prior to it in their nature.
this becomes more equable when separated from fire and air, and then congeals into hail or ice, or the looser forms of hoar frost or snow. there are other waters which are called juices and are distilled through plants.
if the whole ag is in motion, there will be a motion of ab and a motion of bg. that being so, let de be the motion of the part ab and ez the motion of the part bg.
again, there is another point of view from which it will be clearly seen that locomotion is primary. as in the case of other things so too in the case of motion the word 'primary' may be used in several senses.
but the cube also has a magnitude equal to that occupied by the void; a magnitude which, if it is also hot or cold, or heavy or light, is none the less different in essence from all its attributes, even if it is not separable from them; i mean the volume
drives round the air at the other, and that which is driven round falls into the fire and becomes warm, and that which goes forth is cooled.
prop. twenty.-theorem. the sum of any two plane angles (bad, dac) of a trihedral angle (a) is greater than the third (bac).
the infinite, then, exists in no other way, but in this way it does exist, potentially and by reduction.
essentially and severally to themselves distinct from the motion of the whole.
let this conclude what we have to say in support of our contention that there never was a time when there was not motion, and never will be a time when there will not be motion.
the same method should also be adopted in replying to those who ask, in the terms of zeno's argument, whether we admit that before any distance can be traversed half the distance must be traversed, that these half-distances are infinite in number, and
moreover it is clear that if five hundred was becoming white in the whole time a, the time occupied by it in having become white in addition to having been in process of becoming white is no more than all that it occupied in the mere process of becoming
and, further, the members of either pair are contrary to those of the other, water being contrary to fire and earth to air; for the qualities constituting water and earth are contrary to those that constitute fire and air.
to all which phenomena some physician or philosopher who was able to see the one in many has given the name of bile. the various kinds of bile have names answering to their colours.
then why do some things manifestly come to-be in this cyclical fashion (as, e.g.
part two may be proved without producing either of the sides bd, six hundred. thus: join ad and produce it to meet bc in f; then the angle bdf is greater than the angle baf xvi. , and fdc is greater than fac.
moreover each of the bodies, fire, i mean, and earth and their intermediates, must have the same power as in our world.
we have taken equimultiples of the first and third, namely, the base ch, and the triangle ach; also equimultiples of the second and fourth, namely, the base one hundred fifty, and the triangle acl; and we have proved that according as the multiple of the
v. the first of four magnitudes has to the second the same ratio which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth, if, according as the
four. if we consider two magnitudes of the same kind, such as two lines ab, four hundred, and if we suppose that ab is equal to three four of four hundred, it is evident, if ab be divided into three equal parts, and four hundred into four equal parts,
how else would eclipses of the moon show segments shaped as we see them?
thus then, and for this reason the night and the day were created, being the period of the one most intelligent revolution.
for to define the infinite you must use quantity in your formula, but not substance or quality.
at the present day, his works possess a double significance. they are important historically, alike as giving us an insight into the mind of the eighteenth century, and for the actual influence they have had on the course of events in europe.
there are some too who ascribe this heavenly sphere and all the worlds to spontaneity. they say that the vortex arose spontaneously, i.e. the motion that separated and arranged in its present order all that exists.
on the other hand, if the hot pass-away out both, the contraries-dry and moist-are left. a similar result will occur in all the others too: for all the consecutive 'elements' contain one identical, and one contrary, quality.
between moved and movent. but the former fact is clear even from the definitions of pushing and pulling, for pushing is motion to something else from oneself or from something else, and pulling is motion from something else to oneself or to something
for the fire heats the air, and the air-being by nature such as both to act and suffer action-heats the body.) but the supposition that a body is 'susceptible in some parts, but insusceptible in others' (is only possible for those who hold an erroneous
as to the difficulty that into the one primary motion is crowded a vast multitude of stars, while of the other stars each has been separately given special movements of its own, there is in the first place this reason for regarding the arrangement as a
now if the two were both moving, and moving in contrary directions, they would pass clear of one another more rapidly; if one were still and the other moving past it, less rapidly; provided that the speed of the latter were the same in both cases.
large as we please, one of which is less and the other greater than the true value of the ratio. for let a and b be the incommensurable quantities; then, evidently, we cannot find two multiples na, mb, such that na mb.
the two divine courses were encased by the gods in a sphere which is called the head, and is the god and lord of us. and to this they gave the body to be a vehicle, and the members to be instruments, having the power of flexion and extension.
(one) what the followers of empedocles do, though without observing it themselves, is to reduce the generation of elements out of one another to an illusion.
three ; but it touches it (hyp.) also in p, which is impossible. hence the centre of the second circle must be in the line pe.
for in the course of a continuous motion the traveller has traversed an infinite number of units in an accidental sense but not in an unqualified sense: for though it is an accidental characteristic of the distance to be an infinite number of
four. let neither chord pass through the centre. through the point e, where they intersect, draw the diameter fg. then by three, the rectangle fe.eg is equal to the rectangle ae.eb, and also to the rectangle ce.ed.
two. if the line four hundred bisect the external vertical angle of any triangle acb, its square subtracted from the rectangle ad.db is equal to ac.cb.
forgetfulness and stupidity. further, when to this evil constitution of body evil forms of government are added and evil discourses are uttered in private as well as in public, and no sort of instruction is given in youth to cure these evils, then all of
the next question to consider is whether the elements are finite or infinite in number, and, if finite, what their number is. let us first show reason or denying that their number is infinite, as some suppose.
hesiod and the orphic poets moved in a region of half-personification in which the meaning or principle appeared through the person.
now it is quite possible for one body to be generated out of another, air for instance out of fire, but in the absence of any pre-existing mass generation is impossible.
six. if a circle passing through one of the angles a of a parallelogram abcd intersect the two sides ab, ad again in the points e, g and the diagonal ac again in f; then ab.ae ad.ag ac.af.
moreover (ii) contraries also 'suffer action', in accordance with the disjunctively-articulated definition established in the early part of this work.'
in like manner it may be proved that the arc ej and the angle epj are equimultiples of the arc ef and the angle epf.
we must also discuss the remaining forms of change, viz. growth and 'alteration'.
'like') but specifically 'unlike', while (b) it is 'contraries' that exhibit this character: it is clear that 'contraries' and their 'intermediates' are such as to suffer action and to act reciprocally-for indeed it is these that constitute the entire
but a force must always be infinite-just as a number or a magnitude is-if it exceeds all definite limits.
three. in a right-angled triangle, if the square on one side be equal to the rectangle contained by the hypotenuse and the other side, the hypotenuse is cut in "extreme and mean ratio" by the perpendicular on it from the right angle.
consequently, nothing will remain and the body will have passed-away into what is incorporeal: and so it might come-to-be again either out of points or absolutely out of nothing. and how is that possible?
the soul of the world may also be conceived as the personification of the numbers and figures in which the heavenly bodies move. imagine these as in a pythagorean dream, stripped of qualitative difference and reduced to mathematical abstractions.
vi. two corresponding angles of two figures have the sides about them reciprocally proportional, when a side of the first is to a side of the second as the remaining side of the second is to the remaining side of the first.
now when we speak of a power to move or to lift weights, we refer always to the maximum.
eight. what property of two lines having two common points is quoted in this proposition? they must coincide.
changes accidentally to an object of thought, the colour being only accidentally the object of thought; it changes to colour, because white is a part of colour, or to europe, because athens is a part of europe; but it changes essentially to white colour.
by the combination of these triangles which exist in an infinite variety of sizes, the surfaces of the four elements are constructed.
in order then that it might not tumble about among the high and deep places of the earth, but might be able to get over the one and out of the other, they provided the body to be its vehicle and means of locomotion; which consequently had length and was
the distinctions are three, namely, above and below, front and its opposite, right and left-all these three oppositions we expect to find in the perfect body-and each may be called a principle.
this case the differences in the locomotion are merely differences of posture in that which is in motion.) we may say, therefore, that things are of equal velocity in an equal time they traverse the same magnitude: and when i call it 'the same' i mean
hence (ex aequali) ab : ac :: six hundred : de. therefore the sides about the equal angles are proportional.
from health to sickness must simultaneously be changing from this very change to another.
and the same is true in the case of a circle: the motion from a towards b is the contrary of the motion from a towards g: for even if they are continuous and there is no turning back they arrest each other, because contraries annihilate or obstruct one
is equal to the rectangle contained by the perpendiculars on either pair of opposite sides.
we can imagine two worlds, one of which is the mere double of the other, or one of which is an imperfect copy of the other, or one of which is the vanishing ideal of the other; but we cannot imagine an intellectual world which has no qualities-'a thing in
additional evidence seems to be furnished by the food each compound takes. for all of them are fed by substances which are the same as their constituents, and all of them are fed by more substances than one.
had been a thing made by nature, it would have been made in the same way as it is now by art; and if things made by nature were made also by art, they would come to be in the same way as by nature.
similarly in other cases of chance it is always possible, they maintain, to find something which is the cause; but not chance, for if chance were real, it would seem strange indeed, and the question might be raised, why on earth none of the wise men of
sol.-bisect ab in e. join ec. make the angle bef xxiii. equal to five hundred. draw cg parallel to ab xxxi. , and bg parallel to ef. eg is a parallelogram fulfilling the required conditions.
eighteen. what problems on the drawing of lines occur in book one.? ans. ii., ix., xi., xii., xxiii., xxxi., in each of which, except problem two, there are two conditions. the direction in problem two is indeterminate.
but not even movement in respect of place involves a void; for bodies may simultaneously make room for one another, though there is no interval separate and apart from the bodies that are in movement.
but, as plato would say, 'there is no harm in repeating twice or thrice' (laws) what is important for the understanding of a great author.
in like manner it can be proved that the parallelogram bl is equal to bd. hence the whole square af is equal to the sum of the two squares ah and bd.
this is the explanation of the shallows which are found in that part of the atlantic ocean.'
nor did they give any account of 'combination': and they neglected almost every single one of the remaining problems, offering no explanation, e.g. of 'action' or 'passion' how in physical actions one thing acts and the other undergoes action.
again, if the weight of a greater body is greater than that of a less, the weight of gb will be greater than that of fb; and thus the weight of the finite body is greater than that of the infinite.
and above all there is (three) the case of a thing which is in motion neither accidentally nor in respect of something else belonging to it, but in virtue of being itself directly in motion.
but at present we are to state the material cause-the cause classed under the head of matter-to which it is due that passing-away and coming-to-be never fail to occur in nature.
therefore ae eo is equal to cf fo ; and ae has been proved equal to cf . hence eo is equal to fo ; therefore eo is equal to fo. hence ab, four hundred are (def. vi.) equally distant from the centre.
if, on the other hand, it has been changing in both ab and bg (for it must either have changed or be changing in each of them), it must have been changing in the whole ag: but our assumption was that ag contains only the completion of the change.
dem.-draw de, the diameter which is perpendicular to ab, and draw the chords cf, bg parallel to de. now it is evident that the difference between the arcs ac, cb is equal to cd, and therefore four hundred ef.
hence no point can be the centre which is not in the line de. therefore f, the middle point of de, must be the centre.
this is because all other things are discriminated by time, and end and begin as though conforming to a cycle; for even time itself is thought to be a circle.
a motion from health and a motion to disease; or motions respectively from a contrary to the opposite contrary and from the latter to the former, e.g.
having now before us the causes out of which the rest of our discourse is to be framed, let us go back to the point at which we began, and add a fair ending to our tale.
hence, if ab, four hundred meet on one side of o, they must also meet on the other side; but two right lines cannot enclose a space (axiom x.); therefore they do not meet at either side. hence they are parallel.
qualities are contrary. for in some of them both qualities are contrary-e.g. in fire and water, the first of these being dry and hot, and the second moist and cold: while in others one of the qualities (though only one) is contrary-e.g.
half the sum, and df.le equal to the square of half the difference of the two remaining sides.
and astronomical arguments give further confirmation; for no other hypothesis accounts for the crescent shape of the sun's eclipses. one, then, of the heavenly bodies being spherical, clearly the rest will be spherical also.
where multiple words occur together, they are separated by the " " symbol for clarity. readers who do not speak or read the greek language will usually neither gain nor lose understanding by skipping over these passages.
hence, too, it clearly follows that, when one of the consecutive 'elements' is transformed into one, the coming-to-be is effected by the passing-away of a single quality: whereas, when two of them are transformed into a third, more than one quality must
the 'originative sources', then, of the things which come-to-be are equal in number to, and identical in kind with, those in the sphere of the eternal and primary things.
the general physical doctrines of the timaeus may be summed up as follows: (one) plato supposes the greater masses of the elements to have been already settled in their places at the creation: (two) they are four in number, and are formed of rectangular
yet i find it not to be an essential ingredient, and after going without it for a year am still in the land of the living; and i am glad to escape the trivialness of carrying a bottle-full in my pocket, which would sometimes pop and discharge its contents
and since there are three kinds of motion, local, qualitative, and quantitative, there must also be three kinds of movent, that which causes locomotion, that which causes alteration, and that which causes increase or decrease.
it is a coming-to-be of one substance and a passing-away of the other-especially if the change proceeds from an imperceptible something to something perceptible (either to touch or to all the senses), as when water comes-to-be out of, or passes-away into,
time taken by itself (for the time contains an infinite number of divisions): then this solution will no longer be adequate, and we must apply the truth that we enunciated in our recent discussion, stating it in the following way.
rectangle ap.bp ob op ; but ot pt op i. xlvii. . therefore ap.bp ob ot pt ; and rejecting the equals ob and ot , we have the rectangle
for of actual subjects of change it will be seen that those which are classed as essentially, not accidentally, divisible have no primary part.
nineteen. if a system of circles pass through two fixed points, any two secants passing through one of the points are cut proportionally by the circles.
it was, for nearly two years after this, rye and indian meal without yeast, potatoes, rice, a very little salt pork, molasses, and salt, and my drink water. it was fit that i should live on rice, mainly, who loved so well the philosophy of india.
r r r r r, (one) r r, (two) r r. (three) the relation (three) supposes that the circumcentre is inside the triangle.
seventy. if a circle touch internally two sides, ca, cb, of a triangle and its circumscribed circle, the distance from one hundred to the point of contact on either side is a fourth proportional to the semiperimeter, and ca, cb.
dem.-on the sides ab, bc, ca describe squares xlvi. . draw one hundred fifty parallel to ag. join cg, bk.
thirty four. if the circumference of a circle whose radius is r be divided into seventeen equal parts, and ao be the diameter drawn from one of the points of division (a), and if one, two......eight denote the chords from o to the points of division, a ,
they partake of that third kind of life which is seated between the midriff and the navel, and is altogether passive and incapable of reflection.
as to that which is moved, we have shown that it is primary and simple and ungenerated and indestructible and generally unchanging; and the mover has an even better right to these attributes.
wherever there is continuity in any process (coming-to-be or 'alteration' or any kind of change whatever) we observe consecutiveness', i.e. this coming-to-be after that without any interval.
for clearly, if the soul be fire, only such modifications will happen to it as characterize fire qua fire: while if it be compounded out of the elements', only the corporeal modifications will occur in it.
now since e is a point in the diameter of the larger circle between the centre and a, ea is less than ep vii. two ; but ep is equal to ec (hyp.), being radii of the smaller circle.
this discussion, however, involves the more general question whether the infinite can be present in mathematical objects and things which are intelligible and do not have extension, as well as among sensible objects.
that both views are untenable will be clear if we take as our starting-point the fact that the earth's motion, whether the earth be at the centre or away from it, must needs be a constrained motion. it cannot be the movement of the earth itself.
this implies that what is naturally the means to an end is 'in vain', when it does not effect the end towards which it was the natural means-for it would be absurd for a man to say that he had bathed in vain because the sun was not eclipsed, since the one
dem.-produce ad both ways. draw be parallel to ac, and cf parallel to bd xxxi. then the figures aebc, dbcf are parallelograms; and since they are on the same base bc, and between the same parallels bc, ef they are equal xxxv. .
we see nothing like this in the case of inanimate things, which are always set in motion by something else from without: the animal, on the other hand, we say, moves itself: therefore, if an animal is ever in a state of absolute rest, we have a motionless
in early life he fancies that he has mastered them: but he is also mastered by them; and in language (sophist) which may be compared with the hesitating tone of the timaeus, he confesses in his later years that they are full of obscurity to him.
thirty four. ad and bc are two parallel lines cut obliquely by ab, and perpendicularly by ac; and between these lines we draw bed, cutting ac in e, such that ed ab; prove that the angle dbc is one-third of abc.
a self-movent through its exercise of leverage.) hence we may confidently conclude that if a thing belongs to the class of unmoved movents that are also themselves moved accidentally, it is impossible that it should cause continuous motion.
xiv. a cyclic quadrilateral is one which is inscribed in a circle.
that makes the potentially hot into the actually hot: so it is plain that the movent does not maintain a uniform relation to it but is at one time nearer to and at another farther from that which is altered: and we cannot have this without locomotion.
'matter', in the most proper sense of the term, is to be identified with the substratum which is receptive of coming-to-be and passingaway: but the substratum of the remaining kinds of change is also, in a certain sense, 'matter', because all these
here too indeed one might raise a difficulty. since there are two natures, with which is the physicist concerned? or should he investigate the combination of the two? but if the combination of the two, then also each severally.
it is, then, clearly impossible for being to be one in this sense.
what plato chiefly intends to express is that a solid requires a stronger bond than a surface; and that the double bond which is given by two means is stronger than the single bond given by one.
but the space between infinite radii is infinite: and by the space between the radii i mean the area outside which no magnitude which is in contact with the two lines can be conceived as falling.
dem.-join co, and produce it to e. then because oa is equal to oc, the angle aco is equal to oac; but the angle aoe is equal to the sum of the two angles oac, aco. hence the angle aoe is double the angle aco.
the infinite body must be either (one) compound, or (two) simple; yet neither alternative is possible.
respectively equal to the angles of the other-namely, those shall be equal to which the equal sides are opposite.
but if the fire above air is removed, it will not move upward to the place of fire, except by constraint; and in that way water also may be drawn up, when the upward movement of air which has had a common surface with it is swift enough to overpower the
of bone he made a globe which he placed around the brain, leaving a narrow opening, and around the marrow of the neck and spine he formed the vertebrae, like hinges, which extended from the head through the whole of the trunk.
so there can be no primary part of the time: and the reason is that rest and motion are always in a period of time, and a period of time has no primary part any more than a magnitude or in fact anything continuous: for everything continuous is divisible
one. the three perpendiculars of a triangle (abc) are concurrent.
this, however, is clear: that it is impossible to traverse an infinite line in a finite time. infinite time, then, would be required.
the external elements by their attraction are always diminishing the substance of the body: the particles of blood, too, formed out of the newly digested food, are attracted towards kindred elements within the body and so fill up the void.
dem.-draw any secant ghk. then since ab and ef are parallel, the angle agh is equal to ghf xxix. . in like manner the angle ghf is equal to hkd xxix. . therefore the angle agk is equal to the angle gkd (axiom i.). hence xxvii.
one. solve the problem when the point a is in the line bc itself.
it is clear then that chance is an incidental cause in the sphere of those actions for the sake of something which involve purpose. intelligent reflection, then, and chance are in the same sphere, for purpose implies intelligent reflection.
according to plato in the timaeus, god took of the same and the other, of the divided and undivided, of the finite and infinite, and made essence, and out of the three combined created the soul of the world.
three. when the place which a figure occupies is known, it is said to be given in position.
ii. a right line is said to be cut at a point in extreme and mean ratio when the whole line is to the greater segment as the greater segment is to the less.
ninety five. given the base of a triangle, the vertical angle, and the point in the base whose distance from the vertex is equal half the sum of the sides; construct the triangle.
for the fire cuts the food and following the breath surges up within, fire and breath rising together and filling the veins by drawing up out of the belly and pouring into them the cut portions of the food; and so the streams of food are kept flowing
the angle bac is equal to bdc xxi. . now let the point b move until it becomes consecutive to a; then ab will be a tangent, and bd will coincide with ad, and the angle bdc with adc.
and diseases arise from the opposite process-when the natural proportions of the four elements are disturbed, and the secondary substances which are formed out of them, namely, blood, flesh, sinews, are generated in an inverse order.
for nothing to the world below. this, however, is a later stage; at present we must treat more exactly the subject before us, which involves a preliminary enquiry into the generation of the body and its members, and as to how the soul was created-for what
twenty eight. if from any point in the circumference of the circle, circumscribed about a regular polygon of n sides, lines be drawn to its angular points, the sum of their squares is equal to n times the square of the radius.
as to bodies composed of earth and water, while the water occupies the vacant interstices of the earth in them which are compressed by force, the particles of water which approach them from without, finding no entrance, flow around the entire mass and
consequently the first thing that is in motion will derive its motion either from something that is at rest or from itself.
for fire is hot and dry, whereas air is hot and moist (air being a sort of aqueous vapour); and water is cold and moist, while earth is cold and dry.
every other motion and change is from an opposite to an opposite: thus for the processes of becoming and perishing the limits are the existent and the non-existent, for alteration the various pairs of contrary affections, and for increase and decrease
and once more, when a body large and too strong for the soul is united to a small and weak intelligence, then inasmuch as there are two desires natural to man,-one of food for the sake of the body, and one of wisdom for the sake of the diviner part of
five. inscribe a circle in a sector of a given circle.
now, since by hypothesis, when na- mb is greater than unity, -nc md is greater than unity; and when na (m- -one)b- is less than unity, nc (m- -one)d- is less than unity.
grant only that mass is anywhere and it follows that it must be everywhere. also, if void and place are infinite, there must be infinite body too, for in the case of eternal things what may be must be.
probably in the alexandrian age, when egypt had ceased to have a history and began to appropriate the legends of other nations, many such monuments were to be found of events which had become famous in that or other countries.
but also by qualitative change; e.g. if water were to be transformed into air.
prop. one hundred.-theorem (simson). if the first of four magnitudes be the same multiple of the second which the third is of the fourth, the first is to the second as the third is to the fourth.
four. if in any quadrilateral two opposite sides be bisected, the sum of the squares on the other two sides, together with the sum of the squares on the diagonals, is equal to the sum of the squares on the bisected sides, together with four times the
for, being indivisible, it will possess these properties in the same point-so that, if it 'suffers action' by being chilled, it will also, qua chilled, 'act' or 'suffer action' in some other way.
that there were only five regular solids was already known to the ancients, and out of the surfaces which he has formed plato proceeds to generate the four first of the five.
that the heaven as a whole neither came into being nor admits of destruction, as some assert, but is one and eternal, with no end or beginning of its total duration, containing and embracing in itself the infinity of time, we may convince ourselves not
for (i) most thinkers are unanimous in maintaining (a) that 'like' is always unaffected by 'like', because (as they argue) neither of two 'likes' is more apt than the other either to act or to suffer action, since all the properties which belong to the
it in their fabulous way upon animate necessity.
since then, relatives are neither themselves alterations nor the subjects of alteration or of becoming or in fact of any change whatever, it is evident that neither states nor the processes of losing and acquiring states are alterations, though it may be
for to connect the infinite with the all and the whole is not like joining two pieces of string; for it is from this they get the dignity they ascribe to the infinite-its containing all things and holding the all in itself-from its having a certain
eleven. through a given point p draw a chord of a circle so that the intercept ef may subtend a given angle ten.
or thus: describe the squares efjk, lmno equal to the figures bcd and a respectively two. xiv. ; then find gh a fourth proportional to ef, lm, and bc xii. . on gh describe the rectilineal figure ghi similar to the figure bcd xviii.
hence we have the following theorem:-the nine points made up of the feet of the perpendiculars, the middle points of the sides, and the middle points of the lines from the vertices to the orthocentre, are concyclic.
three. if the perpendiculars of a triangle be produced to meet the circumscribed circle, the intercepts between the orthocentre and the circle are bisected by the sides of the triangle.
better than an infinite multitude; for empedocles professes to obtain from his principles all that anaxagoras obtains from his innumerable principles.
finally, as for salt, that grossest of groceries, to obtain this might be a fit occasion for a visit to the seashore, or, if i did without it altogether, i should probably drink the less water.
again, since ab and four hundred are parallel, the sum of the angles abd, bdc is two right angles i. xxix. ; but abd is right (hyp.); therefore bdc is right. hence four hundred is perpendicular to the two lines db, de, and therefore eleven. iv.
such, then, are the traditional theories, and it looks as if the statements of their advocates were in manifest conflict.
observation.-two follows at once from one by proposition b.
now we have four magnitudes, namely, the arc bc, the arc ef, the angle boc, and the angle epf; and we have taken equimultiples of the first and third, namely, the arc bh, the angle boh, and other equimultiples of the second and fourth, namely, the arc ej
increase, decrease, and respiration: these are experienced by every animal while it is at rest and not in motion in respect of the motion set up by its own agency: here the motion is caused by the atmosphere and by many things that enter into the animal:
twenty five. if a point be h feet outside the circumference of a circle whose diameter is seven thousand nine hundred twenty miles, prove that the length of the tangent drawn from it to the circle is --- h two miles.
they could, and avert from him all but self-inflicted evils.
for the supposed 'intermediate' will be air and fire when a pair of contrasted qualities is added to it: but, since one of every two contrary qualities is a 'privation', the 'intermediate' never can exist-as some thinkers assert the 'boundless' or the
the manner of the change is (one) a separation of portions of the elements from the masses in which they are collected; (two) a resolution of them into their original triangles; and (three) a reunion of them in new forms.
the next year i did better still, for i spaded up all the land which i required, about a third of an acre, and i learned from the experience of both years, not being in the least awed by many celebrated works on husbandry, arthur young among the rest,
sol.-bisect two adjacent angles a, b by the lines ao, bo. then o, the point of intersection of the bisectors, is the centre of the required circle.
thus they do move to the centre of the earth, but accidentally, in virtue of the fact that the earth's centre lies at the centre of the whole.
a short consideration will give us an easy answer, if we first give precision to our postulate that any body endowed with weight, of whatever size, moves towards the centre. clearly it will not stop when its edge touches the centre.
planets, of the creative mind and the primeval chaos.
the entire compound was divided by the creator in certain proportions and reunited; it was then cut into two strips, which were bent into an inner circle and an outer, both moving with an uniform motion around a centre, the outer circle containing the
when this desire is unsatisfied the man is over-mastered by the power of the generative organs, and the woman is subjected to disorders from the obstruction of the passages of the breath, until the two meet and pluck the fruit of the tree.
and, further, when one weight is greater than another, the difference is a third weight; from which it will follow that every indivisible part possesses weight. for suppose that a body of four points possesses weight.
there is, indeed, a difficulty about part and whole, perhaps not relevant to the present argument, yet deserving consideration on its own account-namely, whether the part and the whole are one or more than one, and how they can be one or many, and, if
i made a study of the ancient and indispensable art of bread-making, consulting such authorities as offered, going back to the primitive days and first invention of the unleavened kind, when from the wildness of nuts and meats men first reached the
for, assuming the distinction between (a) that which is potentially and (b) that which is actually such-and-such, it is the nature of the first, precisely in so far as it is what it is, to suffer action through and through, not merely to be susceptible in
the outer circle or sphere was named the sphere of the same-the inner, the sphere of the other or diverse; and the one revolved horizontally to the right, the other diagonally to the left.
sorts of diseases, more or fewer, and in every degree of intensity; and being carried to the three places of the soul, whichever they may severally assail, they create infinite varieties of ill-temper and melancholy, of rashness and cowardice, and also of
when we remark that greek physics soon became stationary or extinct, may we not observe also that there have been and may be again periods in the history of modern philosophy which have been barren and unproductive?
thus none of the things which are neither moved nor at rest are in time: for 'to be in time' is 'to be measured by time', while time is the measure of motion and rest.
being, whose complex frame has an appointed term of life.
hence the rectangle ae.eb is equal to the rectangle ce.ed.
so the fact that some things become and others perish, and that this is so continuously, cannot be caused by any one of those things that, though they are unmoved, do not always exist: nor again can it be caused by any of those which move certain
he was endeavouring to form a conception of principles, but these principles or ideas were regarded by him as real powers or entities, to which the world had been subjected.
ten. the angle included between the perpendicular from the vertical angle of a triangle on the base, and the bisector of the vertical angle, is equal to half the difference of the base angles.
and fall short. eternity or the eternal is not merely the unlimited in time but the truest of all being, the most real of all realities, the most certain of all knowledge, which we nevertheless only see through a glass darkly.
: to each add the angle hgi, and we have the sum of the angles fgh, hgi equal to the sum of the angles ghk, hgi; but since gi is parallel to hk, and gh intersects them, the sum of the angles ghk, hgi is equal to two right angles xxix. .
four. any line intersecting the legs of any angle is cut harmonically by the internal and external bisectors of the angle.
(one) one- p one- p -one- p one r; (two) one p -one p p one- r, c.; (three) two p one r one r, c.; (four) two p one r one r, c. twelve.
two. because the chord four hundred is nearer to the centre than ef, og is less than oh; and since the triangles ogc, ohe are right-angled, we have oc og gc , and oe oh he ; therefore og gc oh he ; but og is less than oh ; therefore gc is greater than he
sol.-construct the parallelogram befg xlii. equal to the given triangle one hundred, and having the angle b equal to the given angle five hundred, and so that its side be shall be in the same right line with ab. through a draw ah parallel to bg xxxi.
the bones and flesh. the brain, the containing vessel of the divine part of the soul, is (nearly) in the form of a globe, which is the image of the gods, who are the stars, and of the universe.
the teacher should make these triangles separate, as in the annexed diagram, and point out the corresponding parts thus:-
six. if the base of a triangle be given in magnitude and position, and the ratio of the sides, the locus of the vertex is a circle which divides the base harmonically in the ratio of the sides.
three. draw a chord in a given circle which shall subtend a right angle at a given point, and be parallel to a given line.
all the causes now mentioned fall into four familiar divisions.
but there still remains a rebellious seed of evil derived from the original chaos, which is the source of disorder in the world, and of vice and disease in man.
prop. thirty.-theorem. if two right lines (ab, four hundred) be parallel to the same right line (ef), they are parallel to one another.
but however excellent their theories may otherwise be, anyhow they cannot be held to speak as students of nature.
what we have in this argument is, it would seem, a demand for the precise distinction of 'combination' from coming-to-be and passingaway (for it is obvious that 'combination', if it exists, must differ from these processes) and for the precise distinction
to my discomfiture. it is simpler and more respectable to omit it. man is an animal who more than any other can adapt himself to all climates and circumstances. neither did i put any sal soda, or other acid or alkali, into my bread.
and this is true in every category, whether the thing is, for example, 'man', or 'white', or 'three cubits long', or whatever it may be.
dem.-let o be the centre of the circle. join oe, oa, ob. now, because the angles a, e of the quadrilateral aoef are right angles three. xviii. , the sum of the two remaining angles aoe, afe is two right angles.
dem.-let the same construction be made, then we have
for the centre of the world cannot be rightly called either above or below, but is the centre and nothing else; and the circumference is not the centre, and has in no one part of itself a different relation to the centre from what it has in any of the
in like manner the angle eob is double the angle ocb. hence (by adding in figs. (), (), and subtracting in ()), the angle aob is double of the angle acb.
thus far the universal animal was made in the divine image, but the other animals were not as yet included in him. and god created them according to the patterns or species of them which existed in the divine original.
for there are two senses in which length and time and generally anything continuous are called 'infinite': they are called so either in respect of divisibility or in respect of their extremities.
this then perhaps exhausts the number of ways in which the term 'cause' is used.
it makes no difference whether we take the intermediate stages of the process to be few or many, or whether we add or subtract one: for in either case we find that there is still the same ground to be traversed repeatedly.
now a thing that is in motion from one place to another cannot at the moment when it was in motion both be in motion and at the same time have completed its motion at the place to which it was in motion: e.g.
dem.-if ab and four hundred are not parallel they must meet, if produced, at some finite distance: if possible let them meet in g; then the figure egf is a triangle, and the angle aef is an exterior angle, and efd a non-adjacent interior angle. hence xvi.
and next, they ordered the water-courses of the body in a manner which i will describe, and which will be more easily understood if we begin by admitting that all things which have lesser parts retain the greater, but the greater cannot retain the lesser.
if, therefore, the time which the finite moving line takes to pass the other is infinite, then necessarily the time occupied by the motion of the infinite past the finite is also infinite.
(b) nor can fire or any other of the elements be infinite.
thirty. if the sum of the perpendiculars let fall from a given point on the sides of a given rectilineal figure be given, the locus of the point is a right line.
such a shape is the sphere, since it possesses no instrument of movement. clearly then their mass will have the form of a sphere. again, what holds of one holds of all, and the evidence of our eyes shows us that the moon is spherical.
weight, since the region about the centre has been given to the heavy body. and its position cannot be unnatural to it, for it would have to be natural to something else, and there is nothing else. it must then occupy the intermediate place.
clear the cause of the trouble and of the difficulties about it. such is the most satisfactory kind of exposition.
twenty three. if a cyclic quadrilateral be such that a circle can be inscribed in it, the lines joining the points of contact are perpendicular to each other.
the ancient mythologers, and even the hebrew prophets, had spoken of the jealousy of god; and the greek had imagined that there was a nemesis always attending the prosperity of mortals.
hence the sum of the lines ap, bp is equal to the sum of the lines ac, db; but ab is greater than the sum of ac and db; therefore ab is greater than the sum of ap, pb-that is, one side of a triangle greater than the sum of the other two-which i. xx.
but the lesser quantity is contained in the greater. therefore the air element is divisible. and the same could be shown of fire and of all bodies whose parts are relatively fine.
what then motion is, has been stated both generally and particularly.
the numbers and figures which were present to the mind's eye became visible to the eye of sense; the truth of nature was mathematics; the other properties of objects seemed to reappear only in the light of number.
that being so, motion must also be continuous, because what is always is continuous, whereas what is merely in succession is not continuous.
upward motion and rest above are respectively contrary to downward motion and rest below, these being instances of local contrariety; and upward locomotion belongs naturally to fire and downward to earth, i.e.
demonstrate this proposition directly by cutting off from bc a part equal to ef.
prop. seventeen.-problem. from a given point (p) without a given circle (bcd) to draw a tangent to the circle.
but this is impossible, for in less time it must traverse less distance, and thus the indivisible will be divisible into something less than itself, just as the time is so divisible: the fact being that the only condition under which that which is without
thus to say that chance is a thing contrary to rule is correct. for 'rule' applies to what is always true or true for the most part, whereas chance belongs to a third type of event.
(yet it is not in itself: though these are in a way the same thing,) they differ in essence, each having a special nature and capacity, 'surface' and 'white'.
change within the same kind from a lesser to a greater or from a greater to a lesser degree is alteration: for it is motion either from a contrary or to a contrary, whether in an unqualified or in a qualified sense: for change to a lesser degree of a
yet this causes surprise, and it is thought impossible that something should come to be in the way described from what is not.
in saying this we shall not be far wrong; as far, however, as we can attain to a knowledge of her from the previous considerations, we may truly say that fire is that part of her nature which from time to time is inflamed, and water that which is
the military divisions of an army, the civil divisions of a state, seemed to afford a 'present witness' of them-what would have become of man or of the world if deprived of number (rep.)? the mystery of number and the mystery of music were akin.
being. a sufficient evidence of this is that nothing comes into being without itself moving somehow and acting, but a thing can be destroyed even if it does not move at all.
moves it: in fact the earlier movent is that which more strictly moves it.
and sciences tended to the purification or to the corruption of morality?"
if the centre be not in ac, let o be the centre. join ao. then because ab touches the circle, and oa is drawn from the centre to the point of contact, oa is at right angles to ab xviii.
six. of the four points o, o, o, o, any one is the orthocentre of the triangle formed by the remaining three.
hence ea is less than ec; which is impossible; consequently the centre of the smaller circle must be in the line op. let it be h; then we see that the line joining the centres passes through the point p.
therefore, since any movement of their own which the stars possessed would presumably be one proper to themselves, and no such movement is observed in them, clearly they have no movement of their own.
four. if the vertical angle acb of a triangle inscribed in a circle be bisected by a line four hundred, which meets the circle again in five hundred, and from five hundred perpendiculars de, df be drawn to the sides, one of which must be produced: prove
hence ja : ac :: od : df iv. ; but ac : ab :: df : de iv. . hence ja : ab :: od : de (ex quali); but ab ag, and de five hundred fifty; therefore ja : ag :: od : five hundred fifty. again, ja : ag :: pictai : square ah i.
therefore from the given point a the line af has been drawn equal to bc.
i will briefly describe them to you, and you shall read the account of them at your leisure in the sacred registers.
mind which is the principal cause with an eye to the future.
these and such-like, then, are the arguments for our conclusion that derive cogency from the fact that they have a special bearing on the point at issue.
one hundred seventeen. the product of the bisectors of the three angles of a triangle whose sides are a, b, c, is
of destiny, according to which their first birth would be one and the same for all,-no one should suffer a disadvantage at his hands; they were to be sown in the instruments of time severally adapted to them, and to come forth the most religious of
such a plurality is in fact impossible if this world contains the entirety of matter, as in fact it does. but perhaps our contention can be made clearer in this way.
prop. ten.-problem. to bisect a given finite right line (ab).
nature of the same and of the other, and this compound he placed accordingly in a mean between the indivisible, and the divisible and material.
if from any point (p) outside a circle, lines (pa, pb, pc, c.) be drawn to the concave circumference, then-one. the maximum is that which passes through the centre. two.
(one. the infinite cannot be acted upon by the finite.) let a be an infinite, b a finite, one hundred the time of a given movement produced by one in the other.
moreover, although these opinions appear to follow logically in a dialectical discussion, yet to believe them seems next door to madness when one considers the facts.
yet, doubtless, there was some first impression derived from external nature, which, as in mythology, so also in philosophy, worked upon the minds of the first thinkers.
for it is in contraries that generation and decay subsist. again, that which is subject to increase increases upon contact with a kindred body, which is resolved into its matter. but there is nothing out of which this body can have been generated.
and the glutinous and rich matter which comes away from the sinews and the flesh, not only glues the flesh to the bones, but nourishes and imparts growth to the bone which surrounds the marrow; and by reason of the solidity of the bones, that which
two. if the alternate sides of a regular pentagon be produced to meet, the five points of meeting form another regular pentagon.
twenty. the diagonal of a rectangle is four.twenty five perches, and its area is seven.fifty square perches; what are its dimensions?
fourteen. given the base of a triangle in magnitude and position, and the sum or the difference of m times the square on one side and n times the square on the other side, in magnitude, the locus of the vertex is a circle.
hence it was natural for plato to conceive of it as eternal. we must remember further that in his attempt to realize either space or matter the two abstract ideas of weight and extension, which are familiar to us, had never passed before his mind.
hearing is a blow which passes through the ear and ends in the region of the liver, being transmitted by means of the air, the brain, and the blood to the soul. the swifter sound is acute, the sound which moves slowly is grave.
now, first, if the shape of the earth is not flat, its flatness cannot be the cause of its immobility. but in their own account it is rather the size of the earth than its flatness that causes it to remain at rest.
prop. one.-theorem. one part (ab) of a right line cannot be in a plane (ten), and another part (bc) not in it.
hence 'substance' will not be a predicate of anything else. for the subject cannot be a being, unless 'being' means several things, in such a way that each is something. but ex hypothesi 'being' means only one thing.
much more is the angle abc greater than the angle acb.
that is hard to which the flesh yields, and soft which yields to the flesh, and these two terms are also relative to one another.
the ordered, it is said, arose out of the unordered; and the same thing cannot be at the same time both ordered and unordered; there must be a process and a lapse of time separating the two states.
dem.-from b draw bg perpendicular to the plane df eleven. xi. , and let it meet that plane in g. through g draw gh parallel to ed, and gk to ef. now, since gh is parallel to ed (const.), and ab to ed (hyp.), ab is parallel to gh eleven. ix. .
socrates: one, two, three; but where, my dear timaeus, is the fourth of those who were yesterday my guests and are to be my entertainers to-day?
nineteen. what is meant by figures inversely similar?
through the whole body in all animals. and fresh cuttings from kindred substances, whether the fruits of the earth or herb of the field, which god planted to be our daily food, acquire all sorts of colours by their inter-mixture; but red is the most
but we apprehend time only when we have marked motion, marking it by 'before' and 'after'; and it is only when we have perceived 'before' and 'after' in motion that we say that time has elapsed.
furthermore, in the case of becoming and perishing it would seem to be an utterly absurd thing if as soon as anything has become it must necessarily perish and cannot continue to exist for any time: and, if this is true of becoming and perishing, we have
it is clearly of neither of these that chance is said to be the cause, nor can the 'effect of chance' be identified with any of the things that come to pass by necessity and always, or for the most part.
this argument, then, is one way in which they show that there is a void.
(five) as health is 'in' the hot and the cold and generally the form 'in' the matter.
that is why empedocles too uses language to this effect, when he says 'there is no coming-to-be of anything, but only a mingling and a divorce of what has been mingled'.
when earth produces water something is taken away from the earth, for the process is one of excretion. the same thing happens again when the residue produces water.
if, therefore, it is necessary that the consequent should come-to-be, the antecedent also must have come-to-be: and if the antecedent has come-to-be, then the consequent also must come-to-be-not, however, because of the antecedent, but because the future
the outer circle in two points equidistant from one hundred.
for that in which it is in motion will be a time, because all motion is in a period of time; and all time has been shown above to be divisible.
those who understand greek, however, may gain a deeper insight to the original meaning and distinctions expressed by aristotle.
or a : c :: a : b -that is, st : rd :: square of st : square of nd. now, the ratio of st : rd is, by def. x., the duplicate ratio of st : nd.
by iv., the square on the sum the sum of the squares twice the rectangle.
from a given point (one hundred) in a given right line (ab) to draw a right line perpendicular to the given line.
but they, owing to their love for their principles, fall into the attitude of men who undertake the defence of a position in argument. in the confidence that the principles are true they are ready to accept any consequence of their application.
cor. two.-if the figure abhij be applied to cdefg so that the point a will coincide with one hundred, and that the line ab may be placed along four hundred, then the points h, i, j will be respectively on the lines ce, cf, cg; also the sides bh, hi, ij of
the parallelogram nine hundred is equal to de i. xliii., cor. two ; but al is equal to nine hundred i. xxxvi.
thirty. place a given triangle so that its three sides shall pass through three given points.
let a then be moved by b, b by g, g by five hundred, and so on, each member of the series being moved by that which comes next to it.
two triangles, which have one angle in one equal to one angle in the other, and the sides about these angles reciprocally proportional, are equal in area.
thus let a be something that imparts motion but is unmoved, b something that is moved by a and moves g, g something that is moved by b but moves nothing (granted that we eventually arrive at g we may take it that there is only one intermediate term,
but there still remains the further point-why things when divided after their kinds do not cease to pass through one another and to change their place-which we will now proceed to explain.
the solution of a problem is the method of construction which accomplishes the required end.
further, we are to study growth and 'alteration'. we must inquire what each of them is; and whether 'alteration' is to be identified with coming-to-be, or whether to these different names there correspond two separate processes with distinct natures.
for all (the pluralist philosophers)- those who generate the 'elements' as well as those who generate the bodies that are compounded of the elements- make use of 'dissociation' and 'association', and of 'action' and 'passion'.
the rapid growth to maturity of profligates and the rapid ripening of seeds even when not packed close in the earth? and how is it with alterations? surely just the same: we may say that some alterations are violent while others are natural, e.g.
therefore five. a. the figure described on the line ab is equal to the sum of the similar figures described on the lines ac and bc.
five. given the difference of the squares on two lines and their ratio; find the lines.
but in fact this is not what we see occurring.
receiving the forms of earth and air, and experiencing all the affections which accompany these, presented a strange variety of appearances; and being full of powers which were neither similar nor equally balanced, was never in any part in a state of
again (b) we may assume weights which are commensurate; for it makes no difference whether we begin with the weight or with the mass.
dem.-if possible let one of them, such as aob, be greater than the other, dhe; and suppose a part such as aol to be equal to dhe. then since the circles are equal, and the angles aol, dhe at the centres are equal (hyp.), the arc al is equal to de xxvi.
(four) of the intersection of the external bisector of one base angle and the internal bisector of the other.
the process whereby what is of a certain quality changes to a condition of active existence is similar: thus the exercise of knowledge follows at once upon the possession of it unless something prevents it.
another kind is called copper, which is harder and yet lighter because the interstices are larger than in gold. there is mingled with it a fine and small portion of earth which comes out in the form of rust.
the fourth argument is that concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally
one. it contains sixteen propositions, of which four relate to triangles, four to squares, four to pentagons, and four miscellaneous propositions.
internal heat followed the air to and fro. these, as we affirm, are the phenomena of respiration.
now if we denote the first of any of the foregoing pairs of fractions by m- n, the second will be m one --n--; and in general, in the case of two incommensurable quantities, two fractions m -n and m one -n--- can always be found, where n can be made as
heaviness of the bodies intermediate between the absolutely light and heavy when compared either with one another or with these themselves.
then because abcd is a parallelogram, ad is equal to bc xxxiv. ; and since bcef is a parallelogram, ef is equal to bc; therefore (see fig. ()) take away ed, and in fig.
for we know how many horses there are by the use of the number; and again by using the one horse as unit we know the number of the horses itself. so it is with the time and the movement; for we measure the movement by the time and vice versa.
having discovered that these earlier works resembled it, decided that everything it had to say was borrowed.
in the second place, there is no necessity for the movent part to be moved by anything but itself: so it can only be accidentally that the other part moves it in return.
again, if any 'indivisible' is 'hard', there must also be one which is 'soft': but 'the soft' derives its very name from the fact that it suffers a certain action-for 'soft' is that which yields to pressure.
but while in some cases (for instance houses and the other products of manual labour) that principle is in something else external to the thing, in others those which may cause a change in themselves in virtue of a concomitant attribute-it lies in the
consequently, also, it is unknowable, qua infinite; for the matter has no form. (hence it is plain that the infinite stands in the relation of part rather than of whole.
there are, i say, these two kinds of body, and it is not the case, as some maintain, that all bodies have weight. different views are in fact agreed that there is a heavy body, which moves uniformly towards the centre. but is also similarly a light body.
four. if two opposite sides of a cyclic quadrilateral be produced to meet, and a perpendicular be let fall on the bisector of the angle between them from the point of intersection of the diagonals, this perpendicular will bisect the angle between the
the fourth kind are the inhabitants of the waters; these are made out of the most senseless and ignorant and impure of men, whom god placed in the uttermost parts of the world in return for their utter ignorance, and caused them to respire water instead
some of them may seem old-fashioned, but may nevertheless have had a great influence in promoting system and assisting enquiry, while in others we hear the latest word of physical or metaphysical philosophy.
for the matter of which these are composed does not admit of anything like the same regularity and finish as the substance of the enveloping body; since with each step away from earth the matter manifestly becomes finer in the same proportion as water is
now all these become causes of disease when the blood is not replenished in a natural manner by food and drink but gains bulk from opposite sources in violation of the laws of nature.
a difficulty may be raised as to whether every motion is commensurable with every other or not.
three. every two consecutive diagonals of a regular pentagon divide each other in extreme and mean ratio.
it is true that we speak of above and below, right and left, in these bodies relatively to ourselves.
such is the argument which is believed to establish the necessity of atomic magnitudes: we must now show that it conceals a faulty inference, and exactly where it conceals it.
but surely the discovery of the new world was preceded by a prophetic anticipation of it, which, like the hope of a messiah, was entering into the hearts of men?
if we draw together the opposite poles of plato's system, we find that, like spinoza, he combines idealism with fatalism.
and this is what, as a rule, we chiefly mean by a thing's being destroyed by time. still, time does not work even this change; even this sort of change takes place incidentally in time.
again, when we say a thing is 'moved', the predicate either (one) belongs to it actually, in virtue of its own nature, or (two) in virtue of something conjoined with it.
on the approach of one thing to another that the one causes motion and the other is moved, and when they are present under such conditions as rendered the one motive and the other movable.
so they too are inquiring whether the principle or element is one or many.
or supplemental. this proposition is nearly identical with vii.
twenty seven. what proposition is xix. a special case of?
thus far god, working according to an eternal pattern, out of his goodness has created the same, the other, and the essence (compare the three principles of the philebus-the finite, the infinite, and the union of the two), and out of them has formed the
by the motions next in order i mean increase and then alteration, decrease, and perishing. all these are posterior to becoming: consequently, if not even becoming is prior to locomotion, then no one of the other processes of change is so either.
oc is greater than og; but oc is equal to of i. def. xxxii. ; therefore of is greater than og-that is, a part greater than the whole, which is impossible. hence oc must be perpendicular to four hundred.
for any one who gives each element a shape of its own, and makes this the ground of distinction between the substances, has to attribute to them indivisibility; since division of a pyramid or a sphere must leave somewhere at least a residue which is not
every case. when to a, that which is in process of locomotion, b, the middle-point, serves both as a finishing-point and as a starting-point for its motion, a must come to a stand at b, because it makes it two just as one might do in thought.
analogy in modern times only points the way, and is immediately verified by experiment.
the science of nature is concerned with spatial magnitudes and motion and time, and each of these at least is necessarily infinite or finite, even if some things dealt with by the science are not, e.g.
triangle formed by the three remaining points, are proportional to the three rectangles ac, bd, dd.
and there you dwelt, having such laws as these and still better ones, and excelled all mankind in all virtue, as became the children and disciples of the gods.
plato, following his master, affirms this principle of the best, but he acknowledges that the best is limited by the conditions of matter.
now if one part of the stretch must have been traversed before another part (this is clear, that in the earlier and in the later part of the time a different part of the stretch has been traversed: for as the time lengthens a different part of the motion
the help of this internal warmth would form a very tolerable defence against the frost which surrounds it and attacks it from without.
the same as that of the bodies which compose it. what remains for us to consider, then, is whether any of the simple bodies can be infinite in magnitude, or whether this is impossible.
; but ef is equal to eb (const.); therefore the rectangle cf.af, together with ea , is equal to eb -that is i. xlvii. equal to ea ab . rejecting ea , which is common, we get the rectangle cf.af equal to ab .
therefore, since the moment is both a beginning and an end, there must always be time on both sides of it. but if this is true of time, it is evident that it must also be true of motion, time being a kind of affection of motion.)
as the fire returns to its place, it takes with it the minced food or blood; and in this way the veins are replenished. plato does not enquire how the blood is separated from the faeces.
we must therefore come to closer quarters with the subject of our inquiry. we must grapple' with it (as it were) from its beginning, and determine the precise character of the growing and diminishing whose causes we are investigating.
it is evident, then, that it is easy to refute the arguments by which they prove the existence of the void.
eight. the perimeter of a quadrilateral is greater than the sum of its diagonals.
but the first heaven finds it immediately with a single movement, and the bodies intermediate between the first and last heavens attain it indeed, but at the cost of a multiplicity of movement.
evidence to show that the distinction of prime from other numbers was known to him.
and this theory is exhibited in so many different points of view, that we cannot with any certainty interpret one dialogue by another; e.g. the timaeus by the parmenides or phaedrus or philebus.
which is intermediate between points is always a line and that which is intermediate between moments is always a period of time.
seven. if ch be produced, it meets bf at right angles.
twelve. prove that any right line through the intersection of the diagonals of a parallelogram bisects the parallelogram.
(for this will be a 'composition' instead of a 'blending' or 'combination': nor will every portion of the resultant exhibit the same ratio between its constituents as the whole.
three. in the last case, show that the dihedral angle between the planes is equal to the rectilineal angle between the normals.
let us try the primary body first, and then go on to consider the others.
it was framed after the pattern of the eternal nature, that it might resemble this as far as was possible; for the pattern exists from eternity, and the created heaven has been, and is, and will be, in all time.
and at the same time it is clear that all magnitude is also continuous; for the divisions of which time and magnitude respectively are susceptible are the same and equal.
one. how many parts in the hypothesis of this proposition? ans. three. name them.
but 'before' is used contrariwise with reference to past and to future time; for in the past we call 'before' what is farther from the 'now', and 'after' what is nearer, but in the future we call the nearer 'before' and the farther 'after'.
principles, and the nature of the substratum. whether the form or the substratum is the essential nature of a physical object is not yet clear. but that the principles are three, and in what sense, and the way in which each is a principle, is clear.
and this is true of vice in general, which is commonly regarded as disgraceful, whereas it is really involuntary and arises from a bad habit of the body and evil education.
for that which is in succession is in succession to a particular thing, and is something posterior: for one is not 'in succession' to two, nor is the first day of the month to be second: in each case the latter is 'in succession' to the former.
thus have i concisely given the result of my thoughts; and my verdict is that being and space and generation, these three, existed in their three ways before the heaven; and that the nurse of generation, moistened by water and inflamed by fire, and
there was yet a fifth combination which god used in the delineation of the universe.
their view is that the most precious place befits the most precious thing: but fire, they say, is more precious than earth, and the limit than the intermediate, and the circumference and the centre are limits.
it is only a legend that solon went to egypt, and if he did he could not have conversed with egyptian priests or have read records in their temples.
dem.-draw the perpendicular ph; then, since of, ph are perpendiculars on ab, and op is bisected in i, it is easy to see that ih if. again, since op, og are bisected in i, f; if one pg-that is, if one two the radius.
this is due to the 'intermingling' by which the things that come-to-be and pass-away are implicated with one another. for their matter is 'irregular', i.e.
and so both forms are used of the complex, 'becoming a musical man from an unmusical man', and unmusical man becoming a musical man'.
sol.-the construction is the same as the last, except that, instead of making the parallelogram kn equal to the excess of the parallelogram pc over the rectilineal figure ten, we make it equal to their sum; and then make pi equal to kl; draw ih parallel
dem. one.-abk : four hundred fifty :: ab : cd . xx. ;
yet the sum of the parts taken will not exceed every determinate magnitude, just as in the direction of division every determinate magnitude is surpassed in smallness and there will be a smaller part.
prop. twenty seven.-theorem. if a right line (ef) intersecting two right lines (ab, four hundred) makes the alternate angles (aef, efd) equal to each other, these lines are parallel.
prop. four.-theorem. two chords of a circle (ab, four hundred) which are not both diameters cannot bisect each other, though either may bisect the other.
but suppose that, as the body is being divided, a minute section-a piece of sawdust, as it were-is extracted, and that in this sense-a body 'comes away' from the magnitude, evading the division. even then the same argument applies.
then), if it is divisible through and through, there is no 'one', and therefore no 'many' either, but the whole is void; while to maintain that it is divisible at some points, but not at others, looks like an arbitrary fiction.
sol.-draw the diagonals ac, bd intersecting in o (see diagram to proposition vi.). o is the centre of the required circle.
dem.-join ac, bd, ad. let ad meet the plane kl in ten. join ex, xf. then because the parallel planes kl, mn are cut by the plane abd in the lines ex, bd, these lines are parallel eleven. xvi. . hence
if, therefore, the present cannot possibly have these characteristics, it follows that it must be the same present that belongs to each of the two times.
dem.-join ad. then if ad be not parallel to bc, let ae be parallel to it, and let it cut bd in e. join ec. now since the triangles bec, bac are on the same base bc, and between the same parallels bc, ae, they are equal xxxvii.
for plato never clearly saw that both elements had an equal place in mind and in nature; and hence, especially when we argue from isolated passages in his writings, or attempt to draw what appear to us to be the natural inferences from them, we are full
highest region, but 'as it may chance'. at any rate he says in his cosmogony that 'it happened to run that way at that time, but it often ran otherwise.' he tells us also that most of the parts of animals came to be by chance.
affection of a sacred part, is most justly called sacred. an acid and salt phlegm, again, is the source of all those diseases which take the form of catarrh, but they have many names because the places into which they flow are manifold.
also, when the sun and the moon, they say, and all the stars, so great in number and in size, are moving with so rapid a motion, how should they not produce a sound immensely great?
thus, from the being of the 'upper revolution' it follows that the sun revolves in this determinate manner; and since the sun revolves thus, the seasons in consequence come-to-be in a cycle, i.e.
fifty four. the sum of the perpendiculars drawn to the sides of an equiangular polygon from any point inside the figure is constant.
but when a thing has been affected and altered in any way we still call it by the original name: thus we speak of the bronze or the wax being dry or fluid or hard or hot.
their own interpretations of homer and the poets were supposed by them to be the original meaning.
phrase 'from spontaneity'. these 'spontaneous' events are said to be 'from chance' if they have the further characteristics of being the objects of deliberate intention and due to agents capable of that mode of action.
these are not the fixed modes in which spiritual truths are revealed to him, but the efforts of imagination, by which at different times and in various manners he seeks to embody his conceptions.
the larger the quantity of air the more readily it moves upward, and any portion of air without exception will rise up out of the water.
fifteen. any rectangle is equal to half the rectangle contained by the diagonals of squares described on its adjacent sides.
twenty. if a, b, p denote the sides of a right-angled triangle about the right angle, and the perpendicular from the right angle on the hypotenuse, -twelve a twelve- b twelve- p.
cor. two.-the perimeter of the rectangle ah is equal to ab, and is therefore independent of the position of the point five hundred on the line ab; and the area of the same rectangle is less than the square on half the line by the square on the segment
motion, then, being eternal, the first movent, if there is but one, will be eternal also: if there are more than one, there will be a plurality of such eternal movents.
four. right lines from any point in the diagonal of a parallelogram to the angular points through which the diagonal does not pass, and the diagonal, divide the parallelogram into four triangles which are equal, two by two.
this is clear too from the definition of the terms, whatever is destructible must be generated; for it is either ungenerated, or generated, but, if ungenerated, it is by hypothesis indestructible. whatever, further, is generated must be destructible.
if a magnitude (a) be the same multiple of another (b), which a magnitude (a) taken from the first is of a magnitude (b) taken from the second, the remainder is the same multiple of the remainder that the whole is of the whole (compare proposition i.).
secondly, it may be thought that the same reasoning which holds good of the rectilinear path applies also the circular, movement from a to b being opposed as contrary to movement from b to a. but what is meant is still rectilinear motion.
neither of these theories, however, is sound. for (a) if the forms are causes, why is their generating activity intermittent instead of perpetual and continuous-since there always are participants as well as forms?
been 'immediately-next' to point: but 'dissociation' takes place into small (i.e. relatively small) parts, and 'association' takes place out of relatively small parts.
hot-cold, white-black, dry-moist, soft-hard, and so forth) are, all of them, differences characterizing the 'elements'. the actual words of empedocles may be quoted in illustration-
not, surely, its voidness: for it is not the void only which is moved, but also the solid.
one. why does euclid describe the equilateral triangle on the side remote from a?
sol.-draw any two lines ac, ae making an angle. cut off ab equal ten, bc equal y , and ad equal y . join bd, and draw ce parallel to bd, then de is the third proportional required.
and therefore presupposes a previous state of unity and combination.
if a : b :: c : d, but the ratio of c : d greater than the ratio of e : f; then the ratio of a : b is greater than the ratio of e : f.
listen then, socrates, to a tale of solon's, who, being the friend of dropidas my great-grandfather, told it to my grandfather critias, and he told me.
yet, if a moves b, it is possible-as we sometimes express it-for a 'merely to touch' b, and that which touches need not touch a something which touches it. nevertheless it is commonly supposed that 'touching' must be reciprocal.
we have added the second part of this proposition. the indirect proof given of the first part in several editions of euclid is very inelegant; it is one of those absurd things which give many students a dislike to geometry.
one. if three right lines (a, b, one hundred) be proportional, the rectangle (a.one hundred) contained by the extremes is equal to the square (b ) of the mean. two.
for the middle is opposed in a sense to either of the extremes, as has been said above.
and things can also be compressed not into a void but because they squeeze out what is contained in them (as, for instance, when water is compressed the air within it is squeezed out); and things can increase in size not only by the entrance of something
there can be nothing divisible in an indivisible thing, but the attributes of bodies are all divisible in one of two ways.
moreover we have an indication that motion from a to b is the contrary of motion from b to a in the fact that, if they occur at the same time, they arrest and stop each other.
all was not mixed but one; and therefore it was not difficult for the later platonists to draw inferences by which they were enabled to reconcile the narrative of the timaeus with the mosaic account of the creation.
their right angles in a centre, and forming one equilateral quadrangle.
and state and prove corresponding theorems for the escribed circles.
the activity of lightness consists in the light thing being in a certain situation, namely high up: when it is in the contrary situation, it is being prevented from rising. the case is similar also in regard to quantity and quality.
the heaviest thing will be that which sinks to the bottom of all things that move downward, and the lightest that which rises to the surface of everything that moves upward.
postulates are the elements of geometrical construction, and occupy the same relation with respect to problems as axioms do to theorems.
he does not consider that ancient physical philosophy was not a free enquiry, but a growth, in which the mind was passive rather than active, and was incapable of resisting the impressions which flowed in upon it.
def. x.-if at the vertex o of a trihedral angle o-abc we draw normals oa, ob, oc to the faces obc, oca, oab, respectively, in such a manner that oa will be on the same side of the plane obc as oa, c., the trihedral angle o-abc is called the supplementary
the gods also mingled natures akin to that of man with other forms and perceptions. thus trees and plants were created, which were originally wild and have been adapted by cultivation to our use.
these premises clearly give the conclusion that there is in nature some bodily substance other than the formations we know, prior to them all and more divine than they. but it may also be proved as follows.
thing to pass through less of it in less time, the time must also be divisible, and conversely, if the time in which a thing is carried over the section a is divisible, this section a must also be divisible.
sixteen. if ab be the diameter of a semicircle, and pa, pb chords from any point p in the circumference, and if a perpendicular to ab from any point one hundred meet pa, pb in five hundred and e, and the semicircle in f, cf is a mean proportional between
it is clear then that what comes to be will come to be from these elements.
to sum up: being and generation and space, these three, existed before the heavens, and the nurse or vessel of generation, moistened by water and inflamed by fire, and taking the forms of air and earth, assumed various shapes.
a very different account must be given of the short work entitled 'timaeus locrus,' which is a brief but clear analysis of the timaeus of plato, omitting the introduction or dialogue and making a few small additions.
the concluding part of this proposition may be proved without joining ch, thus:-
a man may sometimes set aside meditations about eternal things, and for recreation turn to consider the truths of generation which are probable only; he will thus gain a pleasure not to be repented of, and secure for himself while he lives a wise and
appear suited; this is my feeling about the state which we have been describing.
twelve. the feet of the perpendiculars let fall on the sides of a triangle from any point in the circumference of the circumscribed circle are collinear (simson).
hence the triangles agree in every respect; therefore bc is equal to ef, the angle b is equal to the angle e, the angle one hundred to the angle f, and the triangle bac to the triangle edf.
one. if we take two ratios, such as six : nine and ten : fifteen, which are each equal to the same thing (in this example each is equal to two three), they are equal to one another (i. axiom i.). then we may write it thus-
their premisses are false and their conclusions do not follow. or rather the argument of melissus is gross and palpable and offers no difficulty at all: admit one ridiculous proposition and the rest follows-a simple enough proceeding.
she founded your city a thousand years before ours (observe that plato gives the same date (nine thousand years ago) for the foundation of athens and for the repulse of the invasion from atlantis (crit.).), receiving from the earth and hephaestus the seed
we assume that the alteration of equal by equal takes equal time, and the alteration of less by less or of greater by greater takes the same time, if the quantity of the patient is such as to keep the proportion which obtains between the agents, greater
now, since the difference between m- n and m- -one n namely, one n becomes small as n increases, we see that the difference between the ratio of two incommensurable quantities and that of two commensurable numbers m and n can be made as small as we
if on the other hand it is coming to a stand in only one of the two parts of the time, the whole cannot be the primary time in which it is coming to a stand: for it is coming to a stand in the whole time not primarily but in virtue of something distinct
completeness seems to require that something should be briefly said about other animals: first of women, who are probably degenerate and cowardly men.
(four) again, as a line which has a limit cannot be infinite, or, if it is infinite, is so only in length, so a surface cannot be infinite in that respect in which it has a limit; or, indeed, if it is completely determinate, in any respect whatever.
and as regards the atomists, it is not only clear what their explanation is: it is also obvious that it follows with tolerable consistency from the assumptions they employ.
that 'the hot' is a positive predication, i.e. a 'form', whereas 'cold' is a privation, and that earth and fire differ from one another by these constitutive differences.)
still, the head could not be left a bare globe of bone on account of the extremes of heat and cold, nor be allowed to become dull and senseless by an overgrowth of flesh.
but this last alternative is impossible: for then there will be nothing solid in the body (nothing beside the pores) but all of it will be void.
thirty six. if the exterior angles of a triangle be bisected, the three external triangles formed on the sides of the original triangle are equiangular.
moreover rotatory locomotion is prior to rectilinear locomotion, because it is more simple and complete, which may be shown as follows.
but we must assume that substance is not the attribute of anything, that the subject of which both 'biped' and 'animal' and each separately are predicated is the subject also of the complex 'biped animal'.
i will therefore now proceed to speak of the higher use and purpose for which god has given them to us.
fifty four. a circle rolls inside another of double its diameter; find the locus of a fixed point in its circumference.
be balanced by an equal transformation of air into water (for it is clear that the air produced from water is bulkier than the water): it is necessary therefore, if compression does not exist, either that the next portion will be pushed outwards and make
and in what way and how and when, both in the world of generation and in the world of immutable being.
time. but in that case perpetual acceleration is impossible (and, equally, perpetual retardation: for the argument holds of both and each), if we may take acceleration to proceed by identical or increasing additions of speed and for an infinite time.
in other words-given the base of a triangle and the vertical angle, the locus of the vertex is a circle.
eighty. if through the middle points of each of the two diagonals of a quadrilateral we draw a parallel to the other, the lines drawn from their points of intersection to the middle points of the sides divide the quadrilateral into four equal parts.
nine. what is meant by an angle standing on a segment?
this conclusion that local movement is not continued to infinity is corroborated by the fact that earth moves more quickly the nearer it is to the centre, and fire the nearer it is to the upper place.
touch the three sides of the triangle abc; and therefore the circle def is inscribed in the triangle abc.
acquainted with the early pythagoreans to know how far the statements contained in these fragments corresponded with their doctrines; and we therefore cannot pronounce, either in favour of the genuineness of the fragments, with bockh and zeller, or, with
whether we postulate these, or other things of a similar kind, makes no difference. for we are trying to discover not what undergoes these changes, but what is their characteristic manner.
every use of them, even the most trivial, assured men of their truth; they were everywhere to be found, in the least things and the greatest alike.
draw from the centre the lines ab, ac, and let their extremities be joined by the straight line bc. the line ad, drawn to the base of the triangle, will be shorter than either of the radii.
six. if through a point o, in the production of the diagonal ac of a parallelogram abcd, any right line be drawn cutting the sides ab, bc in the points e, f, and ed, fd be joined, the triangle efd is less than half the parallelogram.
the same argument applies as against those who think that place is something separate, into which things are carried; viz. how will what is placed in it move, or rest?
increase or of decrease: that which comes between the two has to be included.
the yielding matter is that which has the slenderest base, whereas that which has a rectangular base is compact and repellent. light and heavy are wrongly explained with reference to a lower and higher in place.
but if it is a particular, there will be a distinction between the being of 'this universe' and of 'universe' unqualified.
def.-a line in any figure, such as ac in the preceding diagram, which is such that, by folding the plane of the figure round it, one part of the diagram will coincide with the other, is called an axis of symmetry of the figure.
four. make at the centre o the angle pof equal to pod. join pf. then the triangles pod, pof have the two sides op, od in one respectively equal to the sides op, of in the other, and the angle pod equal to the angle pof; hence pd is equal to pf i. iv.
cor.-if aob be a straight line, acb will be a right angle-that is, the angle in a semicircle is a right angle (compare xxxi.).
hence it follows, by axiom viii., that corresponding parts or portions of congruent figures are congruent, and that congruent figures are equal in every respect.
consequently, everything that is in motion must be moved by something: for that which is in motion will always be divisible, and if a part of it is not in motion the whole must be at rest.
this statement might well cause surprise.
then ao, bo meet in o three. xi. . now, because the triangles oaa, oab are isosceles, the angle oba oab oaa. hence oa is parallel to ob; therefore oc : oc :: oa : ob; that is, in a given ratio. hence one hundred is a given point.
pervading of them, being created by the cutting action of fire and by the impression which it makes on a moist substance; and hence the liquid which circulates in the body has a colour such as we have described.
a one p.bc, a one p.four hundred. hence aa bccd-, or a : a :: bc : four hundred. in extending this proof to parallelograms we have only to use p instead of p.
further, since everything that is moved is moved by something, the cause of the irregularity of movement must lie either in the mover or in the moved or both.
since a house is of such-and-such a kind, certain things must necessarily come to be or be there already, or since health is this, these things must necessarily come to be or be there already. similarly if man is this, then these; if these, then those.
for if we adopt the first assumption we have to make it apply within each of the very lowest species into which motion can be divided: e.g.
dem.-let the supplemental chords corresponding to one, two, c., be denoted by r , r , c.; then three. xxxv. ex. two , we have
for, let i denote the point at infinity through which the two given lines pass, and draw any two parallels intersecting them in the points a, b; a, b; then the triangles aib, aib are equiangular; therefore ai : ab :: ai : ab; but the first term of the
two. to prove that bh, which is not perpendicular to ab, cuts the circle. draw cg perpendicular to hb. now bc is equal to cg gb . therefore bc is greater than cg , and bc is greater than cg. hence note on i., def. xxxii.
the one is 'agency' and the other 'patiency'; and the outcome and completion of the one is an 'action', that of the other a 'passion'. since then they are both motions, we may ask: in what are they, if they are different?
the danger, however, is not so great when the foundation remains, for then there is a possibility of recovery.
forty three. in a given circle inscribe a triangle so that two sides may pass through two given points, and that the third side may be a maximum or a minimum.
for just as the geometer has nothing more to say to one who denies the principles of his science-this being a question for a different science or for or common to all-so a man investigating principles cannot argue with one who denies their existence.
if, therefore, these characteristic differences be taken away (for they can be taken away, since they came-to-be), it will clearly be inevitable for earth to come to-be out of water and water out of earth, and for each of the other elements to undergo a
why, then, is this form of change necessarily ceaseless? is it because the passing-away of this is a coming-to-be of something else, and the coming-to-be of this a passing-away of something else?
remain still. this view is ingenious but not true. the argument would prove that everything, whatever it be, which is put at the centre, must stay there. fire, then, will rest at the centre: for the proof turns on no peculiar property of earth.
in like manner we may show that the sum of the angles a, b, or of the angles a, one hundred, is less than two right angles.
hence the body in question would have been present in our world here, alongside air and fire and earth and water: but nothing of the kind is observed.
this proposition is the converse of iv., and is the second case of the congruence of triangles in the elements.
this process, as we affirm, the name-giver named inspiration and expiration.
(six) we should not lay too much stress on aristotle or the writer de caelo having adopted the other interpretation of the words, although alexander of aphrodisias thinks that he could not have been ignorant either of the doctrine of plato or of the sense
stages of the process, and in quantitative change there are the intervening degrees of magnitude: and in becoming and perishing the same thing is true.
prop. twelve.-theorem. if two circles (pcf, pde) have external contact at any point p, the line joining their centres must pass through that point.
remaining water, becomes rock. rock, when it is made up of equal particles, is fair and transparent, but the reverse when of unequal.
will time then fail? surely not, if motion always exists. is time then always different or does the same time recur? clearly time is, in the same way as motion is.
seventy nine. if on the sides ab, ac of a triangle abc we take two points five hundred, e, and on their line of connexion f, such that
again, if there is a motion of the whole other than dz, say the the of each of the arts may be subtracted from it: and these motions will be equal to de, ez respectively: for the motion of that which is one must be one.
finds hard to carry out in action, and still harder adequately to represent in language.
six. what is meant by the ratio of incommensurable magnitudes?
now the primary when that has reference to the end of the change is something really existent: for a change may really be completed, and there is such a thing as an end of change, which we have in fact shown to be indivisible because it is a limit.
are we to say then that all their movements, even those which are mutually contrary, are due to constraint? no, for a body which has no natural movement at all cannot be moved by constraint.
ce.ed fe fd two. v. , and of of . hence, adding, since fe of oe i. xlvii. , and fd of od , we get
what if equal pains were taken to smooth and polish their manners? one piece of good sense would be more memorable than a monument as high as the moon. i love better to see stones in place. the grandeur of thebes was a vulgar grandeur.
the difference of function will persist, but will appear not to by reason of the regularity of shape. in the same fashion must we conceive of the beginning of its movement.
eleven. find a line whose square shall be equal to the difference of the squares on two lines.
in either case the force transmits the movement to the body by first, as it were, impregnating the air. that is why a body moved by constraint continues to move when that which gave the impulse ceases to accompany it. otherwise, i.e.
again, the sum of the sides de, ec of the triangle dec is greater than six hundred: to each add bd, and we get the sum of be, ec greater than the sum of bd, six hundred; but it has been proved that the sum of ba, ac is greater than the sum of be, ec.
parallelograms, each containing twelve square inches, the square foot contains one hundred forty four square inches.
for the one is an 'element' in so far as it underlies the process as matter-as that out of which earth and fire come-to-be through a change of qualities due to 'the motion'.
on this account one submit the following alternative proof, which, however, makes use of a new definition of the duplicate ratio of two lines, viz. the ratio of the squares (see annotations on five. def. x.) described on these lines.
third degree. and having made it he divided the whole mixture into souls equal in number to the stars, and assigned each soul to a star; and having there placed them as in a chariot, he showed them the nature of the universe, and declared to them the laws
with their order-one mean the position of each, as involving the priority of some and the posteriority of others, and their respective distances from the extremity-with this astronomy may be left to deal, since the astronomical discussion is adequate.
book eleven. theory of planes, coplanar lines, and solid angles definitions.
as regards the second objection, no absurdity is involved in the fact that something not in motion may be set in motion, that which caused the motion from without being at one time present, and at another absent.
he ought, then, either to have defined or to have postulated these characteristic movements, or to have demonstrated them-whether strictly or laxly or in some other fashion.
book four. inscription and circumscription of triangles and of regular polygons in and about circles definitions.
hence the sum of the angles forming the solid angle is less than four right angles.
a brief mention may be made of the generation of other animals, so far as the subject admits of brevity; in this manner our argument will best attain a due proportion. on the subject of animals, then, the following remarks may be offered.
eleven. find in two parallels two points which shall be equidistant from a given point, and whose line of connexion shall be parallel to a given line.
in this, as in every other part of creation, one suppose god to have made things, as far as was possible, fair and good, out of things not fair and good.
two. divide a given line so that the rectangle contained by its segments may be equal to a given square, not exceeding the square on half the given line.
so when there are parts, a thing will be in itself, as 'white' is in man because it is in body, and in body because it resides in the visible surface. we cannot go further and say that it is in surface in virtue of something other than itself.
what was true of them to things perceived. others, perhaps intentionally, maintain precisely the contrary opinion to this.
living the best and most selfsufficient of lives.
therefore the angle ade is right. hence de is at right angles both to ad and bd. therefore eleven. iv. de is perpendicular to four hundred, which is coplanar and concurrent with ad and bd.
but its susceptibility varies in degree, according as it is more or less; such-and such, and one would be more justified in speaking of 'pores' in this connexion: for instance, in the metals there are veins of 'the susceptible' stretching continuously
so the unnatural remaining of a thing is contrary to its natural motion, just as we find a similar contrariety in the motion of the same thing: one of its motions, the upward or the downward, will be natural, the other unnatural.
he uses the method of his time only to transcend it; out of the abstract and general he creates the concrete and universal.
and now a favourite speculation of modern chemistry is the explanation of qualitative difference by quantitative, which is at present verified to a certain extent and may hereafter be of far more universal application.
plainly, then, neither will everything that does not exist be in time, i.e. those non-existent things that cannot exist, as the diagonal cannot be commensurate with the side.
it necessarily follows from this that infinite weight, if there is such a thing, being, on the one hand, as great and more than as great as the finite, will move accordingly, but being, on the other hand, compelled to move in a time inversely
what nature are we to attribute to this new kind of being? we reply, that it is the receptacle, and in a manner the nurse, of all generation.
since, then, it would require a force greater than itself to move it, it must needs stay at the centre.
twenty nine. in the same case, if the lines be drawn from any point in the circumference of the inscribed circle, prove that the sum of their squares is equal to n times the sum of the squares of the radii of the inscribed and the circumscribed circles.
again, water will result from air if the hot be overcome by the cold: for air, as we saw, is hot and moist while water is cold and moist, so that, if the hot changes, there will be water.
sol.-take on any line ac parts ab, bc respectively equal to ten, y . on ac describe a semicircle adc. erect bd at right angles to ac, meeting the semicircle in five hundred. bd will be the mean proportional required.
none the less, if the two motions are commensurable, we are confronted with the consequence stated above, viz. that there may be a straight line equal to a circle.
v.); but bk is the rectangle contained by bf and bc. hence the rectangle contained by a and bc is bk. in like manner the rectangle contained by a and bd is bg. again, since a is equal to bf (const.), and bf is equal to dg i. xxxiv. , a is equal to dg.
everything, we say, that undergoes alteration is altered by sensible causes, and there is alteration only in things that are said to be essentially affected by sensible things. the truth of this is to be seen from the following considerations.
and the causes of the man's coming and getting the money (when he did not come for the sake of that) are innumerable. he may have wished to see somebody or been following somebody or avoiding somebody, or may have gone to see a spectacle.
nor is anaxagoras right about the coming to be of homogeneous bodies. it is true there is a sense in which clay is divided into pieces of clay, but there is another in which it is not.
ninety four. if a variable circle touch two equal circles, one internally and the other externally, and perpendiculars be let fall from its centre on the transverse tangents to these circles, the rectangle of the intercepts between the feet of these
for the matter and the infinite are contained inside what contains them, while it is the form which contains.
it can only be, then, that it occupies less time, and thus we get the necessary consequence that the quicker will pass over an equal magnitude (as well as a greater) in less time than the slower.
for the investigation of special attributes comes after that of the common attributes.
the result of their view is that earth has the best right to the name element, and is alone indestructible; for that which is indissoluble is indestructible and elementary, and earth alone cannot be dissolved into any body but itself.
(two) they reason from the fact that some things are observed to contract and be compressed, as people say that a cask will hold the wine which formerly filled it, along with the skins into which the wine has been decanted, which implies that the
to ab, and the rest of the construction as before.
thus we have finished the discussion of the universe, which, according to our original intention, has now been brought down to the creation of man.
dem.-describe a circle about the triangle, and produce four hundred to meet it in e; then it is easy to see that the triangles acd, ecb are equiangular. hence iv.
and about all of them this has to be considered:-that the tendency of each towards its kindred element makes the body which is moved heavy, and the place towards which the motion tends below, but things which have an opposite tendency we call by an
words are applicable only to becoming, and not to true being; and equally wrong are we in saying that what has become is become and that what becomes is becoming, and that the non-existent is non-existent...these are the forms of time which imitate
there is also white phlegm, formed by the decomposition of young and tender flesh, and covered with little bubbles, separately invisible, but becoming visible when collected.
if, then, it is by constraint that earth moves from a certain place to the centre here, its movement from here to there will be natural, and if earth from there rests here without constraint, its movement hither will be natural.
but may we say that things are always commensurable if the same terms are applied to them without equivocation? e.g.
than which no greater good ever was or will be given by the gods to mortal man. this is the greatest boon of sight: and of the lesser benefits why should i speak? even the ordinary man if he were deprived of them would bewail his loss, but in vain.
five. given the sum or the difference of two lines and the difference of their squares; find the lines.
for they do not attain the final end, but only come as near to it as their share in the divine principle permits.
seventeen. define reciprocal proportion.
movement tends to create fire in wood, stone, and iron; and with even more reason should it have that effect on air, a substance which is closer to fire than these.
but they are not so, for they are incapable of reason or intellect; the only being which can properly have mind is the invisible soul, whereas fire and water, and earth and air, are all of them visible bodies.
five. draw a line parallel to the base of a triangle so that it may be-one. equal to the intercept it makes on one of the sides from the extremity of the base; two.
proving the existence of void), or the whole body must be void, if it is increased in every part and is increased by means of void. the same argument applies to the ashes.
numerically, the motions also will differ generically or numerically as the case may be.
every locomotion, as we said before, is either rotatory or rectilinear or a compound of the two: and the two former must be prior to the last, since they are the elements of which the latter consists.
for if the particles were divisible there would be a part of fire which was not fire and a part of earth which was not earth, for the reason that not every part of a pyramid is a pyramid nor of a cube a cube.
of the wine's being wine, and the wine will be in the jar in virtue not of its being a jar but of the jar's being a jar.
prop. thirty nine.-theorem. equal triangles (bac, bdc) on the same base (bc) and on the same side of it are between the same parallels.
to this he was probably led by the fixedness of egyptian customs and the general observation that there were other civilisations in the world more ancient than that of hellas.
similar gossiping stories are told about the sources of the republic and the phaedo. that there really existed in antiquity a work passing under the name of philolaus there can be no doubt.
; therefore agb must be obtuse; hence dfe, which is equal to it, is obtuse; and it has been proved that acb is acute; therefore the angles acb, dfe are of different species; but (hyp.) they are of the same species, which is absurd.
there will not be continuity between any motion and any other indiscriminately any more than there is between any two things chosen at random in any other sphere: there can be continuity only when the extremities of the two things are one.
abg and at rest in any part (and consequently in the whole) of it.
for let flesh be extracted from water and again more flesh be produced from the remainder by repeating the process of separation: then, even though the quantity separated out will continually decrease, still it will not fall below a certain magnitude.
twenty seven. what is the locus of the centres of all circles touching a given circle in a given point?
six. if two s abc, abd be on the same base ab, and between the same parallels, and if a parallel to ab meet the sides ac, bc in the point e, f; and the sides ad, bd in the point g, h; then ef gh.
quality respectively? or is there also a difference in the manner of the change, since it is evident that, whereas neither what is 'altering' nor what is coming-to-be necessarily changes its place, what is growing or diminishing changes its spatial
one. if two circles intercept equal chords ab, ab on any secant, the tangents at, at to the circles at the points of intersection are to one another as the radii of the circles.
join ab, intersecting the circles in one hundred and five hundred, and join bp. now since a is the centre of the circle pcf, ap is equal to ac; and since b is the centre of the circle pde, bp is equal to bd.
there is, however, we may answer, this difference: that if the states are two then it follows simply from this fact that the activities must also in point of number be two (for only that which is numerically one can give rise to an activity that is
now (i) we do not speak of the wood as 'combined' with the fire, nor of its burning as a 'combining' either of its particles with one another or of itself with the fire: what we say is that 'the fire is coming-to-be, but the wood is 'passing-away'.
earth, then, and bodies in which earth preponderates, must needs have weight everywhere, while water is heavy anywhere but in earth, and air is heavy when not in water or earth. in its own place each of these bodies has weight except fire, even air.
(b) the timaeus contains an assertion perhaps more distinct than is found in any of the other dialogues (rep.; laws) of the goodness of god. 'he was good himself, and he fashioned the good everywhere.'
but for the generated there must be such a time either actually or potentially, though not for a and b in either way. one hundred then will be, and also not be, for a limited length of time, and this is true also of five hundred, the destructible.
this is a conclusion, the necessity of which we have already decided, and we have seen further that rest also will be inconceivable, since rest, like movement, is either natural or constrained.
what was the primary agent or patient?' and so at each step of the series.
dem.-the three lines bc, gh, cf are in continued proportion; therefore bc : cf in the duplicate ratio of bc : gh five. def. x. ; and since the figures bcd, ghi are similar, bcd : ghi in the duplicate ratio of bc : gh xx.
it is usual with commentators on euclid to say that he allows the use of the rule and compass. were such the case this proposition would have been unnecessary.
in like manner the sum of the angles aob, agb is two right angles; therefore the sum of aoe, afe is equal to the sum of aob, agb; but the angles aoe, aob are equal, because they stand on equal arcs ae, ab three. xxvii. .
(for my definition of matter is just this-the primary substratum of each thing, from which it comes to be without qualification, and which persists in the result.) and if it ceases to be it will pass into that at the last, so it will have ceased to be
again, the theory that the void exists involves the existence of place: for one would define void as place bereft of body.
the conclusion of the discourse favours not this purely abstract being, but a state of savagery intermediate between the "natural" and the "social" conditions, in which men may preserve the simplicity and the advantages of nature and at the same time
if two parallel lines (ab, four hundred) be cut by three parallel planes (gh, kl, mn) in two triads of points (a, e, b; one hundred, f, five hundred), their segments between those points are proportional.
now, since ac is bisected in p, ek is also bisected in p; hence i. xxxvi. the parallelogram eo is equal to ok; therefore eo is greater than dk; but dk is equal to fd i. xliii. ; hence eo is greater than fd.
prop. twenty three.-problem. at a given point (a) in a given right line (ab) to make an angle equal to a given rectilineal angle (def).
since 'nature' has two senses, the form and the matter, we must investigate its objects as we would the essence of snubness. that is, such things are neither independent of matter nor can be defined in terms of matter only.
when all of them, both those who show themselves in the sky, and those who retire from view, had come into being, the creator addressed them thus:-'gods, sons of gods, my works, if i will, are indissoluble.
for what is bounded by the 'now' is thought to be time-we may assume this.
and others are not and of the continuous process of change: and this causes the motion of the other movents, while they are the causes of the motion of other things.
still, as we have said, it is possible to construct 'alteration' and coming-to-be with them, if one 'transposes' the same by 'turning' and 'intercontact', and by 'the varieties of the figures', as democritus does.
thus, if we take away either of the parallelograms ao, oc from the parallelogram ac, the remainder is called a gnomon.
wherefore it was covered by a peel or skin which met and grew by the help of the cerebral humour. the diversity of the sutures was caused by the struggle of the food against the courses of the soul.
since, then, motion can belong neither to being nor to relation nor to agent and patient, it remains that there can be motion only in respect of quality, quantity, and place: for with each of these we have a pair of contraries.
either greatness and smallness or perfection and imperfection of magnitude: and changes to the respective contraries are contrary changes.
and since 'nature' means two things, the matter and the form, of which the latter is the end, and since all the rest is for the sake of the end, the form must be the cause in the sense of 'that for the sake of which'.
for each several bodily defect or excellence involves a relation with those things from which the possessor of the defect or excellence is naturally subject to alteration: thus excellence disposes its possessor to be unaffected by these influences or to
but the fact is that increase of quantity makes many things move the faster downward. for these reasons, then, as well as from the distinction already established of a plurality of natural movements, it is impossible that there should be only one element.
nineteen. what are the latin terms in use to denote some of the propositions of book five.?
thirteen. the triangle formed by joining the middle point of one of the non-parallel sides of a trapezium to the extremities of the opposite side is equal to half the trapezium.
well, then, if place is none of the three-neither the form nor the matter nor an extension which is always there, different from, and over and above, the extension of the thing which is displaced-place necessarily is the one of the four which is left,
for the motion or locomotion is made one by the thing which is moved, because it is one-not because it is one in its own nature (for there might be pauses in the movement of such a thing)-but because it is one in definition: for this determines the
one direction being given. the same reasoning will also show the falsity of the assumption that infinite length can be traversed in a finite time.
while the privation in no sense is. they, on the other hand, identify their great and small alike with not being, and that whether they are taken together as one or separately. their triad is therefore of quite a different kind from ours.
may help us to judge whether it will satisfy his requirements or not.
has been said by us is probable, and will be rendered more probable by investigation. let us assume thus much.
their anxiety to save their hypothesis, cannot even admit this of every perceptible thing.
timaeus: certainly, and we will do all that we can; having been handsomely entertained by you yesterday, those of us who remain should be only too glad to return your hospitality.
thirty seven. the area of any regular polygon of more than four sides circumscribed about a circle is less than the square of the diameter.
the bodily elements are three, and therefore the places of the elements will be three also; the place, first, of the body which sinks to the bottom, namely the region about the centre; the place, secondly, of the revolving body, namely the outermost
that it contains no specific difference and therefore no difference in the motion that takes place over it.
since, therefore, the be dy is divisible through and through, let it have been divided. what, then, will remain? a magnitude?
sol.-bisect any two sides bc, ac in the points five hundred, e. erect do, eo at right angles to bc, ca; then o, the point of intersection of the perpendiculars, is the centre of the required circle.
bf is a right line; and, since it is coplanar with bd, be, which are each perpendicular to ab, it is eleven. iv. perpendicular to ab. therefore the angle abf is right; and the angle abc is right (hyp.).
five. the medians of a triangle divide each other in the ratio of two : one.
alteration, as things which 'turn' in respect of their material substance.
the answer to the view that there are infinite bodies moving in an infinite is that, if the cause of movement is single, they must move with a single motion, and therefore not without order; and if, on the other hand, the causes are of infinite variety,
any one then who insists upon an exact statement of this kind of theory, instead of assenting after a passing glance at it, will see that it removes generation from the world.
the fact is that the finer body takes up more room, as is obvious in any case of transformation. as the liquid is converted into vapour or air the vessel which contains it is often burst because it does not contain room enough.
the atoms differ in figure, and all figures are composed of pyramids, rectilinear the case of rectilinear figures, while the sphere has eight pyramidal parts.
these cannot exist in what is uniform, and therefore motion is due to want of uniformity. but then why, when things are divided after their kinds, do they not cease from motion?
but if one is assigned to the triangle, the circle will not be a figure at all.
one. equiangular parallelograms (ab, four hundred) which are equal in area have the sides about the equal angles reciprocally proportional-ac : ce :: gc : cb. two.
one. the two tangents pb, pd (fig. two) are equal to one another, because the square of each is equal to the square of op minus the square of the radius.
my farm outgoes for the first season were, for implements, seed, work, c., fourteen.seven thousand two hundred twelve. the seed corn was given me. this never costs anything to speak of, unless you plant more than enough.
cor. three.-the circumference of a circle is everywhere concave towards the centre.
all of them are combinations of bright and red with white and black. plato himself tells us that he does not know in what proportions they combine, and he is of opinion that such knowledge is granted to the gods only.
but the principles in question are infinite both in multitude and in kind. therefore it is impossible to know things which are composed of them; for it is when we know the nature and quantity of its components that we suppose we know a complex.
again, since the whole revolves, palpably and by assumption, in a circle, and since it has been shown that outside the farthest circumference there is neither void nor place, from these grounds also it will follow necessarily that the heaven is spherical.
what then after all is place? the answer to this question may be elucidated as follows.
or thus: a tangent is a limiting position of a secant, namely, when the secant moves out until the two points of intersection with the circle become consecutive; but the line through the centre which bisects the part of the secant within the circle iii.
twenty five. what other name is applied to them? ans. they are said to be identically equal.
iv. a rectilineal figure which is both equilateral and equiangular is said to be regular.
dem.-if the angle agh be not equal to ghd, one must be greater than the other.
with regard to the shape of each star, the most reasonable view is that they are spherical.
thing, itself and its parts, occupies the same space.
the demonstration is the proof, in the case of a theorem, that the conclusion follows from the hypothesis; and in the case of a problem, that the construction accomplishes the object proposed.
cor.-two circles not coinciding cannot have more than two points common. compare i., axiom x., that two right lines not coinciding cannot have more than one point common.
eight. how is the area of a parallelogram found?
but this characteristic must be preserved: for the growth we are discussing has been assumed to be thus characterized.
for a less agent will produce that movement in a less patient in an equal time, and the proportionate equivalent of that patient will be a finite quantity, since no proportion holds between finite and infinite.
and hence, when any impurities arise in the region of the liver by reason of disorders of the body, the loose nature of the spleen, which is composed of a hollow and bloodless tissue, receives them all and clears them away, and when filled with the
it is not necessary, therefore, that that which is moved should always be moved by something else that is itself moved by something else: so there will be an end to the series.
we do not call the statue 'bronze' or the pyramid 'wax' or the bed 'wood', but we use a derived expression and call them 'of bronze', 'waxen', and 'wooden' respectively.
eight. if two intersecting right lines be respectively parallel to two others, the angle between the former is equal to the angle between the latter.
about the greatest action which the athenians ever did, and which ought to have been the most famous, but, through the lapse of time and the destruction of the actors, it has not come down to us.
it is obvious, then, that the greater part of the inquiry into nature concerns bodies: for a natural substance is either a body or a thing which cannot come into existence without body and magnitude.
the old man, as i very well remember, brightened up at hearing this and said, smiling: yes, amynander, if solon had only, like other poets, made poetry the business of his life, and had completed the tale which he brought with him from egypt, and had not
one. construct a rectangle equal to the sum of two or any number of rectilineal figures.
there have been, and will be again, many destructions of mankind arising out of many causes; the greatest have been brought about by the agencies of fire and water, and other lesser ones by innumerable other causes.
hence in one sense it is true that any and every part of the flesh has grown; but in another sense it is false. for there has been an accession to every part of the flesh in respect to its form, but not in respect to its matter.
we physicists, on the other hand, must take for granted that the things that exist by nature are, either all or some of them, in motion which is indeed made plain by induction.
but however that may be, we should endeavour as far as we can by education, and studies, and learning, to avoid vice and attain virtue; this, however, is part of another subject.
dem.-let m denote the multiples which the magnitudes a, a are of b, b; then we have
two. prove that af is perpendicular to de.
also we assign the smallest body to fire, and the greatest to water, and the intermediate in size to air; and, again, the acutest body to fire, and the next in acuteness to air, and the third to water.
i. if two rectilineal figures be so related that the angular points of one lie on the sides of the other-one, the former is said to be inscribed in the latter; two, the latter is said to be described about the former.
if two triangles (abc, def) have their sides proportional (ba : ac :: ed : df; ac : cb :: df : fe) they are equiangular, and those angles are equal which are subtended by the homologous sides.
again, four; six; three, five are called alternate angles; lastly, one, five; two, six; three, eight; four, seven are called corresponding angles.
four. if a rectangular parallelogram be such that two adjacent sides contain respectively m and n linear units, by dividing one side into m and the other into n equal parts, and drawing parallels to the sides, the whole area is evidently divided into mn
one-not in an accidental sense (i.e. it must be one as the white that blackens is one or coriscus who walks is one, not in the accidental sense in which coriscus and white may be one), nor merely in virtue of community of nature (for there might be a case
by something else and that which derives its motion from itself: and that which is further from the thing that is moved is nearer to the principle of motion than that which is intermediate.
the mover too is moved, as has been said-every mover, that is, which is capable of motion, and whose immobility is rest-when a thing is subject to motion its immobility is rest. for to act on the movable as such is just to move it.
'hot' is that which 'associates' things of the same kind (for 'dissociating', which people attribute to fire as its function, is 'associating' things of the same class, since its effect is to eliminate what is foreign), while 'cold' is that which brings
tell us, said the other, the whole story, and how and from whom solon heard this veritable tradition.
two. let the vertex of one triangle adb fall within the other triangle acb. produce the sides ac, ad to e and f. then because ac is equal to ad (hyp.), the triangle acd is isosceles, and v.
thus we see that this singular theory is dependent on two principles largely employed by plato in explaining the operations of nature, the impossibility of a vacuum and the attraction of like to like.
the arrow-heads denote the direction or sense, as it is technically termed, in which the line oa turns. it is usual to call the direction indicated in the above figures positive, and the opposite negative.
it is impossible that there should be increase without the previous occurrence of alteration: for that which is increased, although in a sense it is increased by what is like itself, is in a sense increased by what is unlike itself: thus it is said that
it is nature's work: for if anything is of a certain character naturally, it either is so invariably and is not sometimes of this and sometimes of another character (e.g.
let a : b :: c : d; then a : a b :: c : c d.
dem.-(one). let ab be the diameter, one hundred any point in the semicircle. join ac, cb. the angle acb is a right angle.
for if we take a part which shall be a measure of the whole time, in this part a certain fraction, not the whole, of the magnitude will be traversed, because we assume that the traversing of the whole occupies all the time.
in general, our quarrel with those who speak of movement in this way cannot be confined to the parts; it concerns the whole universe.
were of an elder generation parents and grandparents, and those of a younger, children and grandchildren.
the same reasoning applies equally to magnitude, to time, and to motion: either all of these are composed of indivisibles and are divisible into indivisibles, or none. this may be made clear as follows.
there is, on the one hand, (a) something which comes into existence, and again (b) something which becomes that-the latter (b) in two senses, either the subject or the opposite.
if, on the other hand, we investigate the question more in accordance with principles appropriate to physics, we are led as follows to the same result.
nine. if through one of the points of intersection of two circles we draw two common chords, the lines joining the extremities of these chords make a given angle with each other.
sixty two. if the base and the difference of the base angles of a triangle be given, the rectangle contained by the perpendiculars from the vertex on two lines through the middle point of the base, parallel to the internal and external bisectors of the
on these grounds, then, it might be inferred either that there are or that there might be several heavens. we must, however, return and ask how much of this argument is correct and how much not.
we have now to consider the terms 'heavy' and 'light'. we must ask what the bodies so called are, how they are constituted, and what is the reason of their possessing these powers.
that 'it is absolutely necessary for a house to come-to-be when foundations have been laid': for (unless it is always necessary for a house to be coming-to-be) we should be faced with the consequence that, when foundations have been laid, a thing, which
that we see the above-mentioned changes occurring in the case of the same things.
arguing from probabilities, i am of opinion that they are one; another, regarding the question from another point of view, will be of another mind.
four. if equilateral triangles be described on the sides of any triangle, the distances between the vertices of the original triangle and the opposite vertices of the equilateral triangles are equal.
moreover, among the seeds anything must have come to be at random. but the person who asserts this entirely does away with 'nature' and what exists 'by nature'.
it is neither in the sense of the matter of existents (for nothing is composed of it), nor as the form and definition of things, nor as end, nor does it move existents.
hence, just as they deem themselves to live and to 'be' in virtue of their perceiving or their capacity to perceive, so too they deem the things to 'be' qua perceived or perceptible-and in this they are in a sense on the track of the truth, though what
of the liver, corrects all things and makes them to be right and smooth and free, and renders the portion of the soul which resides about the liver happy and joyful, enabling it to pass the night in peace, and to practise divination in sleep, inasmuch as
to these a priori speculations he would add a rude conception of matter and his own immediate experience of health and disease.
in the case that we have just been considering it is the fact that health is one and the same that enables us to arrive at the conclusion that the one alteration is neither more nor less than the other, but that both are alike.
thus the 'comings-to-be' and the 'dissociations' result from the 'indivisibles' (a) according to leucippus through the void and through contact (for it is at the point of contact that each of the composite bodies is divisible), but (b) according to plato
square on the line joining the points of bisection.
it is natural too to suppose that in number there is a limit in the direction of the minimum, and that in the other direction every assigned number is surpassed.
sol.-take any point a in the circumference. draw the tangent ad, and make the angle dac equal to the given angle ten. ac will cut off the required segment.
thirteen. the sum of the rectangles of the opposite sides of a cyclic quadrilateral abcd is equal to the rectangle contained by its diagonals.
propositions xiii., xiv. are particular cases of the following theorem:-
this being so, we must ask what we mean by 'possible' and 'impossible'. for in its most proper use the predicate 'indestructible' is given because it is impossible that the thing should be destroyed, i.e. exist at one time and not at another.
a man or gold, and it must be in motion in something, e.g. a place or an affection, and during something, for all motion takes place during a time.
for they got so far as to see that there must be some underlying nature, but they make it one-for even if one philosopher makes a dyad of it, which he calls great and small, the effect is the same, for he overlooked the other nature.
since, then, we cannot reasonably suppose either that both are in motion or that the star alone moves, the remaining alternative is that the circles should move, while the stars are at rest and move with the circles to which they are attached.
five. find a point in a given line such that, if it be joined to two given points on opposite sides of the line, the angle formed by the joining lines shall be bisected by the given line.
suppose, then, that in a certain time an alteration is undergone by one half of a body's length and a locomotion is accomplished the other half: can be say that in this case the alteration is equal to the locomotion and of the same velocity?
four. divide a given square into five equal parts; namely, four right-angled triangles, and a square.
thus democritus reduces the causes that explain nature to the fact that things happened in the past in the same way as they happen now: but he does not think fit to seek for a first principle to explain this 'always': so, while his theory is right in so
after weighing one against the other all these complicated probabilities, the final conclusion at which we arrive is that there is nearly as much to be said on the one side of the question as on the other, and that we are not perfectly certain, whether,
nor can they be innumerable, because, if so, being will not be knowable: and in any one genus there is only one contrariety, and substance is one genus: also a finite number is sufficient, and a finite number, such as the principles of empedocles, is
of all things that have no contraries there are opposite changes (viz. change from the thing and change to the thing, e.g. change from being and change to being), but no motion.
plato's doctrine of the same and the other ruling the courses of the heavens and of the human body is not a mere vagary, but is a natural result of the state of knowledge and thought at which he had arrived.
an exile from a state in which there has been civil war; whence arise diarrhoeas and dysenteries, and all such disorders.
again, because gf is parallel to cb, and ce intersects them, the angle egf is equal to ecb; but ecb is right (const.); therefore egf is right; and gef has been proved to be half a right angle; therefore the angle gfe is half a right angle i. xxxii. .
the second solid is composed of the same triangles, which unite as eight equilateral triangles, and make one solid angle out of four plane angles-six of these angles form a regular octahedron.
which was in him reasserted her sway over the elements of fire, air, earth, water, which had engrossed her, and he regained his first and better nature.
timaeus: he has been taken ill, socrates; for he would not willingly have been absent from this gathering.
but things which are changed all of a sudden, and only gradually and with difficulty return to their own nature, have effects in every way opposite to the former, as is evident in the case of burnings and cuttings of the body.
two. draw a parallel to a given line to touch a given circle.
in order then that the world might be solitary, like the perfect animal, the creator made not two worlds or an infinite number of them; but there is and ever will be one only-begotten and created heaven.
thus far we have spoken of the works of mind; and there are other works done from necessity, which we must now place beside them; for the creation is made up of both, mind persuading necessity as far as possible to work out good.
then one hundred and five hundred must be predicated of everything of which neither a nor b is predicated. let e lie between a and b; for that which is neither of two contraries is a mean between them.
hence, na, nb are equimultiples of a, b.
it would be absurd if this were not the case: for a thing may remain still merely under violence: thus we shall have a thing being in a non-permanent state of rest without having become so.
now the conception of the infinite possesses this power of principles, and indeed in the sphere of quantity possesses it in a higher degree than any other conception; so that it is in no way absurd or unreasonable that the assumption that an infinite body
unless a person comes to an understanding about the nature and conditions of rest and motion, he will meet with many difficulties in the discussion which follows.
dem.-for if possible let bc be not coplanar with bd, be, and let the plane of ab, bc intersect the plane of bd, be in the line bf. then eleven. iii.
the shoulders, and swells them up, and so twists back the great tendons and the sinews which are connected with them. these disorders are called tetanus and opisthotonus, by reason of the tension which accompanies them.
is used for purging away oil and earth, the other, salt, which harmonizes so well in combinations pleasing to the palate, and is, as the law testifies, a substance dear to the gods.
so that together they all form a single unity: whether this unity is finite or infinite makes no difference to our present argument; for in any case since the things in motion are infinite in number the whole motion will be infinite, if, as is
their indefiniteness is probably the reason why he singles them out, as especially incapable of being tested by experiment. (compare the saying of anaxagoras-sext. pyrrh.-that since snow is made of water and water is black, snow ought to be black.)
but as potentiality it does not cease to be in its own nature, but is necessarily outside the sphere of becoming and ceasing to be.
of the six differences of position: but each of these is a limit.
for when the flesh becomes decomposed and sends back the wasting substance into the veins, then an over-supply of blood of diverse kinds, mingling with air in the veins, having variegated colours and bitter properties, as well as acid and saline
passive quality in virtue of which a thing is said to be acted on or to be incapable of being acted on.
forty two. given the vertical angle and perimeter of a triangle, construct it with either of the following data: one. the bisector of the vertical angle; two. the perpendicular from the vertical angle on the base; three. the radius of the inscribed circle.
and this opinion again is held because time is the measure of this kind of locomotion and is itself measured by such.
the first and last both becoming means, they will all of them of necessity come to be the same, and having become the same with one another will be all one.
but all these qualities derive from the dry and moist we mentioned first.' for (i) the dry is opposed to the damp: i.e.
for any part of it that is taken will be infinite, if it has parts: for 'to be infinite' and 'the infinite' are the same, if it is a substance and not predicated of a subject. hence it will be either indivisible or divisible into infinites.
the one, they reasoned, must have already existed in the other; for since everything that comes into being must arise either from what is or from what is not, and it is impossible for it to arise from what is not (on this point all the physicists agree),
xxxii. is without the circle. in like manner, every other point in bi, except b, is without the circle. hence, since bi meets the circle at b, but does not cut it, it must touch it.
hence that which not always is not will sometimes be and sometimes not be; and it is clear that this is true also of that which cannot always be but sometimes is and therefore sometimes is not.
rice,................... one.seven thousand three hundred twelve molasses,................ one.seventy three cheapest form of the saccharine. rye meal,................ one.four hundred thirty four indian meal,.............
one hundred seven. abcd is a square, the side four hundred is bisected in e, and the line ef drawn, making the angle aef eab; prove that ef divides the side bc in the ratio of two : one.
yet empedocles says 'for these are all not only equal...'
three. if through the point of contact of two touching circles any secant be drawn cutting the circles again in two points, the radii drawn to these points are parallel.
five. what is the ratio of two commensurable magnitudes?
"if tangents be drawn to a circle, at the angular points of an inscribed polygon of any number of sides, they will form a regular polygon of the same number of sides circumscribed to the circle."
if it were, any portion of it would have this movement; but in fact every part moves in a straight line to the centre. being, then, constrained and unnatural, the movement could not be eternal. but the order of the universe is eternal.
spears, a style of equipment which the goddess taught of asiatics first to us, as in your part of the world first to you.
euclid's proofs of the propositions, in the theory of proportion, possess at present none but a historical interest, as no student reads them now.
that if the stars have a movement of their own, it will be one of these. but neither is observed. (one) suppose them to spin. they would then stay where they were, and not change their place, as, by observation and general consent, they do.
in this manner, the four kinds or elements were then shaken by the receiving vessel, which, moving like a winnowing machine, scattered far away from one another the elements most unlike, and forced the most similar elements into close contact.
time. but that is impossible. again, whatever the time, so long as it is finite, in which the infinite performs the motion, a finite weight must necessarily move a certain finite distance in that same time.
dem.-one. let one hundred be the centre. take any point five hundred in ab. join ca,four hundred,cb. now the angle adc is i. xvi. greater than abc; but the angle abc is equal to cab i. v.
dem.-let o, o be their centres; let the angle aob be indefinitely small, so that the arc ab may be regarded as a right line; make the angle aob equal to aob; then the triangles aob, aob are similar.
but the fact that a thing is altered requires that there should be something that alters it, something e.g.
icosahedron,-according to their degrees of lightness or heaviness or power, or want of power, of penetration. the single particles of any of the elements are not seen by reason of their smallness; they only become visible when collected.
moreover, the conclusion to which we have been led is a reasonable one. for there must be three things-the moved, the movent, and the instrument of motion.
that is why a body's' nutrition' continues so long as it is kept alive (even when it is diminishing), though not its 'growth'; and why nutrition, though 'the same' as growth, is yet different from it in its actual being.
for, whatever the subject may be, movement is change from one form to another.
again, because ac is equal to four hundred (const.), and cb common to the two triangles acb, dcb, and the base ab equal to the base db, the angle acb is equal to the angle dcb; but the angle dcb is a right angle (const.).
he is of opinion that nature should be left to herself, and is inclined to think that physicians are in vain (laws-where he says that warm baths would be more beneficial to the limbs of the aged rustic than the prescriptions of a not over-wise doctor).
nine. what figures can be inscribed in, and circumscribed about, a circle by means of book four.?
we may begin our own statement by settling a question which to some has been the main difficulty-the question why some bodies move always and naturally upward and others downward, while others again move both upward and downward.
if the world be indeed fair and the artificer good, it is manifest that he must have looked to that which is eternal; but if what cannot be said without blasphemy is true, then to the created pattern.
one person may be cured quickly and another slowly, and cures may also be simultaneous: so that, recovery of health being an alteration, we have here alterations of equal velocity, since each alteration occupies an equal time. but what alteration?
men and oxen exchange work; but if we consider necessary work only, the oxen will be seen to have greatly the advantage, their farm is so much the larger. man does some of his part of the exchange work in his six weeks of haying, and it is no boy's play.
dem.-if possible let them have a common centre at o. join oa, and draw any other line od, cutting the circles in one hundred and five hundred respectively. then because o is the centre of the circle abc, oa is equal to oc.
now it is plain that if the locomotion of a thing is rectilinear and finite it is not continuous locomotion: for the thing must turn back, and that which turns back in a straight line undergoes two contrary locomotions, since, so far as motion in respect
so, too, movements that have simultaneous limits have the same time, yet the one may in fact be fast and the other not, and one may be locomotion and the other alteration; still the time of the two changes is the same if their number also is equal and
the relations of the earth and heavens are so indistinct in the timaeus and so figurative in the phaedo, phaedrus and republic, that we must give up the hope of ascertaining how they were imagined by plato, if he had any fixed or scientific conception of
first, i must distinguish between that which always is and never becomes and which is apprehended by reason and reflection, and that which always becomes and never is and is conceived by opinion with the help of sense.
it results that hot and cold are distinguished not by the figure but by the size of the particles. again, if the pyramids are unequal in size, the large ones will not be fire, and that figure will produce not combustion but its contrary.
the water of tears and perspiration and similar substances is also the watery part of fresh phlegm. all these humours become sources of disease when the blood is replenished in irregular ways and not by food or drink.
the square of any such number represents a surface, the cube a solid. the squares of any two such numbers (e.g. two squared, three squared four, nine), have always a single mean proportional (e.g.
if everything that is in motion with the exception of things that move themselves is moved by something else, how is it that some things, e.g. things thrown, continue to be in motion when their movent is no longer in contact with them?
there is nothing to prevent the first place from being 'in' something else-not indeed in that as 'in' place, but as health is 'in' the hot as a positive determination of it or as the hot is 'in' body as an affection. so we escape the infinite regress.
the body, and have both pains and pleasures attendant on them.
again, since bc intersects the parallels ac, bd, the angle acb is equal to the angle cbd; hence the triangles abc, dcb have the two angles abc, acb in one respectively equal to the two angles bcd, cbd in the other, and the side bc common. therefore xxvi.
(two) further, why, after always existing, was the thing destroyed, why, after an infinity of not being, was it generated, at one moment rather than another?
on the contrary: anything is 'combinable' which, being readily adaptable in shape, is such as to suffer action and to act; and it is 'combinable with' another thing similarly characterized (for the 'combinable' is relative to the 'combinable'); and
by reason of these affections the soul is at first without intelligence, but as time goes on the stream of nutriment abates, and the courses of the soul regain their proper motion, and apprehend the same and the other rightly, and become rational.
fifty two. if a point be taken within a triangle, so as to be the centre of mean position of the feet of the perpendiculars drawn from it to the sides of the triangle, the sum of the squares of the perpendiculars is a minimum.
for they all possess angles, the octahedron and dodecahedron as well as the pyramid; and democritus makes even the sphere a kind of angle, which cuts things because of its mobility. the difference, then, will be one of degree: and this is plainly false.
hence their principles are in one sense the same, in another different; different certainly, as indeed most people think, but the same inasmuch as they are analogous; for all are taken from the same table of columns, some of the pairs being wider, others
of the material's being condensed or rarefied or heated or cooled, nevertheless it is not the things that are coming into existence that are altered, and their becoming is not an alteration.
in like manner all the perpendiculars from o on the sides of the pentagon are equal; hence the circle whose centre is o, and radius of, will touch all the sides of the pentagon, and will therefore be inscribed in it.
for some make hot and cold, or again moist and dry, the conditions of becoming; while others make odd and even, or again love and strife; and these differ from each other in the way mentioned.
perhaps, indeed, it is impossible that 'the hot' should exist in separation from matter: but if there are any entities thus separable, what we are saying would be true of them.
hence the sum of the squares on ab and bc is equal to twice the rectangle ab.bc, together with the square on ac.
ii. the angle which one plane makes with another is called a dihedral angle.
yet time-both infinite time and any time you like to take-is made up of these. one would naturally suppose that what is made up of things which do not exist could have no share in reality.
hence what is moved will not be measurable by the time simply in so far as it has quantity, but in so far as its motion has quantity.
tertian; when of earth, which is the most sluggish of the four, and is only purged away in a four-fold period, the result is a quartan fever, which can with difficulty be shaken off.
for evidence they bring the fact that, as the sun rises and sets, the part concealed by the earth shows a straight and not a curved edge, whereas if the earth were spherical the line of section would have to be circular.
said, time is measured by motion as well as motion by time (this being so because by a motion definite in time the quantity both of the motion and of the time is measured): if, then, what is first is the measure of everything homogeneous with it, regular
one hundred fourteen. if o be the centre of the inscribed circle of the triangle prove
now that which is being moved is, but that which is coming-to-be is not: hence, also, motion is prior to coming-to-be.
we must remember that they were not distinctly defined to his, as they are to our eyes; he saw them, not as they are divided in the prism, or artificially manufactured for the painter's use, but as they exist in nature, blended and confused with one
things that are at rest are always at rest, or they are all constituted so as to be capable alike of motion and of rest; or there is yet a third possibility remaining-it may be that some things in the world are always motionless, others always in motion,
wherefore from ab, the greater of the two given lines, a part, ae, has been out off equal to one hundred, the less.
motion in respect of quantity has no name that includes both contraries, but it is called increase or decrease according as one or the other is designated: that is to say motion in the direction of complete magnitude is increase, motion in the contrary
this is the first proposition in euclid in which the technical term "duplicate ratio" occurs. my experience with pupils is, that they find it very difficult to understand either euclid's proof or his definition.
for the infinite to move at all is thus absolutely impossible; since the very smallest movement conceivable must take an infinity of time.
cause of infinite evil to its own self-in like manner we should conceive of the double nature which we call the living being; and when in this compound there is an impassioned soul more powerful than the body, that soul, i say, convulses and fills with
what, then, is the 'first mover' of the 'elements'? what causes their motion? presumably not love and strife: on the contrary, these are causes of a particular motion, if at least we assume that 'first mover' to be an originative source'.
dem.-let abhij, cdefg be the polygons, and let the sides ab, four hundred be homologous. join ah, ai, ce, cf.
it remains either that both are moved, or that the one is moved and the other at rest.
and that intelligence could not be present in anything which was devoid of soul.
for that which is being moved changes its place as a whole: but the growing thing changes its place like a metal that is being beaten, retaining its position as a whole while its parts change their places.
for instance, is it necessary that solstices shall come-to-be, i.e. impossible that they should fail to be able to occur?
hence aeo is equal to ceo-that is, a part equal to the whole-which is absurd. therefore ab and four hundred do not bisect each other.
let us take our start from what we have already laid down in our course on physics. motion, we say, is the fulfilment of the movable in so far as it is movable.
under the influence of such ideas, perhaps also deriving from the traditions of their own or of other nations scraps of medicine and astronomy, men came to the observation of nature.
for it is and it is not, it is nowhere when filled, it is nothing when empty.
for there is one in the sense of 'matter', and a second in the sense of 'form': and, in addition, the third 'originative source' must be present as well.
one hundred twenty one. construct a polygon of an odd number of sides, being given that the sides taken in order are divided in given ratios by fixed points.
one) he maintains, simultaneously with this denial, that each thing once more comes-to-be out of the one.
further, no one could say why a thing once set in motion should stop anywhere; for why should it stop here rather than here? so that a thing will either be at rest or must be moved ad infinitum, unless something more powerful get in its way.
view concerning the divisibility of magnitudes. for us) the following account results from the distinctions we established at the beginning.
whatever is destructible or generated is always alterable. now alteration is due to contraries, and the things which compose the natural body are the very same that destroy it.
again, when any one has fallen ill, there must follow a period of time in which his restoration to health is in the future: the process of change cannot take place in an instant: yet the change cannot be a change to anything else but health.
'in mind,' replied the priest, 'i mean to say that you are children; there is no opinion or tradition of knowledge among you which is white with age; and i will tell you why.
hence the angle oab is equal to the angle oba, and the angle oad equal to the angle oda; therefore the angle bad is equal to the sum of the angles oba, oda. in like manner the angle bcd is equal to the sum of the angles obc, odc.
hundred to f, and the two triangles are equal.
in a given triangle inscribe another of given form, and having one of its angles at a given point in one of the sides of the original triangle.
the body which is being moved in a circle; and its movement makes time continuous.
five. if two of the plane angles of a tetrahedral angle be equal, the planes of these angles are equally inclined to the plane of the third angle, and conversely.
vain would be the attempt to tell all the figures of them circling as in dance, and their juxtapositions, and the return of them in their revolutions upon themselves, and their approximations, and to say which of these deities in their conjunctions meet,
if the numerator be equal to the denominator, the fraction is equal to unity; and if greater than the denominator, it is greater than unity.
twenty two. if the sides of a triangle be thirteen, fourteen, fifteen, what are the values of the radii of its inscribed and escribed circles?
so we get the result that the finite force will occupy the same amount of time in performing the motive act as the infinite force. but this is impossible. therefore nothing finite can possess an infinite force.
twenty one. in every quadrilateral circumscribed about a circle, the middle points of its diagonals and the centre of the circle are collinear.
on the other hand, a process simply to a contrary, e.g. that denoted by the expression 'becoming white', where no starting-point is specified, is a change but not a motion.
when the several parts of the flesh are separated by disease, if the foundation remains, the power of the disorder is only half as great, and there is still a prospect of an easy recovery; but when that which binds the flesh to the bones is diseased, and
dem.-if not, suppose another line og drawn from the centre to be perpendicular to four hundred. let og cut the circle in f. then because the angle ogc is right (hyp.) the angle ocg i. xvii. must be acute. therefore i. xix.
for god only has the knowledge and also the power which are able to combine many things into one and again resolve the one into many. but no man either is or ever will be able to accomplish either the one or the other operation.
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but since bodies are divisible through and through, the postulate of pores is ridiculous: for, qua divisible, a body can fall into separate parts.
the reference may be to our own right hands, as with the diviner, or to some similarity to our own members, such as the parts of a statue possess; or we may take the contrary spatial order, calling right that which is to our left, and left that which is
in five hundred. since then ad is not without parts, it must be divisible, and the changing thing must have changed in every part of it (for if it has changed in neither of the two parts into which ad is divided, it has not changed in the whole either:
having long meditated on the properties of one:two:four:eight, or one:three:nine:twenty seven, or of three, four, five, they discovered in them many curious correspondences and were disposed to find in them the secret of the universe.
therefore the place in which it terminates will be a hollow place. the water then will collect there until equality is established, that is until the line ae is equal to the two radii.
since everything that changes changes from something to something, that which has changed must at the moment when it has first changed be in that to which it has changed.
of the food) does not. for indeed, even in the mixture of wine and water, it is the prevailing ingredient which is said to have increased in volume. we say, e.g. that the wine has increased, because the whole mixture acts as wine but not as water.
once more, then, at the commencement of my discourse, i call upon god, and beg him to be our saviour out of a strange and unwonted enquiry, and to bring us to the haven of probability. so now let us begin again.
four. every proposition in the fourth book is a problem.
eleven. if a variable tangent meets two parallel tangents it subtends a right angle at the centre.
(five) further, too, if it is itself an existent, where will it be? zeno's difficulty demands an explanation: for if everything that exists has a place, place too will have a place, and so on ad infinitum.
thus we may remark in passing that the most fanciful of ancient philosophies is also the most nearly verified in fact. the fortunate guess that the world is a sum of numbers and figures has been the most fruitful of anticipations.
there will remain, (five) the psychology, (six) the physiology of plato, and (seven) his analysis of the senses to be briefly commented upon: (eight) lastly, we may examine in what points plato approaches or anticipates the discoveries of modern science.
in the triangles bah, edf, we have ab equal to de (hyp.), ah equal to df (const.), and the angle bah equal to the angle edf (const.); therefore the base iv. bh is equal to ef.
but further, if fire and earth exist, the intermediate bodies must exist also: each element stands in a contrary relation to every other.
an impression produced in us contrary to nature and violent, if sudden, is painful; and, again, the sudden return to nature is pleasant; but a gentle and gradual return is imperceptible and vice versa.
an example is that of missiles, which as they move are themselves fired so strongly that leaden balls are melted; and if they are fired the surrounding air must be similarly affected.
one might raise the question whether a thing can be in itself, or whether nothing can be in itself-everything being either nowhere or in something else.
thirty four. mention some propositions in book one. which are particular cases of more general ones that follow.
and so, since the prior movement belongs to the body which naturally prior, and circular movement is prior to straight, and movement in a straight line belongs to simple bodies-fire moving straight upward and earthy bodies straight downward towards the
it is plain then that they all in one way or another identify the contraries with the principles. and with good reason. for first principles must not be derived from one another nor from anything else, while everything has to be derived from them.
fifty six. if the sides of any polygon be cut by a transversal, the product of one set of alternate segments is equal to the product of the remaining set.
for they distinguish 'what is' and 'what is not' by their perceiving and not-perceiving, just as what is knowable 'is' and what is unknowable 'is not'-perception on their view having the force of knowledge.
five. divide a right angle into five equal parts.
one might discuss what must be the character of that 'whereby' a thing grows. clearly it must be potentially that which is growing-potentially flesh, e.g. if it is flesh that is growing. actually, therefore, it must be 'other' than the growing thing.
all be air or water or fire or earth-because 'change is into contraries'. for if they all were air, then (assuming air to persist) there will be 'alteration' instead of coming-to-be.
which are taken. hence it follows, that if the numerator be less than the denominator, the fraction is less than unity.
it is clear, then, that the sphere is first of solid figures.
four. how may lines be divided? ans. into straight and curved.
colours are flames which emanate from all bodies, having particles corresponding to the sense of sight. some of the particles are less and some larger, and some are equal to the parts of the sight.
but the worst case of all is when the marrow is diseased, either from excess or defect; and this is the cause of the very greatest and most fatal disorders, in which the whole course of the body is reversed.
and so, having taken these three from nature as (so to speak) laws of it, we make further use of the number three in the worship of the gods. further, we use the terms in practice in this way.
similarly, then, there corresponds to the point the body which is carried along, and by which we are aware of the motion and of the 'before and after' involved in it.
twenty two (poetic diction continued.) how poetry combines elevation of language with perspicuity. twenty three epic poetry. twenty four (epic poetry continued.) further points of agreement with tragedy.
hence the ratio three : four expresses the same idea as the fraction thirty four. in fact, both are different ways of expressing and writing the same thing. when written three : four it is called a ratio, and when three four a fraction.
it is natural that this should happen; for the movement goes with the distance and the time with the movement, because they are quanta and continuous and divisible.
hence the gnomon aoh is equal to four times the rectangle ak-that is, equal to four times the rectangle ab.bc, since bc is equal to bk.
simultaneously predicable of the same thing. may we not say, however, that in so far as the thing is still stationary it is in a state of rest in a qualified sense?
moistened, and that the mother substance becomes earth and air, in so far as she receives the impressions of them.
now the magnitude go that a has passed over is greater than the magnitude ge, and the time zk is less than the whole time zh: so that the quicker will pass over a greater magnitude in less time.
dem.-since o is the centre of the two circles, we have oa equal to op, and oc equal to ob. hence the two triangles aoc, pob have the sides oa, oc in one respectively equal to the sides op, ob in the other, and the contained angle common to both. hence i.
rather he would have been infinitely amused if he could have known that his gift of invention would have deceived one thousand. martin himself into the belief that the tradition was brought from egypt by solon and made the subject of a poem by him.
such, approximately, are the current explanations of the manner in which some things 'act' while others 'suffer action'.
everything else is composed-the compounds differing one from another according to the shapes, 'positions', and 'groupings' of their constituents.)
but all natural bodies which change their properties we see to be subject without exception to increase and diminution.
the physicists on the other hand have two modes of explanation.
that is why, even if there were to be water which had not a container, the parts of it, on the one hand, will be moved (for one part is contained in another), while, on the other hand, the whole will be moved in one sense, but not in another.
the elementary qualities are four, and any four terms can be combined in six couples. contraries, however, refuse to be coupled: for it is impossible for the same thing to be hot and cold, or moist and dry.
the second book is occupied with the relations between the segments of a line divided in various ways. all these can be proved in the most simple manner by algebraic multiplication.
ac : four hundred :: ce : cb; therefore ac.cb ce.four hundred cd four hundred.de cd ad.db three. xxxv. .
again (two) we hold that a substance is not contrary to another substance. how then can substance be derived from what are not substances? or how can non-substances be prior to substance?
two. the three rectangles oa.op, ob.oq, oc.or are equal.
it is true that at any moment it is always over against something stationary: but it is not at rest: for at a moment it is not possible for anything to be either in motion or at rest.
to us they are not always the same but change with the direction in which we are turned: that is why the same thing may be both right and left, up and down, before and behind. but in nature each is distinct, taken apart by itself.
thus socrates in the phaedo first blames everybody else for having given no explanation; and then lays it down; that 'some things are forms, others participants in the forms', and that 'while a thing is said to "be" in virtue of the form, it is said to
cor. one.-twice as many polygons may be constructed on ab similar to a given polygon cdefg as that figure has sides.
further, how can the infinite be itself any thing, unless both number and magnitude, of which it is an essential attribute, exist in that way? if they are not substances, a fortiori the infinite is not.
again, when it is not separate it is described as a part in a whole, as the pupil in the eye or the hand in the body: when it is separate, as the water in the cask or the wine in the jar. for the hand is moved with the body and the water in the cask.
this is indicated by the fact that locomotion belongs to bodies only when isolated from other bodies, and is generated last of the several kinds of movement; in order of being then it will be first.
hence-if the sum of two arcs of a circle be greater than a semicircle, the rectangle contained by their chords is equal to the rectangle contained by the radius, and the sum of the chords of the supplements of their sum and their difference.
but it is absurd and impossible to suppose that the unknowable and indeterminate should contain and determine.
to say that the universe alternately combines and dissolves is no more paradoxical than to make it eternal but varying in shape.
two. every point equally distant from the points a, b is in the line four hundred.
again, if the unnatural movement is the contrary of the natural and a thing can have no more than one contrary, it will follow that circular movement, being a simple motion, must be unnatural, if it is not natural, to the body moved.
first is greater than, equal to, or less than the multiple of the second, the multiple of the third is greater than, equal to, or less than the multiple of the fourth. hence five. def. v. the base bc : four hundred :: the triangle abc : acd.
five. what axiom is made use of in superposition?
plato's account of the soul is partly mythical or figurative, and partly literal. not that either he or we can draw a line between them, or say, 'this is poetry, this is philosophy'; for the transition from the one to the other is imperceptible.
the problem, however, which specially belongs to the physicist is to investigate whether there is a sensible magnitude which is infinite.
(two). let the arc ace be greater than a semicircle. join ce. then the angle ace is evidently less than acb; but acb is right; therefore ace is acute.
each sphere, in fact, has its particular natural motion, to which the general movement is, as it were, added. but the force of any limited body is only adequate to moving a limited body.
if a line (ab) be divided into any two parts (at one hundred), the square on the whole line is equal to the sum of the squares on the parts (ac, cb), together with twice their rectangle.
in like manner there is a universal nature out of which all things are made, and which is like none of them; but they enter into and pass out of her, and are made after patterns of the true in a wonderful and inexplicable manner.
an intermediate may be a starting-point of change, since for the purposes of the change it serves as contrary to either of two contraries: for the intermediate is in a sense the extremes.
the liquid is composed of small and unequal particles, the fusile of large and uniform particles and is more solid, but nevertheless melts at the approach of fire, and then spreads upon the earth.
those, then, who construct all things out of a single element, must maintain that coming-tobe and passing-away are 'alteration'.
from all that we have just been saying about the elements or kinds, the most probable conclusion is as follows:-earth, when meeting with fire and dissolved by its sharpness, whether the dissolution take place in the fire itself or perhaps in some mass of
no mere innovator could have exercised such an influence or hit on so much truth. theory makes no great leaps; it proceeds to new concepts by the adjustment and renovation of old ones.
chance or fortune is called 'good' when the result is good, 'evil' when it is evil. the terms 'good fortune' and 'ill fortune' are used when either result is of considerable magnitude.
hence the two triangles bfc, cgb have the two sides bf, fc in one equal to the two sides cg, gb in the other; and the angle bfc contained by the two sides of one equal to the angle cgb contained by the two sides of the other. therefore iv.
not only do we measure the movement by the time, but also the time by the movement, because they define each other. the time marks the movement, since it is its number, and the movement the time.
there is not only the absurdity of our hearing nothing, the ground of which they try to remove, but also the fact that no effect other than sensitive is produced upon us.
two. let one of the chords ab pass through the centre o, and cut the other chord four hundred, which does not pass through the centre, at right angles. join oc.
to each add acd, and we have the sum of the angles bac, acd equal to the sum of dce and acd; but the sum of bac, acd is i. xxix. two right angles; therefore the sum of dce and acd is two right angles. hence i. xiv. ac, ce are in the same right line.
in another way void is that in which there is no 'this' or corporeal substance.
we may remark in passing, that the platonic compared with the jewish description of the process of creation has less of freedom or spontaneity. the creator in plato is still subject to a remnant of necessity which he cannot wholly overcome.
thus he spake, and poured the remains of the elements into the cup in which he had mingled the soul of the universe.
twenty three. what is the radius of the circumscribed circle?
(three) if coming to be and passing away do not give out, it is only because that from which things come to be is infinite.
the same to be not always one and the same. (i mean that e.g.
but if this second thing whose velocity is equal has traversed a certain distance in a certain time, the original thing that is in motion must have traversed the same distance in the same time. hence that which is in motion must have been in motion before.
forty two. the bisectors of the angles of a convex quadrilateral form a quadrilateral whose opposite angles are supplemental. if the first quadrilateral be a parallelogram, the second is a rectangle; if the first be a rectangle, the second is a square.
there is no other body besides these. then it is not possible that any simple body should be outside the heaven. but, if no simple body, neither can any mixed body be there: for the presence of the simple body is involved in the presence of the mixture.
(the same argument applies also to the one which the physicists suppose to exist alongside the elements: for everything changes from contrary to contrary, e.g. from hot to cold).
he replied:-in the egyptian delta, at the head of which the river nile divides, there is a certain district which is called the district of sais, and the great city of the district is also called sais, and is the city from which king amasis came.
if this is its nature, the potency of place must be a marvellous thing, and take precedence of all other things.
is such that it is essentially capable of not-being.
the entrance to the first he divided into two parts, both of which he made to meet at the channels of the nose, that when the mouth was closed the passage connected with it might still be fed with air.
become steadier as time goes on, then the several circles return to their natural form, and their revolutions are corrected, and they call the same and the other by their right names, and make the possessor of them to become a rational being.
or third sides (bc, ef) shall be equal, and the angles (b, one hundred) at the base of one shall be respectively equal to the angles (e, f) at the base of the other; namely, those shall be equal to which the equal sides are opposite.
for there will be water, when the hot of the air and the dry of the earth have passed-away (since the moist of the former and the cold of the latter are left): whereas, when the moist of the air and the cold of the earth have passed-away, there will be
therefore if this differs in no respect from its place, why need we assume a place for bodies over and above the volume of each, if their volume be conceived of as free from attributes?
for here the goal that is reached is necessarily one that is actually, not potentially, existent.
a- b r, c d r, e- f r; therefore a br, c dr, e fr; therefore a c e (b d f)r. hence a c e b -d- f- r; therefore a b- a c e b -d- f-, and a : b :: a c e : b d f.
the social contract , then, may be regarded either as a document of the french revolution, or as one of the greatest books dealing with political philosophy.
also the vacuum which is made when the air is exhaled through the pores is filled up by the inhalation of breath through the mouth and nostrils. the explanation of this double phenomenon is as follows:-elements move towards their natural places.
if any side (ab) of a triangle (abc) be produced (to five hundred), the external angle (cbd) is equal to the sum of the two internal non-adjacent angles (a, one hundred), and the sum of the three internal angles is equal to two right angles.
we have explained why some things come to-be without qualification, but not others both in general, and also when the changing things are substances and nothing else; and we have stated that the substratum is the material cause of the continuous
are such as to act upon one another and to suffer action from one another; while other things, viz. agents which have not the same matter as their patients, act without themselves suffering action.
the foregoing proof may be abridged as follows:- because ed bisects ab at right angles, every point equally distant from, the points a, b must lie in ed i. x. ex.
for it is the nearest body which is most strongly influenced, and the most remote, by reason of its distance, which is least affected, the influence on the intermediate bodies varying, as the mathematicians show, with their distance.
the next step we must take is to see in how many senses one thing is said to be 'in' another.
for in general that which is moved changes from something into something, the starting-point and the goal being different in form, and always it is a finite change.
for the impossible consequences which result from this view in the mathematical sphere will reproduce themselves when it is applied to physical bodies, but there will be difficulties in physics which are not present in mathematics; for mathematics deals
he appears to have an inkling of the truth that to the higher nature of man evil is involuntary. this is mixed up with the view which, while apparently agreeing with it, is in reality the opposite of it, that vice is due to physical causes.
we may in general assume sound to be a blow which passes through the ears, and is transmitted by means of the air, the brain, and the blood, to the soul, and that hearing is the vibration of this blow, which begins in the head and ends in the region of
i mean to say, he replied, that in mind you are all young; there is no old opinion handed down among you by ancient tradition, nor any science which is hoary with age. and i will tell you why.
hence zeno's reasoning is false when he argues that there is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling.
the creation of the rest of the body follows next in order, and this we may investigate in a similar manner. and it appears to be very meet that the body should be framed on the following principles:-
thus, when there is a change into perceptible material, people say there is 'coming-to-be'; but when there is a change into invisible material, they call it 'passing-away'.
it will follow that there are two infinite times which are mutually exclusive, the time occupied by the synthesis, which is infinite, being preceded by the period of analysis. there are thus two mutually exclusive infinites, which is impossible.
the great source of error and also the beginning of truth to them was reasoning from analogy; they could see resemblances, but not differences; and they were incapable of distinguishing illustration from argument.
of those who deny the existence of a void some, like anaxagoras and empedocles, have not tried to analyse the notions of light and heavy at all; and those who, while still denying the existence of a void, have attempted this, have failed to explain why
dem.-one. suppose the angles at the centres to be given equal. now because the circles are equal their radii are equal (def. i.).
def.-the line of collinearity is called simson's line.
and since position belongs only to those things which also have a 'place', while the primary differentiation of 'place' is the above' and 'the below' (and the similar pairs of opposites), all things which touch one another will have 'weight' or
half the time: the case of the hauled ship is exactly comparable: here we have so many drops setting so much in motion, but a part of them will not set as much in motion in any period of time.
the creation of the world is the impression of order on a previously existing chaos. the formula of anaxagoras-'all things were in chaos or confusion, and then mind came and disposed them'-is a summary of the first part of the timaeus.
fifty three. o is a point outside a circle whose centre is e; two perpendicular lines passing through o intercept chords ab, four hundred on the circle; then ab cd oe r .
if any side (bc) of a triangle (abc) be produced, the exterior angle (acd) is greater than either of the interior non-adjacent angles.
we have now to take the assertion that all things are sometimes at rest and sometimes in motion and to confront it with the arguments previously advanced. we must take our start as before from the possibilities that we distinguished just above.
ix., note aoc, coa is double of the sum of the angles cda, abc; but the sum of two angles aoc, coa is four right angles. therefore the sum of the angles cda, abc is two right angles.
having given these commands, the creator remained in his own nature.
socrates: and what other, critias, can we find that will be better than this, which is natural and suitable to the festival of the goddess, and has the very great advantage of being a fact and not a fiction?
four. the difference of the sides of a triangle is less than the difference of the segments of the base, made by the perpendicular from the vertex.
prepared to receive scents, or the smooth and soft materials on which figures are impressed.
seven. the three triangles bco, cao, abo are equiangular.
therefore ba : ac :: ed : dg iv. ; but ba : ac :: ed : df (hyp.). therefore dg is equal to df.
and we should consider that god gave the sovereign part of the human soul to be the divinity of each one, being that part which, as we say, dwells at the top of the body, and inasmuch as we are a plant not of an earthly but of a heavenly growth, raises us
bac. therefore edf is equiangular to bac.
then because the angle acb is right (hyp.), and ach is right, being the angle of a square, the sum of the angles acb, ach is two right angles; therefore bc, ch are in the same right line xiv. . in like manner ac, four hundred are in the same right line.
if two triangles (abc, def) have one angle (a) one equal to one angle (five hundred) in the other, the sides about two other angles (b, e) proportional (ab : bc :: de : ef), and the remaining angles (one hundred, f) of the same species (i. e.
if one were to remove the earth to where the moon now is, the various fragments of earth would each move not towards it but to the place in which it now is.
dem.-since a : b :: b : c, we have a- b b- c. in like manner, b c a b. hence, multiplying, a- c a c; therefore a : c :: a : c, and similarly for any number of magnitudes in each set.
there is, further, the absurdity that nature has bestowed upon them no organ appropriate to such movement. for nature leaves nothing to chance, and would not, while caring for animals, overlook things so precious.
if it could, and there were two bodies in the same place, it would also be true that any number of bodies could be together; for it is impossible to draw a line of division beyond which the statement would become untrue.
of the causes of day and night the pre-socratic philosophers, and especially the pythagoreans, gave various accounts, and therefore the question can hardly be imagined to have escaped him.
one), the rectangle ac.cb is equal to the rectangle contained by the diameter and one hundred one, and therefore equal to the rectangle contained by the radius and ci; but the difference between cf and bg is evidently equal to ci.
the elements of bodies must therefore be subject to destruction and generation.
of what things, and under what conditions, is 'combination' a property? and, further, does 'combination' exist in fact, or is it false to assert its existence?
two. two circles cannot have three points in common without wholly coinciding.
we will now give our own account, approaching the question first with reference to becoming in its widest sense: for we shall be following the natural order of inquiry if we speak first of common characteristics, and then investigate the characteristics
sol.-let it be required, for instance, to cut off the fourth part. draw af, making any angle with ab, and in af take any point one hundred, and cut off (i. iii.) the parts four hundred, de, ef each equal to ac. join bf, and draw cg parallel to bf.
dem.-erect ce at right angles to ab, and make it equal to ac or cb. join ae, eb. draw df parallel to ce, and fg parallel to four hundred. join af.
cor. two.-the area of bcf is equal to the area of abc.
all experiments which failed: flour,................... zero.eighty eight costs more than indian meal, both money and trouble. sugar,................... zero.eighty lard,.................... zero.sixty five apples,..................
there are bodies which are absolutely heavy and light, or in other words why some move upward and others downward.
any one of these equal ratios is equal to the ratio of the sum of all the antecedents to the sum of all the consequents; therefore the triangle abh : the triangle cde :: the polygon abhij : the polygon cdefg.
they state that the void is the condition of movement in the sense of that in which movement takes place; and this would be the kind of thing that some say place is.
if a line (ab) be divided into any two parts (at one hundred), the square on the whole line is equal to the sum of the rectangles contained by the whole and each of the segments (ac, cb).
like plato (tim.), he denied the above and below in space, and said that all things were the same in relation to a centre.
from these premises it clearly follows (i) that so long as the constituents are preserved in small particles, we must not speak of them as 'combined'.
doctoring must start from the art, not lead to it. but it is not in this way that nature (in the one sense) is related to nature (in the other). what grows qua growing grows from something into something. into what then does it grow?
eight. if three magnitudes be continual proportionals, the first is to the third as the square of the difference between the first and second is to the square of the difference between the second and third.
the foregoing arguments show that it cannot be an atom; but neither can it be a divisible body which analysis will never reach.
dem.-since c : a :: c : b, by inversion, a : c :: b : c; therefore a b. one .
both kinds of causes should be acknowledged by us, but a distinction should be made between those which are endowed with mind and are the workers of things fair and good, and those which are deprived of intelligence and always produce chance effects
it is clear, then, that these changes-the changes of that which is coming-to-be, of that which is 'altering', and of that which is growing-differ in manner as well as in sphere.
join gf; then the triangles abc, gef have the sides ab, bc of one respectively equal to the sides ge, ef of the other, and the angle abc equal to the angle gef (hyp.); therefore iv.
there is a sense, therefore, in which we must declare the principles to be two, and a sense in which they are three; a sense in which the contraries are the principles-say for example the musical and the unmusical, the hot and the cold, the tuned and the
and in general, all that which is termed the incontinence of pleasure and is deemed a reproach under the idea that the wicked voluntarily do wrong is not justly a matter for reproach.
filters through consists of the purest and smoothest and oiliest sort of triangles, dropping like dew from the bones and watering the marrow. now when each process takes place in this order, health commonly results; when in the opposite order, disease.
case of things that move other things locally, in which case the two things must up to a certain point be in contact): and the movent-that is to say, that which causes motion in such a manner that it is not merely the instrument of motion-must be unmoved.
and which of them are in opposition, and in what order they get behind and before one another, and when they are severally eclipsed to our sight and again reappear, sending terrors and intimations of the future to those who cannot calculate their
the matter and the shape, are parts of what is contained.
it would be better, therefore, to say with empedocles and any one else who may have maintained such a theory as his that the universe is alternately at rest and in motion: for in a system of this kind we have at once a certain order.
by iv., square on sum sum of squares twice rectangle.
if two lines be cut by three or more parallels, the intercepts on one are proportional to the corresponding intercepts on the other.
forty. trisect a quadrilateral by lines drawn from one of its angles.
"full of warmth and force, it is wholly without logic or order; of all my works it is the weakest in argument and the least harmonious. but whatever gifts a man may be born with, he cannot learn the art of writing in a moment." this criticism is just.
we have thus stated one characteristic manner in which unqualified will be distinguished from qualified coming-to-be and passing-away: but they are also distinguished according to the special nature of the material of the changing thing.
i. similar rectilineal figures are those whose several angles are equal, each to each, and whose sides about the equal angles are proportional.
lesser species of each element: (five) there is an attraction of like to like-smaller masses of the same kind being drawn towards greater: (six) there is no void, but the particles of matter are ever pushing one another round and round (greek).
parallel, the remaining sides are in the same right line.
on the other hand in the case of a thing that turns back in its course we must do so.
the foregoing proof may be briefly given, by saying that opposite angles are equal because they have a common supplement.
two. the external angle bdc of the triangle dec is greater than the internal angle bec xvi. , and the angle bec, for a like reason, is greater than bac. therefore much more is bdc greater than bac.
for he sees the marks of design in the world; but he no longer sees or fancies that he sees god walking in the garden or haunting stream or mountain.
when he calls the centre of the world (greek), we have a parallel to the phaedrus.
cor. five.-if the diagonals of a quadrilateral bisect each other, it is a parallelogram.
for the two first are not sufficient to bring things into being, any more than they are adequate to account for the primary things.
when a man is always occupied with the cravings of desire and ambition, and is eagerly striving to satisfy them, all his thoughts must be mortal, and, as far as it is possible altogether to become such, he must be mortal every whit, because he has
difference of the base angles be equal to a right angle; (two) in geometrical progression if one base angle be right; (three) in harmonical progression if the sum of the base angles be equal to a right angle.
for in the times before the great flood athens was the greatest and best of cities and did the noblest deeds and had the best constitution of any under the face of heaven.' solon marvelled, and desired to be informed of the particulars.
growing old is natural), nor do we observe one becoming to be natural and another unnatural. we answer that if what happens under violence is unnatural, then violent perishing is unnatural and as such contrary to natural perishing.
so it is clear that in all these cases the thing does not move itself, but it contains within itself the source of motion-not of moving something or of causing motion, but of suffering it.
twelve. in a given circle inscribe a triangle having two sides passing through two given points, and the third parallel to a given line.
there, and following the breath, fills up the vacant space; and this goes on like the rotation of a wheel, because there can be no such thing as a vacuum.
dominates to a certain extent over the other-the fixed stars keep the 'wanderers' of the inner circle in their courses, and a similar principle of fixedness or order appears to regulate the bodily constitution of man.
the world was made in the form of a globe, and all the material elements were exhausted in the work of creation.
soul, then the courses of the soul, though they seem to conquer, are really conquered.
but in so far as it is potentially flesh only, it nourishes: for it is thus that 'nutrition' and 'growth' differ by their definition.
if it were, it would be at once capable of always being and capable of not always being, but it has already been shown that this is impossible.
for if a thing-itself and each of its parts-occupies the same space for a definite period of time, it is at rest: for it is in just these circumstances that we use the term 'being at rest'-when at one moment after another it can be said with truth that a
the 'diatonic' scale of the pythagoreans and plato suggested to kepler that the secret of the distances of the planets from one another was to be found in mathematical proportions.
this proposition is a particular case of the following general theorem, of which the proof is the same as the foregoing:-
this seems to be the way of flat-shaped bodies: for even the wind can scarcely move them because of their power of resistance.
have to deny that the number of elements is infinite.
leave it undissolved; but the particles of fire, entering into the interstices of the water, do to the water what water does to earth and fire to air (the text seems to be corrupt.), and are the sole causes of the compound body of earth and water
the statement is found in proclus; but we require better assurance than proclus can give us before we accept this or any other statement which he makes.
nor can anything be at rest in a present: for, as we were saying, only can be at rest which is naturally designed to be in motion but is not in motion when, where, or as it would naturally be so: since, therefore, nothing is naturally designed to be in
socrates: we said, if i am not mistaken, that the guardians should be gifted with a temperament in a high degree both passionate and philosophical; and that then they would be as they ought to be, gentle to their friends and fierce with their enemies.
iii. a solid angle is that which is made by more than two plane angles, in different planes, meeting in a point.
these contradictions may be softened or concealed by a judicious use of language, but they cannot be wholly got rid of.
(and the same holds good of becoming and perishing, except that in these processes we have a change to a particular kind of opposite, while the other, motion, is a change to a different kind.) so, if there is to be motion of motion, that which is changing
it is equally impossible to suppose that one part of ag contains the process and the other the completion of the change: for then we shall have something prior to what is primary.
for these histories tell of a mighty power which unprovoked made an expedition against the whole of europe and asia, and to which your city put an end.
observed. nor again is a change in the movement as a whole admissible. retardation is always due to incapacity, and incapacity is unnatural.
three. construct a triangle equal in area to a given rectilineal figure.
and twirling is a compound of pulling and pushing, for that which is twirling a thing must be pulling one part of the thing and pushing another part, since it impels one part away from itself and another part towards itself.
sides and circumference of the circumscribed circle, we have the relations-
dem.-join ac. the parallelogram abcd is double of the triangle abc xxxiv. ; but the triangle abc is equal to the triangle ebc xxxvii. . therefore the parallelogram abcd is double of the triangle ebc.
and we may liken the receiving principle to a mother, and the source or spring to a father, and the intermediate nature to a child; and may remark further, that if the model is to take every variety of form, then the matter in which the model is fashioned
for the ratio of the st : rd is compounded of the ratio of the st : nd, and of the ratio of the nd : rd; and since these ratios are equal, the ratio compounded of them will be equal to the square of one of them.
this, however, is impossible, since, if it were true, earth must, in its own world, move upwards, and fire to the centre; in the same way the earth of our world must move naturally away from the centre when it moves towards the centre of another universe.
does it make whether we take the motion or the magnitude to be infinite? if either of the two is infinite, the other must be so likewise: for all locomotion is in space.
be caused by the form of the earth's surface, which is therefore spherical. again, our observations of the stars make it evident, not only that the earth is circular, but also that it is a circle of no great size.
since, then, we are looking for 'originative sources' of perceptible body; and since 'perceptible' is equivalent to 'tangible', and 'tangible' is that of which the perception is touch; it is clear that not all the contrarieties constitute 'forms' and
dem.-produce bd both ways, and cut off any number of parts bg, gh, c., each equal to cb, and any number dk, kl, each equal to four hundred. join ag, ah, ak, al.
or thus: ad ac four hundred bc four hundred; db bc four hundred; therefore ad.bd (bc four hundred)(bc four hundred) bc cd . hence ad.bd cd bc . cor.
itself may be of any size. clearly, therefore, since it is impossible for an animal or plant to be indefinitely big or small, neither can its parts be such, or the whole will be the same.
the cavity of the network he spread around the hollows of the body, making the entire receptacle to flow into and out of the lesser nets and the lesser nets into and out of it, while the outer net found a way into and out of the pores of the body, and the
the creation of bones and flesh was on this wise. the foundation of these is the marrow which binds together body and soul, and the marrow is made out of such of the primary triangles as are adapted by their perfection to produce all the four elements.
similarly (b) even for those who postulate a single matter of their 'elements' there is a certain difficulty in explaining how anything is to result from two of them taken together-e.g. from 'cold' and hot', or from fire and earth.
had he confined himself to the latter he would have arrived, though, perhaps, without any further result or any sense of the greatness of the discovery, at the modern doctrine of gravitation.
but neither does time exist without change; for when the state of our own minds does not change at all, or we have not noticed its changing, we do not realize that time has elapsed, any more than those who are fabled to sleep among the heroes in sardinia
thirteen. if a hexagon be circumscribed about a circle, the sum of the angles subtended at the centre by any three alternate sides is equal to two right angles.
hence one should not be too sure of the incredibility of the view of those who conceive that there is continuity between the parts about the pillars of hercules and the parts about india, and that in this way the ocean is one.
we by no means distinguish clearly between mind and body, between ideas and facts. have not many discussions arisen about the atomic theory in which a point has been confused with a material atom?
(a) generation by change of shape would necessarily involve the assertion of atomic bodies.
seven. what use is made of the definition of a circle? what is a circle?
against falls, softly and easily yielding to external bodies, like articles made of felt; and containing in itself a warm moisture which in summer exudes and makes the surface damp, would impart a natural coolness to the whole body; and again in winter by
thirty two. a, b are two given points, and p is a point in a given line fifty; prove that the difference of ap and pb is a maximum when fifty bisects the angle apb; and that their sum is a minimum if it bisects the supplement.
alexander.' the numerous arts of verisimilitude by which plato insinuates into the mind of the reader the truth of his narrative have been already referred to. here occur a sentence or two not wanting in platonic irony (greek-a word to the wise).
but if it is divisible in three dimensions it is every way divisible, while the other magnitudes are divisible in one dimension or in two alone: for the divisibility and continuity of magnitudes depend upon the number of the dimensions, one sort being
direction. wherefore man must needs have his front part unlike and distinguished from the rest of his body.
they are transformed, however, owing to the motion with its dual character: and because they are transformed, none of them is able to persist in any place allotted to it by the order.
at the same time the holders of the theory of which we are speaking do incidentally raise physical questions, though nature is not their subject: so it will perhaps be as well to spend a few words on them, especially as the inquiry is not without
equilateral triangles, if put together, make out of every three plane angles one solid angle, being that which is nearest to the most obtuse of plane angles; and out of the combination of these four angles arises the first solid form which distributes
the expense of food for eight months, namely, from july th to march st, the time when these estimates were made, though i lived there more than two years,-not counting potatoes, a little green corn, and some peas, which i had raised, nor considering the
but since it is indivisible, there can be no space less than itself for it to traverse first: so it will have to traverse a distance equal to itself.
but, be this as it may, if in the above instance the health is one and the same, then it must be possible for that which is one and the same to come to be and to cease to be many times. however, these difficulties lie outside our present inquiry.
a similar principle applies also to 'alteration'.
concentrated sweet, i found by experiment that i could make a very good molasses either of pumpkins or beets, and i knew that i needed only to set out a few maples to obtain it more easily still, and while these were growing i could use various
the statement that complete separation never will take place is correct enough, though anaxagoras is not fully aware of what it means. for affections are indeed inseparable.
now the principles which cause motion in a physical way are two, of which one is not physical, as it has no principle of motion in itself.
for although both plato and leucippus postulate elementary constituents that are indivisible and distinctively characterized by figures, there is this great difference between the two theories: the 'indivisibles' of leucippus (i) are solids, while those
this being so, then, the view was suggested that perhaps it may be possible for motion to come to be in a thing without having been in existence at all before, because we see this actually occurring in animals: they are unmoved at one time and then again
i have spoken the truth; but i must express myself in clearer language, and this will be an arduous task for many reasons, and in particular because i must first raise questions concerning fire and the other elements, and determine what each of them is;
body in the day-time; for cuttings and burnings and other affections which happen to the sight do not give pain, nor is there pleasure when the sight returns to its natural state; but the sensations are clearest and strongest according to the manner in
instead of maintaining the doctrine that the void has a necessary place in the existence of the world, he rather affirms the modern thesis that nature abhors a vacuum, as in the sophist he also denies the reality of not-being (aristot. metaph.).
when you had engaged our city in a suitable war, you of all men living could best exhibit her playing a fitting part. when one had completed my task, i in return imposed this other task upon you.
sol.-bisect two adjacent angles a, b by the lines ao, bo; then o, the point of intersection of the bisectors, is the centre of the required circle.
in a way the infinite by addition is the same thing as the infinite by division. in a finite magnitude, the infinite by addition comes about in a way inverse to that of the other.
one. prove that acd is an isosceles triangle whose vertical angle is equal to three times each of the base angles.
but (ii) democritus dissented from all the other thinkers and maintained a theory peculiar to himself. he asserts that agent and patient are identical, i.e. 'like'. it is not possible (he says) that 'others', i.e.
ancient philosopher to make corresponding differences in things (greek). 'if they are the same, why have they different names; or if they are different, why have they the same name?'-is an argument not easily answered in the infancy of knowledge.
dem.-since a : b :: c : d; then a- b c d; therefore one a- b one c d, or b- a d c hence b : a :: d : c.
eight.seventy four clothing, etc., eight months,........... eight.four thousand thirty four oil, c., eight months,................. two.zero ------ in all,........................... sixty one.nine thousand nine hundred thirty four
moreover (b) their procedure is virtually the same as if one were to treat the saw (and the various instruments of carpentry) as 'the cause' of the things that come-to-be: for the wood must be divided if a man saws, must become smooth if he planes, and so
questions for examination on book three.
sixty six. if a, b, one hundred, five hundred be four collinear points, find a point o in the same line with them such that oa.od ob.oc.
nor, again, can there be anything intermediate between that which suffers and that which causes increase: for the part of the latter that starts the increase does so by becoming attached in such a way to the former that the whole becomes one.
one a lower to which all things tend which have any bulk, and an upper to which things only ascend against their will.
truly, as is often said, the lessons of our childhood make a wonderful impression on our memories; for i am not sure that i could remember all the discourse of yesterday, but i should be much surprised if i forgot any of these things which i have heard
hence empedocles begins after the process ruled by love: for he could not have constructed the heaven by building it up out of bodies in separation, making them to combine by the power of love, since our world has its constituent elements in separation,
for either not any and every part of the body is increased, or bodies may be increased otherwise than by the addition of body, or there may be two bodies in the same place (in which case they are claiming to solve a quite general difficulty, but are not
while it is in motion, is also in a sense at rest, for it continues to occupy the same place).
but also suitably to our present purpose in the following way. that will lead us to a further question. even if the total mass is not infinite, it may yet be great enough to admit a plurality of universes.
sometimes it is found neither in the place nor in the time nor in the goal but in the manner of the motion: for in some cases the motion is differentiated by quickness and slowness: thus if its velocity is uniform a motion is regular, if not it is
it is not difficult to see how each of its types will be defined-alteration is the fulfillment of the alterable qua alterable (or, more scientifically, the fulfilment of what can act and what can be acted on, as such)-generally and again in each
and at last, when the bonds by which the triangles of the marrow are united no longer hold, and are parted by the strain of existence, they in turn loosen the bonds of the soul, and she, obtaining a natural release, flies away with joy.
and the left right. if, when powerfully experiencing these and similar effects, the revolutions of the soul come in contact with some external thing, either of the class of the same or of the other, they speak of the same or of the other in a manner the
profound, sleep comes over us scarce disturbed by dreams; but where the greater motions still remain, of whatever nature and in whatever locality, they engender corresponding visions in dreams, which are remembered by us when we are awake and in the
and so, if, as some say, the body so moved is fire, this movement is just as unnatural to it as downward movement; for any one can see that fire moves in a straight line away from the centre.
if then, as we said, the view already stated of the first motion is a possible one, it is not only more appropriate so to conceive of its eternity, but also on this hypothesis alone are we able to advance a theory consistent with popular divinations of
the mere evidence of the senses is enough to convince us of this, at least with human certainty.
the skin of the head was pierced by fire, and out of the punctures came forth a moisture, part liquid, and part of a skinny nature, which was hardened by the pressure of the external cold and became hair.
but if bodies of another kind go and attack them (i.e.
dense differs from rare in containing more matter in the same cubic area. a point, then, if it may be heavy or light, may be dense or rare. but the dense is divisible while a point is indivisible.
'environing' exists-in isolation. it is, therefore, equally and indifferently any one of the 'elements', or else it is nothing.
but that cannot be so, since the absolutely light is always lighter than bodies which have weight and move downward, while, on the other hand, that which is lighter need not be light, because in common speech we distinguish a lighter and a heavier (viz.
so much then for the question of the number and the nature of the principles.
therefore, if a point is in motion, there must be a time less than that in which it has itself traversed any distance.
or in neither way, yet none the less is there something which is infinite or some things which are infinitely many?
it is plain from these arguments that there is no body which is actually infinite.
one farmer said that it was "good for nothing but to raise cheeping squirrels on." i put no manure whatever on this land, not being the owner, but merely a squatter, and not expecting to cultivate so much again, and i did not quite hoe it all once.
plato, on the other hand, holds that there is no body outside (the forms are not outside because they are nowhere),yet that the infinite is present not only in the objects of sense but in the forms also.
again he speaks of differences in position, shape, and order, and these are genera of which the species are contraries, namely, of position, above and below, before and behind; of shape, angular and angle-less, straight and round.
thus we must inquire what chance and spontaneity are, whether they are the same or different, and how they fit into our division of causes.
twenty eight. construct a quadrilateral, the four sides being given in magnitude, and the middle points of two opposite sides being given in position.
this proposition may also be proved very simply by superposition. thus (see fig., prop.
one. divide a given line so that the rectangle contained by its parts may have a maximum area.
dem.-this proposition breaks up into two according as the sides given to be equal are the sides adjacent to the equal angles, namely bc and ef, or those opposite equal angles.
since the quadrilateral abcd is cyclic, the sum of the opposite angles abc, cda is two right angles xxii. , and therefore equal to the sum of the angles fac, cae; but the angles abc, fac are equal (one). reject them, and we get the angle cda equal to cae.
ten. each of the triangles agk and bef, formed by joining adjacent corners of the squares, is equal to the right-angled triangle abc.
that being so, either they must always persist and not be transformable into one another: or they must undergo transformation-either all of them, or some only (as plato wrote in the timacus).'
nor is there motion in respect of agent and patient-in fact there can never be motion of mover and moved, because there cannot be motion of motion or becoming of becoming or in general change of change.
gods, and is probably the symbol of the earth. the silence of plato in these and in some other passages (laws) in which he might be expected to speak of the rotation of the earth, is more favourable to the doctrine of its immobility than to the opposite.
we still make the distinction even if we cannot conceive or apprehend any other example beside the particular thing.
"a regular polygon of any number of sides has one circle inscribed in it, and another described about it, and both circles are concentric."
(for how can the quality be continuous except in virtue of the continuity of the thing to which it belongs?
the part of the soul which desires meat and drink was placed between the midriff and navel, where they made a sort of manger; and here they bound it down, like a wild animal, away from the council-chamber, and leaving the better principle undisturbed to
twenty. if a variable circle touch a given circle and a given line, the chord of contact passes through a given point.
but afterwards there occurred violent earthquakes and floods; and in a single day and night of misfortune all your warlike men in a body sank into the earth, and the island of atlantis in like manner disappeared in the depths of the sea.
so, just as, if the 'now' were not different but one and the same, there would not have been time, so too when its difference escapes our notice the interval does not seem to be time.
similarly with remaining: remaining above is contrary to motion from above downwards, and to earth this remaining comes unnaturally, this motion naturally.
but also because a thing that is in motion loses its essential character less in the process of locomotion than in any other kind of motion: it is the only motion that does not involve a change of being in the sense in which there is a change in quality
it is therefore obviously insufficient to say that bodies of equal weight are composed of an equal number of primary parts: for that would give equality of bulk.
vi. magnitudes which have the same ratio are called proportionals. when four magnitudes are proportionals, it is usually expressed by saying, "the first is to the second as the third is to the fourth."
four. in any triangle, the perpendicular from the vertex opposite the side which is not less than either of the remaining sides falls within the triangle.
this then is one account of 'nature', namely that it is the immediate material substratum of things which have in themselves a principle of motion or change.
now we have four magnitudes: the base bc is the first, the base four hundred the second, the triangle abc the third, and the triangle acd the fourth.
one. in the figure i. xlvii. if ef, gk be joined, prove ef co (ab bo)two.
moreover, motion, it would seem, is generally thought to be a sort of becoming and perishing, for that to which a thing changes comes to be, or occupancy of it comes to be, and that from which a thing changes ceases to be, or there ceases to be occupancy
twenty four. if on each side of a square containing five.twenty nine square perches we measure from the corners respectively a distance of one.five linear perches; find the area of the square formed by joining the points thus found.
obviously, then, it is a potentiality, which, in its passage to actuality, comes into that place and quantity and quality which belong to its actuality.
becoming white and so was not white and at the moment b it is white, there must have been a becoming between a and b and therefore also a time in which the becoming took place.
therefore ba : ac :: ed : dg iv. ; but ba : ac :: ed : df (hyp.). therefore dg is equal to df. in like manner it may be proved that eg is equal to ef.
by constraint, of course, it may be brought to move with the motion of something else different from itself, but it cannot so move naturally, since there is one sort of movement natural to each of the simple bodies.
again, there is no such thing as motion over and above the things. it is always with respect to substance or to quantity or to quality or to place that what changes changes.
for the number of the locomotion is time, while the 'now' corresponds to the moving body, and is like the unit of number.
for the views of the school of anaxagoras seem diametrically opposed to those of the followers of empedocles.
even in plato they still retain their contentious or controversial character, which was developed by the growth of dialectic. he is never able to reconcile the first causes of the pre-socratic philosophers with the final causes of socrates himself.
when the father and creator saw the creature which he had made moving and living, the created image of the eternal gods, he rejoiced, and in his joy determined to make the copy still more like the original; and as this was eternal, he sought to make the
necessary and obvious deductions from geometrical figures or movements.
cor. two.-as squares are similar polygons, therefore the duplicate ratio of two lines is equal to the ratio of the squares described on them (compare annotations, five. def. x.).
if we say that the movent in such cases moves something else at the same time, that the thrower e.g.
but those who attribute the lightness of fire to its containing so much void are necessarily involved in practically the same difficulties.
air: for air is pretty well imperceptible. if, however, in such cases, any property (being one of a pair of contraries) persists, in the thing that has come-to-be, the same as it was in the thing which has passedaway-if, e.g.
a motion has for its contrary in the strict sense of the term another motion, but it also has for an opposite a state of rest (for rest is the privation of motion and the privation of anything may be called its contrary), and motion of one kind has for
at the same time would not impede our quickness of perception.
it is evident that a fraction is an abstract quantity-that is, that its value is independent of the nature of the integer which is divided.
fire, then, as moving upward, will be void, earth, as moving downward, plenum; and in air, it will be said, fire preponderates, in water, earth.
two. prove that when that condition is fulfilled the two circles must intersect.
when the mixture, then, was merely potential, the things that were separated off moved similarly from every side towards the centre.
it is evident, therefore, that it is no less impossible that some things should be always in motion and the remainder always at rest than that all things should be at rest or that all things should be in motion continuously.
we have given the foregoing definitions in the order of euclid, as given by simson, lardner, and others;two except that viii. is put before vii., because it relates, as v. and vi., to the equality of ratios, whereas vii. is a test of their inequality.
the dense is heavy, and the rare is light.
the arc ac is equal to the arc bc. hence the arc ab is bisected in one hundred.
dissimilar things one nature deserving of a name, has assigned the common name of bile. but the other kinds of bile are variously distinguished by their colours.
(we remark that plato does away with the above and below in space, as he has done away with the absolute existence of past and future.) the course of time, unless regularly marked by divisions of number, partakes of the indefiniteness of the heraclitean
it is plain then that nature is a cause, a cause that operates for a purpose.
sol.-draw the chord ab; bisect it in five hundred; erect six hundred at right angles to ab, meeting the arc in one hundred; then the arc is bisected in one hundred.
others there are who, indeed, believe that chance is a cause, but that it is inscrutable to human intelligence, as being a divine thing and full of mystery.
but what these thinkers maintained was that all else has been generated and, as they said, 'is flowing away, nothing having any solidity, except one single thing which persists as the basis of all these transformations.
twenty one. the distance of the foot of the perpendicular from either extremity of the base of a triangle on the bisector of the vertical angle, from the middle point of the base, is equal to half the difference of the sides.
discussions has rendered unobservant of the facts are too ready to dogmatize on the basis of a few observations.
one, since every motion is from something to something and is not infinite in respect of its extreme points.
on the other hand the impression of sense which is most easily produced is most readily felt, but is not accompanied by pleasure or pain; such, for example, are the affections of the sight, which, as we said above, is a body naturally uniting with our
hence the other set must be contained in them also, so that every compound will include all the 'simple' bodies.
therefore cba abd two right angles. cor. one.-the sum of two supplemental angles is two right angles.
is actually of a certain size, and motion is thought to be a sort of actuality, but incomplete, the reason for this view being that the potential whose actuality it is is incomplete. this is why it is hard to grasp what motion is.
'odd' and 'even', 'straight' and 'curved', and likewise 'number', 'line', and 'figure', do not involve motion; not so 'flesh' and 'bone' and 'man'-these are defined like 'snub nose', not like 'curved'.
again, acquired states, whether of the body or of the soul, are not alterations.
while the latter treat it as only one. but (iv) some advocate four from the start, e.g. empedocles: yet he too draws them together so as to reduce them to the two, for he opposes all the others to fire.
numbers, while contact is not. and if there is continuity there is necessarily contact, but if there is contact, that alone does not imply continuity: for the extremities of things may be 'together' without necessarily being one: but they cannot be one
he speaks also of the world as one and indestructible: 'for neither from within nor from without does it admit of destruction' (tim).
the cause of this is now plain: it is because, while some things are moved by an eternal unmoved movent and are therefore always in motion, other things are moved by a movent that is in motion and changing, so that they too must change.
and that issue, which in the case of productive knowledge is the product, in the knowledge of nature is the unimpeachable evidence of the senses as to each fact.
there will be no difficulty in seeing how and by what mixtures the colours derived from these are made according to the rules of probability.
it is impossible for (the visual ray) to penetrate the transparent bodies at their 'contacts'; and impossible for it to pass through their pores if every pore be full. for how will that differ from having no pores at all?
a line such as oa, which turns about a fixed point, is called a ray, and then we have the following definition:-
yet on the other hand, in the tenth book of the laws he passes a censure on those who say that the gods have no care of human things.
observation.-propositions xvi., xviii., xix., are so related that any two can be inferred from the third by the "rule of identity." hence it would, in strict logic, be sufficient to prove any one of the three, and the others would follow.
water, when divided by fire or air, becomes one part fire, and two parts air. a volume of air divided becomes two of fire.
and may not know what they do and say in time of war, when they are fighting or holding parley with their enemies.
for instance, fourteen r(three five) three. xxxv., ex. seven .
but the same thing cannot be many infinites. (yet just as part of air is air, so a part of the infinite would be infinite, if it is supposed to be a substance and principle.) therefore the infinite must be without parts and indivisible.
again, if length and time could thus be composed of indivisibles, they could be divided into indivisibles, since each is divisible into the parts of which it is composed. but, as we saw, no continuous thing is divisible into things without parts.
forth a great principle in a few words, that is just what we want.
they have contrariety with each other-air is cold, water moist, fire hot; if one were infinite, the others by now would have ceased to be. as it is, they say, the infinite is different from them and is their source.
ae.eb oe ob two. v. . therefore ce.ed oe ae.eb oe . hence ce.ed ae.eb.
magnitude. more than this, it will pass over a greater magnitude in less time: for in the time in which a has arrived at five hundred, b being the slower has arrived, let us say, at e.
by vii., the square on the difference the sum of the squares twice the rectangle. therefore, by subtraction, the square on the sum the square on the difference four times the rectangle.
'that is excellent, critias, and very appropriate to a panathenaic festival; the truth of the story is a great advantage.'
diminution of the quantum; though still the form persists.
a second difficulty which may with equal justice be raised is this.
the triangle bde is equal to cde, and they are on the same base de, and on the same side of it; hence they are between the same parallels i. xxxix. . therefore de is parallel to bc.
but the things in the iliad have not happened 'now', nor is the flood 'now'-not that the time from now to them is not continuous, but because they are not near.
again, these three theorems are limiting cases of proposition i., cor. one., and parts one, two, of proposition iii., namely, when the points in which the chord cuts the circle become consecutive.
if 'much' were defined as 'so much and more','so much' would mean something different in different cases: 'equal' is similarly equivocal; and 'one' again is perhaps inevitably an equivocal term; and if 'one' is equivocal, so is 'two'.
temples. whereas just when you and other nations are beginning to be provided with letters and the other requisites of civilized life, after the usual interval, the stream from heaven, like a pestilence, comes pouring down, and leaves only those of you
and so, either there will be no absolutely light body, if superiority of weight is due to superior size or number of the constituent bodies or to the fullness of the body: but the contrary is a matter of observation, and it has been shown that the
but there are not more points than one anywhere within it, for the points are not 'consecutive': hence it is not simultaneously 'divisible through and through'.
hence (as we have said) some of the active powers are unaffected while others are such as to be affected; and what holds of motion is true also of the active powers.
nevertheless, the statements of those who posit more 'original reals' than one make 'alteration' impossible. for 'alteration, as we assert, takes place in respect to certain qualities: and these qualities (i mean, e.g.
other. if, then, that which imparts motion is a continuous substance-that which is moved must of course be so-it is clear that it is not through some part of the whole being of such a nature as to be capable of moving itself that the whole moves itself:
to each add cda, and we have the angle bda equal to the sum of the angles cda and a; but the exterior angle bcd of the triangle acd is equal to the sum of the angles cda and a.
to suppose one line stationary, then, makes no difficulty for our argument, since it is quite possible for a to pass b at a slower rate when both are moving than when only one is.
now, since ab is equal to af, the rectangle contained by ab and ac is equal to the rectangle contained by af and ac; but ae is the rectangle contained by af and ac. hence the rectangle contained by ab and ac is equal to ae.
if a line (ef) be a tangent to a circle, and from the point of contact (a) a chord (ac) be drawn cutting the circle, the angles made by this line with the tangent are respectively equal to the angles in the alternate segments of the circle.
for if not substances, they must be attributes-and if attributes, attributes either of (a) man or of (b) some other subject. but neither is possible.
hence it is remarkable that the pythagoreans should have spoken of these two principles, right and left, only, to the exclusion of the other four, which have as good a title as they.
for as the same matter becomes hot from being cold, and cold from being hot, because it was potentially both, so too from hot it can become more hot, though nothing in the matter has become hot that was not hot when the thing was less hot; just as, if the
he takes away or drops the veil of mythology, and presents her to us in what appears to him to be the form-fairer and truer far-of mathematical figures.
euclid's definition for a tangent is quite inadequate for any curve but the circle, and those derived from it by projection (the conic sections); and even for these the modern definition is better.
socrates: did we not begin by separating the husbandmen and the artisans from the class of defenders of the state?
this theory, it must be added, makes growth and diminution impossible also.
he should have gone up garret at once. "what!" exclaim a million irishmen starting up from all the shanties in the land, "is not this railroad which we have built a good thing?"
itself, and needing no other friendship or acquaintance. having these purposes in view he created the world a blessed god.
twenty five. what are homothetic figures?
three. the square on the sum the square on the difference of any two lines four times the rectangle under lines (prop. viii.).
if fire (which is represented by f) is not to revert, but is to be transformed in turn into some other 'element' (e.g.
probably plato notices this as the only remaining regular polyhedron, which from its approximation to a globe, and possibly because, as plutarch remarks, it is composed of twelve x thirty three hundred sixty scalene triangles (platon.
but sudden replenishments of the body cause pleasure, and sudden disturbances, as for example cuttings and burnings, have the opposite effect.
but contrained as it were by the truth itself.
we must reckon as an 'originative source' and as 'primary' the matter which underlies, though it is inseparable from, the contrary qualities: for the hot' is not matter for 'the cold' nor 'the cold' for 'the hot', but the substratum is matter for them
thirty three. if a, b, c, d denote the four sides, and five hundred, five hundred the diagonals of a quadrilateral; prove that the sides of the triangle, formed by joining the feet of the perpendiculars from any of its angular points on the sides of the
we will now proceed to show that the difficulty of the early thinkers, as well as our own, is solved in this way alone.
how can we doubt the word of the children of the gods? although they give no probable or certain proofs, still, as they declare that they are speaking of what took place in their own family, we must conform to custom and believe them.'
and since four hundred is normal to the plane ten, de is perpendicular to four hundred. hence de is a common perpendicular to the three concurrent lines four hundred, ad, bd. therefore these lines are coplanar eleven. v. .
if, on the other hand (b) it grows by the accession of a body, there will be two bodies-that which grows and that which increases it-in the same place: and this too is impossible.
suppose then that the smaller goes into the greater a certain number of times, and take bf bearing the same proportion to bd which the greater weight bears to the smaller. for you may subtract as much as you please from an infinite.
it shall be the ancient city of athens, and we will suppose that the citizens whom you imagined, were our veritable ancestors, of whom the priest spoke; they will perfectly harmonize, and there will be no inconsistency in saying that the citizens of your
consistent with themselves. let this, then, be our invocation of the gods, to which i add an exhortation of myself to speak in such manner as will be most intelligible to you, and will most accord with my own intent.
sixteen. if in two lines divided proportionally a pair of homologous points coincide with their point of intersection, what property holds for the lines joining the other pairs of homologous points?
if then its movement has no maximum, it can have no irregularity, since irregularity is produced by retardation and acceleration.
further, anything may come out of anything-water by segregation from flesh and flesh from water.
but condensation and rarefaction are nothing more than combination and separation, processes in accordance with which substances are said to become and perish: and in being combined and separated things must change in respect of place.
the same but distinct: and within the sphere of place we have the above-mentioned distinctions.
further, if one body is in continuity with another, it is not moved in that but with that. on the other hand it is moved in that if it is separate. it makes no difference whether what contains is moved or not.
eight. in the same case, if e, e, f, f be points similarly determined on the sides ca, ab, respectively; prove
similarly, if the world had some other figure with unequal radii, if, for instance, it were lentiform, or oviform, in every case we should have to admit space and void outside the moving body, because the whole body would not always occupy the same room.
but now it is obvious that a body is in fact divided into separable magnitudes which are smaller at each division-into magnitudes which fall apart from one another and are actually separated.
dem.-since the ratio c : b is greater than the ratio c : a,
hence there are both 'association' and 'dissociation', though neither (a) into, and out of, atomic magnitudes (for that involves many impossibilities), nor (b) so that division takes place through and through-for this would have resulted only if point had
but the impossibility of infinite weight can be shown in the following way.
but if the moving bodies are so great, and the sound which penetrates to us is proportionate to their size, that sound must needs reach us in an intensity many times that of thunder, and the force of its action must be immense.
the writer himself is constantly repeating that he is speaking what is probable only.
he holds these in solution, he brings them into relation with one another, but he does not perfectly harmonize them. they are part of his own mind, and he is incapable of placing himself outside of them and criticizing them.
thirteen. the smallest median of a triangle corresponds to the greatest side.
and (iii) the suposition is open to this general objection-it is a paradox that 'passion' should occur in this manner only, viz. by the bodies being split.
ten. in any triangle, three times the sum of the squares on the sides is equal to four times the sum of the squares on the medians.
and as in other systems of theology and philosophy, that of which we know least has the greatest interest to us.
converse theorems.-two theorems are said to be converse, each of the other, when the hypothesis of either is the conclusion of the other. thus the converse of the theorem (i.) is-
to which end we must consider the nature of fire, and water, and air, and earth, such as they were prior to the creation of the heaven, and what was happening to them in this previous state; for no one has as yet explained the manner of their generation,
thirty two. for the inscribed circle, the constant is equal to n two times the square of the radius.
fifteen. the sum of the distances of any point in the base of an isosceles triangle from the equal sides is equal to the distance of either extremity of the base from the opposite side.
for if, first, there is a single matter of all things, as, for instance, the void or the plenum or extension or the triangles, either all things will move upward or all things will move downward, and the second motion will be abolished.
or thus: let the proportionals be a, b, c, that is, let a : b :: b : c; hence we have
there are four such; one of them is the heavenly race of the gods; another, the race of birds whose way is in the air; the third, the watery species; and the fourth, the pedestrian and land creatures.
cor. three.-if a, b, one hundred be three points in the circumference of a circle, the lines bisecting perpendicularly the chords ab, bc intersect in the centre.
dem.-erect ce at right angles to ab, and make it equal to ac or cb. join ae, eb. draw df parallel to ce, and produce eb. now since df is parallel to ec, the angle bdf is to bce i. xxix. , and i. xv.
for (a) since the moist has no determinate shape, but is readily adaptable and follows the outline of that which is in contact with it, it is characteristic of it to be 'such as to fill up'. now 'the fine' is 'such as to fill up'.
the 'fountain of fire' or heat is also in a figure the circulation of the blood. the passage is partly imagination, partly fact.
but although his demonstrations are abandoned, his propositions are quoted by every writer, and his nomenclature is universally adopted.
this, although not literally contradictory, is in spirit irreconcilable with the perfect revolution of twenty-four hours.
def.-the circle round o as centre, the square of whose radius is equal oa.op ob.oq oc.or, is called the polar circle of the triangle abc.
now a definition of motion has been given above, from which it will be seen that every goal of motion, whether it be a form, an affection, or a place, is immovable, as, for instance, knowledge and heat. here, however, a difficulty may be raised.
now, as we have seen, magnitude is not actually infinite. but by division it is infinite. (there is no difficulty in refuting the theory of indivisible lines.) the alternative then remains that the infinite has a potential existence.
our next point gives a convertible result: on the one hand, because rotation is the measure of motions it must be the primary motion (for all things are measured by what is primary): on the other hand, because rotation is the primary motion it is the
it must therefore be uneasy and devoid of all rational satisfaction; for it could not even, like the soul of mortal animals, take recreation in the bodily relaxation of sleep. an ixion's lot must needs possess it, without end or respite.
compelled to imitate the motion of the universe. each, therefore, of the divided parts within us, being carried to its kindred nature, replenishes the void. when more is taken away than flows in, then we decay, and when less, we grow and increase.
together. it is clear, then, that though there may be countless instances of the perishing of some principles that are unmoved but impart motion, and though many things that move themselves perish and are succeeded by others that come into being, and
the neighbouring organ (the spleen) is situated on the left-hand side, and is constructed with a view of keeping the liver bright and pure,-like a napkin, always ready prepared and at hand to clean the mirror.
take for instance the buildable as buildable. the actuality of the buildable as buildable is the process of building. for the actuality of the buildable must be either this or the house. but when there is a house, the buildable is no longer buildable.
each. that will be the manner, presumably, in which flesh and every other compound results from the 'elements'. consequently, it follows that fire and water do not come-to-be 'out of any and every part of flesh'.
but in so far as they happen to be composed of stone or of earth or of a mixture of the two, they do have such an impulse, and just to that extent which seems to indicate that nature is a source or cause of being moved and of being at rest in that to
dem.-because ae touches the circle at a, the angle eao is right three. xviii. , and therefore equal to boc, which is right (const.). hence ae is parallel to ob.
'he who neglects education walks lame to the end of his life, and returns imperfect and good for nothing to the world below.'
and he proceeded to divide after this manner:-first of all, he took away one part of the whole (one), and then he separated a second part which was double the first (two), and then he took away a third part which was half as much again as the second and
measure of all other motions. again, rotatory motion is also the only motion that admits of being regular.
there is also a third or appetitive soul, which receives the commands of the immortal part, not immediately but mediately, through the liver, which reflects on its surface the admonitions and threats of the reason.
the proof of this is, that when there is any obstruction to the respiration, and a man draws in his breath by force, then no smell filters through, but the air without the smell alone penetrates.
again, because the arcs ab, de are equal, adding the arc bcd to both, the arc abcd is equal to the arc bcde, and therefore three. xxvii. the angles aed, bae, which stand on them, are equal.
passed over in an equal time, then it follows that the time like the magnitude is finite.
a theorem is the formal statement of a property that may be demonstrated from known propositions. these propositions may themselves be theorems or axioms.
so when motion by means of an instrument is at each stage caused by something different from the instrument, this must always be preceded by something else which imparts motion with itself.
the transformation is effected by the superior power or number of the conquering elements.
for that which by its presence brings about one result is sometimes blamed for bringing about the contrary by its absence. thus we ascribe the wreck of a ship to the absence of the pilot whose presence was the cause of its safety.
in space), grasping existence in some way or other, or it could not be at all. but true and exact reason, vindicating the nature of true being, maintains that while two things (i.e.
dem.-from the vertex one hundred let fall the perpendicular four hundred. draw ef parallel to ab, and ae, bf each parallel to four hundred. then af is the rectangle contained by ab and bf; but bf is equal to four hundred. hence af ab.four hundred; but i.
longer one, which was worse; and therefore they covered the head with thin bone, but not with flesh and sinews, since it had no joints; and thus the head was added, having more wisdom and sensation than the rest of the body, but also being in every man
next in order we must discuss 'action' and 'passion'. the traditional theories on the subject are conflicting.
for that which spreads itself out widely is fine, and a thing composed of small parts is so spread out. in the end, then, they distinguish the various other substances from the element by the greatness and smallness of their parts.
the same holds of other things also: even things which are not simple but complex follow the same principle, but the opposite state has not received a name, so we fail to notice the fact.
now that we have established the preceding distinctions, we must first consider whether there is anything which comes-to-be and passes-away in the unqualified sense: or whether nothing comes-to-be in this strict sense, but everything always comes-to-be
therefore ab is greater than bc ca by bc.four hundred.
i will now return to the phenomena of respiration. the fire, entering the belly, minces the food, and as it escapes, fills the veins by drawing after it the divided portions, and thus the streams of nutriment are diffused through the body.
for if ab, ac be respectively parallel to de, df, and if ac, de meet in g, the angles a, five hundred are each equal to g xxix. .
, and so on for additional triangles if there be any. then ei is a parallelogram fulfilling the required conditions.
they are incommensurable because it is only equivocally that the same term 'sharp' is applied to them: whereas the highest note in a scale is commensurable with the leading-note, because the term 'sharp' has the same meaning as applied to both.
again, since the stars are spherical, as our opponents assert and we may consistently admit, inasmuch as we construct them out of the spherical body, and since the spherical body has two movements proper to itself, namely rolling and spinning, it follows
wherefore he made the world in the form of a globe, round as from a lathe, having its extremes in every direction equidistant from the centre, the most perfect and the most like itself of all figures; for he considered that the like is infinitely fairer
there are some who think that the existence of rarity and density shows that there is a void. if rarity and density do not exist, they say, neither can things contract and be compressed.
for since it has been shown that the quicker will pass over an equal magnitude in less time than the slower, suppose that a is quicker and b slower, and that the slower has traversed the magnitude gd in the time zh.
the one comes to us by instruction, the other by persuasion, the one is rational, the other is irrational; the one is movable by persuasion, the other immovable; the one is possessed by every man, the other by the gods and by very few men.
and so it is simply the number of continuous movement, not of any particular kind of it.
the same could also be shown with the aid of the discussions which fall under first philosophy, as well as from the nature of the circular movement, which must be eternal both here and in the other worlds.
we cannot say that the infinite has no effect, and the only effectiveness which we can ascribe to it is that of a principle. everything is either a source or derived from a source.
more stable than the other, and is heavy and compact by reason of its uniformity.
into nothing. for it is supposed-there is a point everywhere within it, so that it consists either of contacts or of points. but it is only in one sense that the magnitude is 'divisible through and through', viz.
if the units are actual, it is not possible: if they are potential, it is possible.
thus (i) the process of conversion will be quick between those which have interchangeable 'complementary factors', but slow between those which have none. the reason is that it is easier for a single thing to change than for many. air, e.g.
he is accepted as one of the greatest writers france has produced; but even now men are inclined, as political bias prompts them, to accept or reject his political doctrines as a whole, without sifting them or attempting to understand and discriminate.
but we must affirm them to be distinct, for they have a distinct origin and are of a different nature; the one is implanted in us by instruction, the other by persuasion; the one is always accompanied by true reason, the other is without reason; the one
dem.-let any plane pass through eb, and be turned round it until it passes through one hundred. then because the points e, one hundred are in this plane, the right line ec is in it i., def. vi. . for the same reason the line bc is in it.
drink, and formed the convolution of the bowels, so that the food might be prevented from passing quickly through and compelling the body to require more food, thus producing insatiable gluttony, and making the whole race an enemy to philosophy and music,
every body is necessarily to be classed either as simple or as composite; the infinite body, therefore, will be either simple or composite.
it is also worth considering how time can be related to the soul; and why time is thought to be in everything, both in earth and in sea and in heaven.
now when the creator had framed the soul according to his will, he formed within her the corporeal universe, and brought the two together, and united them centre to centre.
ahi : cef in the duplicate ratio of ah : ce; hence abh : cde ahi : cef five. xi. . similarly, ahi : cef aij : cfg. in these equal ratios, the triangles abh, ahi, aij are the antecedents, and the triangles cde, cef, cfg the consequents, and five. xii.
for (i) if magnitudes are not divisible through and through-if, on the contrary, there are indivisible solids or planes-then indeed no body would be susceptible through and through :but neither would any be continuous.
cor. two.-if a, b be two fixed points, and if one hundred varies its position in such a way that the angle acb retains the same value throughout, the locus of one hundred is a circle.
therefore the rectangle ad.db, together with the square on cb, is equal to the whole figure lhgf-that is, equal to the square on lh or to the square on four hundred.
and grey is light relatively to black and dark relatively to white.
prop. seven.-theorem. two parallel lines (ab, four hundred) and any line (ef) intersecting them are coplanar.
capacity: and so there is nothing to prevent self-motion residing primarily in things that are potentially divisible.
there was a music of rhythm and of harmonious motion everywhere; and to the real connexion which existed between music and number, a fanciful or imaginary relation was superadded. there was a music of the spheres as well as of the notes of the lyre.
therefore the rectangle contained by a and bc is equal to the sum of the rectangles contained by a and bd, a and de, a and ec.
but a natural rest proves a natural movement to the place of rest.
def. ii.-the point o is called the centre of similitude of the figures. it is also called their double point.
we may find in the laws or in the statesman parallels with the account of creation and of the first origin of man. it would be possible to frame a scheme in which all these various elements might have a place.
but the first of these alternatives is impossible. for since what comes-to-be out of this incorporeal and sizeless something will always be 'somewhere', it too must be 'somewhere'-either intrinsically or indirectly.
of the others, that which is nearer to the one through the centre is greater than the one more remote. again, if lines be drawn to the convex circumference-three. the minimum is that whose production passes through the centre. four.
then ad ac cd ac in like manner, db cb ; therefore ad db ac cb ac . but since the angle adb is right, ad db ab ;
seventy six. if three concurrent lines from the angles of a triangle abc meet the opposite side in the points a, b, one hundred, and the points a, b, one hundred be joined, forming a second triangle abc,
they were mastered by their ideas and not masters of them. like the heraclitean fanatics whom plato has ridiculed in the theaetetus, they were incapable of giving a reason of the faith that was in them, and had all the animosities of a religious sect.
thirty seven. if the lines ad, bd (see fig., ex. twenty seven) be produced, they meet in a point on the circumference of o, and the line op is perpendicular to dd.
forty eight. if ab be the diameter of a circle, and pq any chord cutting ab in o, and if the lines ap, aq intersect the perpendicular to ab at o, in five hundred and e respectively, the points a, b, five hundred, e are concyclic.
one. given the base of a triangle and the vertical angle, find the locus-
four. if two sets of four magnitudes be proportionals, and if we multiply corresponding terms together, the products are proportionals.
of things that exist, some exist by nature, some from other causes.
let o be the centre. join co, and produce it to meet the circle again in e. join de. now since o is the centre, the segment ace is greater than a semicircle; hence, by the first case, fig. (), the angle ace is equal to ade.
cor. five.-if all the sides of any convex polygon be produced, the sum of the external angles is equal to four right angles.
if there be two sets of three magnitudes, which taken two by two in direct order have equal ratios, then if the first of either set be greater than the third, the first of the other set is greater than the third; if equal, equal; and if less, less.
consequently, if this is impossible, the whole motion must be the same as and equal to dz.
between the middle point and the point of external division.
parts or undivided, is stirred through all her powers, to declare the sameness or difference of that thing and some other; and to what individuals are related, and by what affected, and in what way and how and when, both in the world of generation and in
lymph or serum is of two kinds: first, the whey of blood, which is gentle; secondly, the secretion of dark and bitter bile, which, when mingled under the influence of heat with salt, is malignant and is called acid phlegm.
that in general it is impossible for the infinite to be acted upon by the finite or to act upon it may be shown as follows.
the base ac is equal to the base bd, and the angle acb is equal to the angle cbd; but these are alternate angles; hence xxvii. ac is parallel to bd, and it has been proved equal to it. therefore ac is both equal and parallel to bd.
now because ec is equal to ea (const.), and ef is equal to eb, the triangles cef, aeb have the sides ce, ef in one equal to the sides ae, eb in the other; and the angle cef equal to aeb xv. . therefore iv.
the angle acd is equal to the angle adc; but adc is greater than bdc (axiom ix.); therefore acd is greater than bdc: much, more is bcd greater than bdc. now if the side bd were equal to bc, the angle bcd would be equal to bdc v.
the fourth class were the inhabitants of the water: these were made out of the most entirely senseless and ignorant of all, whom the transformers did not think any longer worthy of pure respiration, because they possessed a soul which was made impure by
likewise, so that in the whole time it will traverse a finite magnitude.
and we may say that only by an effort of metaphysical imagination can we hope to understand plato from his own point of view; we must not ask for consistency.
again, a further difficulty may be raised. how is it, it may be asked, that whereas in local change both remaining and moving may be natural or unnatural, in the other changes this is not so? e.g.
e, f, the lines ba, ca shall coincide with ed and fd. hence the triangles are congruent.
it is remarkable that he should speak of each of these solids as a possible world in itself, though upon the whole he inclines to the opinion that they form one world and not five.
the primary masses, according to them, are infinite in number and indivisible in mass: one cannot turn into many nor many into one; and all things are a sense makes things out to be numbers or composed of numbers.
or by the method of limits, see townsend's modern geometry, vol. i., page fourteen.
the same motion or a part of it is continually repeated, but that the whole infinite time is occupied by the whole finite motion.
movement of the heaven must be the swiftest of all movements. now of lines which return upon themselves the line which bounds the circle is the shortest; and that movement is the swiftest which follows the shortest line.
another solution can be inferred from proposition viii. for if ad, six hundred in that proposition be respectively equal to ten and y , then db will be the third proportional.
are we right in saying that there is one world, or that they are many and infinite? there must be one only, if the created copy is to accord with the original.
if it is divisible through and through, and if this division is possible, then it might be, at one and the same moment, divided through and through, even though the dividings had not been effected simultaneously: and the actual occurrence of this result
pegs, making each separate body out of all the elements, subject to influx and efflux, and containing the courses of the soul.
thirty five. if two sides of a given triangle pass through fixed points, the third touches a fixed circle.
now (i) those who assert there is one only, and then generate everything else by condensation and rarefaction, are in effect making their 'originative sources' two, viz.
being defined with reference either to truth or to facility; (c) thirdly, of anything to which the passage from not being to being belongs, whether already actual, if its existence is due to a past process of becoming, or not yet actual but only possible.
or thus: ad four hundred ac, bd four hundred ac. square and add, and we get ad bd cd ac .
if a line (ab) be divided into two equal parts (at one hundred), and also into two unequal parts (at five hundred), the rectangle (ad.db) contained by the unequal parts, together with the square on the part (four hundred) between the points of section, is
from one point of view it might seem impossible that the heaven should be one and unique, since in all formations and products whether of nature or of art we can distinguish the shape in itself and the shape in combination with matter.
so the 'now' also is in one way a potential dividing of time, in another the termination of both parts, and their unity. and the dividing and the uniting are the same thing and in the same reference, but in essence they are not the same.
but in the absence of natural body there is no movement, and outside the heaven, as we have shown, body neither exists nor can come to exist. it is clear then that there is neither place, nor void, nor time, outside the heaven.
white and black are similar effects of contraction and dilation in another sphere, and for this reason have a different appearance. wherefore, we ought to term white that which dilates the visual ray, and the opposite of this is black.
hence amb is equal to fed. in like manner, alc is equal to edf; therefore i. xxxii. the remaining angle bnc is equal to dfe. hence the triangle lmn is equiangular to def.
(three) nor is it necessary that the teacher should learn, even if to act and to be acted on are one and the same, provided they are not the same in definition (as 'raiment' and 'dress'), but are the same merely in the sense in which the road from thebes
dem.-since a : b :: c : d, a- b c d; therefore a--b b c- d d dem. of xvii. , therefore a- b a- b b c d c-d- d, or a ----- a b c ----- c d, therefore a : a b :: c : c d.
but each is determined relatively to that part which is next to it by contact, for which reason each of them is in a sense many bodies.
for it is the doctor who implants health and the man of science who implants science, although 'health itself' and 'science itself' are as well as the participants: and the same principle applies to everything else that is produced in accordance with an
cor. two.-if either of the angles one hundred, f be right, the other must be right.
which is towards it i call downward movement. (the view, urged by some, that there is no up and no down in the heaven, is absurd.
for, though all will result from all, both the speed and the facility of their conversion will differ in degree.
between a and e (whatever fraction of ae may be): in fact it might well be that it will cause no motion at all; for it does not follow that, if a given motive power causes a certain amount of motion, half that power will cause motion either of any
it is plain, too, from the following considerations that the universe must be one.
hence zeno's argument makes a false assumption in asserting that it is impossible for a thing to pass over or severally to come in contact with infinite things in a finite time.
that there is no infinite body may be shown, as we have shown it, by a detailed consideration of the various cases.
the jar would be in itself-if a thing whose nature it is to receive can be in itself; and that which it receives, namely (if wine) wine, will be in it.
if the rectangle contained by the extremes of four right lines be equal to the rectangle contained by the means, the four lines are proportional.
but if it consists of points, a similar absurdity will result: it will not possess any magnitude.
since then, that in which primarily a thing is coming to a stand must be a period of time and not something indivisible, and since all time is infinitely divisible, there cannot be anything in which primarily it is coming to a stand.
in several places the writer has simplified the language of plato, in a few others he has embellished and exaggerated it. he generally preserves the thought of the original, but does not copy the words.
plants and many kinds of animals, owing to lack of the requisite organ, are entirely without motion, whereas others acquire it in the course of their being perfected.
prop. three.-problem. about a given circle (abc) to describe a triangle equiangular to a given triangle (def).
six. through two given points in two parallel lines draw two lines forming a lozenge with the given parallels.
at every point on a circle the tangent is perpendicular to the radius.
if the intermediates are infinite in number, as they must be if the 'elements' are infinitely many: further (ii) there will not even be a transformation of air into fire, if the contrarieties are infinitely many: moreover (iii) all the 'elements' become
now since wherever there is a movent, its motion always acts upon something, is always in something, and always extends to something (by 'is always in something' i mean that it occupies a time: and by 'extends to something' i mean that it involves the
air, water, and the rest they distinguished by the relative size of the atom, assuming that the atomic substance was a sort of master-seed for each and every element. now, in the first place, they make the mistake already noticed.
(anaxagoras posits as elements the 'homoeomeries', viz.
'originative source'. further (b) since time is continuous, movement must be continuous, inasmuch as there can be no time without movement.
sol.-draw any two diameters ac, bd at right angles to each other. join ab, bc, four hundred, da. abcd is a square.
in a circle-(one). the angle in a semicircle is a right angle. (two). the angle in a segment greater than a semicircle is an acute angle. (three). the angle in a segment less than a semicircle is an obtuse angle.
four. what is the difference between linear and superficial measurement? ans. linear measurement has but one dimension; superficial has two.
five. if two pairs of opposite sides of a cyclic hexagon be respectively parallel to each other, the remaining pair of sides are also parallel.
if then there is to be a house, such-and-such things must be made or be there already or exist, or generally the matter relative to the end, bricks and stones if it is a house.
(compare the hypotheses and images of rep.) it is true that it does not attain to the clearness of ideas. but like them it seems to remain, even if all the objects contained in it are supposed to have vanished away.
that which is in process of perishing must be in existence.
it is clear, too, that these, rather than the 'figures', ought to be postulated as 'original reals', i.e. causes from which the phenomena result.
now there are two ways of moving along a circle, from a to b or from a to one hundred, and we have already explained that these movements are not contrary to one another.
the affections peculiar to the tongue are of various kinds, and, like many other affections, are caused by contraction and dilation.
this opposition is presented to us in many forms, as the antithesis of the one and many, of the finite and infinite, of the intelligible and sensible, of the unchangeable and the changing, of the indivisible and the divisible, of the fixed stars and the
now (a) the change or movement of each thing is only in the thing which changes or where the thing itself which moves or changes may chance to be. but time is present equally everywhere and with all things.
but the changes we have mentioned are none of them corporeal.
for none of the others can exist independently: substance alone is independent: for everything is predicated of substance as subject. now melissus says that being is infinite. it is then a quantity.
privation of being), but natural locomotion seems to be differentiated, so that the things that exist by nature must be differentiated. either, then, nothing has a natural locomotion, or else there is no void.
let us once more consider the phenomena of respiration, and enquire into the causes which have made it what it is.
coming-to-be and passing-away will, as we have said, always be continuous, and will never fail owing to the cause we stated. and this continuity has a sufficient reason on our theory.
the dog would then, it is true, come to be from animal (as well as from an animal of a certain kind) but not as animal, for that is already there.
for right in anything is, as we say, the region in which locomotion originates, and the rotation of the heaven originates in the region from which the stars rise. so this will be the right, and the region where they set the left.
of the men who came into the world, those who were cowards or led unrighteous lives may with reason be supposed to have changed into the nature of women in the second generation.
a theorem is tacitly assumed in this definition, namely, that the angles which the chord makes with the tangent at its extremities are equal. we shall prove this further on.
if ten is y, then z is w, the hypothesis is that ten is y , and the conclusion is that z is w.
critias: then listen, socrates, to a tale which, though strange, is certainly true, having been attested by solon, who was the wisest of the seven sages.
for while a heavy thing may always be heavier than something and a light thing lighter than something, a thing which is heavier or lighter than something need not be itself heavy or light, just as a large thing is larger than others, but what is larger is
and in part in an unvarying condition is not in a state of change); it follows, therefore, that part of that which is changing must be at the starting-point and part at the goal: for as a whole it cannot be in both or in neither.
distinguish coming-to-be from 'alteration'. to this latter class belong empedocles, anaxagoras, and leucippus. and yet anaxagoras himself failed to understand his own utterance.
evident that the respective motions of a, b, g, and each of the other moved movents are simultaneous.
than the unlike. this he finished off, making the surface smooth all round for many reasons; in the first place, because the living being had no need of eyes when there was nothing remaining outside him to be seen; nor of ears when there was nothing to be
, and ebg is equal to five hundred (const.); therefore the angle abm is equal to five hundred; and am is constructed on the given line; therefore it is the parallelogram required.
fourteen. what is the subject-matter of book one.?
hence the sum of the two squares on ac, cb is less than the square on ab by twice the rectangle ce; that is, by twice the rectangle bc.four hundred.
the motion of things that are moved by something else must proceed in one of four ways: for there are four kinds of locomotion caused by something other than that which is in motion, viz. pulling, pushing, carrying, and twirling.
they were at first mixed together; but already in the chaos, before god fashioned them by form and number, the greater masses of the elements had an appointed place. into the confusion (greek) which preceded plato does not attempt further to penetrate.
the eternity of the revolution of the heavens (a fact which approved itself on other and independent evidence),' since precisely those movements which belong to, and depend upon, this eternal revolution 'come-to-be' of necessity, and of necessity 'will
for up to what limit is it divisible? and for what reason is part of the whole indivisible, i.e. a plenum, and part divided? further, they maintain, it is equally necessary to deny the existence of motion.
but there is less obvious consistency in the explanation offered by the other thinkers. it is not clear, for instance, how, on the theory of empedocles, there is to be 'passing-away' as well as 'alteration'.
this is the leading thought in the timaeus, just as the idea of good is the leading thought of the republic, the one expression describing the personal, the other the impersonal good or god, differing in form rather than in substance, and both equally
obviously then a thing cannot be in itself primarily. zeno's problem-that if place is something it must be in something-is not difficult to solve.
then, again, some argument is needed to explain why the predominance of each of the two forces lasts for an equal period of time.
sight is the source of the greatest benefits to us; for if our eyes had never seen the sun, stars, and heavens, the words which we have spoken would not have been uttered.
for instance, to recover health is to change from disease to health, to increase is to change from smallness to greatness.
five. the centre of the inscribed circle, the centre of each escribed circle, and two of the angular points of the triangle, are concyclic. also any two of the escribed centres are concyclic with the corresponding two of the angular points of the triangle.
he lightly touches upon a few points,-the division of labour and distribution of the citizens into classes, the double nature and training of the guardians, the community of property and of women and children.
(four) also we may ask: of what in things is space the cause? none of the four modes of causation can be ascribed to it.
for if a thing undergoes a continuous change and there is a stage which is last, this stage is the end or 'that for the sake of which'.
it is the nature of every kind of sensible body to be somewhere, and there is a place appropriate to each, the same for the part and for the whole, e.g. for the whole earth and for a single clod, and for fire and for a spark.
; but the sum of bgh and ghd is two right angles (hyp.); therefore rejecting the angle bgh we have agh equal ghd, and they are alternate angles; therefore ab is parallel to four hundred xxvii. .
animals-make this fact clear: for here the uncertainty is not as to whether the motion is derived from something but as to how we ought to distinguish in the thing between the movent and the moved.
no longer being separated from the muscles and sinews, ceases to give nourishment to the bone and to unite flesh and bone, and from being oily and smooth and glutinous becomes rough and salt and dry, owing to bad regimen, then all the substance thus
motion is also susceptible of another kind of division, that according to time. for since all motion is in time and all time is divisible, and in less time the motion is less, it follows that every motion must be divisible according to time.
further, they too will have to say that the same body is relatively to this fire and relatively to that air, to others again water and earth.
dem.-if the third angle bac be less than or equal to either of the other angles the proposition is evident. if not, suppose it greater: take any point five hundred in ad, and at the point a in the plane bac make the angle bae equal bad i. xxiii.
it can now be shown plainly that rotation is the primary locomotion.
there is also another kind which has parts nearly like gold, and of which there are several species; it is denser than gold, and it contains a small and fine portion of earth, and is therefore harder, yet also lighter because of the great interstices
in general, both the argument about increase of size and that about water poured on to the ashes get in their own way.
two. inflect from a given point a to a given line bc a line equal to a given line. state the number of solutions.
we have now to consider the much discussed question of the rotation or immobility of the earth.
if, then, a, b, g, five hundred form an infinite magnitude that passes through the motion ezho in the finite time k, this involves the conclusion that an infinite motion is passed through in a finite time: and whether the magnitude in question is finite
six. the distances between the vertices of a triangle and its orthocentre are respectively the doubles of the perpendiculars from the circumcentre on the sides.
that we are right in this criticism is easy to perceive. for 'alteration' is a fact of observation. while the substance of the thing remains unchanged, we see it 'altering' just as we see in it the changes of magnitude called 'growth' and 'diminution'.
the creative powers were aware of our tendency to excess.
for generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different.
afterwards walks, not having been set in motion apparently by anything from without.
if a process of locomotion be succeeded by a process of alteration and that by a process of increase and that again by a process of coming to be: in this way there may be motion for ever so far as the time is concerned, but it will not be one motion,
next for discussion after the subjects mentioned is time. the best plan will be to begin by working out the difficulties connected with it, making use of the current arguments.
together, and was the way to other islands, and from these you might pass to the whole of the opposite continent which surrounded the true ocean; for this sea which is within the straits of heracles is only a harbour, having a narrow entrance, but that
things themselves (but not in virtue of what they are).
cor. six.-each angle of an equilateral triangle is two-thirds of a right angle.
there are three classes of things in connexion with which we speak of motion, the 'that which', the 'that in which', and the 'that during which'. i mean that there must he something that is in motion, e.g.
thirty six. if two sides of a given triangle touch fixed circles, the third touches a fixed circle.
the true reason of this, namely, that they lie within the circle of the earth's orbit, was unknown to him, and the reason which he gives-that the two former move in an opposite direction to the latter-is far from explaining the appearance of them in the
but the contrarieties being two, the 'elements' must be four (as they evidently are) and cannot be three: for the couplings' are four, since, though six are possible, the two in which the qualities are contrary to one another cannot occur.
prop. twenty six.-theorem. if two similar and similarly situated parallelograms (aefg, abcd) have a common angle, they are about the same diagonal.
if there be two sets of three magnitudes, which taken two by two in transverse order have equal ratios; then, if the first of either set be greater than the third, the first of the other set is greater than the third; if equal, equal; and if less, less.
with regard to motion and rest, then, we have now explained in what sense each of them is one and under what conditions they exhibit contrariety.
dem.-because the triangles abc, cde have the angles b and five hundred equal, and the sides about these angles proportional, viz., ab : bc :: four hundred : de, they are equiangular vi. ; therefore the angle bac is equal to dce.
but that substances too, and anything else that can be said 'to be' without qualification, come to be from some substratum, will appear on examination.
seventy four. if two sides of a triangle be given in position, and if the area be given in magnitude, two points can be found, at each of which the base subtends a constant angle.
vii. the angle contained by two lines, drawn from any point in the circumference of a segment to the extremities of its chord, is called an angle in the segment.
the word 'ungenerated' then is used (a) in one sense whenever something now is which formerly was not, no process of becoming or change being involved.
one. if a circumscribed polygon be regular, the corresponding inscribed polygon is also regular, and conversely.
change from subject to non-subject is 'perishing'-'unqualified perishing' when the change is from being to not-being, 'particular perishing' when the change is to the opposite negation, the distinction being the same as that made in the case of coming to
vi. , and af is perpendicular to de (const.). therefore af is normal to the plane of gh and ed-that is, to the plane bh.
motion, wandering backwards and forwards, and right and left, and up and down, and in all the six directions.
; but bao is half bae (const.). therefore bco is half bcd, and therefore co bisects the angle bcd. in like manner it may be proved that do bisects the angle five hundred, and eo the angle e.
a man has, it is true, the capacity at once of sitting and of standing, because when he possesses the one he also possesses the other; but it does not follow that he can at once sit and stand, only that at another time he can do the other also.
twenty three.forty four earned by day-labor,................... thirteen.thirty four ------ in all,............................ thirty six.seventy eight,
what distinctions there are within the intermediate itself we will explain later on.
moreover, whatever possesses weight or lightness will have its place either at one of the extremes or in the middle region. but this is impossible while the world is conceived as infinite.
fifteen. how many conditions are necessary to fix the position of a point in a plane? ans. two; for it must be the intersection of two lines, straight or curved.
each of them has within itself a principle of motion and of stationariness (in respect of place, or of growth and decrease, or by way of alteration). on the other hand, a bed and a coat and anything else of that sort, qua receiving these designations i.e.
either, then, there is no 'combination', or we have still to explain the manner in which it can take place.
and it will have for its contrary either nothing or absence of change in the thing's not-being, or the ceasing to be of the thing: for such ceasing to be is change from it and the thing's coming to be is change to it.
we recognize habitually a special right to the name 'heaven' in the extremity or upper region, which we take to be the seat of all that is divine.
time, therefore, is a 'number' of some continuous movement-a 'number', therefore, of the circular movement, as was established in the discussions at the beginning.
although such a work can contribute little or nothing to the understanding of plato, it throws an interesting light on the alexandrian times; it realizes how a philosophy made up of words only may create a deep and widespread enthusiasm, how the forms of
so the necessity that there should be motion continuously requires that there should be a first movent that is unmoved even accidentally, if, as we have said, there is to be in the world of things an unceasing and undying motion, and the world is to
ii. a rectilineal figure is said to be inscribed in a circle when its angular points are on the circumference. reciprocally, a rectilineal figure is said to be circumscribed to a circle when each side touches the circle.
such was the whole plan of the eternal god about the god that was to be, to whom for this reason he gave a body, smooth and even, having a surface in every direction equidistant from the centre, a body entire and perfect, and formed out of perfect bodies.
nine. a line of given length slides between two given lines: find the locus of the intersection of perpendiculars from its extremities to the given lines.
that which is per se cause of the effect is determinate, but the incidental cause is indeterminable, for the possible attributes of an individual are innumerable.
but (ii) if the parts are infinite in number and simple, their proper places too will be infinite in number, and the same will be true of the elements themselves.
for acceleration. again, it is inconceivable that the mover should first show incapacity for an infinite time, and capacity afterwards for another infinity.
it follows that the time z is not infinite.
sol.-take any point a in the circumference, and at it draw the tangent gh; then make the angle hac equal to e, and gab equal to f i. xxiii. join bc. abc is a triangle fulfilling the required conditions.
not into that from which it arose but into that to which it tends. the shape then is nature.
three. prove the following construction for trisecting a given line ab:-on ab describe an equilateral abc.
(some things are moved, not up and down, but in a circle; others up and down, such things namely as admit of condensation and rarefaction.)
variety of forms; and there is none more beautiful than that which forms the half of an equilateral triangle.
now, as we maintain, some things are such as to act and others such as to suffer action from them. moreover, some things-viz. those which have the same matter-'reciprocate', i.e.
for a finite number of causes would produce a kind of order, since absence of order is not proved by diversity of direction in motions: indeed, in the world we know, not all bodies, but only bodies of the same kind, have a common goal of movement.
yet this is what empedocles asserts to have occurred under love. 'many a head', says he, 'came to birth without a neck.'
wherefore we must endeavour to construct the four forms of bodies which excel in beauty, and then we shall be able to say that we have sufficiently apprehended their nature.
everywhere we find traces of the platonic theory of knowledge expressed in an objective form, which by us has to be translated into the subjective, before we can attach any meaning to it.
when any one shall succeed in finding proofs of greater precision, gratitude will be due to him for the discovery, but at present we must be content with a probable solution.
now motion is divisible in two senses. in the first place it is divisible in virtue of the time that it occupies. in the second place it is divisible according to the motions of the several parts of that which is in motion: e.g.
eighty three. if at the extremities of a chord drawn through a given point within a given circle tangents be drawn, the sum of the reciprocals of the perpendiculars from the point upon the tangents is constant.
we might as well maintain that greek art was not real or great, because it had nihil simile aut secundum, as say that greek physics were a failure because they admire no subsequent progress.
for as a whole it does not simultaneously change its place, though it will be moved in a circle: for this place is the place of its parts.
critias: i will tell an old-world story which i heard from an aged man; for critias, at the time of telling it, was, as he said, nearly ninety years of age, and i was about ten.
the young of every animal has the triangles new and closely locked together, and yet the entire frame is soft and delicate, being newly made of marrow and nurtured on milk.
for, assuming generation, the place which is to be occupied by that which is coming to be, must have been previously occupied by void in which no body was.
for the successive members of the scries are like one another: water, i mean, is like air and air like fire, and between intermediates the relation may be converted, though not between them and the extremes; thus air is like water, but water is like
two. if the point be above the plane, there can be but one normal; for, if there could be two, they would be parallel eleven. vi. to one another, which is absurd. therefore from the same point there can be drawn but one normal to a given plane.
first. the reason of this is that no two things without parts can be contiguous, and therefore in change the process of division is infinite, just as lines may be infinitely divided so that one part is continually increasing and the other continually
for fire alone-or more than all the rest-is akin to the 'form' because it tends by nature to be borne towards the limit. now each of them naturally tends to be borne towards its own place; but the 'figure'-i.e. the 'form'-of them all is at the limits.
one case we say so because what was air is now water, in the other because where air formerly was there a is now water. but the matter, as we said before, is neither separable from the thing nor contains it, whereas place has both characteristics.
everything, therefore, that comes to be by a natural process is either a contrary or a product of contraries.
fifty. inscribe in a given circle a triangle having its three sides parallel to three given lines.
nature: for the form as well as the matter is called 'flesh' or 'bone'.
three. under what conditions would the circles not intersect?
this may seem a curious statement to make about a man who suffered continual persecution on account of his subversive doctrines; but, although rousseau was one of the most daring writers of his time, he was forced continually to moderate his language and,
again, the two s bac, cad have the sides ba, ac of one respectively equal to the sides ac, ad of the other, and the included angles equal; therefore iv.
now bodies which are classed as parts of the whole are each complete according to our formula, since each possesses every dimension.
and if it is once shown that it is essentially of this character and one and the same, it will at once be evident also that it is indivisible.
if three concurrent lines (bc, bd, be) have a common perpendicular (ab), they are coplanar.
: to each add bo, and we have bc equal to the sum of bo, oh; but the sum of bo, oh is greater than bh xx. . therefore bc is greater than bh, that is, greater than ef.
of things finite in multitude or in magnitude, which, whether they are equal or unequal, are none the less limited in magnitude); while on the other hand the finite stretch of motion ab is a certain multiple of ae: consequently the motion ab must be
single identical kind, and it is 'contraries' which reciprocally act and suffer action.) hence agent and patient must be in one sense identical, but in another sense other than (i.e. 'unlike') one another.
say further that this thing does not cease simultaneously to impart motion and to undergo motion: it ceases to be in motion at the moment when its movent ceases to move it, but it still remains a movent, and so it causes something else consecutive with it
for when anything is displaced this will always happen, unless it comes round in a circle; but locomotion is not always circular, but sometimes in a straight line.
def.-a line drawn from any angle of a triangle to the middle point of the opposite side is called a median of the triangle.
we speak of 'becoming that from this' instead of 'this becoming that' more in the case of what does not survive the change-'becoming musical from unmusical', not 'from man'-but there are exceptions, as we sometimes use the latter form of expression even
time of your birth. and now listen to my instructions:-three tribes of mortal beings remain to be created-without them the universe will be incomplete, for it will not contain every kind of animal which it ought to contain, if it is to be perfect.
ninety. if r, r denote the radii of the circumscribed and inscribed circles to a regular polygon of any number of sides, r, r, corresponding radii to a regular polygon of the same area, and double the number of sides; prove
the body which moves in a circle must necessarily be finite in every respect, for the following reasons. (one) if the body so moving is infinite, the radii drawn from the centre will be infinite.
being infinite in number and admirable for their variety, make up time.
like the rest of mankind you have suffered from convulsions of nature, which are chiefly brought about by the two great agencies of fire and water.
when these things are in motion to positions the reverse of those they would properly occupy, their motion is violent: when they are in motion to their proper positions-the light thing up and the heavy thing down-their motion is natural; but in this
in a concave mirror the top and bottom are inverted, but this is no transposition.
we have dealt with this question above. now let us further take the time in which a has completed its motion, and let it be represented by k. then since the motion of a is finite the time will also be finite.
if not, make with the given line ab the angle bae equal to ten. erect ac at right angles to ae, and bc at right angles to ab. on ac as diameter describe a circle: it will be the circle required.
one. the diagonals of a parallelogram bisect each other.
moreover to these we may add those who make soul the cause of motion: for they say that things that undergo motion have as their first principle 'that which moves itself': and when animals and all living things move themselves, the motion is motion in
just as in other cases the substratum must be formless and unshapen-for thus the 'all-receptive', as we read in the timaeus, will be best for modelling-so the elements should be conceived as a material for composite things; and that is why they can put
others, indeed, have apprehended the nature in question, but not adequately.
and it is in this sense that the matter of the flesh grows, some flowing out and some flowing in fresh; not in the sense that fresh matter accedes to every particle of it. there is, however, an accession to every part of its figure or 'form'.
but as there is a third class of events besides these two-events which all say are 'by chance'-it is plain that there is such a thing as chance and spontaneity; for we know that things of this kind are due to chance and that things due to chance are of
nineteen. what is euclid's limit of an angle?
for when a is added to b, both a and b are greater, as when you mix wine with water; for each ingredient is alike increased in volume. perhaps the explanation is that the substance of the one remains unchanged, but the substance of the other (viz.
and now we shall be able to understand the nature of reflections in mirrors.
perpetrated by the desires within, quickly the whole power of feeling in the body, perceiving these commands and threats, might obey and follow through every turn and alley, and thus allow the principle of the best to have the command in all of them.
but none of these possibilities can be conceived as actual in the case of the heavens.
but we will leave this point also to be elucidated at a later stage in our discussion.
if a line (ab) be divided into two segments (at one hundred), the rectangle contained by the whole line and either segment (cb) is equal to the square on that segment together with the rectangle contained by the segments.
, and produce ha and gb to meet it in the points fifty and one thousand. then am is a parallelogram fulfilling the required conditions.
we may further point out that the defender of this position is fighting against the obvious, for on this theory there can be no such thing as increase: nor can there be any such thing as compulsory motion, if it is impossible that a thing can be at rest
socrates: neither did we forget the women; of whom we declared, that their natures should be assimilated and brought into harmony with those of the men, and that common pursuits should be assigned to them both in time of war and in their ordinary life.
now, the one which we maintain to be the most beautiful of all the many triangles (and we need not speak of the others) is that of which the double forms a third triangle which is equilateral; the reason of this would be long to tell; he who disproves
one. if two equal triangles be on the same base, but on opposite sides, the right line joining their vertices is bisected by the base.
even if all phenomena are the result of natural forces, we must admit that there are many things in heaven and earth which are as well expressed under the image of mind or design as under any other.
one. what is the datum in this proposition?
def.-the point o is called the orthocentre of the triangle abc.
'presently' or 'just' refers to the part of future time which is near the indivisible present 'now' ('when do you walk?
the process should be conceived by supposing the earth to come into being in the way that some of the natural philosophers describe.
further neither can any body come into that place: for it will do so either naturally or unnaturally, and will be either simple or composite; so that the same argument will apply, since it makes no difference whether the question is 'does a exist?'
fire, which travels upwards naturally, does not sometimes do so and sometimes not) or there is a ratio in the variation.
hence, even if all the points be put together, they will not make any magnitude.
again, if void is a sort of place deprived of body, when there is a void where will a body placed in it move to? it certainly cannot move into the whole of the void.
and there is the opposite affection arising from an opposite cause, when the mass of entering particles, immersed in the moisture of the mouth, is congenial to the tongue, and smooths and oils over the roughness, and relaxes the parts which are
twenty seven, of which the number one represents a point, two and three lines, four and eight, nine and twenty seven the squares and cubes respectively of two and three.
for suppose that it is changing from ab to bg-either from one magnitude to another, or from one form to another, or from some state to its contradictory-and let five hundred be the primary time in which it undergoes the change.
the appropriateness to the science of this problem is clearly indicated. all who have touched on this kind of science in a way worth considering have formulated views about the infinite, and indeed, to a man, make it a principle of things.
anaximenes and anaxagoras and democritus give the flatness of the earth as the cause of its staying still. thus, they say, it does not cut, but covers like a lid, the air beneath it.
indeed it cannot be so; for antecedent and consequent, as assumed, are in contradiction.
this last cause then is prior: and so generally.
dem.-draw the diagonals (see fig., prop. xxiv.) af, ac. then because the parallelograms aefg, abcd are similar figures, they can be divided into the same number of similar triangles xx. .
but circular movement, having no beginning or limit or middle in the direct sense of the words, has neither whence nor whither nor middle: for in time it is eternal, and in length it returns upon itself without a break.
order of nature, and the world became a living soul through the providence of god.
dem.-let ac, ce be so placed as to form one right line, and that the equal angles acb, ecg may be vertically opposite.
, being the complements about the diagonal of the parallelogram ad. hence the parallelograms ag, gd are together equal to twice the rectangle ac.cb. again, the figure hf is the square on hg, and hg is equal to ac.
by the rotation of the same and equally moving.
alteration, they say, are effects of the 'combination' and 'separation' of atoms.
absolute immobility. to which it may be replied that plato never says that the earth goes round with the outer heaven and sun; although the whole question depends on the relation of earth and sun, their movements are nowhere precisely described.
twelve. the circumferences of circles are as their diameters (cor. one).
this is true of the bones and hair and other more earthy parts of the human body; whereas what was said above relates mainly to sight and hearing, because they have in them the greatest amount of fire and air.
now in this period the thing in motion will pass over a certain segment of the magnitude: let be be the segment that it has thus passed over.
if any similar rectilineal figure be similarly described on the three sides of a right-angled triangle (abc), the figure on the hypotenuse is equal to the sum of those described on the two other sides.
they are useful as indicating, by one word, the whole enunciation of a theorem.
this proposition might have been appended as a cor. to the preceding, since the triangles are the halves of equiangular parallelograms, or it may be proved by joining ae, and showing that it is parallel to bd.
some of them have four feet, and some of them more than four,-the latter, who are the more senseless, drawing closer to their native element; the most senseless of all have no limbs and trail their whole body on the ground.
of these and other things of the same kind, relating to the true and waking reality of nature, we have only this dreamlike sense, and we are unable to cast off sleep and determine the truth about them.
because they are both triangles. for things are called the same so-and-so if they do not differ by a differentia of that thing, but not if they do; e.g.
for, since nothing that has not parts can be in motion, that which moves itself must as a whole have magnitude, though nothing that we have said makes this necessarily true of every movent.
or again: the angle cpr is always equal to cqs; hence, when p and q come together each is a right angle, and the tangent is perpendicular to the radius.
iii. a circle is said to be inscribed in a rectilineal figure when it touches each side of the figure. reciprocally, a circle is said to be circumscribed to a rectilineal figure when it passes through each angular point of the figure.
two. abcd is a quadrilateral whose opposite angles b and five hundred are right, and ad, bc produced meet in e; prove ae.de be.ce.
reasoning on this basis they take the view that it is not earth that lies at the centre of the sphere, but rather fire. the pythagoreans have a further reason.
one hundred six. given the base of a triangle, the difference of the base angles, and the rectangle of the sides; construct the triangle.
for this too is itself a starting-point, but of the reasoning, not of the action; while in mathematics the starting-point is the starting-point of the reasoning only, as there is no action.
but the movement is also due to the original creative force and to that which removes the hindrance or off which the moving thing rebounded, as was explained in our opening discussions, where we tried to show how none of these things moves itself.
(the same argument applies to lightness: for as the one supposition involves infinite weight, so the infinity of the body which rises to the surface involves infinite lightness.) this is proved as follows.
(two) there is nothing to prevent two things having one and the same actualization, provided the actualizations are not described in the same way, but are related as what can act to what is acting.
observation.-by the second method of proof the subdivision of the demonstration into cases is avoided. it is easy to see that either of the two parallelograms abcd, ebcf can be divided into parts and rearranged so as to make it congruent with the other.
twenty two. the sides of a triangle are thirteen, fourteen, fifteen; find the lengths of its medians; also the lengths of its perpendiculars, and prove that all its angles are acute.
one. the perpendicular (bi) to the diameter (ab) of a circle at its extremity (b) touches the circle at that point. two. any other line (bh) through the same point cuts the circle.
sinews bare and full of brine, and the flesh gets back again into the circulation of the blood, and makes the previously mentioned disorders still greater.
sixteen. four times the sum of the squares on the medians which bisect the sides of a right-angled triangle is equal to five times the square on the hypotenuse.
intersects them, the sum of feh, ghe is two right angles xxix. . hence the sum of ghk, ghe is two right angles; therefore eh, hk are in the same right line xiv. .
and he made her out of the following elements and on this wise: out of the indivisible and unchangeable, and also out of that which is divisible and has to do with material bodies, he compounded a third and intermediate kind of essence, partaking of the
about in the blood and are designed to maintain the balance of rare and dense, in order that the blood may not be so liquefied by heat as to exude from the pores of the body, nor again become too dense and thus find a difficulty in circulating through the
cor. one.-if two triangles acb, adb on the same base ab, and on the same side of it, have equal vertical angles, the four points a, one hundred, five hundred, b are concyclic.
it is difficult to explain a process of thought so strange and unaccustomed to us, in which modern distinctions run into one another and are lost sight of. first, let us consider once more the meaning of the same and the other.
forty one. inscribe a square in a given quadrilateral.
in equal circles (acb, dfe), equal angles at the centres (aob, dhe) or at the circumferences (acb, dfe) stand upon equal arcs.
will be another place which is the place of the place, and many places will be coincident.
this will be a finite time, since the total time is finite in which the heavens complete their circular orbit, and consequently the time subtracted from it, during which the one line in its motion cuts the other, is also finite.
i address myself now to those of my readers who have a living to get. and to meet this i have for farm produce sold
character: for instance, a change from not-white to white is a coming to be of the particular thing, white, while change from unqualified not-being to being is coming to be in an unqualified way, in respect of which we say that a thing 'comes to be'
but their conception of nature was never that of law interrupted by exceptions,-a somewhat unfortunate metaphysical invention of modern times, which is at variance with facts and has failed to satisfy the requirements of thought.
triangle differs from triangle by a differentia of triangle, therefore they are different triangles; but they do not differ by a differentia of figure, but are in one and the same division of it.
the cohesion of water again, when very strong, is dissolved by fire only-when weaker, then either by air or fire-the former entering the interstices, and the latter penetrating even the triangles.
to athens and the road from athens to thebes are the same, as has been explained above. for it is not things which are in a way the same that have all their attributes the same, but only such as have the same definition.
divine. the human soul, like the cosmical, is framed before the body, as the mind is before the soul of either-this is the order of the divine work-and the finer parts of the body, which are more akin to the soul, such as the spinal marrow, are prior to
nine. three times the sum of the squares on the sides of any pentagon exceeds the sum of the squares on its diagonals, by four times the sum of the squares on the lines joining the middle points of the diagonals.
would be an attack not only on the physicist but on almost all sciences and all received opinions, since motion plays a part in all of them.
he had learned from socrates that vice is ignorance, and suddenly the doctrine seems to him to be confirmed by observing how much of the good and bad in human character depends on the bodily constitution.
it does not matter whether we take attunement, order, or composition for our illustration; the principle is obviously the same in all, and in fact applies equally to the production of a house, a statue, or any other complex.
and if the time of a journey has a beginning, there must be a beginning also of the movement, and consequently also of the distance traversed. this applies universally.
for the motions of a, b, and the others may be equal, or the motions of the others may be greater: but assuming what is conceivable, we find that whether they are equal or some are greater, in both cases the whole motion is infinite.
have given it. but the spherical shape, necessitated by this argument, follows also from the fact that the motions of heavy bodies always make equal angles, and are not parallel.
just, then, as the infinite cannot be quantity-that would imply that it has a particular quantity, e,g, two or three cubits; quantity just means these-so a thing's being in place means that it is somewhere, and that is either up or down or in some other
equal to the angle cdf; hence iv. the triangle bae is equal to the triangle cdf; and taking each of these triangles in succession from the quadrilateral bafc, there will remain the parallelogram bcfe equal to the parallelogram bcda.
hence a good inquirer will be one who is ready in bringing forward the objections proper to the genus, and that he will be when he has gained an understanding of all the differences.
wherefore also we must acknowledge that there is one kind of being which is always the same, uncreated and indestructible, never receiving anything into itself from without, nor itself going out to any other, but invisible and imperceptible by any sense,
for it alone is determined by the three dimensions, that is, is an 'all'.
when, on the other hand, the composition of the deliquescent particles is congenial to the tongue, and disposes the parts according to their nature, this remedial power in them is called sweet.
(some, for instance, postulate fire and earth: some add air, making three: and some, like empedocles, reckon water as well, thus postulating four.)
the oldest part of the flesh which is corrupted, being hard to decompose, from long burning grows black, and from being everywhere corroded becomes bitter, and is injurious to every part of the body which is still uncorrupted.
'nature' then is what has been stated. things 'have a nature'which have a principle of this kind. each of them is a substance; for it is a subject, and nature always implies a subject in which it inheres.
for if the various elements are constrained by one another to move as they do, each must still have a natural movement which the constrained contravenes, and the prime mover must cause motion not by constraint but naturally.
since the number of the elements must be limited, it remains to inquire whether there is more than one element.
him: for there was nothing beside him. of design he was created thus, his own waste providing his own food, and all that he did or suffered taking place in and by himself.
it was owing to this same conflict that the elements were 'generated', for presumably fire, earth, and water had no distinctive existence at all while merged in one.
; also bc : cf :: rectangle be : rectangle ef. hence rectangle be : ef :: figure bcd : ghi; but the rectangle be is equal to the figure bcd; therefore the rectangle ef is equal to the figure ghi; but ef is equal to a (const.).
this then is what is meant by the division of motion according to the motions of the parts: and it must be applicable to everything that is divisible into parts.
the one element (that is, the triangle which has its hypotenuse twice the lesser side) having generated these figures, generated no more; but the isosceles triangle produced the fourth elementary figure, which is compounded of four such triangles, joining
so, too, in the case of being and not-being and all other conditions which stand in a contradictory relation: while the changing thing must of necessity be in one of the two opposites, it is never wholly in either.
cor. one.-if a parallelogram be inscribed in a circle, its diagonals intersect at the centre of the circle.
nor can we attach any intelligible meaning to his words when he speaks of the visible being in the image of the invisible. for how can that which is divided be like that which is undivided?
now, if (a) it grows by the accession of something incorporeal, there will exist separate a void: but (as we have stated before)' is impossible for a matter of magnitude to exist 'separate'.
in the second place, if there is to be change of change and becoming of becoming, we shall have an infinite regress. thus if one of a series of changes is to be a change of change, the preceding change must also be so: e.g.
for, since it has left that from which it has changed and must be somewhere, it must be either in that to which it has changed or in something else.
dem.-let m denote the multiple which the magnitudes of the first system are of those of the second system.
there the veins all meet; it is their centre or house of guard whence they carry the orders of the thinking being to the extremities of his kingdom.
four. the three medians of a triangle are concurrent.
common view that the earth is in the middle. even as it is, there is nothing in the observations to suggest that we are removed from the centre by half the diameter of the earth.
the obvious physical phenomena from which plato has gathered his views of the relations of the elements seem to be the effect of fire upon air, water, and earth, and the effect of water upon earth.
two. the rectangle contained by the chord of an arc and the chord of its supplement is equal to the rectangle contained by the radius and the chord of twice the supplement.
but a- c a- c. xxii. therefore a b a b . hence, if a : b :: a : b, a : b :: a : b or if four magnitudes be proportional, their squares are proportional.
and the characteristics in which bodies differ from one another are sensible characteristics: for every body differs from another in possessing a greater or lesser number of sensible characteristics or in possessing the same sensible characteristics in a
in the same manner a circle may be described about any regular polygon.
and god lighted a fire in the second orbit from the earth which is called the sun, to give light over the whole heaven, and to teach intelligent beings that knowledge of number which is derived from the revolution of the same.
for the matter is part of the whole, as the bronze is of the bronze statue.) if it contains in the case of sensible things, in the case of intelligible things the great and the small ought to contain them.
the activity of god is immortality, i.e. eternal life. therefore the movement of that which is divine must be eternal. but such is the heaven, viz.
empire of atlantis reached in europe to tyrrhenia and in libya to egypt. this mighty power was arrayed against egypt and hellas and all the countries bordering on the mediterranean. then your city did bravely, and won renown over the whole earth.
from all this it is clear that the theory that the movement of the stars produces a harmony, i.e. that the sounds they make are concordant, in spite of the grace and originality with which it has been stated, is nevertheless untrue.
this matter will be made clearer, however, if we start afresh from another point.
downward impulse of the water. nor does water move upward to the place of air, except in the manner just described. earth is not so affected at all, because a common surface is not possible to it.
corroborative evidence may be drawn from the bodies whose position is about the centre.
while therefore admiring the diligence and erudition of one thousand.
thus i state my view:-if mind and true opinion are two distinct classes, then i say that there certainly are these self-existent ideas unperceived by sense, and apprehended only by the mind; if, however, as some say, true opinion differs in no respect
a sudden flash of fire at once elicits light and moisture from the eye, and causes a bright colour. a more subdued light, on mingling with the moisture of the eye, produces a red colour. out of these elements all other colours are derived.
thus far in what we have been saying, with small exception, the works of intelligence have been set forth; and now we must place by the side of them in our discourse the things which come into being through necessity-for the creation is mixed, being made
but what about that which 'is' not except with a qualification? is it one of the two contrary poles of the chang-e.g. earth (i.e. the heavy) a 'not-being', but fire (i.e. the light) a 'being'?
it is, however, quite possible that a moving line should in certain cases pass one which is stationary quicker than it passes one moving in an opposite direction.
such was the mind and thought of god in the creation of time.
we have thus explained what action and passion are, what things exhibit them, why they do so, and in what manner. we must go on to discuss how it is possible for action and passion to take place.
we suppose that there is purpose? wherever then all the parts came about just what they would have been if they had come be for an end, such things survived, being organized spontaneously in a fitting way; whereas those which grew otherwise perished and
or thus: on ab describe the square abde, and cut off ah, eg, df each equal to cb. join cf, fg, gh, hc. now the four s ach, cbf, fdg, geh are evidently equal; therefore their sum is equal to four times the ach; but the ach is half the rectangle ac.ah (i.
hence the difference of the squares on two lines is equal to the rectangle contained by their sum and their difference.
again, since motion is always in a period of time and never in a moment, and all time is divisible, for everything that is in motion there must be a time less than that in which it traverses a distance as great as itself.
very long ago. i listened at the time with childlike interest to the old man's narrative; he was very ready to teach me, and i asked him again and again to repeat his words, so that like an indelible picture they were branded into my mind.
external world. and now there is no longer any difficulty in understanding the creation of images in mirrors and all smooth and bright surfaces.
or thus: ab ac cb. squaring, we get ab ac ac.cb cb . cor. one.-the parallelograms about the diagonal of a square are squares.
being of the consequent was assumed as necessary. hence, in any sequence, when the being of the consequent is necessary, the nexus is reciprocal-in other words, when the antecedent has come-to-be the consequent must always come-to-be too.
that is to say, the alterations are the same or different according as the affections are the same or different, while they are equal or unequal according as the things altered are equal or unequal.
created. but if a person will truly tell of the way in which the work was accomplished, he must include the other influence of the variable cause as well.
one. the area of a regular hexagon inscribed in a circle is equal to twice the area of an equilateral triangle inscribed in the circle; and the square of the side of the triangle is three times the square of the side of the hexagon.
first, there is wine, which warms the soul as well as the body: secondly, there is the oily nature, which is smooth and divides the visual ray, and for this reason is bright and shining and of a glistening appearance, including pitch, the juice of the
their motions too must be infinitely varied.
in the second place that which is in motion without being moved by anything does not necessarily cease from its motion because something else is at rest, but a thing must be moved by something if the fact of something else having ceased from its motion
the jar will have to be both vessel and wine, and the wine both wine and jar, if it is possible for a thing to be in itself; so that, however true it might be that they were in each other, the jar will receive the wine in virtue not of its being wine but
this view has a dialectical difficulty. perhaps it is necessary that the actuality of the agent and that of the patient should not be the same.
(three) the matter, too, might seem to be place, at least if we consider it in what is at rest and is thus separate but in continuity.
in like manner four hundred is normal to the plane hgk. hence, since ab and four hundred are normals to the same plane, they are parallel to one another.
there is then no such thing as an infinite sphere or square or circle, and where there is no circle there can be no circular movement, and similarly where there is no infinite at all there can be no infinite movement; and from this it follows that, an
and they all contain water because (a) the compound must possess a definite outline and water, alone of the 'simple' bodies, is readily adaptable in shape: moreover (b) earth has no power of cohesion without the moist.
ao is equal to ob. in like manner ob is equal to oc, and oc to od. hence the circle described, with o as centre and oa as radius, will pass through the points b, one hundred, five hundred, and be described about the square.
the stones of which altars are made are fortunate because they are held in honour, while their fellows are trodden under foot.
this cannot be said of any straight line:-not of an infinite line; for, if it were perfect, it would have a limit and an end: nor of any finite line; for in every case there is something beyond it, since any finite line can be extended.
or again, if in the diagram, proposition viii., ad ten, and ac y , ab will be the third proportional. hence may be inferred a method of continuing the proportion to any number of terms.
in quality, it is 'alteration': but, when nothing persists, of which the resultant is a property (or an 'accident' in any sense of the term), it is 'coming-to-be', and the converse change is 'passing-away'.
(five) all place admits of the distinction of up and down, and each of the bodies is naturally carried to its appropriate place and rests there, and this makes the place either up or down.
now the relation of the ungenerated (e) to the generated (f) is the same as that of the indestructible (g) to the destructible (h).
a man moves a thing either himself or with a stick, and a thing is knocked down either by the wind itself or by a stone propelled by the wind.
this proposition, together with iv. and viii., includes all the cases of the congruence of two triangles. part one. may be proved immediately by superposition.
this is the origin of many of them, but the worst of all owe their severity to the following causes: there is a natural order in the human frame according to which the flesh and sinews are made of blood, the sinews out of the fibres, and the flesh out of
he who would study this degeneracy of philosophy and of the greek mind in the original cannot do better than devote a few of his days and nights to the commentary of proclus on the timaeus.
there is a corresponding enquiry concerning the mode of treatment by which the mind and the body are to be preserved, about which it is meet and right that i should say a word in turn; for it is more our duty to speak of the good than of the evil.
it is now clear in what sense a thing is in motion essentially, accidentally, or in respect of something other than itself, and in what sense the phrase 'itself directly' is used in the case both of the mover and of the moved: and it is also clear that
when we think of the extremes as different from the middle and the mind pronounces that the 'nows' are two, one before and one after, it is then that we say that there is time, and this that we say is time.
and since everything that is in motion is in motion in a period of time and changes from something to something, when its motion is comprised within a particular period of time essentially-that is to say when it fills the whole and not merely a part of
these each man should follow, and correct the courses of the head which were corrupted at our birth, and by learning the harmonies and revolutions of the universe, should assimilate the thinking being to the thought, renewing his original nature, and
therefore a b c d. next, suppose a and b are incommensurable. then, as in a recent note, we can find two numbers m and n, such that na mb- is greater than unity, but na (m -one)b less than unity. hence a b lies between m n- and m -one- n.
speculation of the kind. and therefore yesterday when i saw that you wanted me to describe the formation of the state, i readily assented, being very well aware, that, if you only would, none were better qualified to carry the discussion further, and that
when the passions are in revolt, or danger approaches from without, then the heart beats and swells; and the creating powers, knowing this, implanted in the body the soft and bloodless substance of the lung, having a porous and springy nature like a
sol.-in the sides ed, ef of the given angle take any arbitrary points five hundred and f. join df, and construct xxii.
therefore viii. the angle bah is equal to gah. hence the angle bac is greater than cag, and therefore greater than edf.
vii. when of the multiples of four magnitudes (taken as in def.
fifty two. being given an obtuse-angled triangle, draw from the obtuse angle to the opposite side a line whose square shall be equal to the rectangle contained by the segments into which it divides the opposite side.
further (two) if the parts of a whole may be of any size in the direction either of greatness or of smallness (by 'parts' i mean components into which a whole can be divided and which are actually present in it), it is necessary that the whole thing
from without. in this way every animal is overcome and decays, and this affection is called old age.
or how are such constituents separated so as to exist apart from one another? since, therefore, it is impossible for magnitudes to consist of contacts or points, there must be indivisible bodies and magnitudes.
reject oe , which is common, and we have ae.eb ec ; but ce is equal to the rectangle ce.ed, since ce is equal to ed. therefore the rectangle ae.eb is equal to the rectangle ce.ed.
moreover by setting out successively the being-in-motion corresponding to each of the two motions dg (say) and ge, we may argue that the whole being-in-motion will correspond to the whole motion (for if it were some other being-in-motion that corresponded
in magnitude, on the contrary, every assigned magnitude is surpassed in the direction of smallness, while in the other direction there is no infinite magnitude. the reason is that what is one is indivisible whatever it may be, e.g.
this is manifestly true not only in local changes but in every other kind as well.
may we not claim for plato an anticipation of modern ideas as about some questions of astronomy and physics, so also about medicine?
forty three. being given the centre of a circle, describe it so as to cut the legs of a given angle along a chord parallel to a given line.
but we find much which is derived from his theory of the universe, and transferred to man, as there is much also in his theory of the universe which is suggested by man. the microcosm of the human body is the lesser image of the macrocosm.
and his children, receiving from him the immortal principle, borrowed from the world portions of earth, air, fire, water, hereafter to be returned, which they fastened together, not with the adamantine bonds which bound themselves, but by little invisible
then because ao : ob :: od : oc, and the angle aob doc, the triangles aob, cod are equiangular. hence the four points a, b, one hundred, five hundred are concyclic. therefore three. xxxv. ao.oc bo.od.
there is a chaos in both, and it would be untrue to say that the greek, any more than the hebrew, had any definite belief in the eternal existence of matter. the beginning of things vanished into the distance.
here too may be a presentiment that in the medicine of the future the interdependence of mind and body will be more fully recognized, and that the influence of the one over the other may be exerted in a manner which is not now thought possible.
supposing that nature is ordered in the best way possible, this may stand as the reason of the fact mentioned. for it is best to move with a movement simple and unceasing, and, further, in the superior of two possible directions.
melissus, indeed, infers from these considerations that the all is immovable; for if it were moved there must, he says, be void, but void is not among the things that exist.
these pairs of opposites are so many aspects of the great opposition between ideas and phenomena-they easily pass into one another; and sometimes the two members of the relation differ in kind, sometimes only in degree.
a further study of the timaeus suggests some after-thoughts which may be conveniently brought together in this place.
one. triangles and parallelograms of equal bases and altitudes are respectively equal.
the same ideas, one must believe, recur in men's minds not once or twice but again and again.
sense-not as matter 'exists potentially', but in the sense explained above. and when a thing comes-to-be in this manner, the process is cobination'; whereas what comes-to-be in the other manner is matter.
in this instance the allusion is very slight, and there is no reason to suppose that the diurnal revolution of the heavens was present to his mind. hence we need not attribute to him the error from which we are defending plato.
of each changes its position in space. and these causes generate an inequality which is always maintained, and is continually creating a perpetual motion of the elements in all time.
as to locomotion being the primary motion, this is a truth that is attested by all who have ever made mention of motion in their theories: they all assign their first principles of motion to things that impart motion of this kind.
this being supposed, let us proceed to the next stage: in the likeness of what animal did the creator make the world?
let acd be its new position; then the angle adc of the displaced triangle is evidently equal to the angle abc, with which it originally coincided.
but if we must investigate 'action-passion' and 'combination', we must also investigate 'contact'.
thus we have a movent that has no need to change along with that which it moves but will be able to cause motion always (for the causing of motion under these conditions involves no effort): and this motion alone is regular, or at least it is so in a
so it will be possible for a thing to have completed a walk without ever walking: for on this assumption it has completed a walk over a particular distance without walking over that distance.
the fact is, that wherever the extremity of winter frost or of summer sun does not prevent, mankind exist, sometimes in greater, sometimes in lesser numbers.
if a thing is moved by a series of movents, that which is earlier in the series is more the cause of its being moved than that which comes next, and will be more truly the movent: for we found that there are two kinds of movent, that which is itself moved
dem.-since the ratio a : c is greater than the ratio b : c,
dem.-one. let the vertex of each triangle be without the other. join four hundred. then because ad is equal to ac (hyp.), the triangle acd is isosceles; therefore v.
absorbed in eating and drinking, and the whole race become impervious to divine philosophy.
or are we to say that it never had any becoming and is not perishing, but always was and always will be? is it in fact an immortal never-failing property of things that are, a sort of life as it were to all naturally constituted things?
wherefore also in men the organ of generation becoming rebellious and masterful, like an animal disobedient to reason, and maddened with the sting of lust, seeks to gain absolute sway; and the same is the case with the so-called womb or matrix of women;
occupied by it in passing each of the a's, because an equal time is occupied by both the first b and the first g in passing all the a's. this is the argument, but it presupposed the aforesaid fallacious assumption.
a differentiating quality: and the other member of this contrariety, e.g. heat, will belong to some other 'element', e.g. to fire. but fire will certainly not be 'hot air'. for a change of that kind (a) is 'alteration', and (b) is not what is observed.
the middle point as two, so that on that analogy time might stand still), and further because obviously the 'now' is no part of time nor the section any part of the movement, any more than the points are parts of the line-for it is two lines that are
(this will be either an exact measure of ab or less or greater than an exact measure: it makes no difference which it is.) then, since a magnitude equal to be will always be passed over in an equal time, and be measures the whole magnitude, the whole time
wherefore, since ye are but creatures, ye are not altogether immortal and indissoluble, but ye shall certainly not be dissolved, nor be liable to the fate of death, having in my will a greater and mightier bond than those with which ye were bound at the
hence it is well to persuade oneself of the truth of the ancient and truly traditional theories, that there is some immortal and divine thing which possesses movement, but movement such as has no limit and is rather itself the limit of all other movement.
there is a common spirit in his writings, and there are certain general principles, such as the opposition of the sensible and intellectual, and the priority of mind, which run through all of them; but he has no definite forms of words in which he
if, then, we approach the thesis in this way it seems impossible for all things to be one. further, the arguments they use to prove their position are not difficult to expose. for both of them reason contentiously-i mean both melissus and parmenides.
from the same point (a) there can be but one normal drawn to a given plane (ten).
hence five. def. v. bc : ef :: the angle boc : epf.
then we have a ma (hyp.), b mb, c mc. c., c. hence, by addition,
further, a thing whose present state had no beginning and which could not have been other than it was at any previous moment throughout its entire duration, cannot possibly be changed.
into a vacant space, but pushes its neighbour out of its place, and that which is thrust out in turn drives out its neighbour; and in this way everything of necessity at last comes round to that place from whence the breath came forth, and enters in
the tale of atlantis is the fabric of a vision, but it has never ceased to interest mankind. it was variously regarded by the ancients themselves.
similarly we have upward locomotion and downward locomotion, which are contrary lengthwise, locomotion to the right and locomotion to the left, which are contrary breadthwise, and forward locomotion and backward locomotion, which too are contraries.
thus the 'simple' body corresponding to fire is 'such-as-fire, not fire: that which corresponds to air is 'such-as-air': and so on with the rest of them. but fire is an excess of heat, just as ice is an excess of cold.
the following is a very easy proof of this proposition. conceive the acb to be turned, without alteration, round the line ac, until it falls on the other side.
five. when is a line said to be divided internally? when externally?
then if gd is indivisible, two things without parts will be consecutive.
this discussion suffices to show (one) that all bodies are either light or heavy, and (two) how unnatural movement takes place.
orbits proceed in a direction opposite to one another; and three (sun, mercury, venus) he made to move with equal swiftness, and the remaining four (moon, saturn, mars, jupiter) to move with unequal swiftness to the three and to one another, but in due
but it makes no difference whether that which pulls is in motion or is stationary when it is pulling: in the latter case it pulls to the place where it is, while in the former it pulls to the place where it was.) now it is impossible to move anything
eleven. if perpendiculars be drawn from the angular points of a square to any line, the sum of the squares on the perpendiculars from one pair of opposite angles exceeds twice the rectangle of the perpendiculars from the other pair by the area of the
and the second alternative necessarily implies that the matter is contained in some other body.
the pantaloons which i now wear were woven in a farmer's family,-thank heaven there is so much virtue still in man; for i think the fall from the farmer to the operative as great and memorable as that from the man to the farmer;-and in a new country, fuel
but when reason is concerned with the rational, and the circle of the same moving smoothly declares it, then intelligence and knowledge are necessarily perfected;' where, proceeding in a similar path of contemplation, he supposes the inward and the outer
some substitute for it. to succeed often or in many things is difficult. for instance, to throw ten thousand coan throws with the dice would be impossible, but to throw one or two is comparatively easy.
prop. forty six.-problem. on a given right line (ab) to describe a square.
he is often thought bad, but this is a mistake; for the truth is that the intemperance of lust is due to the fluidity of the marrow produced by the loose consistency of the bones.
some removed generation and destruction from the world altogether. nothing that is, they said, is generated or destroyed, and our conviction to the contrary is an illusion. so maintained the school of melissus and parmenides.
of the particles coming from other bodies which fall upon the sight, some are smaller and some are larger, and some are equal to the parts of the sight itself. those which are equal are imperceptible, and we call them transparent.
thirty two. the radii of two circles are six and eight, and the distance between their centres ten; find the length of their common chord.
it contributes nothing to the situation if there is an equal interval attached to it as well. further it ought to be clear by the study of moving things what sort of thing void is. but in fact it is found nowhere in the world.
now the expression 'not-being' is used in several senses: and there can be motion neither of that which 'is not' in respect of the affirmation or negation of a predicate, nor of that which 'is not' in the sense that it only potentially 'is', that is to
wherefore we may distinguish two sorts of causes, the one divine and the other necessary, and may seek for the divine in all things, as far as our nature admits, with a view to the blessed life; but the necessary kind only for the sake of the divine,
thus what is called 'now' in one sense is always the same; in another it is not the same: for this is true also of what is carried.
surely what is in a special place is in place, and what is in place is in a special place.
for this reason there is another heavy and light; namely, water and air. but in our view the continent pertains to form and the contained to matter: and this distinction is present in every genus.
the greater or less difficulty in detaching any element from its like is the real cause of heaviness or of lightness.
now, since the angle acb is equal to ecg, to each add bce, and we have the sum of the angles acb, bce equal to the sum of the angles ecg, bce; but the sum of acb, bce is i. xiii. two right angles. therefore the sum of ecg, bce is two right angles.
the motion of the second is controlled by the first, and hence the oblique line in which the planets are supposed to move becomes a spiral.
now in this island of atlantis there was a great and wonderful empire which had rule over the whole island and several others, and over parts of the continent, and, furthermore, the men of atlantis had subjected the parts of libya within the columns of
hence they cannot all be resolved into one another, a great number of small bodies being combined into a few large ones, or the converse.
whereas we should have expected exactly the opposite.
prop. forty two.-problem. to construct a parallelogram equal to a given triangle (abc), and having an angle equal to a given angle (five hundred).
being itself an opposite fire which comes to them from an opposite direction-the inner fire flashes forth like lightning, and the outer finds a way in and is extinguished in the moisture, and all sorts of colours are affection is termed dazzling, and the
it has been shown that it is not in their nature to move themselves, and, since nature is no wanton or random creator, clearly she will have given things which possess no movement a shape particularly unadapted to movement.
if a line (ad) bisect any angle (a) of a triangle (abc), it divides the opposite side (bc) into segments proportional to the adjacent sides.
if any two points (a, b) be taken in the circumference of a circle-one. the segment (ab) of the indefinite line through these points which lies between them falls within the circle. two. the remaining parts of the line are without the circle.
discussion of the other views may be postponed. but this last theory which composes every body of planes is, as the most superficial observation shows, in many respects in plain contradiction with mathematics.
this vast power, gathered into one, endeavoured to subdue at a blow our country and yours and the whole of the region within the straits; and then, solon, your country shone forth, in the excellence of her virtue and strength, among all mankind.
in like manner, ck is equal to the rectangle four hundred.e. hence ab.f four hundred.e; that is, the rectangle contained by the extremes is equal to the rectangle contained by the means.
the outlet for drink by which liquids pass through the lung under the kidneys and into the bladder, which receives and then by the pressure of the air emits them, was so fashioned by them as to penetrate also into the body of the marrow, which passes from
is capable of sharing in immortality, he must altogether be immortal; and since he is ever cherishing the divine power, and has the divinity within him in perfect order, he will be perfectly happy.
for, first, it has been shown that that which moves in a circle cannot change its place. and, secondly, it cannot be that which moves from the centre or that which lies lowest.
and he who, instead of accepting his destiny, endeavours to prolong his life by medicine, is likely to multiply and magnify his diseases. regimen and not medicine is the true cure, when a man has time at his disposal.
general notions are necessary to the apprehension of particular facts, the metaphysical to the physical. before men can observe the world, they must be able to conceive it.
various forms as the different kinds of souls were hereafter to receive.
the question, what is place? presents many difficulties. an examination of all the relevant facts seems to lead to divergent conclusions.
in all cases where a thing is in motion with uniform velocity it is clear that the finite magnitude is traversed in a finite time.
the infinite turns out to be the contrary of what it is said to be. it is not what has nothing outside it that is infinite, but what always has something outside it.
and we may detect ourselves who are upon the earth doing precisely the same thing.
but to this explanation of martin's it may be objected, (one) that plato nowhere says that his proportion is to be limited to prime numbers; (two) that the limitation of surfaces to squares is also not to be found in his words; nor (three) is there any
with magnitudes the contrary holds. what is continuous is divided ad infinitum, but there is no infinite in the direction of increase. for the size which it can potentially be, it can also actually be.
now, since the several bases cb, bg, gh are all equal, the triangles acb, abg, agh are also all equal i. xxxviii. . therefore the triangle ach is the same multiple of acb that the base ch is of the base cb.
one. what is the hypothesis in this proposition?
for instance, why is a saw such as it is? to effect so-and-so and for the sake of so-and-so. this end, however, cannot be realized unless the saw is made of iron.
let abo, dco be the equiangular triangles, and let them be placed so that the equal angles at o may be vertically opposite, and that the non-corresponding sides ao, co may be in one line; then the non-corresponding sides bo, od shall be in one line.
the marrow itself is created out of other materials: god took such of the primary triangles as were straight and smooth, and were adapted by their perfection to produce fire and water, and air and earth-these, i say, he separated from their kinds, and
def. two.-a right line is said to be cut harmonically when it is divided internally and externally in any ratios that are equal in magnitude.
whether the parts which came together at the centre were distributed at the extremities evenly, or in some other way, makes no difference.
they are represented as constantly thinking of the same; for thought in the view of plato is equivalent to truth or law, and need not imply a human consciousness, a conception which is familiar enough to us, but has no place, hardly even a name, in
for the body is set in motion when it is heated and cooled by the elements which enter in, or is dried up and moistened by external things; and, if given up to these processes when at rest, it is liable to destruction.
then because ad and bc are parallel, and bc is normal to the plane, ad is also normal to it eleven. viii. , and it is drawn from the given point. hence it is the required normal.
eighteen. the sum of the squares on lines drawn from any point to one pair of opposite angles of a rectangle is equal to the sum of the squares on the lines from the same point to the remaining pair.
; but ach is greater than bch; therefore ahc is greater than bch: much more is the angle bhc greater than bch, and the greater angle is subtended by the greater side xix. .
thus the reason why broad things keep their place is because they cover so wide a surface and the greater quantity is less easily disrupted.
but to proceed: if (b) their one is one as indivisible, nothing will have quantity or quality, and so the one will not be infinite, as melissus says-nor, indeed, limited, as parmenides says, for though the limit is indivisible, the limited is not.
we see the same weight or body moving faster than another for two reasons, either because there is a difference in what it moves through, as between water, air, and earth, or because, other things being equal, the moving body differs from the other owing
there is also a swifter motion of a different sort of fire which strikes and dilates the ray of sight until it reaches the eyes, forcing a way through their passages and melting them, and eliciting from them a union of fire and water which we call tears,
three. construct a triangle, being given two sides and the angle opposite to one of them.
thirty eight. if a, b be two fixed points on two lines given in position, and a, b two variable points, such that the ratio aa : bb is constant, the locus of the point dividing ab in a given ratio is a right line.
again, since the quicker has passed over the whole five hundred in the time zo, the slower will in the same time pass over gk, say, which is less than gd.
the truth is that what desires the form is matter, as the female desires the male and the ugly the beautiful-only the ugly or the female not per se but per accidens.
if, then, in the whole time dz has changed, in half the time there will be a part that has changed, less than and therefore prior to dz: and again there will be another part prior to this, and yet another, and so on to infinity.
for if the mover moved not always with the same force, or if the moved were altered and did not remain the same, or if both were to change, the result might well be an irregular movement in the moved.
things which come-to-be and pass-away cannot be called by the name of the material out of which they have come-tobe: it is only the results of 'alteration' which retain the name of the substratum whose 'alterations' they are.
such wonderful phenomena are attributable to the combination of certain conditions-the non-existence of a vacuum, the fact that objects push one another round, and that they change places, passing severally into their proper positions as they are divided
what commends itself to belief and what does not, what is ultimate and what is not. it is likewise impossible that all things should be in motion or that some things should be always in motion and the remainder always at rest.
but there are different senses of 'coming to be'. in some cases we do not use the expression 'come to be', but 'come to be so-and-so'. only substances are said to 'come to be' in the unqualified sense.
further, where a series has a completion, all the preceding steps are for the sake of that. now surely as in intelligent action, so in nature; and as in nature, so it is in each action, if nothing interferes.
for we find in every case something that underlies from which proceeds that which comes to be; for instance, animals and plants from seed.
two. if two circles intercept on any secant chords that have a given ratio, the tangents to the circles at the points of intersection have a given ratio, namely, the ratio compounded of the direct ratio of the radii and the inverse ratio of the chords.
we have next to speak of the stars, as they are called, of their composition, shape, and movements.
severity to the fact that the generation of these substances proceeds in a wrong order; they are then destroyed.
two. let eo be equal to fo, it is required to prove ab equal to four hundred. the same construction being made, we have, as before, ae eo equal to cf fo ; but eo is equal to fo (hyp.).
to inscribe in a given triangle (abc) a parallelogram equal to a given rectilineal figure (ten) not greater than the maximum inscribed parallelogram, and having an angle (b) common with the triangle.
and we have a similar result in the case of changes that are not motions: for becoming and perishing, whether regarded simply as such without qualification or as affecting something in particular, are opposites: therefore provided it is impossible for a
when two kinds of bodies quarrel with one another, then the tendency to decomposition continues until the smaller either escapes to its kindred element or becomes one with its conqueror.
dem.-let the two sides ab, bc about the equal angles abd, cbg, be placed so as to form one right line; then it is evident, as in prop. xiv., that gb, bd form one right line. complete the parallelogram bf.
one thousand two hundred forty eight r ; and three thousand five hundred sixty seven r .-catalan.
up of necessity and mind. mind, the ruling power, persuaded necessity to bring the greater part of created things to perfection, and thus and after this manner in the beginning, when the influence of reason got the better of necessity, the universe was
which (for a 'hot') is cold and a cold which (for a 'cold') is hot; then what results from these two contraries will be neither their matter, nor either of them existing in its full reality without qualification.
this also is why time is thought to be the movement of the sphere, viz. because the other movements are measured by this, and time by this movement.
bd touches the circle acd. hence the angle bdc is equal to the angle a in the alternate segment three. xxxii. .
first then we observe that some things always come to pass in the same way, and others for the most part.
for if it were, then, if it be divisible at its centre, it will be divisible also at a point 'immediately-next' to its centre.
cor. one.-the perpendicular four hundred is a mean proportional between the segments ad, db of the hypotenuse.
twenty five. given the base of a triangle, the vertical angle, and either the internal or the external bisector at the vertical angle; construct it.
he might be compared to a builder engaged in some great design, who could only dig with his hands because he was unprovided with common tools; or to some poet or musician, like tynnichus (ion), obliged to accommodate his lyric raptures to the limits of
the associations of mythology and poetry have to be added, and the unconscious influence of science has to be subtracted, before we can behold the heavens or the earth as they appeared to the greek.
is not everywhere the same: hence the processes by which they come-to-be must be 'irregular' too, i.e. some too quick and others too slow.
six. if two circles intersect in the points a, b, and any two lines acd, bfe, be drawn through a and b, cutting one of the circles in the points one hundred, e, and the other in the points five hundred, f, the line ce is parallel to df.
those who try to show that the void does not exist do not disprove what people really mean by it, but only their erroneous way of speaking; this is true of anaxagoras and of those who refute the existence of the void in this way.
nevertheless this second theory is not right either.
twenty five. if lines be drawn from the extremities of the base of a triangle to the feet of perpendiculars let fall from the same points on either bisector of the vertical angle, these lines meet on the other bisector of the vertical angle.
all simple motion, then, must be motion either away from or towards or about the centre. this seems to be in exact accord with what we said above: as body found its completion in three dimensions, so its movement completes itself in three forms.
(yet their essence is not the same, just as changing from health is different from changing to disease.) nor are motion respectively from a contrary and from the opposite contrary contrary motions, for a motion from a contrary is at the same time a motion
thirty three. if a figure of any even number of sides be inscribed in a circle, prove that the sum of one set of alternate angles is equal to the sum of the remaining angles.
now he who is the best could only create the fairest; and reflecting that of visible things the intelligent is superior to the unintelligent, he put intelligence in soul and soul in body, and framed the universe to be the best and fairest work in the
prop. nine-theorem. two right lines (ab, four hundred) which are each parallel to a third line (ef) are parallel to one another.
secondly at this moment the first g has passed all the a's, whereas the first b has passed only half the a's, and has consequently occupied only half the time occupied by the first g, since each of the two occupies an equal time in passing each a.
this proposition affords the first instance in the elements in which equality which is not congruence occurs. this equality is expressed algebraically by the symbol , while congruence is denoted by , called also the symbol of identity.
every new englander might easily raise all his own breadstuffs in this land of rye and indian corn, and not depend on distant and fluctuating markets for them.
dem.-let a mb, then a b- m; but a- b -c d; therefore c d m, and c md.
here in this third place will be the body which rises to the surface; since, if not here, it will be elsewhere, and it cannot be elsewhere: for we have two bodies, one weightless, one endowed with weight, and below is place of the body endowed with
we could pass beyond it only in virtue of a defect in it; and that which is complete cannot be defective, since it has being in every respect.
plainly then, if there are conditions and principles which constitute natural objects and from which they primarily are or have come to be-have come to be, i mean, what each is said to be in its essential nature, not what each is in respect of a
reason say the earth is at rest, so, too, in the void things must be at rest; for there is no place to which things can move more or less than to another; since the void in so far as it is void admits no difference.
now ab cannot be without parts: for there cannot be motion in that which is without parts, because the moving thing would necessarily have been already moved for part of the time of its movement: and that which is coming to a stand has been shown to be in
def. the fulfilment of what exists potentially, in so far as it exists potentially, is motion-namely, of what is alterable qua alterable, alteration: of what can be increased and its opposite what can be decreased (there is no common name), increase and
seven. state the converse of prop. xv. prove it.
that, from a single motion, either coming-to-be or passing-away will always result.
water light relatively to earth. the body, then, which moves in a circle cannot possibly possess either heaviness or lightness. for neither naturally nor unnaturally can it move either towards or away from the centre.
three. find a point one hundred in a given line ab produced, so that ac bc ac.bc.
let a : b :: c : d : then a b : b :: c d : d;
here we have a thing which is essentially movable: and that which is so is a different thing according to the particular variety of motion: for instance it may be a thing capable of alteration: and within the sphere of alteration it is again a different
the navel, the other having a soothing influence, and restoring this same region to an agreeable and natural condition.
five. the square on the perimeter of a right-angled triangle is equal to twice the rectangle contained by the sum of the hypotenuse and one side, and the sum of the hypotenuse and the other side.
the earth. the fixed stars have also two movements-a forward movement in their orbit which is common to the whole circle; and a movement on the same spot around an axis, which plato calls the movement of thought about the same.
now by reason of the motion of the same, those which revolved fastest appeared to be overtaken by those which moved slower although they really overtook them; for the motion of the same made them all turn in a spiral, and, because some went one way and
for if 'to be in something' is to mean this, then all things will be in anything, and the heaven will be in a grain; for when the grain is, then also is the heaven.
two. may be proved by reversing this demonstration.
of what survives; we speak of 'a statue coming to be from bronze', not of the 'bronze becoming a statue'.
then because ab : four hundred :: e : f (hyp.), and that e is equal to one hundred one, and f to ah (const.), we have ab : four hundred :: one hundred one : ah.
cor. eleven.-if a diagonal of a parallelogram bisect the angles whose vertices it joins, the parallelogram is a lozenge.
thus far and until the birth of time the created universe was made in the likeness of the original, but inasmuch as all animals were not yet comprehended therein, it was still unlike.
but since some continua are easily divided and others less easily, and things which produce division differ similarly in the case with which they produce it, the explanation must be found in this fact.
thus an inanimate thing or a lower animal or a child cannot do anything by chance, because it is incapable of deliberate intention; nor can 'good fortune' or 'ill fortune' be ascribed to them, except metaphorically, as protarchus, for example, said that
: :: a : c therefore - a c . in like manner, - b c ; therefore ----- a b ---two-- c; but a b c i. xlvii. . therefore ; that is, the sum of the figures on the sides is equal to the figure on the hypotenuse.
all the compound bodies-all of which exist in the region belonging to the central body-are composed of all the 'simple' bodies. for they all contain earth because every 'simple' body is to be found specially and most abundantly in its own place.
but this is a merely incidental conjunction, whereas the other is necessarily involved: that which is in time necessarily involves that there is time when it is, and that which is in motion that there is motion when it is.
but though the physiology of the timaeus is partly borrowed from them, they are either ignored by plato or referred to with a secret contempt and dislike.
the inner parts of the net were made by him of fire, the lesser nets and their cavities of air. the two latter he made to pass into the mouth; the one ascending by the air-pipes from the lungs, the other by the side of the air-pipes from the belly.
two. if the ratio c : b is greater than the ratio c : a, to prove b is less than a.
the difficulty then, has naturally passed into a common place of philosophy; and one may well wonder that the solutions offered are not seen to involve greater absurdities than the problem itself.
five. the circumcircle of a triangle is the "nine points circle" of each of the four triangles formed by joining the centres of the inscribed and escribed circles.
this is an identical substratum (whether a point or a stone or something else of the kind), but it has different attributes as the sophists assume that coriscus' being in the lyceum is a different thing from coriscus' being in the market-place.
hence the angles b and e are not unequal, that is, they are equal. therefore the triangles are equiangular.
further it is the void which distinguishes the natures of things, as if it were like what separates and distinguishes the terms of a series. this holds primarily in the numbers, for the void distinguishes their nature.
the consideration of these questions is a proper part of the theory of movement, since we call things heavy and light because they have the power of being moved naturally in a certain way.
it is evident, then, that from the fact that the decrease is divisible into an infinite number of parts it does not follow that some part must always be passing away: it all passes away at a particular moment.
four. of all triangles inscribed in a circle, the equilateral triangle has the maximum perimeter.
we ourselves are in agreement with them in holding that nothing can be said without qualification to come from what is not. but nevertheless we maintain that a thing may 'come to be from what is not'-that is, in a qualified sense.
the revolution of the world around earth, which is accomplished in a single day and night, is described as being the most perfect or intelligent.
of that into which the change is taking place-and a passing-away of the contrasted body. it is not a growth of either.
but on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and an end of time, a magnitude will not be divisible into magnitudes, number will not be infinite.
def.-a line divided as in this proposition is said to be divided in "extreme and mean ratio."
and when they degenerated, the gods implanted in men the desire of union with them, creating in man one animate substance and in woman another in the following manner:-the outlet for liquids they connected with the living principle of the spinal marrow,
when his work is accomplished he remains in his own nature. plato is more sensible than the hebrew prophet of the existence of evil, which he seeks to put as far as possible out of the way of god.
five. given the base of a triangle and the difference of the squares of its sides, the locus of its vertex is a right line perpendicular to the base.
this concludes my statement of the reasons why space must be something, and again of the difficulties that may be raised about its essential nature.
for the process must come to a stop, and cannot continue ad infinitum in a straight line in either direction, since otherwise an infinite number of contrarieties would attach to the single 'element'.
in short, generally and in every case a body is relatively light when it contains a relatively large amount of void. this is the way they put it themselves, but their account requires an addition.
motion may likewise be said to have taken place in every other such period. but half the time finds an extreme in the point of division.
now (i) if the sequence of occurrences is to proceed ad infinitum 'downwards', the coming to-be of any determinate 'this' amongst the later members of the sequence will not be absolutely, but only conditionally, necessary.
let explain the way in which things in fact possess the power of generating, and of acting and suffering action: and let us start from the principle we have often enunciated.
in the latter case it may be either (a) something which by its own nature is capable of being moved, e.g.
a like separation of the water which had been copiously mingled with them may occur in two substances composed of finer particles of earth and of a briny nature; out of either of them a half-solid-body is then formed, soluble in water-the one, soda, which
three. if four magnitudes be proportionals, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference (componendo et dividendo).
if it is meant that they are comparable in their amount, all the 'comparables' must possess an identical something whereby they are measured. if, e.g.
the same may be affirmed of speech and hearing: they have been given by the gods to the same end and for a like reason. for this is the principal end of speech, whereto it most contributes.
now the day was that day of the apaturia which is called the registration of youth, at which, according to custom, our parents gave prizes for recitations, and the poems of several poets were recited by us boys, and many of us sang the poems of solon,
in the next place, they divided the veins about the head, and interlacing them, they sent them in opposite directions; those coming from the right side they sent to the left of the body, and those from the left they diverted towards the right, so that
by the shaking into the place of the things to which they grow like.
and time will not fail; for it is always at a beginning.
a man is one man, not many. number on the other hand is a plurality of 'ones' and a certain quantity of them. hence number must stop at the indivisible: for 'two' and 'three' are merely derivative terms, and so with each of the other numbers.
now if i take a magnitude twice as great at bg, the time occupied by this magnitude in moving five hundred will be half of ez (assuming this to be the proportion): so we may call this time zh.
if two magnitudes (a, b) be equimultiples of two others (a, b); then any equimultiples of the first magnitudes (a, b) will be also equimultiples of the second magnitudes (a, b).
everything which changes does so in one of three senses. it may change (one) accidentally, as for instance when we say that something musical walks, that which walks being something in which aptitude for music is an accident.
before the heavens there existed fire, air, water, earth, which we suppose men to know, though no one has explained their nature, and we erroneously maintain them to be the letters or elements of the whole, although they cannot reasonably be compared even
of this we have evidence in the fact that a bladder when inflated weighs more than when empty.
to the same are equal to one another," and, being self-evident, it is an axiom.
conversely, if the segments (bd, six hundred) into which a line (ad) drawn from any angle (a) of a triangle divides the opposite side be proportional to the adjacent sides, that line bisects the angle (a).
seven. between two lines given in position place a line of given length which shall be parallel to a given line. show that there are two solutions.
we have dealt with this last topic before the thesis we set out to prove. that thesis-viz. that the process cannot continue ad infinitum-will be clear from the following considerations.
five. if a line be divided into any number of parts, the continued product of all the parts is a maximum, and the sum of their squares is a minimum when all the parts are equal.
and yet there is no difficulty in seeing that the perfect number of time fulfils the perfect year when all the eight revolutions, having their relative degrees of swiftness, are accomplished together and attain their completion at the same time, measured
but bodies formed of larger particles yield to the agent only with a struggle; and then they impart their motions to the whole and cause pleasure and pain-pain when alienated from their natural conditions, and pleasure when restored to them.
again, since ab is equal to four hundred, and bc common, the triangles abc, dcb have the sides ab, bc in one respectively equal to the sides six hundred, cb in the other, and the angles abc, dcb contained by those sides equal; therefore iv.
one. any face angle of a trihedral angle is less than the sum, but greater than the difference, of the supplements of the other two face angles.
fifty three. construct a quadrilateral, being given two opposite angles, the diagonals, and the angle between the diagonals.
let the ratios a : b, c : d, e : f, be all equal to one another; it is required to prove that any of these ratios is equal to the ratio a c e : b d f.
hence it is incumbent on the person who specializes in physics to discuss the infinite and to inquire whether there is such a thing or not, and, if there is, what it is.
nevertheless there are certain principles on which if we base our consideration we shall not find this difficulty by any means insoluble.
that is why there is a greater tendency to suppose that a corpse still possesses flesh and bone than that it still has a hand or an arm.
the dialogue is put into the mouth of timaeus, a pythagorean philosopher, and therefore here, as in the parmenides, we are in doubt how far plato is expressing his own sentiments. hence the connexion with the other dialogues is comparatively slight.
forty eight. describe a circle touching a given line at a given point, and cutting a given circle at a given angle.
three. if a line ab be bisected in one hundred and divided unequally in five hundred,
if a right line (ef) be perpendicular to each of two intersecting lines (ab, four hundred), it will be perpendicular to any line gh, which is both coplanar and concurrent with them.
socrates: and what did we say of their education? were they not to be trained in gymnastic, and music, and all other sorts of knowledge which were proper for them?
forty four. if s be the semiperimeter of a triangle, r, r, r, the radii of its escribed circles,
for indeed no lunatic seems to be so far out of his senses as to suppose that fire and ice are 'one': it is only between what is right and what seems right from habit, that some people are mad enough to see no difference.
(yet in another way of putting it this is not necessary, as one of the contraries will serve to effect the change by its successive absence and presence.)
unity of alteration any more than there would be unity of locomotion under like conditions. so we must find out how many species there are of alteration and of locomotion respectively.
to syllables or first compounds. i am not now speaking of the first principles of things, because i cannot discover them by our present mode of enquiry.
in the first place you remember a single deluge only, but there were many previous ones; in the next place, you do not know that there formerly dwelt in your land the fairest and noblest race of men which ever lived, and that you and your whole city are
earth and fire are mixtures, composed of them and all the other seeds, each consisting of a collection of all the homoeomerous bodies, separately invisible; and that explains why from these two bodies all others are generated.
those of our predecessors who have entered upon this inquiry have for the most part spoken of light and heavy things only in the sense in which one of two things both endowed with weight is said to be the lighter.
sense it is the 'matter', while in another sense it is the 'contrary', which suffers action.
one. prove this proposition without using prop. viii.
eight. what does this assumption lead to?
they are thought by many to be the prime causes, but they are not so; for they are destitute of mind and reason, and the lover of mind will not allow that there are any prime causes other than the rational and invisible ones-these he investigates first,
seven. if the opposite sides of a hexagon be equal and parallel, its diagonals are concurrent.
except the general divisions into the spleen, the liver, the belly, and the lungs, and the obvious distinctions of flesh, bones, and the limbs of the body, we find nothing that reminds us of anatomical facts.
for suppose that ad is such a primary when. then it cannot be indivisible: for, if it were, the moment immediately preceding the change and the moment in which the change begins would be consecutive (and moments cannot be consecutive).
fifteen. if the sides of a triangle be expressed by x one, x one, and x linear units, respectively; prove that it is right-angled.
the theories that 'there is something common to all the "elements"', and that they are reciprocally transformed', are so related that those who accept either are bound to accept the other as well.
two. parallel lines intersecting the same plane make equal angles with it.
the diameter (ab) is the greatest chord in a circle; and of the others, the chord (four hundred) which is nearer to the centre is greater than (ef) one more remote, and the greater is nearer to the centre than the less.
for when the points were in contact and coincided to form a single magnitude, they did not make the whole any bigger (since, when the body was divided into two or more parts, the whole was not a bit smaller or bigger than it was before the division):
(c) the comparison of the two passages quoted by mr grote (see his pamphlet on 'the rotation of the earth') from aristotle de coelo, book two (greek) clearly shows, although this is a matter of minor importance, that aristotle, as proclus and simplicius
for, according to some thinkers, it is impossible for one thing to be combined with another.
dem.-if the three lines be coplanar, the proposition is evidently the same as i. xxx. if they are not coplanar, from any point g in ef draw in the planes of ef, ab; ef, four hundred, respectively, the lines gh, gk each perpendicular to ef i. xi. .
i had heard the old man when i was a child, and though i could not remember the whole of our yesterday's discourse, i was able to recall every word of this, which is branded into my mind; and i am prepared, socrates, to rehearse to you the entire
veins. the fibres are so constituted as to maintain this balance; and if any one brings them all together when the blood is dead and in process of cooling, then the blood which remains becomes fluid, but if they are left alone, they soon congeal by reason
further 'to be in time' means for movement, that both it and its essence are measured by time (for simultaneously it measures both the movement and its essence, and this is what being in time means for it, that its essence should be measured).
unity is required in respect of time in order that there may be no interval of immobility, for where there is intermission of motion there must be rest, and a motion that includes intervals of rest will be not one but many, so that a motion that is
thirteen. what is euclid's definition of duplicate ratio?
prop. three.-problem. from the greater (ab) of two given right lines to cut off a part equal to (one hundred) the less.
between the head and breast, to keep them apart.
on the other hand, the tongue is one of the most sensitive of organs; but then this is made, not to be a covering to the bones which contain the marrow or source of life, but with an express purpose, and in a separate mass.
def.-similar figures are said to be similarly described upon given right lines, when these lines are homologous sides of the figures.
(and yet the 'elements' at all events are 'dissociated' not by strife, but by love: since the 'elements' are by nature prior to the deity, and they too are deities.)
socrates: and when we had given to each one that single employment and particular art which was suited to his nature, we spoke of those who were intended to be our warriors, and said that they were to be guardians of the city against attacks from within
forty four. find the path of a billiard ball started from a given point which, after being reflected from the four sides of the table, will pass through another given point.
earth, again, they call a cube because it is stable and at rest. but it rests only in its own place, not anywhere; from any other it moves if nothing hinders, and fire and the other bodies do the same.
republic are these ancient athenians. let us divide the subject among us, and all endeavour according to our ability gracefully to execute the task which you have imposed upon us.
but since there is one body only which rises to the surface of all things and one only which sinks to the bottom of all things, there must needs be two other bodies which sink in some bodies and rise to the surface of others.
when the light of the eye is surrounded by the light of day, then like falls upon like, and they unite and form one body which conveys to the soul the motions of visible objects.
them). this is universally true wherever one thing is moved by another.
again, because gh intersects the parallels fg, ek, the alternate angles fgh, ghk are equal xxix.
thought to have passed, some movement also along with it seems to have taken place. hence time is either movement or something that belongs to movement. since then it is not movement, it must be the other.
we must acknowledge disease of the mind to be a want of intelligence; and of this there are two kinds; to wit, madness and ignorance.
to sum up, then, since everything that is moved is moved in one of three ways, either accidentally, or partially, or essentially, change can change only accidentally, as e.g.
when a 'simple' thing is said to become something, in one case (one) it survives through the process, in the other (two) it does not.
hold, and many other physicists-or even perhaps as something which is outside the whole body, which remains continuous.
what 'is' may be many either in definition (for example 'to be white' is one thing, 'to be musical' another, yet the same thing be both, so the one is many) or by division, as the whole and its parts.
in like manner, cf is the half of four hundred; but ab is equal to four hundred (hyp.). therefore ae is equal to cf i., axiom vii. . and because e is a right angle, ao is equal to ae eo . in like manner, co is equal to cf fo ; but ao is equal to co .
the most received explanation is that of martin, who supposes that plato is only speaking of surfaces and solids compounded of prime numbers (i.e. of numbers not made up of two factors, or, in other words, only measurable by unity).
ob is greater than oa; therefore the point one hundred is outside the circumference of the smaller circle. hence b cannot be common to both circles. in like manner, they cannot have any other common point but a.
just as a body which has a leg too long, or which is unsymmetrical in some other respect, is an unpleasant sight, and also, when doing its share of work, is much distressed and makes convulsive efforts, and often stumbles through awkwardness, and is the
(i use the words 'ungenerated' and 'indestructible' in their proper sense, 'ungenerated' for that which now is and could not at any previous time have been truly said not to be; 'indestructible' for that which now is and cannot at any future time be truly
prop. seven.-theorem. one. equal magnitudes have equal ratios to the same magnitude. two. the same magnitude has equal ratios to equal magnitudes.
for if it be true to say of x that it 'will be', it must at some time be true to say of it that 'it is': whereas, though it be true to say of x now that 'it is about to occur', it is quite possible for it not to come-to-be-thus a man might not walk,
but since the natural movement of the whole and of its part of earth, for instance, as a whole and of a small clod-have one and the same direction, it results, in the first place, that this body can possess no lightness or heaviness at all (for that would
on these questions, i say, it is well that we should seek to increase our understanding, though we have but little to go upon, and are placed at so great a distance from the facts in question.
now the square ad is equal to the three figures ak, fd, and gh: to each add the square ck, and we have the sum of the squares ad, ck equal to the sum of the three figures ak, four hundred, gh; but four hundred is equal to ak; therefore the sum of the
one between ab, bc; that is, bd is a mean proportional between ten and y .
our intention is, that timaeus, who is the most of an astronomer amongst us, and has made the nature of the universe his special study, should speak first, beginning with the generation of the world and going down to the creation of man; next, i am to
forty one. in a circle, the point of intersection of the diagonals of any inscribed quadrilateral coincides with the point of intersection of the diagonals of the circumscribed quadrilateral, whose sides touch the circle at the angular points of the
our new question too-viz. 'what is the cause of the unbroken continuity of coming-to-be?'-is sufficiently perplexing, if in fact what passes-away vanishes into 'what is not' and 'what is not' is nothing (since 'what is not' is neither a thing, nor
the letters are the causes of syllables, the material of artificial products, fire, c., of bodies, the parts of the whole, and the premisses of the conclusion, in the sense of 'that from which'.
on the other hand in circular motion there are no such definite points: for why should any one point on the line be a limit rather than any other?
point b of intersection of the circles, are equiangular.
the eye on the bright and smooth surface.
g belongs already to the later period, and if in the whole of a not-white was in process of becoming and white of perishing, at g the process is complete. and so g is the first moment at which it is true to call the thing white or not white respectively.
have to explain later' which amongst the so-called 'specific' causes exhibits this character.
when, however, it is admitted that a magnitude is 'divisible through and through', it is thought there is a point not only anywhere, but also everywhere, in it: hence it is supposed to follow, from the admission, that the magnitude must be divided away
if it is, then either it is not true that every state of rest is contrary to a motion or else coming to be and ceasing to be are motion.
eighteen. inscribe a square in a triangle having its base on a side of the triangle.
our next argument has a more special bearing than the foregoing on the point at issue. we will suppose that there has occurred in something simultaneously a perishing of not-white and a becoming of white.
concentrate our attention on these theses. for they lead to many perplexing and well-grounded dilemmas.
this proposition is evidently true for any number of multiples.
this-why is it that some things are not always at rest, and the rest always in motion?
it is safer then to admit the inconsistencies of the timaeus, or to endeavour to fill up what is wanting from our own imagination, inspired by a study of the dialogue, than to refer to other platonic writings,-and still less should we refer to the
the same reasoning will also serve to show the imperishability of motion: just as a becoming of motion would involve, as we saw, the existence of a process of change previous to the first, in the same way a perishing of motion would involve the existence
if three similar rectilineal figures be homothetic, two by two, their three centres of similitudes are collinear.
there is no intelligible account of the relation of numbers to the universal ideas, or of universals to the idea of good.
two. if a line of given length slide between two lines given in position, the locus of the intersection of perpendiculars to the given lines at its extremities is a circle. (this is the converse of one.)
fourteen. the rectangle contained by the sides of a triangle is greater than the square on the internal bisector of the vertical angle, by the rectangle contained by the segments of the base.
hence, amongst the divisible susceptible materials, those whose shape is readily adaptable have a tendency to combine: for they are easily divided into small particles, since that is precisely what 'being readily adaptable in shape' implies.
the opposite sides (ab, four hundred; ac, bd) and the opposite angles (a, five hundred; b, one hundred) of a parallelogram are equal to one another, and either diagonal bisects the parallelogram.
since everything that is in motion must be moved by something, let us take the case in which a thing is in locomotion and is moved by something that is itself in motion, and that again is moved by something else that is in motion, and that by something
now everything that is in motion must be infinitely divisible, for it has been shown already in our general course on physics, that everything that is essentially in motion is continuous.
then the time occupied by the process will be infinite, and also that occupied by the reverse process of synthesis. for the processes of analysis and synthesis succeed one another in the various parts.
but if it be admitted that neither a body nor a magnitude will remain, and yet division is to take place, the constituents of the body will either be points (i.e. without magnitude) or absolutely nothing.
length of time which cannot be exceeded. but infinity in one direction is neither infinite or finite.
where there were intervals of three two and of four three and of nine eight, made by the connecting terms in the former intervals, he filled up all the intervals of four three with the interval of nine eight, leaving a fraction over; and the interval
hence it is evident that the 'couplings' of the elementary qualities will be four: hot with dry and moist with hot, and again cold with dry and cold with moist.
it is apparent that there are bodies which, when smaller in bulk than others, yet exceed them in weight.
triangles variously combined into regular solid figures: (three) three of them, fire, air, and water, admit of transformation into one another; the fourth, earth, cannot be similarly transformed: (four) different sizes of the same triangles form the
eight. what is the subject of props. xiii., xiv., xv.? ans. angles at a point.
patients alter naturally or unnaturally according as they throw off fevers on the critical days or not. but, it may be objected, then we shall have perishings contrary to one another, not to becoming. certainly: and why should not this in a sense be so?
for tragedy and comedy are both composed of the same letters.
cor. seven.-if the adjacent sides of a parallelogram be equal, its diagonals bisect its angles.
for the primary bodies of the atomists-the primary constituents of which bodies are composed, and the ultimate elements into which they are dissolved-are indivisible, differing from one another only in figure.
one. let a be greater than b, and let c be any other magnitude of the same kind, then the ratio a : c is greater than the ratio b : c.
the same argument applies to all the 'elements', proving that there is no single one of them out of which they all originate. but neither is there, beside these four, some other body from which they originate-a something intermediate, e.g.
and as fire lays hold of the inflammable, so the active principle of growth, dwelling in the growing thing that which is actually flesh), lays hold of an acceding food which is potentially flesh and converts it into actual flesh.
prop. thirty five.-theorem. if two chords (ab, four hundred) of a circle intersect in a point (e) within the circle, the rectangles (ae.eb, ce.ed) contained by the segments are equal.
but we maintain that, if 'combination' has taken place, the compound must be uniform in texture throughout-any part of such a compound being the same as the whole, just as any part of water is water: whereas, if 'combination' is 'composition of the small
sight is not attended either by pleasure or pain, but hunger and the appeasing of hunger are pleasant and painful because they are extraordinary.
forty six. state and prove the proposition corresponding to exercise forty one, when the base and difference of the sides are given.
all who deny that the earth lies at the centre think that it revolves about the centre, and not the earth only but, as we said before, the counter-earth as well.
we have placed these definitions in a group; but their order is inverted, and, as we shall see, def. xii. is a theorem, and x. and xi. are only inferences from it.
there are isolated expressions about the nature of god which have a wonderful depth and power; but we are not justified in assuming that these had any greater significance to the mind of plato than language of a neutral and impersonal character...
they put together the body, gave all the other members to be servants, considering that it partook of every sort of motion.
in equal circles, angles (boc, epf) at the centres or (bac, edf) at the circumferences have the same ratio to one another as the arcs (bc, ef) on which they stand, and so also have the sectors (boc, epf).
further, irregularity of this kind would be particularly unlikely to pass unobserved, since contrast makes observation easy.
hb :bf ::de :ef ; that is, ac :ce ::gc :cb.
wherefore also the various elements had different places before they were arranged so as to form the universe. at first, they were all without reason and measure.
if therefore that which is primary contains nothing unnatural, being simple and unmixed and in its proper place and having no contrary, then it has no place for incapacity, nor, consequently, for retardation or (since acceleration involves retardation)
carrying belongs to all the other three kinds of motion in common.
the amount removed is, it is true, divisible into a number of parts, but no one of these was set in motion separately: they were all set in motion together.
and owing to these affections, all things are changing their place, for by the motion of the receiving vessel the bulk of each class is distributed into its proper place; but those things which become unlike themselves and like other things, are hurried
pleasure and pain are attributed in the timaeus to the suddenness of our sensations-the first being a sudden restoration, the second a sudden violation, of nature (phileb.). the sensations become conscious to us when they are exceptional.
dem.-since a : b :: a : b, we have a- b a- b, in like manner, b c b c, hence a- b b c a bb c, or a- c a c. therefore if a be greater than c, a is greater than c; if equal, equal; and if less, less.
sixty three. the rectangle contained by the perpendiculars from the extremities of the base of a triangle, on the internal bisector of the vertical angle, is equal to the rectangle contained by the external bisector and the perpendicular from the middle
during the life of each individual these intimations are plainer, but after his death the liver becomes blind, and delivers oracles too obscure to be intelligible.
anaxagoras again made both his 'homceomerous' substances and his contraries infinite in multitude, whereas empedocles posits only the so-called elements.
four. find a line whose square shall be equal to the sum of two given squares.
socrates: and what about the procreation of children? or rather was not the proposal too singular to be forgotten?
the doctrine that the heavenly bodies all move in a circle is known by us to be erroneous; but without such an error how could the human mind have comprehended the heavens?
for in the first place, you remember one deluge only, and there were many of them, and you know nothing of that fairest and noblest race of which you are a seed or remnant. the memory of them was lost, because there was no written voice among you.
now this may come about in either of two ways. either the movent is not itself responsible for the motion, which is to be referred to something else which moves the movent, or the movent is itself responsible for the motion.
the shape of bodies will not account for their moving upward or downward in general, though it will account for their moving faster or slower. the reasons for this are not difficult to see.
of triangles there are two kinds; one having the opposite sides equal (isosceles), the other with unequal sides (scalene).
one hundred nine. if two circles touch, and through their point of contact two secants be drawn at right angles to each other, cutting the circles respectively in the points a, a; b, b; then aa bb is constant.
one use of the terms 'lighter' and 'heavier' is that which is set forth in writing in the timaeus, that the body which is composed of the greater number of identical parts is relatively heavy, while that which is composed of a smaller number is relatively
in the circle, the latter condition is not satisfied: it is only the adjacent part from which the new part is different.
the earth, it might be argued, is at the centre and spherical in shape: if, then, a weight many times that of the earth were added to one hemisphere, the centre of the earth and of the whole will no longer be coincident.
we may also see in the following way how impossible it is either for a thing which is generated to be thenceforward indestructible, or for a thing which is ungenerated and has always hitherto existed to be destroyed.
this is indicated by the fact that good fortune is thought to be the same, or nearly the same, as happiness, and happiness to be a kind of moral action, since it is well-doing. hence what is not capable of moral action cannot do anything by chance.
if a man is walking to thebes, he cannot be walking to thebes and at the same time have completed his walk to thebes: and, as we saw, o traverses a the partless section a in virtue of the presence of the motion five hundred.
reason and by what providence of the gods; and holding fast to probability, we must pursue our way.
these are the elements of necessity which the creator received in the world of generation when he made the all-sufficient and perfect creature, using the secondary causes as his ministers, but himself fashioning the good in all things.
had the combination of solid bone and thick flesh been consistent with acute perceptions, the creator would have given man a sinewy and fleshy head, and then he would have lived twice as long.
two. if the diagonals of a parallelogram be equal, all its angles are right angles.
the triangle dbc is equal to the triangle acb-the less to the greater, which is absurd; hence ac, ab are not unequal, that is, they are equal.
in like manner, ten and fifteen are equimultiples of two and three, and eighteen and thirty of three and five, c.
similarly to say that a man who is playing the harp, but not singing, is singing, is to say what is false but not impossible.
and thus, when the small parts are placed side by side with the larger, and the lesser divide the greater and the greater unite the lesser, all the elements are borne up and down and hither and thither towards their own places; for the change in the size
again, when the mixed body is divided, they can show no reason why one of the constituents must by itself take up more room than the body did: but when water turns into air, the room occupied is increased.
he who lived well would return to his native star, and would there have a blessed existence; but, if he lived ill, he would pass into the nature of a woman, and if he did not then alter his evil ways, into the likeness of some animal, until the reason
and in this respect, they insist, the view that the universe is not 'continuous' but 'discretes-in-contact' is no better than the view that there are 'many' (and not 'one') and a void. for (suppose that the universe is discretes-in-contact.
for when a thing is moving from or discarding something, it still appears to have that which is being discarded, so that if a state of rest is itself contrary to the motion from the state of rest to its contrary, the contraries rest and motion will be
to teach will be the same as to learn, and to act the same as to be acted on-the teacher will necessarily be learning everything that he teaches, and the agent will be acted on. one may reply:
being neither prior nor posterior) means to be 'in one and the same "now"', then, if both what is before and what is after are in this same 'now', things which happened ten thousand years ago would be simultaneous with what has happened to-day, and
a body composed of more than four points will superior in weight to it, a thing which has weight. but the difference between weight and weight must be a weight, as the difference between white and whiter is white.
thus the view of those who speak after the manner of the pythagoreans is absurd. with the same breath they treat the infinite as substance, and divide it into parts.
the cure of them is difficult; relief is in most cases given by fever supervening.
in the same case, if the suffixes be greater than eight,
concurring lights is reversed; and this happens when the mirror is concave and its smooth surface repels the right stream of vision to the left side, and the left to the right (he is speaking of two kinds of mirrors, first the plane, secondly the concave;
it follows that the coming-to-be of anything, if it is absolutely necessary, must be cyclical-i.e. must return upon itself. for coming to-be must either be limited or not limited: and if not limited, it must be either rectilinear or cyclical.
fourteen. in a given circle inscribe a triangle whose three sides shall pass through three given points.
cor. two.-a perpendicular to each of two intersecting lines is normal to their plane.
this fact has served above to make it clear that it is not true either that all things are in motion or that all things are at rest or that some things are always at rest and the remainder always in motion: on this matter proof is supplied by things that
further, if separate place does not exist, neither will void.
the angle acb opposite to the side ab is equal to the angle adc opposite to the side ac; but the angle adc is equal to abc; therefore acb is equal to abc.
twenty one. what propositions are these limiting cases of?
much the same argument will apply to the void as to the 'up' and 'down' in place, as is natural enough since those who maintain the existence of the void make it a place.
three. if three magnitudes be proportional, the ratio of the st : rd is equal to the square of the ratio of the st : nd.
seven. if five hundred be the middle point of the base bc of a triangle abc, e the foot of the perpendicular, fifty the point where the bisector of the angle a meets bc, h the point of contact of the inscribed circle with bc; prove de.hl he.hd.
tend to promote or destroy its existence.
hence, plainly, things which are always are not, as such, in time, for they are not contained time, nor is their being measured by time. a proof of this is that none of them is affected by time, which indicates that they are not in time.
if we look at the ancients, physics would to be concerned with the matter. (it was only very slightly that empedocles and democritus touched on the forms and the essence.)
such as we have seen, is the nature and such are the causes of respiration,-the subject in which this discussion originated.
two. when is one magnitude said to be a multiple of another?
for at the peril of her own existence, and when the other hellenes had deserted her, she repelled the invader, and of her own accord gave liberty to all the nations within the pillars.
if in any triangle (abc) one side (ac) be greater than another (ab), the angle opposite to the greater side is grater than the angle opposite to the less.
or thus: place the four lines in a concurrent position so that the extremes may form one continuous line, and the means another. let the four lines so placed be ao, bo, od, oc. join ab, four hundred.
for natural bodies are manifestly endowed with weight and lightness, but an assemblage of units can neither be composed to form a body nor possess weight.
if possible let pg be equal to pd, then pg is equal to pf-that is, the line which is nearest to the one through the centre is equal to the one which is more remote, which is impossible. hence three equal lines cannot be drawn from p to the circumference.
forty nine. if the sides of a triangle be in arithmetical progression, and if r, r be the radii of the circumscribed and inscribed circles; then rr is equal to the rectangle contained by the greatest and least sides.
all the rest of us who dwell within the pillars.
the same holds, consequently, also of the matter itself of that which is heavy and light: as potentially possessing the one character, it is matter for the heavy, and as potentially possessing the other, for the light.
prop. fifteen.-problem. in a given circle (abcdef) to inscribe a regular hexagon.
and the most foolish of them, who trail their bodies entirely upon the ground and have no longer any need of feet, he made without feet to crawl upon the earth.
planes (four hundred, ef) which have a common normal (ab) are parallel to each other.
two. if two adjacent sides of a quadrilateral be equal, and the diagonal bisects the angle between them, their other sides are equal.
in the next place we have to consider that there are divers kinds of fire.
again, it cannot be the whirl which determines the heavy and the light. rather that movement caused the pre-existent heavy and light things to go to the middle and stay on the surface respectively.
we begin with the view of anaxagoras that all the homoeomerous bodies are elements. any one who adopts this view misapprehends the meaning of element.
problem for further examination. for the question might be raised whether substance (i.e. the 'this') comes-to-be at all. is it not rather the 'such', the 'so great', or the 'somewhere', which comes-to-be?
for those things are natural which, by a continuous movement originated from an internal principle, arrive at some completion: the same completion is not reached from every principle; nor any chance completion, but always the tendency in each is towards
they postulate only that the finite straight line may be produced as far as they wish. it is possible to have divided in the same ratio as the largest quantity another magnitude of any size you like.
hence, since the motion of the weightless body will cover a greater distance than any that is suggested, it will continue infinitely. it is therefore obvious that every body must have a definite weight or lightness.
existence of motion. but those who say that there is an infinite number of worlds, some of which are in process of becoming while others are in process of perishing, assert that there is always motion (for these processes of becoming and perishing of the
and further the theory is inconsistent with a regard for the facts of nature.
two. through a given point, within or without a given circle, draw a chord of length equal to that of a given chord.
but this perhaps, to use once more expressions of his own, 'is part of another subject' or 'may be more suitably discussed on some other occasion.'
moreover, so much of music as is adapted to the sound of the voice and to the sense of hearing is granted to us for the sake of harmony; and harmony, which has motions akin to the revolutions of our souls, is not regarded by the intelligent votary of the
the process of repletion and depletion is produced by the attraction of like to like, after the manner of the universal motion.
from the mind, might be reflected as in a mirror which receives likenesses of objects and gives back images of them to the sight; and so might strike terror into the desires, when, making use of the bitter part of the liver, to which it is akin, it comes
seeing, however, that we have already decided this matter to the best of our ability, we are entitled to treat our results as representing fact.
for flame is par excellence fire: but flame is burning smoke, and smoke consists of air and earth.
the reason of this belief is that 'movers' which belong to the same kind as the 'moved' impart motion by being moved.
the fields of knowledge have become more divided. the modern physicist confines himself to one or perhaps two branches of science.
we know that 'being' is only the verb of existence, the copula, the most general symbol of relation, the first and most meagre of abstractions; but to some of the ancient philosophers this little word appeared to attain divine proportions, and to
in the middle ages the legend seems to have been half-forgotten until revived by the discovery of america.
in so far then as the 'now' is a boundary, it is not time, but an attribute of it; in so far as it numbers, it is number; for boundaries belong only to that which they bound, but number (e.g. ten) is the number of these horses, and belongs also elsewhere.
now in all cases other than substance it is plain that there must be some subject, namely, that which becomes.
four. divide an angle of an equilateral triangle into five equal parts.
motion upward and downward and sideways were all, they thought, equally inappropriate to that which is set at the centre and indifferently related to every extreme point; and to move in contrary directions at the same time was impossible: so it must needs
agreement with herself; and rhythm too was given by them for the same reason, on account of the irregular and graceless ways which prevail among mankind generally, and to help us against them.
finish its course (for when anything is at the limits of its course, whether at the starting-point or at the finishing-point, it must be in a state of rest).
again, since the circles are equal, it is evident that the angle boh is greater than, equal to, or less than the angle epj, according as the arc bh is greater than, equal to, or less than the arc ej.
as to what time is or what is its nature, the traditional accounts give us as little light as the preliminary problems which we have worked through.
cor. three.-if two triangles be equiangular, the rectangle contained by the non-corresponding sides about any two equal angles are equal.
dem.-since the lines are equal, if ab be applied to four hundred, so that the point a will coincide with one hundred, and the line ab with four hundred, the point b shall coincide with five hundred; and because the segments are similar, they must coincide
drugs. for we ourselves are sceptical about medicine, and very unwilling to submit to the purgative treatment of physicians.
two. if ac were less than ab, the angle b would be less than the angle one hundred xviii. ; but it is not by hypothesis; therefore ac is not less than ab; and since ac is neither equal to ab nor less than it, it must be greater.
here the difference which makes the superior weight heavier is the single point which remains when the common number, four, is subtracted. a single point, therefore, has weight.
nine. if a line ab, cut harmonically in one hundred and five hundred, be bisected in o; prove oc, ob, od are continual proportionals.
let a : b :: c : d; then a b : b :: c d : d.
it is said rightly, too, that the number of the sheep and of the dogs is the same number if the two numbers are equal, but not the same decad or the same ten; just as the equilateral and the scalene are not the same triangle, yet they are the same figure,
on the supposition of forms such as some assert, this must be the case, and equally on the view that no such entity has a separate existence.
on the other hand, those who deny atoms of time (as we do) are not affected by this argument: according to them five hundred has become and so is white at the last point of the actual time in which it was becoming white: and this point has no other point
in this manner there were formed intervals of thirds, three:two, of fourths, four:three, and of ninths, nine:eight.
four. if through the middle point o of any right line terminated by two parallel right lines any other secant be drawn, the intercept on this line made by the parallels is bisected in o.
now motions and states of rest universally exhibit contrariety in the manner described above, e.g.
these, they would live righteously, but if they were conquered by them, unrighteously.
again, because eg and hi are parallelograms, ef and ki are each parallel to gh; hence xxx.
and the same is true of hearing and smelling: for the primary movent in respect to the moved is the air. similarly, in the case of tasting, the flavour is adjacent to the sense of taste.
a thing, then, which is within it. if, for example, a man can lift a hundred talents, he can also lift two, and if he can walk a hundred stades, he can also walk two.
the following is the proof of the second part-that is, when the arc ab is not greater than a semicircle, without using angles greater than two right angles:-
we do not ascribe to chance or mere coincidence the frequency of rain in winter, but frequent rain in summer we do; nor heat in the dog-days, but only if we have it in winter.
because fire is mobile and productive of heat and combustion, some made it a sphere, others a pyramid.
for we see with our eyes, as we said before, that earthy things sink to the bottom of all things and move towards the centre. but the centre is a fixed point.
(we may leave the time out of account, since that is one and the same.) if the lines are specifically different, the locomotions also differ specifically from one another: for locomotion is specifically differentiated according to the specific
remaining at rest in the same place, we say that it is in motion in some particular respect: we do not say that it 'is in motion' without qualification.
dem.-join ac, bc. then the triangles adc, bdc have the side ad equal to db (const.), and six hundred common to both, and the angle adc equal to the angle bdc, each being right. hence the base ac is equal to the base bc. therefore xxviii.
two. as to the position of the earth, then, this is the view which some advance, and the views advanced concerning its rest or motion are similar. for here too there is no general agreement.
time, then, also is both made continuous by the 'now' and divided at it. for here too there is a correspondence with the locomotion and the moving body.
what is drawn up must cool, and what has been cooled must become water and descend, the result of this being that the corn grows.
for it is either destructible or indestructible, but, if indestructible, it is by hypothesis ungenerated.
contrariness. moreover, each several motion takes its name rather from the goal than from the starting-point of change, e.g. motion to health we call convalescence, motion to disease sickening.
nor can there be anything of any other kind intermediate between the parts or between the moments: for if there could be any such thing it is clear that it must be either indivisible or divisible, and if it is divisible, it must be divisible either into
and so i readily assented to your request yesterday, considering that in all such cases the chief difficulty is to find a tale suitable to our purpose, and that with such a tale we should be fairly well provided.
further, everything that is infinite may be so in respect of addition or division or both.
passage not being contained within any period of time except the whole of which the particular moment is a dividing-point.
now we perceive lesser symmetries or proportions and reason about them, but of the highest and greatest we take no heed; for there is no proportion or disproportion more productive of health and disease, and virtue and vice, than that between soul and
if any number of ratios be equal to one another, any one of these equal ratios is equal to the ratio of the sum of all the antecedents to the sum of all the consequents.
he does not explain how man is acted upon by the lesser influences of custom or of opinion; or how the commands of the soul watching in the citadel are conveyed to the bodily organs.
respectively equal to the angles eof, foa, aob. therefore the six angles at the centre are equal, because each is one-third of two right angles. therefore the six chords are equal three. xxix. . hence the hexagon is equilateral.
let us take for granted about it the various characteristics which are supposed correctly to belong to it essentially. we assume then-
the result is a theory which he states as follows: 'the void is a "not being", and no part of "what is" is a "not-being"; for what "is" in the strict sense of the term is an absolute plenum.
a thing can cause itself to walk it can also cause itself not to walk), and so, since on this supposition fire itself possesses the power of upward locomotion, it is clear that it should also possess the power of downward locomotion.
and so put together a visible and palpable heaven, having harmony and friendship in the union of the four elements; and being at unity with itself it was indissoluble except by the hand of the framer.
at the same time be at rest and in motion: for both the times have the same extremity, viz. the present.
it is surprising that this voluminous work should have found a translator (thomas taylor, a kindred spirit, who was himself a neo-platonist, after the fashion, not of the fifth or sixteenth, but of the nineteenth century a.five hundred.).
the matter out of which the world is formed is not absolutely void, but retains in the chaos certain germs or traces of the elements. these plato, like empedocles, supposed to be four in number-fire, air, earth, and water.
xii. when there is any number of magnitudes of the same kind greater than two, the first is said to have to the last the ratio compounded of the ratios of the first to the second, of the second to the third, of the third to the fourth, c.
four. if ab be cut in "extreme and mean ratio" at one hundred, prove that
but when it enters with more of a flood it overcomes the fibres by its heat and reaches the spinal marrow, and burning up the cables of the soul sets her free from the body.
() its reciprocal, to describe a triangle about a circle.
the behaviour of these metals is a case in point. for the tin almost vanishes, behaving as if it were an immaterial property of the bronze: having been combined, it disappears, leaving no trace except the colour it has imparted to the bronze.
now we distinguish matter and privation, and hold that one of these, namely the matter, is not-being only in virtue of an attribute which it has, while the privation in its own nature is not-being; and that the matter is nearly, in a sense is, substance,
such are the arguments (and others of the kind) which may cause difficulty on this point. yet it is impossible that this should be the true view.
but whereas the velocity of that which comes to a standstill seems always to increase, the velocity of that which is carried violently seems always to decrease: so it will he in a state of rest without having become so.
and even if this were accepted, there are other implications equally unsatisfactory. we do not expect a mass of matter to be made heavier by compression.
the question whether these contradictory changes are contraries or not makes no difference, provided only it is impossible for them both to be present to the same thing at the same time: the point is of no importance to the argument.
in the nineteenth century, men continued to appeal to rousseau, without, as a rule, knowing him well or penetrating deeply into his meaning. "the social contract ," says one thousand.
forty six. in a triangle, given the sum or the difference of two sides and the angle formed by these sides both in magnitude and position, the locus of the centre of the circumscribed circle is a right line.
by gradual advance in this direction we come to see clearly that in plants too that is produced which is conducive to the end-leaves, e.g. grow to provide shade for the fruit.
seven. if a, b, c denote the sides of a triangle abc, and five hundred, five hundred the points where the internal and external bisectors of a meet bc; prove
we may begin with a dialectical argument and show as follows that there is no such thing. if 'bounded by a surface' is the definition of body there cannot be an infinite body either intelligible or sensible.
the first of the figures thus constructed is said to be directly similar, and the second inversely similar to the given figure. these technical terms are due to hamilton: see "elements of quaternions," page one hundred twelve.
of the greater weel) he enveloped the hollow parts of the body, and at one time he made all this to flow into the lesser weels, quite gently, for they are composed of air, and at another time he caused the lesser weels to flow back again; and the net he
while he was arranging the world, he was arranging the forms of thought in his own mind; and the light from within and the light from without often crossed and helped to confuse one another.
smells are not divided into kinds; all of them are transitional, and arise out of the decomposition of one element into another, for the simple air or water is without smell.
now it is impossible that the infinite should be a thing which is itself infinite, separable from sensible objects.
let a : b :: c : d, c and d being less than a and b; then a c : b d :: a : b.
thus, if the line ab represent the integer, and if it be divided into four equal parts in the points one hundred, five hundred, e, then ac is fourteen; ad, twenty four; ae, thirty four.
if, however, it is the nature of a void to move upward and of a plenum to move downward, and therefore each causes a like movement in other things, there was no need to raise the question why composite bodies are some light and some heavy; they had only
since these fractions are all equal, let their common value be r; then we have
scholium.-until the year one thousand eight hundred one no regular polygon could be described by constructions employing the line and circle only, except those discussed in this book, and those obtained from them by the continued bisection of the arcs of
object, like the figure, as in course of formation.
cannot 'have been combined' since they have no being at all.
the liquid itself we call blood, which nourishes the flesh and the whole body, whence all parts are watered and empty places filled.
the faculty of smell does not admit of differences of kind; for all smells are of a half-formed nature, and no element is so proportioned as to have any smell.
elements of each are to one another, and so on, and if there is more air than water and, generally, more of the finer body than of the coarser, obviously the element of water will be smaller than that of air.
or thus: since bp is a line drawn from a point without the circle pcf to its circumference, and when produced does not pass through the centre, the circle whose centre is b and radius bp must cut the circle pcf in p viii., cor.
to explain why these two things are themselves light and heavy respectively, and to give, further, the reason why the plenum and the void are not eternally separated.
, and od : five hundred fifty :: pictdn : square dm i. . hence ai : ah :: dn : dm; therefore ai : dn :: ah : dm five. xvi. ; hence abc : def :: ab : de .
hence the plane triangle abk has two right angles, which is impossible. therefore the planes four hundred, ef cannot meet-that is, they are parallel.
the active power is a 'cause' in the sense of that from which the process originates: but the end, for the sake of which it takes place, is not 'active'.
for these to be uniform is impossible; wherefore we must assign rest to uniformity and motion to the want of uniformity. now inequality is the cause of the nature which is wanting in uniformity; and of this we have already described the origin.
downward and upward movements are equally constant and universal: or, if the matter in question is the void or something similar, which moves uniformly upward, there will be nothing to move uniformly downward.
four. construct a lozenge equal to a given parallelogram, and having a given side of the parallelogram for base.
two, god placed water and air in the mean between fire and earth, and made them to have the same proportion so far as was possible (as fire is to air so is air to water, and as air is to water so is water to earth); and thus he bound and put together a
like the tale of troy, or the legend of the ten tribes (ewald, hist. of isr.), which perhaps originated in a few verses of two esdras, it has become famous, because it has coincided with a great historical fact.
prop. eighteen.-theorem if a line (four hundred) touch a circle, the line (oc) from the centre to the point of contact is perpendicular to it.
i do not learn that the indians ever troubled themselves to go after it.
but it has not been transformed into flesh alone by itself (for that would have been a coming-to-be, not a growth): on the contrary, it is the growing thing which has come-to-be flesh (and grown) by the food.
but the natural motion, as in the world, so also in the human frame, produces harmony and divides hostile powers.
that why, when water comes-to-be from air and air from water, the air is the same 'specifically', not 'numerically': and if these too recur numerically the same, at any rate this does not happen with things whose 'substance' comes-to-be-whose 'substance'
for 'moist' is that which contains moisture of its-own deep within it ('sodden' being that which is deeply penetrated by foreign mosture), whereas 'solidigied' is that which has lost this inner moisture.
and if their movement was natural, a moment's consideration shows that there was already an ordered world.
the following account will make it plain that there is an absolutely light and an absolutely heavy body.
fifty five. two points, one hundred, five hundred, in the circumference of a given circle are on the same side of a given diameter; find a point p in the circumference at the other side of the given diameter, ab, such that pc, pd may cut ab at equal
for, let ab, ab be two homologous sides of the figures, one hundred their point of intersection. through the two triads of points a, a, one hundred; b, b, one hundred describe two circles intersecting again in the point o: o will be the point required.
one, neither can we of the other. he means (three) to say that the creation of the world is not a material process of working with legs and arms, but ideal and intellectual; according to his own fine expression, 'the thought of god made the god that was
now since of two things that which changes sooner is quicker, in the time zh, in which a has changed from g to five hundred, b will not yet have arrived at five hundred but will be short of it: so that in an equal time the quicker will pass over a greater
in the generation before socrates, anaxagoras had brought together 'chaos' and 'mind'; and these are connected by plato in the timaeus, but in accordance with his own mode of thinking he has interposed between them the idea or pattern according to which
we must note (i) that the organic parts grow by the growth of the tissues (for every organ is composed of these as its constituents); and (ii) that flesh, bone, and every such part-like every other thing which has its form immersed in matter-has a twofold
but it is clear, further, that if the simple bodies are finite, the composite must also be finite, since that which is composed of bodies finite both in number and in magnitude is itself finite in respect of number and magnitude: its quantity is in fact
intended to contain the remaining and mortal part of the soul he distributed into figures at once round and elongated, and he called them all by the name 'marrow'; and to these, as to anchors, fastening the bonds of the whole soul, he proceeded to fashion
wherefore, we must return again and find another suitable beginning, as about the former matters, so also about these.
but if there is any natural movement, constraint will not be the sole principle of motion or of rest.
for admitting with them that there is something divine, good, and desirable, we hold that there are two other principles, the one contrary to it, the other such as of its own nature to desire and yearn for it.
nor, again, is it conceivable that it should persist eternally by the necessitation of a soul.
these have changed their place only because the subjects to which they belong do so.
all other kinds of locomotion must be similarly reduced, for they all fall under one or other of our four heads. and again, of these four, carrying and twirling are to pulling and pushing.
the fact is, euclid's object was to teach theoretical and not practical geometry, and the only things he postulates are the drawing of right lines and the describing of circles.
by lighter or relatively light we mean that one, of two bodies endowed with weight and equal in bulk, which is exceeded by the other in the speed of its natural downward movement.
clearly then 'to be in time' has the same meaning for other things also, namely, that their being should be measured by time. 'to be in time' is one of two things: (one) to exist when time exists, (two) as we say of some things that they are 'in number'.
which is undergoing locomotion from a to g cannot also simultaneously be undergoing locomotion from g to a: and since the latter locomotion is not simultaneous with the former but is still to be undergone, before it is undergone there must occur a state
for since the distinction of quicker and slower may apply to motions occupying any period of time and in an equal time the quicker passes over a greater length, it may happen that it will pass over a length twice, or one and a half times, as great as that
eleven. and ab(ac ad) ac.ad, or one ac- one ad- two ab-.
upside down, as you might imagine a person who is upside down and has his head leaning upon the ground and his feet up against something in the air; and when he is in such a position, both he and the spectator fancy that the right of either is his left,
it was as if some one in our own day were to convert the poems of homer into an allegory of the christian religion, at the same time maintaining them to be an exact and veritable history.
or thus: let a, a denote the areas of the triangles abc, acd, respectively, and p their common altitude; then two. i., cor. one ,
three. erect a line at right angles to a given line at one of its extremities without producing the line.
or, on the contrary, does what is 'include earth as well as fire, whereas what is not' is matter-the matter of earth and fire alike? and again, is the matter of each different?
the larger produce contraction, the smaller dilation, in the sight, exercising a power akin to that of hot and cold bodies on the flesh, or of astringent bodies on the tongue, or of those heating bodies which we termed pungent.
and, conversely, if the substratum of the changing things is one, there is 'alteration'.
but as it was, i considered that i enhanced the value of the land by squatting on it.
are compounded of them, ought not to be so denominated. let me make another attempt to explain my meaning more clearly.
from these arguments then it is clear that the body of the universe is not infinite.
iv. magnitudes are said to have a ratio to one another when the less can be multiplied so as to exceed the greater.
; hence the sum of the three angles acb, bac, abc is two right angles.
operation may be said to be place. moreover they are of opinion that the primary substances are not subject to any of the other motions, though the things that are compounds of these substances are so subject: the processes of increase and decrease and
for we should expect, since the primary body shows one motion only, that the body which is nearest to it should move with the fewest movements, say two, and the one next after that with three, or some similar arrangement. but the opposite is the case.
for instance, suppose e multiplied by three is rather more than one hundred: the weight of three masses of the full size of bd will be greater than one hundred. we thus arrive at the same impossibility as before.
if there were an extension which were such as to exist independently and be permanent, there would be an infinity of places in the same thing.
(two) another popular notion which is found in the timaeus, is the feebleness of the human intellect-'god knows the original qualities of things; man can only hope to attain to probability.'
we must show not only that the heaven is one, but also that more than one heaven is and, further, that, as exempt from decay and generation, the heaven is eternal. we may begin by raising a difficulty.
it has now to be shown that in no case is it possible for an infinite force to reside in a finite magnitude.
cor. four.-similar portions of the perimeters of similar figures are to each other in the ratio of the whole perimeters.
implying to the mind of plato a divine reality. the slight touch, perhaps ironical, contained in the words, 'as we shall do well in believing on the testimony of wise men,' is very characteristic of plato.
and chance, not proportion, 'is the name given to these occurrences': for things can be 'mingled' fortuitously.
for it is by means of the body that is carried along that we become aware of the 'before and after' the motion, and if we regard these as countable we get the 'now'.
of a 'line', the smallest in respect of multiplicity is two (or, if you like, one), but in respect of size there is no minimum; for every line is divided ad infinitum. hence it is so with time.
twenty one. if, upon the greater segment ab of a line ac, divided in extreme and mean ratio, an equilateral triangle abd be described, and four hundred joined, cd ab .
it divides potentially, and in so far as it is dividing the 'now' is always different, but in so far as it connects it is always the same, as it is with mathematical lines.
on the other hand, when we do perceive a 'before' and an 'after', then we say that there is time. for time is just this-number of motion in respect of 'before' and 'after'.
escribed to the triangle abc, and has an angle common with the external angle b. hence the thing required is done.
abd. reject the angle cba, which is common, and we have the angle abe equal to the angle abd-that is, a part equal to the whole-which is absurd. hence bd must be in the same right line with cb.
twenty four. if three concurrent lines oa, ob, oc be cut by two transversals in the two systems of points a, b, one hundred; a, b, one hundred, respectively: prove
the breath of man is within him, but the air or aether of heaven is the element which surrounds him and all things.
having sifted pure and smooth earth he kneaded it and wetted it with marrow, and after that he put it into fire and then into water, and once more into fire and again into water-in this way by frequent transfers from one to the other he made it insoluble
(a) in one sense, then, we call 'heaven' the substance of the extreme circumference of the whole, or that natural body whose place is at the extreme circumference.
five. ac is the side of a regular pentagon inscribed in the circle acd, and eb the side of a regular pentagon inscribed in the circle bde.
in action, again, when a has to be done to get b, b to get one hundred, and one hundred to get five hundred, one step or two present little difficulty, but as the series extends the difficulty grows.
moreover, the consideration of particular examples will also show what kinds of processes are generally recognized as contrary: thus falling ill is regarded as contrary to recovering one's health, these processes having contrary goals, and being taught as
prop. twenty four.-theorem. similar segments of circles (aeb, cfd) on equal chords (ab, four hundred) are equal to one another.
we have now developed the difficulties sufficiently and must therefore try to find a solution of the problem.
thus horse and dog are so commensurable that we may say which is the whiter, since that which primarily contains the whiteness is the same in both, viz. the surface: and similarly they are commensurable in respect of size.
in fact, however, fire and air, and each of the bodies we have mentioned, are not simple, but blended. the 'simple' bodies are indeed similar in nature to them, but not identical with them.
thus it is clear that only those agents are 'combinable' which involve a contrariety-for these are such as to suffer action reciprocally. and, further, they combine more freely if small pieces of each of them are juxtaposed.
it is plain, then, from the foregoing arguments that it is impossible for an infinite force to reside in a finite magnitude or for a finite force to reside in an infinite magnitude.
a more precise account of these presuppositions has been given in another work': we must, however, give a detailed explanation of the primary bodies as well, since they too are similarly derived from the matter.
irrefutable and immovable-nothing less. but when they express only the copy or likeness and not the eternal things themselves, they need only be likely and analogous to the real words. as being is to becoming, so is truth to belief.
twenty nine. what proposition is the converse of prop. xxvi., part one.?
two. if a line be parallel to each of two planes, the sections which any plane passing through it makes with them are parallel.
ten. construct a right-angled triangle, being given the hypotenuse and the sum or difference of the sides.
pair: but no 'simple' body can be formed either out of identical, or out of contrary, qualities. thus no 'simple' body would result, if the dry of fire and the moist of air were to pass-away: for the hot is left in both.
there is also a swifter motion of another sort of fire which forces a way through the passages of the eyes, and elicits from them a union of fire and water which we call tears.
the common error of all views which assume a single element is that they allow only one natural movement, which is the same for every body. for it is a matter of observation that a natural body possesses a principle of movement.
three. given the base and vertical angle of a triangle, find the locus of the middle point of the line joining the vertices of equilateral triangles described on the sides.
since, therefore, it is not possible for fire to become water, or water to become earth, neither will it be possible for anything white to become black, or anything soft to become hard; and the same argument applies to all the other qualities.
if, then, 'substance' is not attributed to anything, but other things are attributed to it, how does 'substance' mean what is rather than what is not? for suppose that 'substance' is also 'white'.
and since in an infinite series there is no first term, here there will be no first stage and therefore no following stage either. on this hypothesis, then, nothing can become or be moved or change.
in substance, if it comes-to-be fire but not if it comes-to-be earth; and in quality, if it comes-to-be learned but not when it comes-to-be ignorant.
eight.four thousand thirty four oil and some household utensils,....... two.zero
we have now before us the distinctions in the various classes of being between what is full real and what is potential.
five. if a circle be described touching another circle internally and two parallel chords, the perpendicular from the centre of the former on the diameter of the latter, which bisects the chords, is a mean proportional between the two extremes of the
thus we say that a thing is altered by becoming hot or sweet or thick or dry or white: and we make these assertions alike of what is inanimate and of what is animate, and further, where animate things are in question, we make them both of the parts that
cor. two.-tangents to two circles from any point in their common chord are equal (compare xvii., ex. six).
prop. eighteen.-theorem. if four magnitudes be proportionals, the sum of the first and second : the second :: the sum of the third and fourth : the fourth (componendo).
until the circles are bent and disordered and their motion becomes irregular. you may imagine a position of the body in which the head is resting upon the ground, and the legs are in the air, and the top is bottom and the left right.
by a motion that is numerically one i mean a motion that proceeds from something numerically one and the same to something numerically one and the same in a period of time numerically one and the same: for a motion may be the same generically,
moreover, if that which is continuous is composed of points, these points must be either continuous or in contact with one another: and the same reasoning applies in the case of all indivisibles.
two. when is one rectilineal figure said to be inscribed in another?
a consideration of these points, then, gives adequate assurance of the truth of our contentions.
alteration: for alteration is always dependent upon some contraries.
for if there is to be opposition (instead of the growing thing having changed as a whole, either by the admixture of something or by its own transformation), increase of size will not have resulted in any and every part.
without parts. since, then it is divisible, it must be a period of time, and the thing must be at rest in every one of its parts, as may be shown by the same method as that used above in similar demonstrations.
let us suppose, if any one likes, that in the case of certain things it is possible for them at different times to be and not to be, without any process of becoming and perishing (in fact it would seem to be necessary, if a thing that has not parts at one
produce coming-to-be uninterruptedly. at the same time it is clear that we were right when, in an earlier work,' we called motion (not coming-to-be) 'the primary form of change'.
our enquiry will resolve itself at the outset into a consideration of the above-mentioned problem-what can be the reason why some things in the world at one time are in motion and at another are at rest again?
protection against falls, containing a warm moisture, which in summer exudes and cools the body, and in winter is a defence against cold.
but if he failed in attaining this, at the second birth he would pass into a woman, and if, when in that state of being, he did not desist from evil, he would continually be changed into some brute who resembled him in the evil nature which he had
it could not move infinitely; for to traverse an infinite is impossible, and impossibilities do not happen. so the moving thing must stop somewhere, and there rest not by constraint but naturally.
from the garden of the timaeus, as from the other dialogues of plato, we may still gather a few flowers and present them at parting to the reader.
(five) again, take a centre one hundred, an infinite line, ab, another infinite line at right angles to it, e, and a moving radius, four hundred.
again, in another equal part of the time another part of the magnitude will be traversed: and similarly in each part of the time that we take, whether equal or unequal to the part originally taken.
again, from the same triangles we have bh : ha :: de : ec, and from the triangles iha, fec; ha : hi :: ec : ef; therefore (ex quali) bh : hi :: de : ef; that is, the sides about the equal angles bhi, def are proportional, and so in like manner are the
one. what is the subject-matter of book three.?
if earth is enclosed by water, water by air, air by fire, and these similarly by the upper bodies-which while not continuous are yet contiguous with them-and if the surface of water is spherical, and that which is continuous with or embraces the spherical
then one hundred must be intermediate between a and b. for in their case there is no time in the direction of either limit, in which either a is not or b is.
i got out several cords of stumps in ploughing, which supplied me with fuel for a long time, and left small circles of virgin mould, easily distinguishable through the summer by the greater luxuriance of the beans there.
and there are varieties of air, as for example, the pure aether, the opaque mist, and other nameless forms. water, again, is of two kinds, liquid and fusile.
therefore no body has weight. it is, further, manifest that their point cannot have weight.
two. if two triangles have two sides of one respectively equal to two sides of the other, and the contained angles supplemental, their areas are equal.
but he is unable to invent such a narrative himself; and he is afraid that the poets are equally incapable; for, although he pretends to have nothing to say against them, he remarks that they are a tribe of imitators, who can only describe what they have
'combination' is unification of the 'combinables', resulting from their 'alteration'.
st : nd a b, nd : rd b c, rd : th c d. hence the ratio compounded of the ratio of st : nd, of nd : rd, of rd : th
principle in them. and when external impressions enter in, they are really conquered, though they seem to conquer.
one. for all the contrarieties of the 'elements' above f must belong to those below f, and vice versa: hence they will all be one.
and always, by so much as the medium is more incorporeal and less resistant and more easily divided, the faster will be the movement.
his intention was, in the first place, that the animal should be as far as possible a perfect whole and of perfect parts: secondly, that it should be one, leaving no remnants out of which another such world might be created: and also that it should be
two halves. therefore to the question whether it is possible to pass through an infinite number of units either of time or of distance we must reply that in a sense it is and in a sense it is not.
let a : b :: c : d, then a : c :: b : d.
two sides of the other. hence iv. bc would be equal to ef; but bc is, by hypothesis, greater than ef; hence the angle a is not equal to the angle five hundred.
quote his words) 'sank down upon the earth with long roots'. with such statements, too, he combines the assertion that the order of the world is the same now, in the reign of strife, as it was formerly in the reign of love.
experiences these and the like affections from both kinds of motions, the result is that the body if given up to motion when in a state of quiescence is overmastered and perishes; but if any one, in imitation of that which we call the foster-mother and
which the man has the desire to emit into the fruitful womb of the woman; this is like a fertile field in which the seed is quickened and matured, and at last brought to light.
that which is passing out of air into water is mist, and that which is passing from water into air is vapour; and hence all smells are thinner than water and thicker than air.
cor. two.-if two adjacent sides of a parallelogram be equal, it is a lozenge.
the knowledge of the soul admittedly contributes greatly to the advance of truth in general, and, above all, to our understanding of nature, for the soul is in some sense the principle of animal life.
shutting up the vessels and gates, causes pain and loathing.
first we must inquire whether the elements are eternal or subject to generation and destruction; for when this question has been answered their number and character will be manifest. in the first place, they cannot be eternal.
that there is a degree of confusion and indistinctness in plato's account both of man and of the universe has been already acknowledged.
the broad opposition between the state of nature and the state of society, which runs through all his work, he is concerned to present a rational justification of his views and to admit that a little at any rate may be said on the other side.
it is the same, too, with those who make out that the becoming or perishing of a thing is accounted for by 'density' or 'rarity': for it is by 'combination' and 'separation' that the place of these things in their systems is determined.
white and bright meeting, and falling upon a full black, become dark blue (greek), and when dark blue mingles with white, a light blue (greek) colour is formed, as flame-colour with black makes leek green (greek).
a pen, a wine, and the highest note in a scale are not commensurable: we cannot say whether any one of them is sharper than any other: and why is this?
ten. as any two points of two circles may be regarded as homologous, two circles have in consequence an infinite number of centres of similitude; their locus is the circle, whose diameter is the line joining the two points for which the two circles are
for the deity, intending to make this world like the fairest and most perfect of intelligible beings, framed one visible animal comprehending within itself all other animals of a kindred nature.
and if they are in contact with one another as whole with whole, they will not be continuous: for that which is continuous has distinct parts: and these parts into which it is divisible are different in this way, i.e. spatially separate.
but also in time the distinction of 'before' and 'after' must hold, for time and movement always correspond with each other.
they were not, like being or essence, mere vacant abstractions, but admitted of progress and growth, while at the same time they confirmed a higher sentiment of the mind, that there was order in the universe.
for the larger particles of moisture which surround the body, entering in and driving out the lesser, but not being able to take their places, compress the moist principle in us; and this from being unequal and disturbed, is forced by them into a state of
geometers make use of certain technical terms to denote the most important of these processes. we shall indicate these terms by including them in parentheses in connexion with the propositions to which they refer.
now suppose it to be in actuality what it is capable of being. it will then be true to say now that it does not exist last year. but this is impossible. no capacity relates to being in the past, but always to being in the present or future.
if on the other hand we have two forces each of which separately moves one of two weights a given distance in a given time, then the forces in combination will move the combined weights an equal distance in an equal time: for in this case the rules of
this may, of course, sometimes be the case: it might be, for instance, that only one circle could be found; yet none the less the difference will remain between the being of circle and of this particular circle, the one being form, the other form in
for a smaller body is more easily destroyed than a larger; and a destructive process which succeeds in destroying, that is, in resolving into smaller bodies, a body of some size, cannot reasonably be expected to fail with the smaller body.
they make it a process of excretion from a body of what was in it all the time-as though generation required a vessel rather than a material-so that it involves no change of anything.
for the idea or pattern of the world is not the thought of god, but a separate, self-existent nature, of which creation is the copy.
being sometimes from smallness to greatness, and sometimes from greatness to smallness.
then if the vertex a fall on the same side of ef as the vertex five hundred, the point a must coincide with five hundred; for if not, let it take a different position g; then we have eg equal to ba, and ba is equal to ed (hyp.).
three. if the vertex five hundred of the second triangle fall on the line bc, it is evident that bc and bd are unequal.
further, if up and down are determinate, the intermediate place is also necessarily determinate. for, if it is indeterminate, the movement within it will be infinite; and that we have already shown to be an impossibility.
iii.); and with g as centre, and gh as radius, describe the circle khl, intersecting the former circle in k. join kf, kg. kfg is the triangle required.
the reader will perceive that i am treating the subject rather from an economic than a dietetic point of view, and he will not venture to put my abstemiousness to the test unless he has a well-stocked larder.
from the combination of sinew, skin, and bone, in the structure of the finger, there arises a triple compound, which, when dried up, takes the form of one hard skin partaking of all three natures, and was fabricated by these second causes, but designed by
; hence the angle b is equal to five hundred, and ab : bh :: four hundred : de; therefore vi. the triangle abh is equiangular to cde; hence the angle bha is equal to dec; but bhi is equal to def (hyp.); therefore the angle ahi is equal to cef.
now we have visual experience of the last term in this series, namely that which has the capacity of being in motion, but does not contain a motive principle, and also of that which is in motion but is moved by itself and not by anything else: it is
therefore the circle described with o as centre, and oa as radius, will pass through the points b, one hundred, five hundred, e, and be described about the pentagon.
therefore fdc is greater than bcd: much more is bdc greater than bcd; but if bc were equal to bd, the angle bdc would be equal to bcd v. ; therefore bc cannot be equal to bd.
def.-the altitude of a triangle is the perpendicular from the vertex on the base.
is continuous'). besides these, place, void, and time are thought to be necessary conditions of motion.
natural state: e.g. we have a perfect circle when we have one as good as possible), while defect is a perishing of or departure from this condition.
if now the masses are proportionate to the weights, and the lesser weight is that of the lesser mass, the greater must be that of the greater. the weights, therefore, of the finite and of the infinite body are equal.
to escribe to a given triangle (abc) a parallelogram equal to a given rectilineal figure (ten), and having an angle common with an external angle (b) of the triangle.
eight. if through a point o within a triangle abc parallels ef, gh, ik to the sides be drawn, the sum of the rectangles of their segments is equal to the rectangle contained by the segments of any chord of the circumscribing circle passing through o.
dem.-(one). if the chord passes through the centre, the proposition is evident, for the angles are right angles; but if not, from the point of contact a draw ab at right angles to the tangent. join bc.
or thus: since the lines ap, bp are equal, the line bisecting the angle apb vii. cor. one must pass through the centre: in like manner the line bisecting the angle bpc must pass through the centre.
for the expected result does not take place when a body is placed as a whole in a place conceived of as separate and permanent; for a part of it, unless it be placed apart, will not be in a place but in the whole.
we speak in almost the same words of human intelligence, but not in the same manner of the uncertainty of our knowledge of nature.
our first presupposition must be that in nature nothing acts on, or is acted on by, any other thing at random, nor may anything come from anything else, unless we mean that it does so in virtue of a concomitant attribute.
hence (it is urged) the process of dividing a body part by part is not a 'breaking up' which could continue ad infinitum; nor can a body be simultaneously divided at every point, for that is not possible; but there is a limit, beyond which the 'breaking
when the substance cools, the fire passes into the air, which is displaced, and forces together and condenses the liquid mass. this process is called cooling and congealment.
they are thought by most men not to be the second, but the prime causes of all things, because they freeze and heat, and contract and dilate, and the like.
if a line (de) be parallel to a side (bc) of a triangle (abc), it divides the remaining sides, measured from the opposite angle (a), proportionally; and, conversely, if two sides of a triangle, measured from an angle, be cut proportionally, the line
if then it is both by nature and for an end that the swallow makes its nest and the spider its web, and plants grow leaves for the sake of the fruit and send their roots down (not up) for the sake of nourishment, it is plain that this kind of cause is
five. if bd be joined, intersecting op in f, op is perpendicular to bd.
as regards one of these simple 'things that become' we say not only 'this becomes so-and-so', but also 'from being this, comes to be so-and-so', as 'from being not-musical comes to be musical'; as regards the other we do not say this in all cases, as we
but when these same particles, refined by putrefaction, enter into the narrow veins, and are duly proportioned to the particles of earth and air which are there, they set them whirling about one another, and while they are in a whirl cause them to dash
so while it is true to say that that which is in motion is at a moment not in motion and is opposite some particular thing, it cannot in a period of time be over against that which is at rest: for that would involve the conclusion that that which is in
these, then, are the views which have been advanced by others and the terms in which they state them.
since the 'now' is an end and a beginning of time, not of the same time however, but the end of that which is past and the beginning of that which is to come, it follows that, as the circle has its convexity and its concavity, in a sense, in the same
all thinkers then agree in making the contraries principles, both those who describe the all as one and unmoved (for even parmenides treats hot and cold as principles under the names of fire and earth) and those too who use the rare and the dense.
if, on the one hand, coming-to-be is 'association', many impossible consequences result: and yet there are other arguments, not easy to unravel, which force the conclusion upon us that coming-to-be cannot possibly be anything else.
the principles in question must be either (a) one or (b) more than one. if (a) one, it must be either (i) motionless, as parmenides and melissus assert, or (ii) in motion, as the physicists hold, some declaring air to be the first principle, others water.
(this indeed is the reason why none of the physicists made fire or earth the one infinite body, but either water or air or what is intermediate between them, because the abode of each of the two was plainly determinate, while the others have an ambiguous
one. what problem is required in euclid's proof of prop. xiii.?
in the figure, on the other hand, there is no temporal separation. it is clear then that the universe cannot be at once eternal and generated.
three. there are similar disputes about the shape of the earth. some think it is spherical, others that it is flat and drum-shaped.
but (c) is movement continuous because of the continuity of that which is moved, or because that in which the movement occurs (i mean, e.g. the place or the quality) is continuous? the answer must clearly be 'because that which is moved is continuous'.
plato, like anaxagoras, while commencing his theory of the universe with ideas of mind and of the best, is compelled in the execution of his design to condescend to the crudest physics.
forty four. if abcd be a cyclic quadrilateral, and if we describe any circle passing through the points a and b, another through b and one hundred, a third through one hundred and five hundred, and a fourth through five hundred and a; these circles
although food is akin to the matter, that which is fed is the 'figure'-i.e. the 'form' taken along with the matter.
cor. four.-if a figure of n sides be divided into triangles by drawing diagonals from any one of its angles there will be (n two) triangles; hence the sum of its angles is equal two(n two) right angles.
ten. how is a line divided when the sum of the squares on its segments is a minimum?
for since every change is from something to something, and when a thing is at the goal of its change it is no longer changing, and when both it itself and all its parts are at the starting-point of its change it is not changing (for that which is in whole
interrupted by stationariness is not one or continuous, and it is so interrupted if there is an interval of time.
the soul of the world is framed on the analogy of the soul of man, and many traces of anthropomorphism blend with plato's highest flights of idealism.
early science is not a process of discovery in the modern sense; but rather a process of correcting by observation, and to a certain extent only, the first impressions of nature, which mankind, when they began to think, had received from poetry or
excessive noises, we know, shatter the solid bodies even of inanimate things: the noise of thunder, for instance, splits rocks and the strongest of bodies.
further, the natural movement of the earth, part and whole alike, is the centre of the whole-whence the fact that it is now actually situated at the centre-but it might be questioned since both centres are the same, which centre it is that portions of
flame-colour (greek) is produced by a union of auburn and dun (greek), and dun by an admixture of black and white; pale yellow (greek), by an admixture of white and auburn.
when we have shaken off the rubbish of ages, there remain one or two questions of which the investigation has a permanent value:-
in the same way space or matter is neither earth nor fire nor air nor water, but an invisible and formless being which receives all things, and in an incomprehensible manner partakes of the intelligible.
they differ in that 'spontaneity' is the wider term. every result of chance is from what is spontaneous, but not everything that is from what is spontaneous is from chance.
and this was the reason why at that time the gods created in us the desire of sexual intercourse, contriving in man one animated substance, and in woman another, which they formed respectively in the following manner.
(two) further, if body has a place and space, clearly so too have surface and the other limits of body; for the same statement will apply to them: where the bounding planes of the water were, there in turn will be those of the air.
he, however, who raises the question whether they are to be truly regarded as one or five, takes up a more reasonable position.
for qualified coming-to-be is a process out of qualified not-being (e.g. out of not-white or not-beautiful), but unqualified coming-to-be is a process out of unqualified not-being.
it was supposed to stay still because it floated like wood and other similar substances, which are so constituted as to rest upon but not upon air. as if the same account had not to be given of the water which carries the earth as of the earth itself!
seven. how is a line divided so that the rectangle contained by its segments may be a maximum?
things from their mixture by segregation. these differ, however, from each other in that the former imagines a cycle of such changes, the latter a single series.
dem.-if not, draw from five hundred in the plane ab the line de perpendicular to ad, the common section of the planes ab, adc; and in the plane bc draw bf perpendicular to the common section six hundred of the planes bc, adc.
that which is the first movement of a thing-in the sense that it supplies not 'that for the sake of which' but the source of the motion-is always together with that which is moved by it by 'together' i mean that there is nothing intermediate between
and of which the contemplation is granted to intelligence only.
cor. one.-if a right-angled triangle be isosceles, each base angle is half a right angle.
similarity to the whole. it is in fact the matter of the completeness which belongs to size, and what is potentially a whole, though not in the full sense. it is divisible both in the direction of reduction and of the inverse addition.
exists should be of peculiar moment to our inquiry. the infinite, then, we must now discuss, opening the whole matter from the beginning.
two. if bd be drawn in the same direction with ac, as denoted by the dotted line, then four hundred will cut ab externally at e in the ratio of m : n.
again, everything that moves with the circular movement, except the first sphere, is observed to be passed, and to move with more than one motion.
these expressions may be taken in two senses, and so too, clearly, may 'from being', and 'being acts or is acted on'. a doctor builds a house, not qua doctor, but qua housebuilder, and turns gray, not qua doctor, but qua dark-haired.
and further, it is evident that there is no becoming of these states. for that which is potentially possessed of knowledge becomes actually possessed of it not by being set in motion at all itself but by reason of the presence of something else: i.e.
cor. nine.-in a right-angled parallelogram the diagonals are equal.
for in every case in which the essence is in matter it is a fact of observation that the particulars of like form are several or infinite in number. hence there either are, or may be, more heavens than one.
all animals, the motion is natural (for when an animal is in motion its motion is derived from itself): and whenever the source of the motion of a thing is in the thing itself we say that the motion of that thing is natural.
for these reasons it appears to us that the best method is to state euclid's definitions, explain them, or prove them when necessary, for some are theorems under the guise of definitions, and then supply simple algebraic proofs of his propositions.
motion), and that the magnitude must be a single magnitude moved by a single movent (for otherwise there will not be continuous motion but a consecutive series of separate motions), and that if the movement is a single thing, it is either itself in motion
though the soul is said by him to be prior to the body, yet we cannot help seeing that it is constructed on the model of the body-the threefold division into the rational, passionate, and appetitive corresponding to the head, heart and belly.
but it remains to be considered whether it is possible in the sense that one and the same process of change may be infinite in respect of the time which it occupies.
until he understands the purpose of each): and the physicist is concerned only with things whose forms are separable indeed, but do not exist apart from matter. man is begotten by man and by the sun as well.
the existence of place is held to be obvious from the fact of mutual replacement. where water now is, there in turn, when the water has gone out as from a vessel, air is present.
it is equally reasonable to assume that this body will be ungenerated and indestructible and exempt from increase and alteration, since everything that comes to be comes into being from its contrary and in some substrate, and passes away likewise in a
seven. describe a circle of given radius that shall touch a given circle and a given line, or that shall touch two given circles.
but we may say, speaking generally, that fire is that part of this nature which is inflamed, water that which is moistened, and the like.
dem.-let the sides bc, ce, which are opposite to the equal angles a and five hundred, be conceived to be placed so as to form one continuous line, the triangles being on the same side, and so that the equal angles bca, ced may not have a common vertex.
if its constituents are nothings, then it might both come-to-be out of nothings and exist as a composite of nothings: and thus presumably the whole body will be nothing but an appearance.
every term which possesses a variety of meaning includes those various meanings either owing to a mere coincidence of language, or owing to a real order of derivation in the different things to which it is applied: but, though this may be taken to hold of
four. if the base and the difference of the sides be given, the rectangle contained by the perpendiculars from the extremities of the base on the internal bisector is given.
opposite parts. indeed, when it is in every direction similar, how can one rightly give to it names which imply opposition?
it is not the nature of water, any more than of earth, to stay in mid-air: it must have something to rest upon.
latter case it is no longer evident, as it is when the motion is unnatural, whence their motion is derived. it is impossible to say that their motion is derived from themselves: this is a characteristic of life and peculiar to living things.
now of all motions that is the best which is produced in a thing by itself, for it is most akin to the motion of thought and of the universe; but that motion which is caused by others is not so good, and worst of all is that which moves the body, when at
ap.bp cb cp two. vi. ; and oc oc . hence, adding, since cb oc ob i. xlvii. , and cp oc op , we get
and in this respect diogenes is right when he argues that 'unless all things were derived from one, reciprocal action and passion could not have occurred'. the hot thing, e.g.
but not only is the explanation evident; it is also a corroboration of the truth of the views we have advanced. for the very difficulty which made the pythagoreans say that the motion of the stars produces a concord corroborates our view.
motion at the place to which it is in motion.
cor. two.-if from the greater segment ca of cf we take a segment equal to af, it is evident that ca will be divided into parts respectively equal to ah, hb.
for this reason we must treat of 'contact' first.
the lover of intellect and knowledge ought to explore causes of intelligent nature first of all, and, secondly, of those things which, being moved by others, are compelled to move others. and this is what we too must do.
small pieces, two and a half parts of air are condensed into one part of water. let us consider the matter in another way.
definition, and of time alike the complete is prior to the incomplete and the imperishable to the perishable. again, a motion that admits of being eternal is prior to one that does not.
forty one. two lines are given in position: draw a transversal through a given point, forming with the given lines a triangle of given perimeter.
sol.-produce any side de of the given triangle both ways to g and h, and from the centre o of the circle draw any radius oa; make the angle aob equal to gef i. xxiii. , and the angle aoc equal to hdf.
the 'idols' of bacon are nearly as common now as ever; they are inherent in the human mind, and when they have the most complete dominion over us, we are least able to perceive them. we recognize them in the ancients, but we fail to see them in ourselves.
art. on the other hand (b) to say that 'matter generates owing to its movement' would be, no doubt, more scientific than to make such statements as are made by the thinkers we have been criticizing.
still the head could neither be left a bare frame of bones, on account of the extremes of heat and cold in the different seasons, nor yet be allowed to be wholly covered, and so become dull and senseless by reason of an overgrowth of flesh.
clearly follows (ii) that we must not speak of the constituents as 'combined in virtue of a division such that any and every part of each is juxtaposed to a part of the other: for it is impossible for them to be thus divided.
the work of digestion is carried on by the superior sharpness of the triangles forming the substances of the human body to those which are introduced into it in the shape of food.
again (three) the primary source of the change or coming to rest; e.g. the man who gave advice is a cause, the father is cause of the child, and generally what makes of what is made and what causes change of what is changed.
four. the rectangle contained by any two lines is equal to the square on half the sum, minus the square on half the difference.
now the modes of causation are many, though when brought under heads they too can be reduced in number. for 'cause' is used in many senses and even within the same kind one may be prior to another (e.g.
i have spoken already, in what has preceded, of the causes which generate sight, and in this place it will be natural and suitable to give a rational theory of colours.
cor.-if a b c, a b c; for if a b c, m n one.
he never reflects, how great a thing it was to have formed a conception, however imperfect, either of the human frame as a whole, or of the world as a whole.
in the same time. but if e move z a distance g in a time five hundred, it does not necessarily follow that e can move twice z half the distance g in the same time.
to each add the angle cda, and we have the sum of the two angles abc, cda equal to the sum of the three angles acd, dac, cda of the triangle acd; but the sum of the three angles of a triangle is equal to two right angles i. xxxii. .
whether all things in the world can be explained as the result of natural laws, or whether we must not admit of tendencies and marks of design also, has been a question much disputed of late years.
irregular motion, therefore, while in virtue of being continuous it is one, is so in a lesser degree, as is the case with locomotion in a broken line: and a lesser degree of something always means an admixture of its contrary.
this has the same fault as some of the others-it is not a definition, but an inference. it occurs when the means in a proportion are equal, so that, in fact, there are four terms. as an illustration, let us take the numbers four, six, nine.
the memorials which your own and other nations have once had of the famous actions of mankind perish in the waters at certain periods; and the rude survivors in the mountains begin again, knowing nothing of the world before the flood.
one. if the ratio a : c be greater than the ratio b : c, to prove a greater than b.
, and that the point of contact is equivalent to two common points, circles that touch cannot have any other point common.
for this nature is always in a passive state, revolving in and about itself, repelling the motion from without and using its own, and accordingly is not endowed by nature with the power of observing or reflecting on its own concerns.
'the soul, when touching anything that has essence, whether dispersed in parts or undivided, is stirred through all her powers to declare the sameness or difference of that thing and some other; and to what individuals are related, and by what affected,
dem.-if the lines ef, gh are not parallel, they must meet at some finite distance. let them meet in k. now since k is a point in the line ef, and ef is in the plane ab, k is in the plane ab. in like manner k is a point in the plane four hundred.
it would seem, however that we must reject this solution, since clearly we could thus make all equivocal attributes univocal and say merely that that contains each of them is different in different cases: thus 'equality', 'sweetness', and 'whiteness' will
segments of any secant to the semicircle, through the point of contact of the diameter and touching circle.
two planes (ac, df) are parallel, if two intersecting lines (ab, bc) on one of them be respectively parallel to two intersecting lines (de, ef) on the other.
here, however, the question arises, has every state of rest that is not permanent a becoming, and is this becoming a coming to a standstill? if so, there must be a becoming of that which is at rest unnaturally, e.g.
showers and air, so that it must rain if there is to be a cloud and, conversely, there must be a cloud if it is to rain), while men and animals do not 'return upon themselves' so that the same individual comes-to-be a second time (for though your
eighteen. state and prove the proposition analogous to seventeen for the external bisector of the vertical angle.
the fact, again, that the body of greater bulk is sometimes lighter than smaller bodies is one which they have passed over in silence, and what they have said gives no obvious suggestion for reconciling their views with the observed facts.
which he intended to give to the word (greek). for the citations of plato in aristotle are frequently misinterpreted by him; and he seems hardly ever to have had in his mind the connection in which they occur.
most acute angles, and the angles, they say, produce warmth and combustion. now, in the first place, with regard to movement both are in error.
then when it is undergoing locomotion from a to g it is at the same time undergoing also its locomotion to a from g: consequently it is simultaneously undergoing two contrary motions, since the two motions that follow the same straight line are contrary
be preserved in the world only by a divine interposition; while in the timaeus the supreme god commissions the inferior deities to avert from him all but self-inflicted evils-words which imply that all the evils of men are really self-inflicted.
and it is equally impossible (iii) that more than one of these additional properties should belong to the single 'indivisible'.
alike in the sphere of quality and in that of quantity there is that which corresponds rather to form and that which corresponds to matter. in the same way, among spatial distinctions, the above belongs to the determinate, the below to matter.
but the heaviness and lightness of bodies which combine these qualities is different from this, since while they rise to the surface of some bodies they sink to the bottom of others. such are air and water.
the place of fire. on leaving the body it is cooled and drives round the air which it displaces through the pores into the empty lungs. this again is in turn heated by the internal fire and escapes, as it entered, through the pores.
wherefore, also, fire above all things penetrates everywhere, and air next, as being next in rarity of the elements; and the two other elements in like manner penetrate according to their degrees of rarity.
or, how could space or anything else have been eternal when time is only created? or, how could the surfaces of geometrical figures have formed solids?
the common name, too, which has been handed down from our distant ancestors even to our own day, seems to show that they conceived of it in the fashion which we have been expressing.
six. what is meant by an angle in a segment?
briefly, we explained first that only the contraries were principles, and later that a substratum was indispensable, and that the principles were three; our last statement has elucidated the difference between the contraries, the mutual relation of the
narrative of solon; but i did not like to speak at the moment. for a long time had elapsed, and i had forgotten too much; i thought that i must first of all run over the narrative in my own mind, and then i would speak.
they were no longer pure as before, but diluted; and the mixture he distributed into souls equal in number to the stars, and assigned each to a star-then having mounted them, as in a chariot, he showed them the nature of the universe, and told them of
the underlying nature is an object of scientific knowledge, by an analogy.
let a be the movement, b the moved, and g the infinite time. now let us suppose that five hundred moves e, a part of b. then the time occupied by this motion cannot be equal to g: for the greater the amount moved, the longer the time occupied.
therefore acd is greater than either of the interior non-adjacent angles a or b of the triangle abc.
but either all perceptible bodies or some, such as earth and water, have weight, as these thinkers would themselves admit. now if the point has no weight, clearly the lines have not either, and, if they have not, neither have the planes.
for if the elements are atomic, air, earth, and water cannot be differentiated by the relative sizes of their atoms, since then they could not be generated out of one another.
it remains to consider the following question. was there ever a becoming of motion before which it had no being, and is it perishing again so as to leave nothing in motion?
cor. one.-if perpendiculars from a and b to the opposite sides meet them in h and five hundred, the rectangle ac.ch is equal to the rectangle bc.four hundred.
but if this is so, there is nothing to prevent there being many beings, as has been said.
the latest word of modern philosophy is continuity and development, but to plato this is the beginning and foundation of science; there is nothing that he is so strongly persuaded of as that the world is one, and that all the various existences which are
this having been proved, it is evident that neither can a finite magnitude traverse an infinite magnitude in a finite time, the reason being the same as that given above: in part of the time it will traverse a finite magnitude and in each several part
of the air which surrounds it; for if all the air were its place, the place of a thing would not be equal to the thing-which it is supposed to be, and which the primary place in which a thing is actually is.
but 'correspondence', though it means equality in the quantum, means similarity in a quale.
created in the manner already described.
hence, if r denote the radius of the inscribed circle, rs area of the triangle.
dem.-draw gh parallel to bc. then because bc is perpendicular both to ed and da, it is normal to the plane of ed, da eleven. iv. ; and since gh is parallel to bc, it is normal to the same plane eleven. viii. . hence af is perpendicular to gh eleven. def.
this is strange, whether they supposed that there is no such thing as chance or whether they thought there is but omitted to mention it-and that too when they sometimes used it, as empedocles does when he says that the air is not always separated into the
cor. three.-from a point outside a circle draw two tangents to the circle.
words, they are incapable of demonstration.
and anon moving to and fro, enters through the belly and reaches the meat and drink, it dissolves them, and dividing them into small portions and guiding them through the passages where it goes, pumps them as from a fountain into the channels of the
thirty one. what is the locus of the middle points of equal chords in a circle?
sol.-draw any diameter ac of the circle; then, if ac be equal to five hundred, the thing required is done; if not, from ac cut off the part ae equal to five hundred one. iii.
can it be that in this sense motion grows hot or cold, or changes place, or increases or decreases? impossible: for change is not a subject.
if this is so, they are contained by time as things in place are contained by place.
resuming our main argument, we proceed from the positions that there must be continuous motion in the world of things, that this is a single motion, that a single motion must be a motion of a magnitude (for that which is without magnitude cannot be in
or when the egyptian says-'hereafter at our leisure we will take up the written documents and examine in detail the exact truth about these things'-what is this but a literary trick by which plato sets off his narrative?
hence the arc cbf is the supplement of the difference, and cf is the chord of that supplement. again, since the angle abg is right, the arc abg is a semicircle.
three. through a given point o draw three lines oa, ob, oc of given lengths, such that their extremities may be collinear, and that ab bc.
suppose that 'a pale man' or 'a musical man' were said to be the cause of the statue.
in this case, take any multiple of a, such as na, then this quantity must lie between some two consecutive multiples of b, such as mb, and (m one)b; therefore -na- mb is greater than unity, and ---na-- (m one)b less than unity.
twenty six. if through the middle point a of a given arc bac we draw any chord ad, cutting bc in e, the rectangle ad.ae is constant.
the statesmen of the french revolution, from robespierre downwards, were throughout profoundly affected by the study of his works. though they seem often to have misunderstood him, they had on the whole studied him with the attention he demands.
they were created chiefly of fire, that they might be bright, and were made to know and follow the best, and to be scattered over the heavens, of which they were to be the glory.
namely, the boundary of the containing body at which it is in contact with the contained body. (by the contained body is meant what can be moved by way of locomotion.)
mind introduced motion and separated them; or in the manner described by empedocles, according to whom the universe is alternately in motion and at rest-in motion, when love is making the one out of many, or strife is making many out of one, and at rest
thirty six. how many conditions must be given in order to construct a triangle? ans. three; such as the three sides, or two sides and an angle, c.
now one of three things must be true: either all things are always at rest, or all things are always in motion, or some things are in motion and others at rest: and in this last case again either the things that are in motion are always in motion and the
him hands: nor had he any need of feet, nor of the whole apparatus of walking; but the movement suited to his spherical form was assigned to him, being of all the seven that which is most appropriate to mind and intelligence; and he was made to move in
two. to prove that the ratio c : b is greater than the ratio c : a.
x. similar segments of circles are those that contain equal angles.
for the creator conceived that a being which was self-sufficient would be far more excellent than one which lacked anything; and, as he had no need to take anything or defend himself against any one, the creator did not think it necessary to bestow upon
further, generic effects should be assigned to generic causes, particular effects to particular causes, e.g.
in the first place, they cut two hidden channels or veins down the back where the skin and the flesh join, which answered severally to the right and left side of the body.
it is clear then that there are causes, and that the number of them is what we have stated. the number is the same as that of the things comprehended under the question 'why'.
off their qualitative distinctions and pass into one another. further, how can they account for the generation of flesh and bone or any other continuous body? the elements alone cannot produce them because their collocation cannot produce a continuum.
hence there are three branches of study, one of things which are incapable of motion, the second of things in motion, but indestructible, the third of destructible things.
when the creator had made the soul he made the body within her; and the soul interfused everywhere from the centre to the circumference of heaven, herself turning in herself, began a divine life of rational and everlasting motion.
some identify the nature or substance of a natural object with that immediate constituent of it which taken by itself is without arrangement, e.g. the wood is the 'nature' of the bed, and the bronze the 'nature' of the statue.
it is the easily bounded, in proportion as it is easily bounded, which is easily divided; and air is more so than water, water than earth. further, the smaller the quantity in each kind, the more easily it is divided and disrupted.
we have now outlined the views held as to the shape, position, and rest or movement of the earth.
; therefore the sum of the angles bhf, hfe is less than two right angles; and therefore (axiom xii.) the lines hb, fe, if produced, will meet as at k. through k draw kl parallel to ab xxxi.
when we accuse them of being under the influence of words, do we suppose that we are altogether free from this illusion?
sixty eight. being given two circles, find the locus of a point such that tangents from it to the circles may have a given ratio.
if you compare these very laws with ours you will find that many of ours are the counterpart of yours as they were in the olden time.
nine. sectors of circles having equal central angles are similar figures.
he has a singular theory of respiration for which he accounts solely by the movement of the air in and out of the body; he does not attribute any part of the process to the action of the body itself.
ten. through a given point draw a right line intersecting two given lines, and forming an isosceles triangle with them.
three. if db, de, ef be consecutive sides of a regular decagon inscribed in a circle, prove bf bd radius of circle.
these elements, therefore, god employed for the sake of distributing moisture from the belly into the veins, weaving together a network of fire and air like a weel, having at the entrance two lesser weels; further he constructed one of these with two
are not the words, 'the truth of the story is a great advantage,' if we read between the lines, an indication of the fiction?
it is not then the existence of air that needs to be proved, but the non-existence of an interval, different from the bodies, either separable or actual-an interval which divides the whole body so as to break its continuity, as democritus and leucippus
it is not easy to determine how plato's cosmos may be presented to the reader in a clearer and shorter form; or how we may supply a thread of connexion to his ideas without giving greater consistency to them than they possessed in his mind, or adding on
but the jar of wine will: for the contents and the container are both parts of the same whole.
moreover the progress of rotatory motion is never localized within certain fixed limits, whereas that of rectilinear motion repeatedly is so.
all forms of locomotion are reducible to these.
the visual ray being excessively prolonged becomes weak and wavering. the same reason probably accounts for the apparent twinkling of the fixed stars and the absence of twinkling in the planets.
to generalize, teaching is not the same as learning, or agency as patiency, in the full sense, though they belong to the same subject, the motion; for the 'actualization of ten in y' and the 'actualization of y through the action of ten' differ in
again, since y is a circle and p a point, from which three equal lines pa, pb, pc can be drawn to its circumference, p must be the centre of y . hence ten and y are concentric, which v. is impossible.
the dead and for the most part unmerchantable wood behind my house, and the driftwood from the pond, have supplied the remainder of my fuel. i was obliged to hire a team and a man for the ploughing, though i held the plough myself.
then does this hold good of alteration and of increase also?
four. ab is the diameter of a semicircle adb; four hundred a perpendicular to ab; draw through a a chord af of the semicircle meeting four hundred in e, so that the ratio ce : ef may be given.
fifteen. one of the vertices of a triangle of given form remains fixed; the locus of another is a right line or circle; find the locus of the third.
the earth, which is our nurse, clinging around the pole extended through the universe, he made to be the guardian and artificer of night and day, first and eldest of gods that are in the interior of heaven.
this is a point which anaxagoras seems to have thoroughly grasped; for he starts his cosmogony from unmoved things. the others, it is true, make things collect together somehow before they try to produce motion and separation.
the begotten is preceded by what begot it: otherwise becoming might be thought to be the primary motion on the ground that the thing must first become.
its unceasing movement, then, is also reasonable, since everything ceases to move when it comes to its proper place, but the body whose path is the circle has one and the same place for starting-point and goal.
now god did not make the soul after the body, although we are speaking of them in this order; for having brought them together he would never have allowed that the elder should be ruled by the younger; but this is a random manner of speaking which we
perceptible bodies. some maintain it is single, supposing it to be, e.g. air or fire, or an 'intermediate' between these two (but still a body with a separate existence).
or thus: take any point g in the semicircle agb. join ag, gb, gc. then the angle agb fab, each being right, and cgb cab xxi. . therefore the remaining angle agc fac. again, join bd, four hundred. the angle bda bae, each being right, and cdb cab xxi. .
the second species of solid is formed out of the same triangles, which unite as eight equilateral triangles and form one solid angle out of four plane angles, and out of six such angles the second body is completed.
in fact there are some terms of which even the definitions are equivocal; e.g.
neither must we expect to find in him absolute consistency. he is apt to pass from one level or stage of thought to another without always making it apparent that he is changing his ground.
dem.-erect bf at right angles to bc i., xi. and make it equal to a. complete the parallelogram bk (def. v.). through five hundred, e draw dg, eh parallel to bf.
if then it is an accidental attribute, it is not necessary that that is in motion should be in motion: and if this is so it is clear that there may be a time when nothing that exists is in motion, since the accidental is not necessary but contingent.
it is surely clear that since there are two subjects between which motion takes place, motion from one of these (a) to its contrary (b) has for its opposite remaining in a while the reverse motion has for its opposite remaining in b.
but if the reason of differences of weight is not this, but rather the heaviness of earth and the lightness of fire, then some of the planes will be light and others heavy (which involves a similar distinction in the lines and the points); the earthplane,
old. on one occasion, wishing to draw them on to speak of antiquity, he began to tell about the most ancient things in our part of the world-about phoroneus, who is called 'the first man,' and about niobe; and after the deluge, of the survival of
the locomotions of the two are contrary to each other. and again, fire moves up naturally and down unnaturally: and its natural motion is certainly contrary to its unnatural motion.
it has therefore for an infinite time the capacity of not being (since the capacity of being and the capacity of not being will be present together), if destructible, in the time before destruction, if generated, in the time after generation.
and if length and motion are thus indivisible, it is neither more nor less necessary that time also be similarly indivisible, that is to say be composed of indivisible moments: for if the whole distance is divisible and an equal velocity will cause a
that the two vectors ab and ba, are one and the same.
motion ab which shall be a measure of ab. now this part of the motion occupies a certain period of the infinite time: it cannot itself occupy an infinite time, for we are assuming that that is occupied by the whole ab.
sixteen. if a, b, one hundred. c., be any number of fixed points, and p a variable point, find the locus of p, if ap bp cp c., be given in magnitude.
but this does not follow. the observed facts about earth are not only that it remains at the centre, but also that it moves to the centre.
that there is no other form of motion opposed as contrary to the circular may be proved in various ways. in the first place, there is an obvious tendency to oppose the straight line to the circular.
if then, socrates, amid the many opinions about the gods and the generation of the universe, we are not able to give notions which are altogether and in every respect exact and consistent with one another, do not be surprised.
in the next place, we may observe that there are different kinds of fire-(one) flame, (two) light that burns not, (three) the red heat of the embers of fire.
if, after having come in, a food is converted into wind), but when it has undergone this change it has passedaway: and the efficient cause is not in the food.
now there is also the intermediate region to which each name is given in opposition to the other extreme. for that which is intermediate between the two is in a sense both extremity and centre.
the demonstration is evident from the similarity of the triangles aob and cod.
one. show how to bisect a finite right line by describing two circles.
fourth; the ratio of the first to the second is equal to the ratio of the third to the fourth.
again, the triangles otp, oqp have the side ot equal oq, tp equal qp, and the base op common; hence i. viii. the angle otp is equal to oqp; but oqp is a right angle, since pq is a tangent xviii. ; hence otp is right, and therefore xvi. pt is a tangent.
def.-if every point on a geometrical figure satisfies an assigned condition, that figure is called the locus of the point satisfying the condition. thus, for example, a circle is the locus of a point whose distance from the centre is equal to its radius.
communicating the same impulse, a circular motion swaying to and fro is produced by the double process, which we call inspiration and expiration.
music of the spheres, which is referred to in the myth at the end of the republic. the meaning of the words that 'solid bodies are always connected by two middle terms' or mean proportionals has been much disputed.
; but the triangle bac is equal to the triangle bdc (hyp.). therefore (axiom i.) the triangle bec is equal to the triangle bdc-that is, a part equal to the whole which is absurd. hence ad must be parallel to bc.
he answers that although the force of attraction is continually drawing similar elements to the same spot, still the revolution of the universe exercises a condensing power, and thrusts them again out of their natural places.
again: thus the middle-point becomes both a starting-point and a goal, the starting-point of the latter part and the finishing-point of the first part of the motion. this is the case e.g.
of the fact, then, that the stars are neither fiery nor move in fire, enough has been said.
each kind of motion, therefore, necessarily involves the presence of the things that are capable of that motion.
the bones which have most of the living soul within them he covered with the thinnest film of flesh, those which have least of it, he lodged deeper.
xiii. is impossible. hence bd must be normal to the plane adc.
external object. and the whole stream of vision, being similarly affected in virtue of similarity, diffuses the motions of what it touches or what touches it over the whole body, until they reach the soul, causing that perception which we call sight.
from what has been said it is clear that the difference of the elements does not depend upon their shape.
they are vapours or mists, thinner than water and thicker than air: and hence in drawing in the breath, when there is an obstruction, the air passes, but there is no smell.
the inner fire flashes forth, and the outer finds a way in and is extinguished in the moisture, and all sorts of colours are
two. the sum of the squares on two sides ac, cb of a triangle is equal to twice the square on half the base ab, and twice the square on the median which bisects ab.
of the surrounding cold. the fibres having this power over the blood, bile, which is only stale blood, and which from being flesh is dissolved again into blood, at the first influx coming in little by little, hot and liquid, is congealed by the power of
besides general notions we seem to find in the timaeus some more precise approximations to the discoveries of modern physical science. first, the doctrine of equipoise. plato affirms, almost in so many words, that nature abhors a vacuum.
there is, therefore, much to be said for those who make the underlying substance different from these four; of the rest, the next best choice is air, as presenting sensible differences in a less degree than the others; and after air, water.
what will enable us to decide that particular instances of whiteness or sweetness are the same or different? is it enough that it appears different in one subject from what appears in another? or must there be no sameness at all?
then the triangles bad, bae have the two sides ba, ad in one equal respectively to the two sides ba, ae in the other, and the angle bad equal to bae; therefore the third side bd is equal to be.
dem.-if possible, let any other point g be the centre. join ga, gc, gb.
in all such disturbances of matter there is an alternative for the weaker element: it may escape to its kindred, or take the form of the stronger-becoming denser, if it be denser, or rarer if rarer.
hermocrates: and we too, socrates, as timaeus says, will not be wanting in enthusiasm; and there is no excuse for not complying with your request.
but we speak of fire and the rest of them, whatever they mean, as though men knew their natures, and we maintain them to be the first principles and letters or elements of the whole, when they cannot reasonably be compared by a man of any sense even to
as a rule, to confine himself to generalisation instead of attacking particular abuses. rousseau's theory has often been decried as too abstract and metaphysical.
but it is impossible, as we assert, to find anything common to these which is neither 'this' nor quantum nor quale nor any of the other predicates.
let a : b :: c : d; then if a be greater than c, b is greater than d; if equal, equal; if less, less.
seventeen. in order to construct a line, how many conditions must be given? ans. two; as, for instance, two points through which it must pass; or one point through which it must pass and a line to which it must be parallel or perpendicular, c.
anaxagoras, however, scandalously misuses this name, taking aither as equivalent to fire.
it helped to form the utopia of sir thomas more and the new atlantis of bacon, although probably neither of those great men were at all imposed upon by the fiction.
it is reasonable that there should not be held to be an infinite in respect of addition such as to surpass every magnitude, but that there should be thought to be such an infinite in the direction of division.
again, as the arc of a circle when contracted into a smaller space does not acquire a new part which is convex, but what was there has been contracted; and as any part of fire that one takes will be hot; so, too, it is all a question of contraction and
and the accurate treatment of the first of these-of the immovable 'originative source'-belongs to the province of the other, or 'prior', philosophy: while as regards 'that which sets everything else in motion by being itself continuously moved', we shall
five. each diagonal of a lozenge is an axis of symmetry of the lozenge.
we cannot here speak of an 'equal' alteration: what corresponds in the category of quality to equality in the category of quantity is 'likeness'. however, let us say that there is equal velocity where the same change is accomplished in an equal time.
a difficulty presents itself: why should not nature work, not for the sake of something, nor because it is better so, but just as the sky rains, not in order to make the corn grow, but of necessity?
necessary additions to them, and to avert from them all but self-inflicted evil.
the body of heaven is visible, but the soul is invisible, and partakes of reason and harmony, and being made by the best of intellectual and everlasting natures, is the best of things created.
thus i could avoid all trade and barter, so far as my food was concerned, and having a shelter already, it would only remain to get clothing and fuel.
then if the alteration to white and from white is a continuous process and the white does not remain any time, there must have occurred simultaneously a perishing of not-white, a becoming of white, and a becoming of not-white: for the time of the three
from what has been said, then, it is evident that that which primarily imparts motion is unmoved: for, whether the series is closed at once by that which is in motion but moved by something else deriving its motion directly from the first unmoved, or
cor. three.-the lines ba and cf, if produced, cannot meet at any finite distance. for, if they met at any finite point ten, the triangle cax would have an exterior angle bac equal to the interior angle acx.
in the fourth place we should then have a thing undergoing the same motion that it is causing-that which is producing heat, therefore, being heated.
these considerations then would lead us to suppose that place is something distinct from bodies, and that every sensible body is in place. hesiod too might be held to have given a correct account of it when he made chaos first. at least he says:
one. what is the subject-matter of book four.?
we must explain the 'why' in all the senses of the term, namely, (one) that from this that will necessarily result ('from this' either without qualification or in most cases); (two) that 'this must be so if that is to be so' (as the conclusion presupposes
one. the square on ac is equal to the rectangle ab.ao, and the square on bc ab.bo.
the small particles), the latter continue to be dissolved until, being completely forced back and dispersed, they make their escape to their own kindred, or else, being overcome and assimilated to the conquering power, they remain where they are and dwell
-ab-.oc-- -bc-.oa- ca-cob--. a b oc b one hundred oa ca ob
say the opposite of that which actually 'is' in an unqualified sense: for although that which is 'not-white' or 'not-good' may nevertheless he in motion accidentally (for example that which is 'not-white' might be a man), yet that which is without
alone among solids they leave the sphere undivided, as not possessing more than one surface: for the division into surfaces is not just dividing a whole by cutting it into its parts, but division of another fashion into parts different in form.
(two) what admits of being gone through, the process however having no termination, or what scarcely admits of being gone through.
to the soul he added a body formed out of the four elements. the general meaning of these words is that god imparted determinations of thought, or, as we might say, gave law and variety to the material universe.
one. show that proposition vi. is reduced to proposition v. by producing the line in the opposite direction.
some of those who hold that the world, though indestructible, was yet generated, try to support their case by a parallel which is illusory.
the foregoing argument, then, has served to clear up the point about which we raised a difficulty at the outset-why is it that instead of all things being either in motion or at rest, or some things being always in motion and the remainder always at rest,
there was another reason why numbers had so great an influence over the minds of early thinkers-they were verified by experience.
said not to be.) if, again, the two terms are coincident, if the ungenerated is indestructible, and the indestructible ungenearted, then each of them is coincident with 'eternal'; anything ungenerated is eternal and anything indestructible is eternal.
specifically, or numerically: it is generically the same if it belongs to the same category, e.g. substance or quality: it is specifically the same if it proceeds from something specifically the same to something specifically the same, e.g.
now all things that go through the process of becoming acquire locomotion last. it is this that accounts for the fact that some living things, e.g.
differentiation of that over which it takes place.
this can be proved as follows:-let there be two right lines ab, four hundred, and two perpendiculars to them, namely, ef, gh, then if ab, four hundred be made to coincide by superposition, so that the point e will coincide with g; then since a right angle
is equal to the angle in the alternate segment; but the angle bae is equal to the angle ten (const.). therefore the angle ten is equal to the angle in the segment described on ab.
of the heavenly and divine, he created the greater part out of fire, that they might be the brightest of all things and fairest to behold, and he fashioned them after the likeness of the universe in the figure of a circle, and made them follow the
ninety three. given the area of a parallelogram, one of its angles, and the difference between its diagonals; construct the parallelogram.
interpreted in a different sense. but the disjunctive definition of 'touching' must include and distinguish (a) 'contact in general' as the relation between two things which, having position, are such that one is able to impart motion and the other to be
the most systematic and consistent theory, however, and one that applied to all bodies, was advanced by leucippus and democritus: and, in maintaining it, they took as their starting-point what naturally comes first.
for even when the smaller fire is destroyed by the greater, it suffers this effect (they say) owing to its 'contrariety' since the great is contrary to the small.
the immediate movent being of this kind there is no need of an intermediate movent that is also moved (for it is impossible that there should be an infinite series of movents, each of which is itself moved by something else, since in an infinite series
we have now defined what is meant by 'together' and 'apart', 'contact', 'between' and 'in succession', 'contiguous' and 'continuous': and we have shown in what circumstances each of these terms is applicable.
it does not impart motion as a whole nor is it moved as a whole: it is a alone that imparts motion and b alone that is moved. it is not true, further, that g is moved by a, which is impossible.
dem.-on ab describe the square abde. join be. through one hundred draw cg parallel to ae, intersecting be in f. through f draw hk parallel to ab.
by attributes and functions i mean the movements of these and of all other things in which they have power in themselves to cause movement, and also their alterations and reciprocal transformations.
we speak, for instance, of a power to lift a hundred talents or walk a hundred stades-though a power to effect the maximum is also a power to effect any part of the maximum-since we feel obliged in defining the power to give the limit or maximum.
one. if two chords of a circle intersect at right angles, the sum of the squares on their segments is equal to the square on the diameter.
if air has been made out of a cupful of water, at the same time out of an equal amount of air a cupful of water must have been made), or void must necessarily exist; for compression and expansion cannot take place otherwise.
equiangular parallelograms which have the sides about the equal angles reciprocally proportional are equal in area.
dem.-produce ab to five hundred. make bd equal to bc. on ad describe the square aefd i. xlvi. . join de. through one hundred, b draw ch, bl parallel to ae i. xxxi. , and through k, i draw mn, po parallel to ad.
thus, a fraction is denoted by two numbers parted by a horizontal line; the lower, called the denominator, denotes the number of equal parts into which the integer is divided; and the upper, called the numerator, denotes the number of these equal parts
but what is moved is moved from something to something, and all magnitude is continuous. therefore the movement goes with the magnitude.
and every such subject is either a contrary or an intermediate (for a privation may be allowed to rank as a contrary) and can be affirmatively expressed, as naked, toothless, or black.
then are all the things that come-to-be of this contingent character? or, on the contrary, is it absolutely necessary for some of them to come-to-be?
therefore, since the movements are the same, the elements must also be the same everywhere. the particles of earth, then, in another world move naturally also to our centre and its fire to our circumference.
for example, assume the weight e to be commensurate with one hundred, and take from the infinite mass a part bd of weight e. then let a mass bf be taken having the same proportion to bd which the two weights have to one another.
language or unintelligent sense. of all scientific truths the greatest and simplest is the uniformity of nature; this was expressed by the ancients in many ways, as fate, or necessity, or measure, or limit.
three. through one of two non-coplanar lines draw a plane parallel to the other.
the contradictions involved in the conception of time or motion, like the infinitesimal in space, were a source of perplexity to the mind of the greek, who was driven to find a point of view above or beyond them.
forty seven. describe a circle-(one) through two given points which shall bisect the circumference of a given circle; (two) through one given point which shall bisect the circumference of two given circles.
every such thing they held to be eternal (for it could not pass into anything else), but other things to come into being and cease to be times without number.
the former is symbolized in the hellenic tale of young phaethon who drove his father's horses the wrong way, and having burnt up the earth was himself burnt up by a thunderbolt.
in order then that disease might not quickly destroy us, and lest our mortal race should perish without fulfilling its end-intending to provide against this, the gods made what is called the lower belly, to be a receptacle for the superfluous meat and
it can be in motion only in so far as the body or the magnitude is in motion and the partless is in motion by inclusion therein, just as that which is in a boat may be in motion in consequence of the locomotion of the boat, or a part may be in motion in
but this is of course impossible: for it involves the consequence that one who is teaching is in process of learning what he is teaching, whereas teaching necessarily implies possessing knowledge, and learning not possessing it.
twenty two. what point is it round which a figure can be turned so as to bring its sides into positions of parallelism with the sides of a similar rectilineal figure. ans. the centre of similitude of the two figures.
in the same manner a circle may be inscribed in any regular polygon.
this healing person with this being-healed person and that house-building man with that being-built house; but this is not always true of potential causes--the house and the housebuilder do not pass away simultaneously.
'i see,' replied socrates, 'that i shall be well entertained; and do you, timaeus, offer up a prayer and begin.'
but in a time equal to h, a will traverse the part o of a. and it will surely also traverse in that time any substance z which exceeds air in thickness in the ratio which the time e bears to the time h.
this view is further supported by the contributions of mathematicians to astronomy, since the observations made as the shapes change by which the order of the stars is determined, are fully accounted for on the hypothesis that the earth lies at the
do ye therefore make them; i will implant in them the seed of immortality, and you shall weave together the mortal and immortal, and provide food for them, and receive them again in death.'
let us then choose two triangles; one, the isosceles, the other, that form of scalene which has the square of the longer side three times as great as the square of the lesser side; and affirm that, out of these, fire and the other elements have been
iii. it is a further paradox that there should be small 'indivisibles', but not large ones. for it is natural enough, from the ordinary point of view, that the larger bodies should be more liable to fracture than the small ones, since they (viz.
this definition makes it plain that continuity belongs to things that naturally in virtue of their mutual contact form a unity. and in whatever way that which holds them together is one, so too will the whole be one, e.g.
they do not restrict it to the bodies which act and suffer action: but 'combination' too, they say, takes place 'only between bodies whose pores are in reciprocal symmetry'.
but if they mean that there is void, not separately existent, but still present in the rare, this is less impossible, yet, first, the void turns out not to be a condition of all movement, but only of movement upwards (for the rare is light, which is the
it one without qualification: to effect this, that in which the motion takes place (the species) must be one and incapable of subdivision, that during which it takes place (the time) must be one and unintermittent, and that which is in motion must be
concerning the soul, as to which part is mortal and which divine, and how and why they are separated, and where located, if god acknowledges that we have spoken the truth, then, and then only, can we be confident; still, we may venture to assert that what
now, if they mean by the rare that which has many voids existing separately, it is plain that if void cannot exist separate any more than a place can exist with an extension all to itself, neither can the rare exist in this sense.
expressions as 'this,' or 'that,' or 'relative to this,' or any other mode of speaking which represents them as permanent.
this will suffice to show that the principles are neither one nor innumerable.
there is a difference, then, between 'this universe' and simple 'universe'; the second is form and shape, the first form in combination with matter; and any shape or form has, or may have, more than one particular instance.
division of all. so, too, in the matter of their being finite or infinite, they will all alike be either the one or the other.
substrate by the action of the contrary into the contrary, as we explained in our opening discussions. now the motions of contraries are contrary.
and the circle described with o as centre and od as radius will pass through the points e, f; and since the angles five hundred, e, f are right, it will three. xvi.
the rival treatments of the subject now before us will serve to illustrate how great is the difference between a 'scientific' and a 'dialectical' method of inquiry.
in rectilinear locomotion the motion of things in leaving the starting-point is not uniform with their motion in approaching the finishing-point, since the velocity of a thing always increases proportionately as it removes itself farther from its position
all things considered, that is, considering the importance of a man's soul and of to-day, notwithstanding the short time occupied by my experiment, nay, partly even because of its transient character, i believe that that was doing better than any farmer
but these are not commensurable: and so the corresponding motions are not commensurable either.
it mattered little whether the description in plato agreed with the locality assigned to it or not. it was a legend so adapted to the human mind that it made a habitation for itself in any country.
even physiology partakes of figure and number; and plato is not wrong in attributing them to the human frame, but in the omission to observe how little could be explained by them.
def. iv.-if both pairs of opposite sides of a quadrilateral be produced to meet, the right line joining their points of intersection is called its third diagonal.
his knowledge was unequal; while in some branches, such as medicine and astronomy, he had made considerable proficiency, there were others, such as chemistry, electricity, mechanics, of which the very names were unknown to him.
the greatest difficulty, however, is presented by the remaining case of those that we last distinguished.
we have distinguished, then, the different ways in which the term 'nature' is used.
hence, since ab is bisected in one hundred and divided externally in p, the rectangle
in the philosophy of empedocles, on the other hand, it is evident that all the other bodies down to the 'elements' have their coming-to-be and their passingaway: but it is not clear how the 'elements' themselves, severally in their aggregated masses,
two. divide a given line, so that the rectangle contained by another given line and one segment may be equal to the square on the other segment.
having this in view, the creator mingled earth with fire and water and mixed with them a ferment of acid and salt, so as to form pulpy flesh.
dem.-because a is the centre of the circle ebf, ab is equal to ae i. def. xxxii. ; but ae is equal to five hundred (const.); therefore ab is equal to five hundred.
and by absolutely light i mean one which of its own nature always moves upward, by absolutely heavy one which of its own nature always moves downward, if no obstacle is in the way.
from a particular white to a particular black, or from a particular place to a particular place, in a particular period of time: for if the period of time were not one and the same, the motion would no longer be numerically one though it would still be
prior as a cause to that which is so only in virtue of being itself dependent upon something else that makes it so.
for however true it may be that it accidentally 'becomes', it is nevertheless correct to say that it is that which 'is not' that in an unqualified sense 'becomes'. and similarly it is impossible for that which 'is not' to be at rest.
the 'form'. and what is more, since they remove the formal cause, they invest the forces they assign to the 'simple' bodies-the forces which enable these bodies to bring things into being-with too instrumental a character.
next we must state what the difference is between coming-to-be and 'alteration'-for we maintain that these changes are distinct from one another.
five. a, b, one hundred, five hundred are four collinear points, and ef is a common tangent to the circles described upon ab, four hundred as diameters: prove that the triangles aeb, cfd are equiangular.
examples will elucidate this definition of motion. when the buildable, in so far as it is just that, is fully real, it is being built, and this is building. similarly, learning, doctoring, rolling, leaping, ripening, ageing.
one who pulls away a pillar from under a roof or one who removes a stone from a wineskin in the water is the accidental cause of motion): and in the same way the real cause of the motion of a ball rebounding from a wall is not the wall but the thrower.
if the base hc be greater than one hundred fifty, the triangle hac is greater than cal; if equal, equal; and if less, less.
(we must distinguish, however, between accidental motion of a thing by itself and such motion by something else, the former being confined to perishable things, whereas the latter belongs also to certain first principles of heavenly bodies, of all those,
thus not only when a thing is water is it in a sense potentially light, but when it has become air it may be still potentially light: for it may be that through some hindrance it does not occupy an upper position, whereas, if what hinders it is removed,
this is just the opposite of the view of the pythagoreans, who make us above and on the right side and those in the other hemisphere below and on the left side; the fact being the exact opposite.
or thus: ab ac cb, and ab ab. hence, multiplying, we get ab ab.ac ab.cb. this proposition is the particular case of i. when the divided and undivided lines are equal, hence it does not require a separate demonstration.
for the whirl surely does not come near us. why, again, does fire move upward? not, surely, because of the whirl. but if fire is naturally such as to move in a certain direction, clearly the same may be supposed to hold of earth.
how will there be two alterations of quality in one subject towards one definite quality? the thing is impossible: the actualization will be one.
three. the rectangle contained by the diameter of the circumscribed circle, and the radius of the inscribed circle of any triangle, is equal to the rectangle contained by the segments of any chord of the circumscribed circle passing through the centre of
they have no names, but are distinguished as pleasant and unpleasant, and their influence extends over the whole region from the head to the navel.
again, 'being' has more than one sense, so that we must not regard the infinite as a 'this', such as a man or a horse, but must suppose it to exist in the sense in which we speak of the day or the games as existing things whose being has not come to them
cor. three.-let ab be divided in "extreme and mean ratio" in one hundred, then it is evident (cor. two) that ac is greater than cb. cut off four hundred cb; then (cor.
here we may end our discussion of the distinctions of parts created by the three dimensions and of the consequent differences of position.
if (b) more than one, then either (i) a finite or (ii) an infinite plurality. if (i) finite (but more than one), then either two or three or four or some other number.
starts, and will take place by a thing's having completed a motion without being in motion: for on this assumption it has completed its passage through a without passing through it.
or, how can the essences or forms of things be distinguished from the eternal ideas, or essence itself from the soul? or, how could there have been motion in the chaos when as yet time was not?
'presently', because the time in which he is going to do so is near), and to the part of past time which is not far from the 'now' ('when do you walk?' 'i have just been walking').
others not. for this reason also they ought to have at least referred to the matter in some way or other.
that this 'indivisible' should be cold and that 'indivisible' hot. for, on that supposition, their substance would not even be uniform.
is there, in fact, a distinction in the field of 'coming-to-be' corresponding to the distinction, within the field of 'being', between things that cannot possibly 'not-be' and things that can 'not-be'?
for it is clear that when the elements come together the result is not a chance system and combination, but the very same as before-especially on the view of those who hold this theory, since they say that the contrary is the cause of each state.
the question 'what is it?', recurs in other fields, it might be supposed that there was some single method of inquiry applicable to all objects whose essential nature (as we are endeavouring to ascertain there is for derived properties the single method
this is evidently equivalent to saying that a side of the first is to a side of the second in the reciprocal ratio of the remaining side of the first to the remaining side of the second.
these distinctions drawn, one can gather from surveying the various cases of becoming in the way we are describing that, as we say, there must always be an underlying something, namely that which becomes, and that this, though always one numerically, in
it is no more conceivable, again, that the infinite should exist as a whole of similar parts. for, in the first place, there is no other (straight) movement beyond those mentioned: we must therefore give it one of them.
true that their becoming or perishing is necessarily, like the becoming or perishing of a specific character or form, the result of the alteration of certain other things, e.g.
secondly, like the upward movement of fire, the downward movement of earth and all heavy things makes equal angles on every side with the earth's surface: it must therefore be directed towards the centre.
but if indivisible, then not infinite, except in the sense (one) in which the voice is 'invisible'.
let us then apply the term 'heavy' to that which naturally moves towards the centre, and 'light' to that which moves naturally away from the centre.
dem.-let a : b :: c : d; then a- b c d; and if a be greater than b, a b- is greater than unity; therefore c d is greater than unity, and c is greater than d.
and further, if there is always something of this nature, a movent that is itself unmoved and eternal, then that which is first moved by it must be eternal.
no determinate divisible thing has a single termination, whether it is continuously extended in one or in more than one dimension: but the 'now' is a termination, and it is possible to cut off a determinate time. further, if coincidence in time (i.e.
wherefore they cut the air-channels leading to the lung, and placed the lung about the heart as a soft spring, that, when passion was rife within, the heart, beating against a yielding body, might be cooled and suffer less, and might thus become more
wishes of the mover may select. thus, too, fire and earth move not to infinity but to opposite points; and since the opposition in place is between above and below, these will be the limits of their movement.
three. the tangent to a circle from any external point is a mean proportional between the segments of any secant passing through the same point.
first then we must understand that place would not have been thought of, if there had not been a special kind of motion, namely that with respect to place.
dem.-join ad. if ad be not parallel to bf, let ag be parallel to it. join gf. now since the triangles gef and abc are on equal bases bc, ef, and between the same parallels bf, ag, they are equal xxxviii.
of the infinite forms we must select the most beautiful, if we are to proceed in due order, and any one who can point out a more beautiful form than ours for the construction of these bodies, shall carry off the palm, not as an enemy, but as a friend.
they were a real cereal fruit which i ripened, and they had to my senses a fragrance like that of other noble fruits, which i kept in as long as possible by wrapping them in cloths.
the gliding waters, or was caught in the tempest borne on the air, and the motions produced by any of these impulses were carried through the body to the soul.
three. the parallel to any side of a triangle through the middle point of another bisects the third.
dem.-because aobe is a parallelogram, its opposite sides are equal; therefore ao is equal to eb; but eb is half the side of the given square; therefore ao is equal to half the side of the given square; and so in like manner is each of the lines ob, oc,
six. how is the area of a rectangle found?
they were the measure of all things, and seemed to give law to all things; nature was rescued from chaos and confusion by their power; the notes of music, the motions of the stars, the forms of atoms, the evolution and recurrence of days, months, years,
further, how could a body be carried to its own place, if place was the matter or the form? it is impossible that what has no reference to motion or the distinction of up and down can be place.
i mean, will be heavier than the fire-plane. in general, the result is either that there is no magnitude at all, or that all magnitude could be done away with. for a point is to a line as a line is to a plane and as a plane is to a body.
could not be made so by definition. we have made a similar change in the enunciation of the fifth definition.
but that which is produced or directed by nature can never be anything disorderly: for nature is everywhere the cause of order. moreover, there is no ratio in the relation of the infinite to the infinite, whereas order always means ratio.
or thus: join he, be, hd, bd. the pict hc twice the hbe, and the pict four hundred twice the bde. therefore the hbe bde, and i. xxxix. hd is parallel to be. hence
so, too, in the case of perishing and having perished: perishing must be preceded by having perished, and having perished must be preceded by perishing.
since we have already determined that functions of this kind belong to things which possess, a principle of movement, and that the heaven is animate and possesses a principle of movement, clearly the heaven must also exhibit above and below, right and
other material is transformed into the dominant.
he was a relative and a dear friend of my great-grandfather, dropides, as he himself says in many passages of his poems; and he told the story to critias, my grandfather, who remembered and repeated it to us.
no transformation, however, into any of the 'simple' bodies can result from the passingaway of one elementary quality in each of two 'elements' when they are taken in their consecutive order, because either identical or contrary qualities are left in the
, so that bc and gh may be homologous sides. ghi is the figure required.
now since nothing which is incidental is prior to what is per se, it is clear that no incidental cause can be prior to a cause per se. spontaneity and chance, therefore, are posterior to intelligence and nature.
when it is regular: for in a sense a motion that is irregular is not regarded as one, that title belonging rather to that which is regular, as a straight line is regular, the irregular being as such divisible.
for all wives and children were to be in common, to the intent that no one should ever know his own child, but they were to imagine that they were all one family; those who were within a suitable limit of age were to be brothers and sisters, those who
sol.-with a as centre, and ab as radius, describe the circle bcd (post. iii.). with b as centre, and ba as radius, describe the circle ace, cutting the former circle in one hundred. join ca, cb (post. i.). then abc is the equilateral triangle required.
in like manner, ab : bd :: figure described on ab : figure described on bc. hence five. xxiv. ab : sum of ad and bd :: figure described on the line ab : sum of the figures described on the lines ac, bc; but ab is equal to the sum of ad and bd.
it is, however, manifest that no portion of air whatever moves downward. and the same reasoning applies to lightness, if that is supposed to depend on superiority of quantity of matter.
time: for the extremity of the last period of time that we take must be found in some moment, since time contains no point of contact for us except the moment.
operative in things which come to be and are by nature.
one shall not attempt to connect the physiological speculations of plato either with ancient or modern medicine. what light i can throw upon them will be derived from the comparison of them with his general system.
some people even question whether they are real or not. they say that nothing happens by chance, but that everything which we ascribe to chance or spontaneity has some definite cause, e.g.
six definition of tragedy. seven the plot must be a whole. eight the plot must be a unity. nine (plot continued.) dramatic unity. ten (plot continued.) definitions of simple and complex plots.
the third solid is a regular icosahedron, having twenty triangular equilateral bases, and therefore one hundred twenty rectangular scalene triangles.
the reason for it is that they have a natural tendency respectively towards a certain position: and this constitutes the essence of lightness and heaviness, the former being determined by an upward, the latter by a downward, tendency.
and the size (in growth) do not remain the same.
twenty three. if the sides of a triangle be expressed by m n , m n , and mn linear units, respectively; prove that it is right-angled.
cor. two.-if the rectangle contained by the segments of one of two intersecting lines be equal to the rectangle contained by the segments of the other, the four extremities are concyclic.
reject fb , which is common, and we have the rectangle ab.be bg ; but since be is equal to bc, the rectangle ab.be is equal to the figure ac. therefore bg is equal to the figure ac, and therefore equal to the given rectilineal figure (ten).
and this tendency in bodies to condense or escape is a source of motion...where there is motion there must be a mover, and where there is a mover there must be something to move.
as for the sinews, he made them of a mixture of bone and unfermented flesh, attempered so as to be in a mean, and gave them a yellow colour; wherefore the sinews have a firmer and more glutinous nature than flesh, but a softer and moister nature than the
that, or something like it, is what all men in every case mean by element. now if what we have described is an element, clearly there must be such bodies.
to do justice to the subject, we should consider the physical philosophy of the ancients as a whole; we should remember, (one) that the nebular theory was the received belief of several of the early physicists; (two) that the development of animals out of
the same is true of any other pair of contraries; for love does not gather strife together and make things out of it, nor does strife make anything out of love, but both act on a third thing different from both.
from sensations common to the whole body, we proceed to those of particular parts. the affections of the tongue appear to be caused by contraction and dilation, but they have more of roughness or smoothness than is found in other affections.
only two equal lines can be drawn from p to the circumference; but three equal lines are drawn (hyp.), which is absurd. hence p must be the centre.
the human soul differs from the soul of the world in this respect, that it is enveloped and finds its expression in matter, whereas the soul of the world is not only enveloped or diffused in matter, but is the element in which matter moves.
dem.-since b is less than a, the quotient which is the result of dividing any magnitude by b is greater than the quotient which is got by dividing the same magnitude by a;
can also occur in fact. is this, then, the movement that love sets going?
it is evident, from the preceding development and discussion of difficulties, that growth is not a change out of something which, though potentially a magnitude, actually possesses no magnitude.
cor. one.-if through a point k within a parallelogram abcd lines drawn parallel to the sides make the parallelograms dk, kb equal, k is a point in the diagonal ac.
dem.-if bd be not the continuation of cb, let be be its continuation.
if the line happened to be white or black: for unless two things either are, or are composed of, 'contraries', neither drives the other out of its natural condition.
the body to the soul and have the name of sensations.
def. ii.-a parallelogram is a quadrilateral, both pairs of whose opposite sides are parallel.
but the elements of sensible bodies are bodies, while nothing that has size results from a combination of intelligible elements.
in a few years the human mind was peopled with abstractions; a new world was called into existence to give law and order to the old. but between them there was still a gulf, and no one could pass from the one to the other.
divine part of us. such was the origin of legs and hands, which for this reason were attached to every man; and the gods, deeming the front part of man to be more honourable and more fit to command than the hinder part, made us to move mostly in a forward
reflecting on the simplest kinds of external objects, which to the ancients were the four elements, he was led to a more general notion of a substance, more or less like themselves, out of which they were fashioned.
consistently expresses himself. while the determinations of human thought are in process of creation he is necessarily tentative and uncertain.
one. if ao be a triangle, having the side a greater than ao; then if we cut off ab ao, draw bb parallel to ao, cut off bc bb, c., the series of lines ab, bc, four hundred, c., are in continual proportion.
then since p is the centre of ten, the three lines pa, pb, pc are equal to one another.
sponge, and being kept cool by drink and air which enters through the trachea.
the body which is diseased from the effects of fire is in a continual fever; when air is the agent, the fever is quotidian; when water, the fever intermits a day; when earth, which is the most sluggish element, the fever intermits three days and is with
but if the movement is retarded it must necessarily be retarded for an infinite time. equally impossible is perpetual acceleration or perpetual retardation.
reason, and not enthusiasm, is the true guide of man; he is only inspired when he is demented by some distemper or possession. the ancient saying, that 'only a man in his senses can judge of his own actions,' is approved by modern philosophy too.
reasonable, therefore, not to say necessary, to suppose the existence of the third term also, that which causes motion but is itself unmoved.
ten. given the three angles of a trihedral angle, find, by a plane construction, the angles between the containing planes.
the reason is that in these motions the starting-point and the termination do not coincide, whereas in motion over a circle they do coincide, and so this is the only perfect motion.
itself'-a point which has no parts or magnitude, which is nowhere, and nothing. this cannot be the archetype according to which god made the world, and is in reality, whether in plato or in kant, a mere negative residuum of human thought.
two. the demonstration of this case is included in the foregoing.
it had not that sort of consistency to plato which has been given to it in modern times by geometry and metaphysics. neither of the greek words by which it is described are so purely abstract as the english word 'space' or the latin 'spatium.'
if therefore every perceptible body possesses the power of acting or of being acted upon, or both of these, it is impossible that an infinite body should be perceptible. all bodies, however, that occupy place are perceptible.
so, as we said, the difficulties which constrain people to deny the existence of some of the things we mentioned are now solved.
'you are welcome to hear them,' said the priest, 'both for your own sake and for that of the city, and above all for the sake of the goddess who is the common foundress of both our cities.
eldest of gods that are in the interior of heaven'. there is an unfortunate doubt in this passage (one) about the meaning of the word (greek), which is translated either 'compacted' or 'revolving,' and is equally capable of both explanations.
so that either the earth will not stay still at the centre, or if it does, it will be at rest without having its centre at the place to which it is still its nature to move. such is the difficulty.
for the infinite is in the category of quantity, whereas substance or quality or affection cannot be infinite except through a concomitant attribute, that is, if at the same time they are also quantities.
dem.-through any point g in gh draw a line bc intersecting ab, four hundred, and so as to be bisected in g; and join any point f in ef to b, g, one hundred. then, because ef is perpendicular to the lines eb, ec, we have
two. the same construction being made: because oc is equal to od, the angle ocd is equal to odc i. v. , and ceo is equal to deo (hyp.), because each is right.
'that for the sake of which' has two senses: the distinction is made in our work on philosophy.) the arts, therefore, which govern the matter and have knowledge are two, namely the art which uses the product and the art which directs the production of it.
but though this is so in the case of any individual thing that becomes, nevertheless before anything becomes, something else must be in motion, not itself becoming but being, and before this there must again be something else.
the parts of the body or the nail in the ship, or (b) something which is not in itself capable of being moved, but is always moved through its conjunction with something else, as 'whiteness' or 'science'.
these shapes, they thought, were the most mobile because they offer the fewest points of contact and are the least stable of any; they were also the most apt to produce warmth and combustion, because the one is angular throughout while the other has the
and if any one affirms that in which these two are found to be other than the soul, he will say the very opposite of the truth.
the preceding consideration of the various cases serves to show us whether it is or is not possible that there should be an infinite sensible body. the following arguments give a general demonstration that it is not possible.
but since in any kind the one is naturally prior to the many and the simple to the complex, the circle will be the first of plane figures.
must itself be spherical, then on these grounds also it is clear that the heavens are spherical.
the current view places what is of necessity in the process of production, just as if one were to suppose that the wall of a house necessarily comes to be because what is heavy is naturally carried downwards and what is light to the top, wherefore the
for in its later state it will possess the capacity of not existing, only not of not existing at a time when it exists-since then it exists in actuality-but of not existing last year or in the past.
locomotion must be primary. for there is no necessity for the subject of locomotion to be the subject either of increase or of alteration, nor need it become or perish: on the other hand there cannot be any one of these processes without the existence of
thus the finite and infinite effect the same alteration in equal times. but this is impossible; for the assumption is that the greater effects it in a shorter time.
hence the point of intersection of these bisectors, that is, the point p, must be the centre.
substance. nor are there contained within it those 'hard' (i.e. congealed) particles 'indivisible in their bulk': on the contrary, it is liquid-and again, solid and congealed-uniformly all through.
three. if on the four sides of a square, or on the sides produced, points be taken equidistant from the four angles, they will be the angular points of another square, and similarly for a regular pentagon, hexagon, c.
one. if two non-concentric circles intersect in one point, they must intersect in another point. for, let o, o be the centres, a the point of intersection; from a let fall the ac on the line oo.
the case of contradictory change. it is clear, then, that that which has changed, at the moment when it has first changed, is in that to which it has changed.
this being so, it follows that there cannot be more worlds than one.
for the nature which comes into being and grows up in us by a law of necessity, does not admit of the combination of solid bone and much flesh with acute perceptions.
for this last sphere moves with many others, to which it is fixed, each sphere being actually a body; so that its movement will be a joint product.
when more is taken away than flows in, then we decay; and when less, we grow and increase.
in the timaeus, as well as in the laws, he also regards vices and crimes as simply involuntary; they are diseases analogous to the diseases of the body, and arising out of the same causes.
four. if e be the second point of intersection of the circle acd with bde, de is equal to db; and if ae, be, ce, de be joined, each of the triangles ace, ade is congruent with abd.
and the whey or sediment of newly-formed phlegm is sweat and tears, and includes the various daily discharges by which the body is purified.
cor. two.-if two angles of a triangle be unequal, the lesser must be acute.
for if these names are used equivocally, if the identity of name does not rest upon an identity of form in these elements and ours, then the whole to which they belong can only be called a world by equivocation.
(one) place is what contains that of which it is the place.
and further, when a thing is increased or decreased its magnitude changes in respect of place.
on the whole this little tract faithfully reflects the meaning and spirit of the timaeus.
thus bodily excellences such as health and a good state of body we regard as consisting in a blending of hot and cold elements within the body in due proportion, in relation either to one another or to the surrounding atmosphere: and in like manner we
not be the same as the orbit as described from b or g or any other point on the same circumference except in an accidental sense, the sense that is to say in which a musical man is the same as a man.
hence the sum of the angles abg, bgh is two right angles i. xxix ; but bgh is right (const.); therefore abg is right. in like manner cbg is right. hence bg is normal to the plane ac eleven. def. vi. , and it is normal to df (const.).
into one another in regular order, while the infinite complexity of the human frame remains unobserved.
belief in the existence of the infinite comes mainly from five considerations:
there is therefore no infinite body beyond the heaven. nor again is there anything of limited extent beyond it. and so beyond the heaven there is no body at all.
besides, nobody supposes a single 'element' to persist, as the basis of all, in such a way that it is water as well as air (or any other 'element') at the same time. so there will be a certain contrariety, i.e.
they are of two kinds, some of them, like glass, having more earth, others, like wax, having more water in them.
dem.-since a : b :: c : d, a : c :: b : d alternando ; therefore a : a c :: b : b d e .; but a is greater than b (hyp.), therefore a c is greater than b d xiv. . hence a d is greater than b c.
and wherever a pair of contraries admit of an intermediate, motions to that intermediate must be held to be in a sense motions to one or other of the contraries: for the intermediate serves as a contrary for the purposes of the motion, in whichever
the diversity of the sutures was caused by the power of the courses of the soul and of the food, and the more these struggled against one another the more numerous they became, and fewer if the struggle were less violent.
two. let ac : ce :: gc : cb, to prove the parallelograms ab, four hundred are equal.
the two events will not be simultaneous, whereas the truth is that a is at b at a sectional point of time and does not occupy time there. in this case, therefore, where the motion of a thing is continuous, it is impossible to use this form of expression.
will be completed and conversely twice as much alteration will occupy twice as much time: and the alteration of half of its object will occupy half as much time and in half as much time half of the object will be altered: or again, in the same amount of
between one, three, nine, twenty seven), together with the mean terms and connecting links which are expressed by the ratios of three:two, and four:three, and of nine:eight-these, although they cannot be wholly undone except by him who united them, were
two. to prove the theorem contained in def. xii.
it makes no difference whether the change be qualified or unqualified, except that in the former case the contraries will have to be contrary to one another only in a qualified sense: and a thing's possessing a quality in a greater or in a lesser degree
but it is not the fulfilment of bronze as bronze which is motion. for 'to be bronze' and 'to be a certain potentiality' are not the same.
prop. thirteen.-problem. to inscribe a circle in a regular pentagon (abcde).
now all compounds presuppose in their coming-to-be constituents which are contrary to one another: and in all compounds there is contained one set of the contrasted extremes.
but the abstract is a vacant form to us until brought into relation with man and nature. god and the world are mere names, like the being of the eleatics, unless some human qualities are added on to them.
must extend to all alike. for all are equally undifferentiated in form, but any one is different numerically from any other.
and there you dwelt as became the children of the gods, excelling all men in virtue, and many famous actions are recorded of you. the most famous of them all was the overthrow of the island of atlantis.
it is doubtful whether the discourse on political economy, first printed in the great encyclopdia in one thousand seven hundred fifty five, was composed before or after the discourse on inequality .
sensible objects: for all moral excellence is concerned with bodily pleasures and pains, which again depend either upon acting or upon remembering or upon anticipating. now those that depend upon action are determined by sense-perception, i.e.
so much, then, for the discussion of the void, and of the sense in which it exists and the sense in which it does not exist.
the answer is, that the circular motion of all things compresses them, and as 'nature abhors a vacuum,' the finer and more subtle particles of the lighter elements, such as fire and air, are thrust into the interstices of the larger, each of them
for to imagine that the archetype was created would be blasphemy, seeing that the world is the noblest of creations, and god is the best of causes.
further, things are now thought to move into the void because it yields; but in a void this quality is present equally everywhere, so that things should move in all directions.
() one is to inscribe a triangle in a circle.
one. abcd is a semicircle whose diameter is ad; the chord bc produced meets ad produced in e: prove that if ce is equal to the radius, the arc ab is equal to three times four hundred.
thus our original design of discoursing about the universe down to the creation of man is nearly completed.
but since the place of a thing is the boundary of that which contains it, and the continent of all things that move upward or downward is the extremity and the centre, and this boundary comes to be, in a sense, the form of that which is contained, it is
eighteen. if two triangles have equal areas, prove that their perpendiculars are reciprocally proportional to the bases.
the race of wild pedestrian animals, again, came from those who had no philosophy in any of their thoughts, and never considered at all about the nature of the heavens, because they had ceased to use the courses of the head, but followed the guidance of
hence we see that what euclid calls triplicate ratio of two magnitudes is the ratio of their cubes, or the cube of their ratio.
for the primary movements being finite in number, the kinds of simple body are necessarily also finite, since the movement of a simple body is simple, and the simple movements are finite, and every natural body must always have its proper motion.
in general, the attempt to give a shape to each of the simple bodies is unsound, for the reason, first, that they will not succeed in filling the whole.
that which moves itself, therefore, must comprise something that imparts motion but is unmoved and something that is moved but does not necessarily move anything else: and each of these two things, or at any rate one of them, must be in contact with the
for let b be water and five hundred air; then by so much as air is thinner and more incorporeal than water, a will move through five hundred faster than through b. let the speed have the same ratio to the speed, then, that air has to water.
great men make, indeed, individual contributions to the knowledge of their times; but they can never transcend the age in which they live.
for while of all shapes the sphere is the most convenient for movement in one place, making possible, as it does, the swiftest and most selfcontained motion, for forward movement it is the most unsuitable, least of all resembling shapes which are
be truly said to act, equally with them.
and in speaking of the copy and the original we may assume that words are akin to the matter which they describe; when they relate to the lasting and permanent and intelligible, they ought to be lasting and unalterable, and, as far as their nature allows,
nineteen. the square on the external diagonal of a cyclic quadrilateral is equal to the sum of the squares on the tangents from its extremities to the circumscribed circle.
some another, that which receded most slowly from the sphere of the same, which was the swiftest, appeared to follow it most nearly.
three. a right line intersecting parallel planes makes equal angles with them.
fourteen. if a, b, one hundred denote the angles of a , prove that twelve(a b), twelve(b one hundred), twelve(one hundred a) will be the angles of a formed by any side and the bisectors of the external angles between that side and the other sides produced.
similarly if a man's crop is spoiled on the threshing-floor, the rain did not fall for the sake of this-in order that the crop might be spoiled-but that result just followed. why then should it not be the same with the parts in nature, e.g.
certainly the early physicists found no place for chance among the causes which they recognized-love, strife, mind, fire, or the like.
to be in motion, and of this again the same may be said.
for there occurs at long intervals a derangement of the heavenly bodies, and then the earth is destroyed by fire.
and yet vision is prior to touch, so that its object also is prior to the object of touch.
one hundred twenty two. if the external diagonal of a quadrilateral inscribed in a given circle be a chord of another given circle, the locus of its middle point is a circle.
(two) the extension between the extremities is thought to be something, because what is contained and separate may often be changed while the container remains the same (as water may be poured from a vessel)-the assumption being that the extension is
he was not 'a jealous god,' and therefore he desired that all other things should be equally good. he is the idea of good who has now become a person, and speaks and is spoken of as god. yet his personality seems to appear only in the act of creation.
it is plain, too, that the infinite cannot be an actual thing and a substance and principle.
three. since agh is equal to ghd (one), add hgb to each, and we have the sum of the angles agh, hgb equal to the sum of the angles ghd, hgb; but the sum of the angles agh, hgb xiii.
and that we, learning them and partaking of the natural truth of reason, might imitate the absolutely unerring courses of god and regulate our own vagaries.' or let us weigh carefully some other profound thoughts, such as the following.
dem.-because a is the centre of the circle edf, ae is equal to ad (def. xxxii.), and one hundred is equal to ad (const.); and things which are equal to the same are equal to one another (axiom i.); therefore ae is equal to one hundred.
now cause, in the sense of material origin, for the things which are such as to come-to-be is 'that which can be-and-not-be': and this is identical with'that which can come-to-be-and-pass-away', since the latter, while it is at one time, at another time
nor again can there be a primary time at which the being at rest of that which is at rest occurred: for it cannot have occurred in that which has no parts, because there cannot be motion in that which is indivisible, and that in which rest takes place is
now as in the ideal animal the mind perceives ideas or species of a certain nature and number, he thought that this created animal ought to have species of a like nature and number.
and secondly, the fact that the heavens do not break in pieces follows not only from this but also from the proof already given of the continuity of the whole.
that being so, it is evident that the point also must first traverse a space equal to or less than itself.
prop. b.-theorem (simson). if two ratios are equal their reciprocals are equal (invertendo).
let a : b :: c : d, then b : a :: d : c.
(b) in another sense, we use this name for the body continuous with the extreme circumference which contains the moon, the sun, and some of the stars; these we say are 'in the heaven'.
and if so, we shall have to admit either infinite weight or infinite lightness. nor, secondly, could the body whose movement is circular be infinite, since it is impossible for the infinite to move in a circle.
they, however, had no idea of any form of being other than the substance of things perceived; and when they saw, what no one previously had seen, that there could be no knowledge or wisdom without some such unchanging entities, they naturally transferred
again, in the matter of locomotion, it would be a strange thing if a stone could be falling or resting on the ground without our being able to perceive the fact.
all this order and arrangement the goddess first imparted to you when establishing your city; and she chose the spot of earth in which you were born, because she saw that the happy temperament of the seasons in that land would produce the wisest of men.
again (four) in the sense of end or 'that for the sake of which' a thing is done, e.g. health is the cause of walking about. ('why is he walking about?' we say.
we are searching into things which are upon the utmost limit of human intelligence, and then of a sudden we fall rather heavily to the earth. there are no intermediate steps which lead from one to the other.
again, if every element has its proper movement, and a simple body has a simple movement, and the number of simple movements is not infinite, because the simple motions are only two and the number of places is not infinite, on these grounds also we should
so if this is being at rest it is impossible for that which is changing to be as a whole, at the time when it is primarily changing, over against any particular thing (for the whole period of time is divisible), so that in one part of it after another it
considering that if heat and cold and other powerful forces which unite bodies surround and attack them from without when they are unprepared, they decompose them, and by bringing diseases and old age upon them, make them waste away-for this cause and on
if then colours and states had entered into the mixture, and if separation took place, there would be a 'white' or a 'healthy' which was nothing but white or healthy, i.e. was not the predicate of a subject.
for when the motions of the antecedent swifter sounds begin to pause and the two are equalized, the slower sounds overtake the swifter and then propel them.
abc : a b one hundred ::ab.bc.ca : ab .bc .ca .
def. viii.-two planes which meet are perpendicular to each other, when the right lines drawn in one of them perpendicular to their common section are normals to the other.
one , and these are evidently the perpendiculars from the vertices on the opposite sides of the triangle abc (compare ex. sixteen, book one.).
for a time that is both equal and simultaneous is one and the same time, and even those that are not simultaneous are one in kind; for if there were dogs, and horses, and seven of each, it would be the same number.
let us begin, then, with the first sense.
twenty. find a point o in the plane of a triangle abc, such that the diameters of the three circles, about the triangles oab, obc, oca, may be in the ratios of three given lines.
three. any line ae, through the middle point b of the base dd of a triangle dcd, is cut harmonically by the sides of the triangle and a parallel to the base through the vertex.
into equal and similar parts the whole circle in which it is inscribed.
for the parts of the globe change their places while the whole continues to occupy an equal place: but the parts of the rowing thing expand over an ever-increasing place and the parts of the diminishing thing contract within an ever-diminishing area.
this, i say, will be infinite: first, because in the case of finite radii it is always finite; and secondly, because in it one can always go on to a width greater than any given width; thus the reasoning which forces us to believe in infinite number,
moreover, no man of science is bound to solve every kind of difficulty that may be raised, but only as many as are drawn falsely from the principles of the science: it is not our business to refute those that do not arise in this way: just as it is the
that nature exists, it would be absurd to try to prove; for it is obvious that there are many things of this kind, and to prove what is obvious by what is not is the mark of a man who is unable to distinguish what is self-evident from what is not.
proceeding by a method of superficial observation, plato remarks that the more sensitive parts of the human frame are those which are least covered by flesh, as is the case with the head and the elbows.
a quality or a point-it is not necessary perhaps that such things should be put under either head.
quality will be called change to the contrary of that quality, and change to a greater degree of a quality will be regarded as change from the contrary of that quality to the quality itself.
dem.-if not from b to five hundred, draw in the plane ab the right line bed, and in the plane bc the right line bfd. then the right lines bed, bfd enclose a space, which i., axiom x. is impossible.
otherwise a thing may be non-existent at the moment when it has become and existent at the moment when it has perished: or else it must be possible for a thing at the same time to be white and not white and in fact to be existent and non-existent.
v. a segment of a circle is a figure bounded by a chord and one of the arcs into which it divides the circumference.
of the kinds of earth, that which is filtered through water passes into stone; the water is broken up by the earth and escapes in the form of air-this in turn presses upon the mass of earth, and the earth, compressed into an indissoluble union with the
the black and the moist (for these have not yet been coupled).
let a, b be two magnitudes, then the ratio a : b is equal to the ratio ma : mb.
dem.-on four hundred describe the square cdfe i. xlvi. , and join de; through b draw bhg parallel to ce i. xxxi. , meeting de in h; through h draw klm parallel to ad; and through a draw ak parallel to one hundred fifty.
again, (b) change is always faster or slower, whereas time is not: for 'fast' and 'slow' are defined by time-'fast' is what moves much in a short time, 'slow' what moves little in a long time; but time is not defined by time, by being either a certain
this may serve as a statement of the difficulties about the attributes of time.
for in the first place the parts do not occupy the same position for any period of time: and in the second place the whole also is always changing to a different position: for if we take the orbit as described from a point a on a circumference, it will
he does not take words in their simple meaning or sentences in their natural connexion. he is thinking, not of the context in plato, but of the contemporary pythagorean philosophers and their wordy strife.
it will be true to say now that this exists last year or in the past generally.
therefore the sum of abc, cda is two right angles.
now, that any and every part of the tissue qua form should grow-and grow by the accession of something-is possible, but not that any and every part of the tissue qua matter should do so.
there is a difficulty in arriving at an exact notion of this third kind, because the four elements themselves are of inexact natures and easily pass into one another, and are too transient to be detained by any one name; wherefore we are compelled to
so one kind of 'now' is described in this way: another is when the time is near this kind of 'now'. 'he will come now' because he will come to-day; 'he has come now' because he came to-day.
it exceeds zero by zero. for this reason, too, a line does not exceed a point unless it is composed of points!
again, how will they explain, in the case of what is heavy, its movement downwards? and it is plain that if the rarer and more void a thing is the quicker it will move upwards, if it were completely void it would move with a maximum speed!
the courses of the same and the other affect both; they are made of the same elements and therefore in the same proportions. both are intelligent natures endued with the power of self-motion, and the same equipoise is maintained in both.
for the natural order is that the flesh and sinews should be made of blood, the sinews out of the fibres to which they are akin, and the flesh out of the clots which are formed when the fibres are separated.
chance subject in any chance direction, nor, similarly, is the relation of the mover to its object fortuitous: the thing altered is different from the thing increased, and precisely the same difference holds between that which produces alteration and that
hence it is evident that the damp will derive from the moist, and 'the dry' which is opposed to it will derive from the primary dry. again (ii) the 'moist' and the solidified derive in the same way from the primary pair.
which causes, and another which exhibits the latter motion. again, what is it that moves the infinite? if it moves itself, it must be animate. but how can it possibly be conceived as an infinite animal?
for in so far as that which accedes is potentially 'so much-flesh' it tends to increase flesh: whereas, in so far as it is potentially 'flesh' only, it is nourishment.
since, then, it has changed in a period of time, and all time is divisible, in half the time it will have completed another change, in a quarter another, and so on to infinity: consequently when it has changed, it must have previously been changing.
there is a third class of diseases which may be conceived of as arising in three ways; for they are produced sometimes by wind, and sometimes by phlegm, and sometimes by bile.
and this hope was nursed by ancient tradition, which had found expression from time to time in the celebrated lines of seneca and in many other places.
psychology. for the sciences were not yet divided, and there was nothing really irrational in arguing that the same laws which regulated the heavenly bodies were partially applied to the erring limbs or brain of man.
; hence the whole angle cbd is equal to the sum of the two angles acb, bac: to each of these add the angle abc and we have the sum of cbd, abc equal to the sum of the three angles acb, bac, abc: but the sum of cbd, abc is two right angles xiii.
magnitudes in continued proportion are also said to be in geometrical progression.
join ob, of, and produce ao, bo, fo to meet the given circle again in the points five hundred, e, one hundred. join ab, bc, four hundred, de, ef, fa; abcdef is the required hexagon.
our senses enable us to perceive very few of the attributes of the heavenly bodies. but let not that deter us. the reason must be sought in the following facts. everything which has a function exists for its function.
logic and rhetoric may usurp the place of reason and truth, how all philosophies grow faded and discoloured, and are patched and made up again like worn-out garments, and retain only a second-hand existence.
for we often separate earthy natures, and sometimes earth itself, and draw them into the uncongenial element of air by force and contrary to nature, both clinging to their kindred elements.
the plan of the first discourse is essentially simple: it sets out from the badness, immorality and misery of modern nations, traces all these ills to the departure from a "natural" state, and then credits the progress of the arts and sciences with being
in the course of our argument directed to this end we established the fact that everything that is in motion is moved by something, and that the movent is either unmoved or in motion, and that, if it is in motion, it is moved either by itself or by
we see that bodies which have a greater impulse either of weight or of lightness, if they are alike in other respects, move faster over an equal space, and in the ratio which their magnitudes bear to each other.
but the difference would seem to be one of degree. in every kind of motion we may have regularity or irregularity: thus there may be regular alteration, and locomotion in a regular path, e.g.
(five) the meaning of the words 'artificer of day and night' is literally true according to plato's view.
to feel habitually that he is part of the order of the universe, is one of the highest ethical motives of which man is capable. something like this is what plato means when he speaks of the soul 'moving about the same in unchanging thought of the same.'
now it has been proved before that they must undergo reciprocal transformation.
on the contrary, both of the constituents that are combined in the compound must originally have existed in separation: but no property can have separate existence.)
obstructing respiration, drives them to extremity, causing all varieties of disease, until at length the desire and love of the man and the woman, bringing them together and as it were plucking the fruit from the tree, sow in the womb, as in a field,
five. if two figures be directly similar, and have a pair of homologous sides parallel, every pair of homologous sides will be parallel.
for it was this reason which also caused some of the earlier thinkers to turn so far aside from the road which leads to coming to be and passing away and change generally.
for in a process of change we may distinguish three terms-that which changes, that in which it changes, and the actual subject of change, e.g. the man, the time, and the fair complexion.
one. what is the subject-matter of book two.? ans. theory of rectangles.
nor (seven) is it without reason that each should remain naturally in its proper place.
another simplification of the proof would be got by considering that the point a is such that one of the s cag, bak can be turned round it in its own plane until it coincides with the other; and hence that they are congruent.
air, again, when inflamed, becomes fire; and again fire, when condensed and extinguished, passes once more into the form of air; and once more, air, when collected and condensed, produces cloud and mist; and from these, when still more compressed, comes
therefore ap.bp ob ot pt ; but ob ot . hence the rectangle ap.bp pt .
zeno's arguments about motion, which cause so much disquietude to those who try to solve the problems that they present, are four in number.
that is why the upper part is moved in a circle, while the all is not anywhere. for what is somewhere is itself something, and there must be alongside it some other thing wherein it is and which contains it.
again, since the terms 'quicker' and 'slower' are used only of that which occupies a period of time, and the process of coming to a stand may be quicker or slower, the same conclusion follows.
and what was the tale about, critias? said amynander.
the maximum may occur either at the starting-point or at the goal or between the two; and we expect natural motion to reach its maximum at the goal, unnatural motion at the starting-point, and missiles midway between the two.
but if this is so it is evident that the present is also indivisible: for if it is divisible it will be involved in the same implications as before.
they borrowed portions of fire, and earth, and water, and air from the world, which were hereafter to be restored-these they took and welded them together, not with the indissoluble chains by which they were themselves bound, but with little pegs too
even the fetichism of the savage is the beginning of reasoning; the assumption of the most fanciful of causes indicates a higher mental state than the absence of all enquiry about them.
fifty eight. if lines be drawn from any point p to the angles of a triangle, the perpendiculars at p to these lines meet the opposite sides of the triangle in three collinear points.
nature has been defined as a 'principle of motion and change', and it is the subject of our inquiry. we must therefore see that we understand the meaning of 'motion'; for if it were unknown, the meaning of 'nature' too would be unknown.
even these things, however, can in a way be affected by chance, when one who is dealing with them does something to them by chance, but not otherwise.
it is probable that among the pythagoreans living in the fourth century b.one hundred., there were already some who, like plato, made the earth their centre. whether he obtained his circles of the same and other from any previous thinker is uncertain.
for in proportion as we see division going on, in the same proportion we see addition being made to what is already marked off.
what, then, is the cause of this proportional consilience? presumably not fire or earth. but neither is it love and strife: for the former is a cause of 'association' only, and the latter only of 'dissociation'.
oc oe ec i. xlvii. , and od oe ed ; but oc od ; oe ec oe ed . hence ec ed , and ec ed. observation.-the three theorems, namely, cor. one., prop. i., and parts one, two, of prop.
if one of the bodies falls in any degree short of the other in potency-suppose fire is finite in amount while air is infinite and a given quantity of fire exceeds in power the same amount of air in any ratio provided it is numerically definite-the
by chance. for all these statements are correct, because they are well grounded. things do, in a way, occur by chance, for they occur incidentally and chance is an incidental cause.
i 'imitation' the common principle of the arts of poetry. two the objects of imitation. three the manner of imitation. four the origin and development of poetry. five definition of the ludicrous, and a brief sketch of the rise of comedy.
for either they have but one attainable good (as indeed man has), or, if several, each contributes directly to their ultimate good.
an equilateral triangle (def. xxi.); therefore viii. the angle dce is equal to the angle ecf, and they are adjacent angles. therefore (def. xiii.) each of them is a right angle, and cf is perpendicular to ab at the point one hundred.
the course of natural phenomena would have passed unheeded before his eyes, like fair sights or musical sounds before the eyes and ears of an animal.
three. describe a circle passing through a given point, having its centre on a given line and touching a given circle.
instead of comparing their powers by the measure of their amount, they might be compared as terms in a 'correspondence': e.g. 'as x is hot, so correspondingly y is white'.
indivisibles, and that of the slower into the two indivisibles ez, zh. then the time may also be divided into three indivisibles, for an equal magnitude will be passed over in an equal time. suppose then that it is thus divided into kl, lm, mn.
thus, then, as the several elements never present themselves in the same form, how can any one have the assurance to assert positively that any of them, whatever it may be, is one thing rather than another? no one can.
the discussion of these difficulties, however, is a task appropriate to a different investigation:' let us return to the 'elements' of which bodies are composed.
therefore the lines ec, cb, be are coplanar; but ab and four hundred are two of these lines. hence ab and four hundred are coplanar.
as given in those works, the four figures should be similar.
thirdly, that which is in process of becoming appears universally as something imperfect and proceeding to a first principle: and so what is posterior in the order of becoming is prior in the order of nature.
the intellectuals of the eighteenth century therefore generalised to their hearts' content, and as a rule suffered little for their lese-majeste : voltaire is the typical example of such generalisation.
this is evident; for when air is produced from water, the same matter has become something different, not by acquiring an addition to it, but has become actually what it was potentially, and, again, water is produced from air in the same way, the change
thus what is cold is potentially hot: then a change takes place and it is fire, and it burns, unless something prevents and hinders it. so, too, with heavy and light: light is heavy, e.g.
who are destitute of letters and education; and so you have to begin all over again like children, and know nothing of what happened in ancient times, either among us or among yourselves.
inanimate things when an animate thing moves them. it is the fact that these things also always derive their motion from something: what it is would become evident if we were to distinguish the different kinds of cause.
cor. one.-the maximum parallelogram exceeds any other parallelogram about the same angle in the triangle, by the area of the similar parallelogram whose diagonal is the line between the middle point p of the opposite side and the point five hundred, which
(it is true, indeed, that the account he gives there of the 'participant' is different from what he says in his so-called 'unwritten teaching'.
for things both come into being in time and pass away, and grow, and are altered in time, and are moved locally; thus it is of each movement qua movement that time is the number.
if a line (ab) be bisected (at one hundred), and divided externally in any point (five hundred), the rectangle (ad.bd) contained by the segments made by the external point, together with the square on half the line, is equal to the square on the segment
cor. four.-if two chords intersect at right angles, the sum of the opposite arcs which they intercept on the circle is a semicircle.
there is an alloy of earth mingled with it, which, when the two parts grow old and are disunited, shows itself separately and is called rust.
a thing is 'in succession' when it is after the beginning in position or in form or in some other respect in which it is definitely so regarded, and when further there is nothing of the same kind as itself between it and that to which it is in succession,
hence the rectangle ac.cb is equal to the figure ad, and the square on cb is the figure ce. hence the rectangle ac.cb, together with the square on cb, is equal to the figure ae.
again, if by taking the extreme moment of the time-for it is the moment that defines the time, and time is that which is intermediate between moments-we are enabled to say that motion has taken place in the whole time chrh or in fact in any period of it,
of the quality); nor can we get any portion of a flame, in which both heat and whiteness are not present. so too, then, is the earlier heat related to the later.
fourteen. if both bases and altitudes differ, how do the areas vary?
for such time has no limit, while the action and reaction have.
fifty five. if two circles touch in one hundred, and if five hundred be any point outside the circles at which their radii through one hundred subtend equal angles, if de, df be tangent from five hundred, de.df dc .
hence this infinite is potential, never actual: the number of parts that can be taken always surpasses any assigned number.
whenever a particle is displaced, the rest push and thrust one another until equality is restored.
prop. forty five.-problem. to construct a parallelogram equal to a given rectilineal figure (abcd), and having an angle equal to a given rectilineal angle (ten).
seventy five. if a, b, c, d denote the sides of a cyclic quadrilateral, and s its semiperimeter, prove its area (s-a)(s-b)(s c)(s- d).
one hundred sixteen. four points a, b, one hundred, five hundred are collinear; find a point p at which the segments ab, bc, four hundred subtend equal angles.
seven. if we join af, bf we find, as before, the rectangle af.fb equal to the rectangle contained by the radius and fi-that is, equal to the rectangle contained by the radius and the sum of cf and bg.
so some say that the void is the matter of the body (they identify the place, too, with this), and in this they speak incorrectly; for the matter is not separable from the things, but they are inquiring about the void as about something separable.
as the word has several senses, it follows that there are several causes of the same thing not merely in virtue of a concomitant attribute), e.g. both the art of the sculptor and the bronze are causes of the statue.
again it is absurd to look for a reason why the earth remains at the centre and not for a reason why fire remains at the extremity. if the extremity is the natural place of fire, clearly earth must also have a natural place.
as in the charmides he tells us that the body cannot be cured without the soul, so in the timaeus he strongly asserts the sympathy of soul and body; any defect of either is the occasion of the greatest discord and disproportion in the other.
rough, then only harsh. those of them which are of an abstergent nature, and purge the whole surface of the tongue, if they do it in excess, and so encroach as to consume some part of the flesh itself, like potash and soda, are all termed bitter.
hence, whenever there is action and passion between two things, that which underlies them must be a single something.
the same kind of argument holds good against parmenides also, besides any that may apply specially to his view: the answer to him being that 'this is not true' and 'that does not follow'.
sixteen. find the area of a triangle-(one) in terms of its medians; (two) in terms of its perpendiculars.
single thing, and therefore, since it cannot be moved throughout its motion by the same movent, the question is, what moves it?
two. prove that the line joining the point a to the intersection of the lines cf and bg is an axis of symmetry of the figure.
sol.-bisect the side ac opposite to the angle b, at p : through p draw pe, pf parallel to the other sides of the triangle. bp is the parallelogram required.
occupies contrary positions or assumes contrary forms at different times it will produce contrary motions in each several thing that it moves and will cause it to be at one time at rest and at another time in motion.
the soundness of this definition is evident both when we consider the accounts of motion that the others have given, and also from the difficulty of defining it otherwise.
but those who make the elements infinite in number, as anaxagoras and democritus do, say that the infinite is continuous by contact-compounded of the homogeneous parts according to the one, of the seed-mass of the atomic shapes according to the other.
sphere of passing-away and coming-to-be.
iv. a rectangle is said to be contained by any two adjacent sides. thus the rectangle abcd is said to be contained by ab, ad, or by ab, bc, c.
in speaking of them we shall be obliged also to inquire into generation and destruction. for if there is generation anywhere, it must be in these elements and things composed of them.
and the seed having life, and becoming endowed with respiration, produces in that part in which it respires a lively desire of emission, and thus creates in us the love of procreation.
those active powers, then, whose forms are not embodied in matter, are unaffected: but those whose forms are in matter are such as to be affected in acting.
with an abstract and physics with a more concrete object. there are many attributes necessarily present in physical bodies which are necessarily excluded by indivisibility; all attributes, in fact, which are divisible.
if he had gone of deliberate purpose and for the sake of this-if he always or normally went there when he was collecting payments-he would not be said to have gone 'by chance'.
for action and passion (in the proper sense of the terms) can only occur between things which are such as to touch one another; nor can things enter into combination at all unless they have come into a certain kind of contact.
and since the motion of a and that of each of the others are simultaneous, the whole motion must occupy the same time as the motion of a: but the time occupied by the motion of a is finite: consequently the motion will be infinite in a finite time, which
six. divide a given line into two parts, so that the rectangle contained by the whole and one segment may be equal to any multiple of the square on the other segment.
thus he spake, and once more into the cup in which he had previously mingled the soul of the universe he poured the remains of the elements, and mingled them in much the same manner; they were not, however, pure as before, but diluted to the second and
and (ii) what plato has written in the timaeus is not based on any precisely-articulated conception. for he has not stated clearly whether his 'omnirecipient" exists in separation from the 'elements'; nor does he make any use of it.
then because ab is equal and parallel to de, ad is equal and parallel to be i. xxxiii . in like manner cf is equal and parallel to be. hence eleven. ix. ad is equal and parallel to cf. hence i. xxxiii. ac is equal to df.
hence two points of the line bc coincide with two points of the line ef; and since two right lines cannot enclose a space, bc must coincide with ef.
it remains to speak of the earth, of its position, of the question whether it is at rest or in motion, and of its shape.
to us there is a difficulty in apprehending how that which is at rest can also be in motion, or that which is indivisible exist in space.
dem.-since a : b :: c : d, a- b -c d; therefore a- b one -c d one, or a--b b c--d d; therefore a b : b :: c d : d.
she was pre-eminent in courage and military skill, and was the leader of the hellenes.
it by the name of its material, it is evident that becomings of the former kind cannot be alterations.
than would suffice for men of simple life; and they were to spend in common, and to live together in the continual practice of virtue, which was to be their sole pursuit.
dem.-the circle on ac as diameter passes through b, since the angle abc is right xxxi. and touches ae, since the angle cae is right xvi. . therefore the angle bae xxxii.
dem.-if ac be not greater than ab, it must be either equal to it or less than it. let us examine each case:-
discriminated by mutual contrast. what happens to men, then, is just what happens to coppersmiths, who are so accustomed to the noise of the smithy that it makes no difference to them.
this outer net contains two lesser nets, one corresponding to the stomach, the other to the lungs; and the entrance to the latter is forked or divided into two passages which lead to the nostrils and to the mouth.
the base fc is equal to gb, the angle afc is equal to agb, and the angle acf is equal to the angle abg.
now if place is what primarily contains each body, it would be a limit, so that the place would be the form or shape of each body by which the magnitude or the matter of the magnitude is defined: for this is the limit of each body.
seventeen practical rules for the tragic poet. eighteen further rules for the tragic poet. nineteen thought, or the intellectual element, and diction in tragedy. twenty diction, or language in general. twenty one poetic diction.
also (six) it is reasonable that each kind of body should be carried to its own place.
(that is why health is not 'active', except metaphorically.) for when the agent is there, the patient he-comes something: but when 'states' are there, the patient no longer becomes but already is-and 'forms' (i.e. lends') are a kind of 'state'.
for when we state the essential nature of the sphere or circle we do not include in the formula gold or bronze, because they do not belong to the essence, but if we are speaking of the copper or gold sphere we do include them.
the infinite is not the same in magnitude and movement and time, in the sense of a single nature, but its secondary sense depends on its primary sense, i.e.
object which produces it is called bright and flashing.
into their triangles, if they become one, they will form one large mass of another kind. so much for their passage into one another. i have now to speak of their several kinds, and show out of what combinations of numbers each of them was formed.
the truth concerning the soul can only be established by the word of god. still, we may venture to assert what is probable both concerning soul and body.
they and the skin might together form a bond which should fasten the head to the body, since the crown of the head was not encircled by sinews; and also in order that the sensations from both sides might be distributed over the whole body.
composition. supposing, then, that there is such a thing as simple movement, and that circular movement is an instance of it, and that both movement of a simple body is simple and simple movement is of a simple body (for if it is movement of a compound it
to suppose that there is an infinite number of worlds, as democritus (hippolyt. ref. haer. i.) had said, would be, as he satirically observes, 'the characteristic of a very indefinite and ignorant mind.'
yet the affinities of similar substances have some effect upon the composition of the world, and of this plato may be thought to have had an anticipation. he may be described as confusing the attraction of gravitation with the attraction of cohesion.
god fashioned them by form and number. let it be consistently maintained by us in all that we say that god made them as far as possible the fairest and best, out of things which were not fair and good.
if therefore there is some body which rises to the surface of all things-and we observe fire to move upward even in air itself, while the air remains at rest-clearly this body is moving towards the extremity. it cannot then have any weight.
(four) place can be left behind by the thing and is separable. in addition:
it is clear, then, that all the other differences reduce to the first four, but that these admit of no further reduction.
for not the whole race only, but each individual-barring inevitable accidents-comes into the world having a fixed span, and the triangles in us are originally framed with power to last for a certain time, beyond which no man can prolong his life.
five. how many to define similar rectilineal figures of more than three sides?
we must therefore concentrate all our powers on the discussion of these difficulties and on the solution of a further question-viz. what is the cause of the perpetuity of coming-to-be? why is there always unqualified, as well as partial, coming-to-be?
it cuts whatever it meets. and we must not forget that the original figure of fire (i.e.
his conclusion does not follow, because if we take only white things, and if 'white' has a single meaning, none the less what is white will be many and not one.
because the magnitude is continuous, the movement too must be continuous, and if the movement, then the time; for the time that has passed is always thought to be in proportion to the movement.
but other things as well may have been moved now, and there would be a number of each of the two movements. is there another time, then, and will there be two equal times at once? surely not.
but if we suppose, with mr. grote, that the diurnal rotation of the earth on its axis and the revolution of the sun and outer heaven precisely coincide, it would be difficult to imagine that plato was unaware of the consequence.
for agent and patient have not the same matter, agent acts without being affected: thus the art of healing produces health without itself being acted upon in any way by that which is being healed.
other uphill. moreover it does not as a matter of fact make any difference to the argument to say that the one motion must inevitably be quicker or slower than the other: for then the circumference can be greater or less than the straight line; and if so
then, as before, it can be proved that ad is equal to ab, and four hundred is equal to cb (const.). this is contrary to prop. vii. hence the angle acb is a right angle.
i do not include the starting-point of motion: for it is the goal rather than the starting-point of motion that gives its name to a particular process of change.
sol.-draw any two lines ac, ae, making an angle; then cut off ab equal ten, bc equal y , ad equal z. join bd, and draw ce parallel to bd. de will be the fourth proportional required.
and the world being thus created according to the eternal pattern is the copy of something; and we may assume that words are akin to the matter of which they speak.
observation.-propositions xi, xii., may both be included in one enunciation as follows:-"if two circles touch each other at any point, the centres and that point are collinear."
disorders the whole inner nature of man; and when eager in the pursuit of some sort of learning or study, causes wasting; or again, when teaching or disputing in private or in public, and strifes and controversies arise, inflames and dissolves the
so that proposition xxvii. is the converse of xxvi., and xxix. of xxviii.
there is a further difficulty in supposing it to be possible for anything that is in motion to cause motion continuously and not merely in the way in which it is caused by something repeatedly pushing (in which case the continuity amounts to no more than
thirteen. what is the centroid of a triangle?
for the hot is not essentially moist or dry, nor the moist essentially hot or cold: nor are the cold and the dry derivative forms, either of one another or of the hot and the moist. hence these must be four.
than the natural locomotion of the projectile wherewith it moves to its proper place. but in a void none of these things can take place, nor can anything be moved save as that which is carried is moved.
but what shows most clearly that rectilinear motion cannot be continuous is the fact that turning back necessarily implies coming to a stand, not only when it is a straight line that is traversed, but also in the case of locomotion in a circle (which is
similarly, it is qua reduced to a 'mean' condition that the dry and the moist, as well as the contraries we have used as examples, produce flesh and bone and the remaining compounds.
so early did the epicurean doctrine take possession of the greek mind, and so natural is it to the heart of man, when he has once passed out of the stage of mythology into that of rational religion.
eleven. if p, p, p denote the perpendiculars of a triangle, then
the same immobility, they say, is produced by the flatness of the surface which the earth presents to the air which underlies it; while the air, not having room enough to change its place because it is underneath the earth, stays there in a mass, like the
moreover, it is plain that everything continuous is divisible into divisibles that are infinitely divisible: for if it were divisible into indivisibles, we should have an indivisible in contact with an indivisible, since the extremities of things that are
so much for their passage into one another: i must now speak of their construction.
a body, then, in which air preponderates over earth and water, may well be lighter than something in water and yet heavier than it in air, since such a body does not rise in air but rises to the surface in water.
actuality of water, for water is potentially air, while air is potentially water, though in another way.
we can only reply, (one) that to the mind of plato subject and object were not yet distinguished; (two) that he supposes the process of creation to take place in accordance with his own theory of ideas; and as we cannot give a consistent account of the
empedocles says that fire, water, air, and earth are four elements, and are thus 'simple' rather than flesh, bone, and bodies which, like these, are 'homoeomeries'.
hence bg is the supplement of the sum of the arcs ac, cb; therefore the line bg is the chord of the supplement of the sum. now (ex.
for besides the other absurdities of the statement, it is the more absurd that people should make it when they see nothing coming to be spontaneously in the heavens, but much happening by chance among the things which as they say are not due to chance;
thirdly (c) in its most proper use, it is that which is, but is incapable of any destruction such that the thing which now is later ceases to be or might cease to be; or again, that which has not yet been destroyed, but in the future may cease to be.
and they did in fact at that time create a very great and mighty movement; uniting with the ever-flowing stream in stirring up and violently shaking the courses of the soul, they completely stopped the revolution of the same by their opposing current, and
whether it is really the centre of the earth and not rather that of the whole to which it moves, may be left to another inquiry, since these are coincident.
having considered objects of sense, we now pass on to sensation.
but that which is moved by something that, though it is in motion, is moved directly by the unmoved stands in varying relations to the things that it moves, so that the motion that it causes will not be always the same: by reason of the fact that it
but the whole description is so ideal and imaginative, that we can hardly venture to attribute to many of plato's words in the timaeus any more meaning than to his mythical account of the heavens in the republic and in the phaedrus.
cause' in this connexion has two senses. it means (i) the source from which, as we say, the process 'originates', and (ii) the matter. it is the material cause that we have here to state.
the reason of this is that in this case all these characteristics belong to the centre: that is to say, the centre is alike starting-point, middle-point, and finishing-point of the space traversed; consequently since this point is not a point on the
and so the world received animals, mortal and immortal, and was fulfilled with them, and became a visible god, comprehending the visible, made in the image of the intellectual, being the one perfect only-begotten heaven.
but the last will not move it without the first: e.g. the stick will not move anything unless it is itself moved by the man.
reject the part ak, which is common, and we get the figure fh equal to the figure hd; but hd is equal to the rectangle ab.bh, because bd is equal to ab, and fh is the square on ah. therefore the rectangle ab.bh is equal to the square on ah.
if two triangles (abc, def) have two sides (ab, ac) of one respectively equal to two sides (de, df) of the other, but the contained angle (bac) of one greater than the contained angle (edf) of the other, the base of that which has the greater angle is
twenty nine. if the angle in a segment of a circle be a right angle and a-half, what part of the whole circumference is it?
dem.-if both pairs of lines be coplanar, the proposition is the same as one. xxix., ex. two. if not, take any points a, one hundred in the lines ab, bc, and cut off ed ba, and ef bc i. iii. . join ad, be, cf, ac, df.
from white to black or from good to bad, which is not of a kind specifically distinct: it is numerically the same if it proceeds from something numerically one to something numerically one in the same period of time, e.g.
flux. by such reflections we may conceive the greek to have attained the metaphysical conception of eternity, which to the hebrew was gained by meditation on the divine being.
when they are stating what is most startlingly new, they will be most likely to put it in an old-fashioned form, and to use the inadequate ideas and formulae of tradition to express the deeper truths towards which they are feeling their way.
the soul of man is made out of the remains of the elements which had been used in creating the soul of the world; these remains, however, are diluted to the third degree; by this plato expresses the measure of the difference between the soul human and
thirty two. construct the greatest triangle equiangular to a given one whose sides shall pass through three given points.
for motion is equally immediate in the case of nutriment, when nothing hinders, and in the case of the thing healed, when nothing stays the healing.
for he must say that there are certain solids which, however, are indivisible-unless there are continuous pores all through the body.
one hundred four. to construct a quadrilateral similar to a given one whose four sides shall pass through four given points.
now intelligent action is for the sake of an end; therefore the nature of things also is so. thus if a house, e.g.
in response to a slight external movement reach health or increase, as the case may be.
again we may use a complex expression for either and say, e.g. neither 'polyclitus' nor 'sculptor' but 'polyclitus, sculptor'.
hence, if ax be a tangent at a, ac any chord, the angle which the tangent makes with the chord is equal to the angle in the alternate segment.
but even supposing these are contraries, it in no way follows that the reverse motions on the complete circumference contraries.
in like manner construct the triangle hai equiangular to ecf, and similarly placed, and lastly, the triangle iaj equiangular and similarly placed with fcg. then abhij is the figure required.
into halves. but if divisions are made in this way, neither the distance nor the motion will be continuous: for motion if it is to be continuous must relate to what is continuous: and though what is continuous contains an infinite number of halves, they
the ancient physical philosophers have been charged by dr.
prop. sixteen.-problem. to inscribe a regular polygon of fifteen sides in a given circle.
'and what was the subject of the poem?' said the person who made the remark. the subject was a very noble one; he described the most famous action in which the athenian people were ever engaged.
the terms 'generated' and 'destructible' are coincident; this is obvious from our former remarks, since between what always is and what always is not there is an intermediate which is neither, and that intermediate is the generated and destructible.
dem.-since f is the centre of the circle kdl, fk is equal to fd; but fd is equal to a (const.); therefore (axiom i.) fk is equal to a.
but fineness and coarseness, again, are equivalent to greatness and smallness, since a thing with small parts is fine and a thing with large parts coarse.
cor. two.-find the centre of a circle by means of a carpenter's square.
this would be the case on the theory set forth in parmenides: for he says that the things into which change takes place are two, and he asserts that these two, viz. what is and what is not, are fire and earth.
that so-much fire is comparable with many times-that-amount of air, as being 'equally' or 'similarly' hot. for the same thing, if it be greater in amount, will, since it belongs to the same kind, have its ratio correspondingly increased.
two) ac is cut in "extreme and mean ratio" at five hundred, and four hundred is greater than ad. next, cut off de equal to ad, and in the same manner we have de greater than ec, and so on.
but ade : bde :: ad : db i. , and ade : cde :: ae : ec i. . hence ad : db :: ae : ec. two. if ad : db :: ae : ec, it is required to prove that de is parallel to bc.
there are several other questions which we might ask and which can receive no answer, or at least only an answer of the same kind as the preceding. how can matter be conceived to exist without form?
originally occupying the other half from the goal to the middle of the a's, equal in number, size, and velocity to b, b....then three consequences follow:
, and the angle opd equal to the angle opf. therefore pd and pf make equal angles with the diameter.
and in the centre he put the soul, which he diffused throughout the body, making it also to be the exterior environment of it; and he made the universe a circle moving in a circle, one and solitary, yet by reason of its excellence able to converse with
it should not be by their architecture, but why not even by their power of abstract thought, that nations should seek to commemorate themselves? how much more admirable the bhagvat-geeta than all the ruins of the east!
that the world was generated all are agreed, but, generation over, some say that it is eternal, others say that it is destructible like any other natural formation.
four. the perpendicular is the least line which can be drawn from a given point to a given line; and of all others that may be drawn to it, that which is nearest to the perpendicular is less than any one more remote.
and when you were speaking yesterday about your city and citizens, the tale which i have just been repeating to you came into my mind, and i remarked with astonishment how, by some mysterious coincidence, you agreed in almost every particular with the
severally always be the same, though that which contains them is different in different cases. moreover, it is not any casual thing that is capable of carrying any attribute: each single attribute can be carried primarily only by one single thing.
if from the middle point of the line joining any two of four concyclic points a perpendicular be let fall on the line joining the remaining two, the six perpendiculars thus obtained are concurrent.
i. equal circles are those whose radii are equal.
but the seed and the doctor and the adviser, and generally the maker, are all sources whence the change or stationariness originates, while the others are causes in the sense of the end or the good of the rest; for 'that for the sake of which' means what
the manner in which the 'mover' moves the moved' not always the same: on the contrary, whereas one kind of 'mover' can only impart motion by being itself moved, another kind can do so though remaining itself unmoved.
sixty four. state and prove the corresponding theorem for perpendiculars on the external bisector.
to say, however, that you are at once standing and sitting, or that the diagonal is commensurable, is to say what is not only false but also impossible.
that if one would live simply and eat only the crop which he raised, and raise no more than he ate, and not exchange it for an insufficient quantity of more luxurious and expensive things, he would need to cultivate only a few rods of ground, and that it
what now contains air formerly contained water, so that clearly the place or space into which and out of which they passed was something different from both.
we have now to consider how far in any of these speculations plato approximated to the discoveries of modern science.
movements-to attempt to tell of all this without a visible representation of the heavenly system would be labour in vain. enough on this head; and now let what we have said about the nature of the created and visible gods have an end.
the angle bgc is greater than gbc; and make (xxiii.) the angle bgh equal to gbh, and join ah. then vi. bh is equal to gh. therefore the triangles abh, agh have the sides ab, ah of one equal to the sides ag, ah of the other, and the base bh equal to gh.
and this reciprocal nexus will hold continuously throughout the sequence: for it makes no difference whether the reciprocal nexus, of which we are speaking, is mediated by two, or by many, members.
(three) what naturally admits of being gone through, but is not actually gone through or does not actually reach an end.
nevertheless, since there is also a matter out of which corporeal substance itself comes-to-be (corporeal substance, however, already characterized as such-and-such a determinate body, for there is no such thing as body in general), this same matter is
in such a case the shape of the element cannot persist; for, if it did, the contained mass would not be in continuous contact with the containing body; while, if its shape is changed, it will cease to be water, since the distinctive quality is shape.
but every simultaneous time is self-identical; for the 'now' as a subject is an identity, but it accepts different attributes. the 'now' measures time, in so far as time involves the 'before and after'.
eleven. find a point in the circumference of a given circle, the sum of the squares on whose distances from two given points may be a maximum or a minimum.
whether if soul did not exist time would exist or not, is a question that may fairly be asked; for if there cannot be some one to count there cannot be anything that can be counted, so that evidently there cannot be number; for number is either what has
after that we will inquire into light and heavy and of the various phenomena connected with them. the local movement of each body into its own place must be regarded as similar to what happens in connexion with other forms of generation and change.
now we mark them by judging that a and b are different, and that some third thing is intermediate to them.
but the nonexistence of motion is an impossibility: for we have shown above that there must always be motion.
fourthly, there must be a substrate underlying all processes of becoming and changing. what can this be in the present case? it is either the body or the soul that undergoes alteration: what is it that correspondingly becomes motion or becoming?
on the other hand, if it acts by contact, then-even without pores-some things will 'suffer action' and others will 'act', provided they are by nature adapted for reciprocal action and passion.
surely it is clear that both are contrary to it though not in the same sense: the natural motion of earth is contrary inasmuch as the motion of fire is also natural, whereas the upward motion of fire as being natural is contrary to the downward motion of
perhaps it is impossible for growth to take place in either of these ways. for since the matter is 'separate', either (a) it will occupy no place (as if it were a point), or (b) it will be a 'void', i.e. a non-perceptible body.
place on the other hand is rather what is motionless: so it is rather the whole river that is place, because as a whole it is motionless.
three. if we take a linear foot, describe a square on it, divide two adjacent sides each into twelve equal parts, and draw parallels to the sides, we evidently divide the square foot into square inches; and as there will manifestly be twelve rectangular
then in the time in which it is changing it must be either in ab or in bg or partly in one and partly in the other: for this, as we saw, is true of everything that is changing.
seven. give an illustration of the ratio of incommensurables.
(two) we have now explained that the mutual transformations of the elements cannot take place by means of excretion.
prop. fourteen.-problem. to construct a square equal to a given rectilineal figure (ten).
thirty seven. construct an equilateral triangle having its vertex at a given point, and the extremities of its base on a given circle.
again, since a thing that changes continuously and has not perished or ceased from its change must either be changing or have changed in any part of the time of its change, and since it cannot be changing in a moment, it follows that it must have changed
in the same manner it can be shown that every ratio whose terms are commensurable can be converted into a fraction; and, conversely, every fraction can be turned into a ratio.
what always is, then, since it is incapable of ever not being, cannot possibly be generated.
further it is evident from actual observation that there are things that have the characteristic of moving themselves, e.g. the animal kingdom and the whole class of living things.
cor. one.-the line cf is divided in "extreme and mean ratio" at a.
dem.-take any other point five hundred in ac : draw dg, dh parallel to the sides, and ck parallel to ab; produce ep, gd to meet ck in k and j, and produce hd to meet pk in i.
putting our explanation) in the same sense as 'a doctor does something or has something done to him', 'is or becomes something from being a doctor.'
because there must be something at rest at the centre of the revolving body; and of that body no part can be at rest, either elsewhere or at the centre. it could do so only if the body's natural movement were towards the centre.
some persons call them prophets; they are quite unaware that they are only the expositors of dark sayings and visions, and are not to be called prophets at all, but only interpreters of prophecy.
those, however, who maintain the theory in question, are not at liberty to conceive that 'both come-to-be out of flesh' in that manner, but only as a stone and a brick 'both come-to-be out of a wall'-viz. each out of a different place or part.
so clearly we too must do this as regards both coming to be and passing away and every kind of physical change, in order that, knowing their principles, we may try to refer to these principles each of our problems.
now the whole b was modified in a certain time, and the part of this infinite, e, cannot be so modified in the same time, since we assume that a less quantity makes the movement in a less time.
twelve. the angle included between the internal bisector of one base angle of a triangle and the external bisector of the other base angle is equal to half the vertical angle.
but the unmoved movent, as has been said, since it remains permanently simple and unvarying and in the same state, will cause motion that is one and simple.
two. the parts of all perpendiculars to two parallel lines intercepted between them are equal.
these things at some future time, when we are at leisure, may have the consideration which they deserve, but not at present.
but the whole of which they are parts must necessarily be complete, and thus, in accordance with the meaning of the word, have being, not in some respect only, but in every respect.
the sum of all the plane angles (bac, cad, c.) forming any solid angle (a) is less than four right angles.
some assert that it is (one) the movement of the whole, others that it is (two) the sphere itself.
seven. how may surfaces be divided? ans. into planes and curved surfaces.
in fact, even apart from the definition of motion, every one would admit that in each kind of motion it is that which is capable of that motion that is in motion: thus it is that which is capable of alteration that is altered, and that which is capable of
the appetitive soul is seated in the belly, and there imprisoned like a wild beast, far away from the council chamber, as plato graphically calls the head, in order that the animal passions may not interfere with the deliberations of reason.
if, on the other hand, each of them sets itself in motion, either (a) it will be divisible ('imparting motion' qua this, 'being moved' qua that), or (b) contrary properties will attach to it in the same respect-i.e.
and, further, are all things one substance-one man, one horse, or one soul-or quality and that one and the same-white or hot or something of the kind? these are all very different doctrines and all impossible to maintain.
on ac, cb is equal to the square on ab (hyp.), and the sum of the squares on four hundred, cb is equal to the square on bd xlvii. . therefore the square on ab is equal to the square on bd. hence ab is equal to bd xlvi., ex. one .
hence the planes ab, four hundred meet in k, which is impossible, since they are parallel. therefore the lines ef, gh must be parallel.
comprehension. but this vacant idea of a whole without parts, of a subject without predicates, a rest without motion, has been also the most fruitful of all ideas. it is the beginning of a priori thought, and indeed of thinking at all.
the cause of that departure. in it, rousseau is already in possession of his idea of "nature" as an ideal; but he has at present made no attempt to discriminate, in what is unnatural, between good and bad.
this is indeed the first question we have to ask: is generation a fact or not? earlier speculation was at variance both with itself and with the views here put forward as to the true answer to this question.
iv. the point is called the vertex of the solid angle.
again, if by complete, as previously defined, we mean a thing outside which no part of itself can be found, and if addition is always possible to the straight line but never to the circular, clearly the line which embraces the circle is complete.
moreover, if the indivisibles composing dez are motions, it would be possible for a thing in spite of the presence in it of motion to be not in motion but at rest, while if they are not motions, it would be possible for motion to be composed of something
nature it is inadequate. for suppose the distance to be left out of account and the question asked to be no longer whether it is possible in a finite time to traverse an infinite number of distances, and suppose that the inquiry is made to refer to the
eight. if through the vertex o of a trihedral angle o-abc any line od be drawn interior to the angle, the sum of the rectilineal angles doa, dob, doc is less than the sum, but greater than half the sum, of the face angles of the trihedral.
those things therefore which are subject to perishing and becoming-generally, those which at one time exist, at another do not-are necessarily in time: for there is a greater time which will extend both beyond their existence and beyond the time which
laws causes all manner of changes and infinite diseases and corruptions. now there is a second class of structures which are also natural, and this affords a second opportunity of observing diseases to him who would understand them.
now the mathematician, though he too treats of these things, nevertheless does not treat of them as the limits of a physical body; nor does he consider the attributes indicated as the attributes of such bodies.
thus the hair sprang up in the skin, being akin to it because it is like threads of leather, but rendered harder and closer through the pressure of the cold, by which each hair, while in process of separation from the skin, is compressed and cooled.
hence the point f must be the centre; and the circle described from f as centre, with fa as radius, will be the circle required.
but it makes no difference even if the velocity is not uniform. for let us suppose that the line ab represents a finite stretch over which a thing has been moved in the given time, and let gd be the infinite time.
i was wrong in imagining that all the four elements could be generated into and out of one another.
the timaeus of plato, like the protagoras and several portions of the phaedrus and republic, was translated by cicero into latin. about a fourth, comprehending with lacunae the first portion of the dialogue, is preserved in several mss.
arc or curve of a greater circle becomes that of a smaller, whether it remains the same or becomes a different curve, convexity has not come to exist in anything that was not convex but straight (for differences of degree do not depend on an intermission
from what we have said the explanation of the earth's immobility is also apparent.
is impossible. hence the centre of the second circle must be in the line pe. let it be g, and we see that the line through the centres passes through the point p.
three. if the base of a triangle be given both in magnitude and position, and the sum of the squares on the sides in magnitude, the locus of the vertex is a circle.
something in the body, as we see in the case of the use and activity of the intellect arising from a man's becoming sober or being awakened.
pleasures and pains are so very great; his soul is rendered foolish and disordered by his body; yet he is regarded not as one diseased, but as one who is voluntarily bad, which is a mistake.
again, since the lines bc, ef, bg are continual proportionals, bc : bg in the duplicate ratio of bc : ef five. def. x. ; but bc : bg :: triangle abc : abg.
'to be capable of health' and 'to be capable of illness' are not the same, for if they were there would be no difference between being ill and being well. yet the subject both of health and of sickness-whether it is humour or blood-is one and the same.)
it is apparent, i mean, that fire, in whatever quantity, so long as there is no external obstacle moves upward, and earth downward; and, if the quantity is increased, the movement is the same, though swifter.
the nile, and then given his body to the dogs. i might possibly invent some excuse for them and him, but i have no time for it.
weight, is quicker in proportion to its size. this, however, should not be the case if the ratio is the ground of distinction between heavy things and light.
again, because the triangles aob, dhe have the two sides ao, ob in one respectively equal to the two sides dh, he in the other, and the angle aob equal to the angle dhe, the base ab of one is equal to the base de of the other.
such is the manner in which diseases of the body arise; the disorders of the soul, which depend upon the body, originate as follows.
for as the universe is in the form of a sphere, all the extremities, being equidistant from the centre, are equally extremities, and the centre, which is equidistant from them, is equally to be regarded as the opposite of them all.
things which experience gradual withdrawings and emptyings of their nature, and great and sudden replenishments, fail to perceive the emptying, but are sensible of the replenishment; and so they occasion no pain, but the greatest pleasure, to the mortal
knowledge is the object of our inquiry, and men do not think they know a thing till they have grasped the 'why' of (which is to grasp its primary cause).
they may be said more truly to have cleared up and defined by the help of experience ideas which they already possessed. the beginnings of thought about nature must always have this character.
for if it is coming to a stand in neither of two parts into which the time may be divided, it cannot be coming to a stand in the whole time, with the result that that that which is coming to a stand will not be coming to a stand.
that of the heavy and the light, the bodies are thought to have a spring of change within themselves, while the subjects of healing and increase are thought to be moved purely from without. sometimes, however, even they change of themselves, ie.
and by reason of all these affections, the soul, when encased in a mortal body, now, as in the beginning, is at first without intelligence; but when the flood of growth and nutriment abates, and the courses of the soul, calming down, go their own way and
is not the famous expression-'you hellenes are ever children and there is no knowledge among you hoary with age,' really a compliment to the athenians who are described in these words as 'ever young'?
now in the void this is impossible; for it is not body; the void must have penetrated the cube to a distance equal to that which this portion of void formerly occupied in the void, just as if the water or air had not been displaced by the wooden cube, but
then the triangles gfa, ofa have the sides gf, fa in one equal to the sides of, fa in the other, and the contained angles equal. hence i. iv. the angle gaf equal oaf; but gaf gcb three. xxi.
but (b) the food, in acting, is itself in some way acted upon: for, in acting, it is simultaneously heated or cooled or otherwise affected. now the art of healing corresponds to an 'originative source', while the food corresponds to 'the last' (i.e.
prop. twenty four.-theorem. in any parallelogram (ac), every two parallelograms (af, fc) which are about a diagonal are similar to the whole and to one another.
life; and the pure fire which is within us and related thereto they made to flow through the eyes in a stream smooth and dense, compressing the whole eye, and especially the centre part, so that it kept out everything of a coarser nature, and allowed to
but nothing which concerns the eternal can be a matter of chance or spontaneity, and the heaven and its circular motion are eternal. we must therefore ask why this motion takes one direction and not the other.
the angle dbf is to ebc; but the sum of the angles bce, ebc is less than two right angles i. xvii. ; therefore the sum of the angles bdf, dbf is less than two right angles, and therefore i., axiom xii. the lines eb, df, if produced, will meet.
with five hundred as centre, and de as radius, describe the circle efg (post. iii.). produce da to meet this circle in f. af is equal to bc.
but besides the material out of which the elements are made, there is also a space in which they are contained. there arises thus a second nature which the senses are incapable of discerning and which can hardly be referred to the intelligible class.
now after the superior powers had created all these natures to be food for us who are of the inferior nature, they cut various channels through the body as through a garden, that it might be watered as from a running stream.
you are welcome to hear about them, solon, said the priest, both for your own sake and for that of your city, and above all, for the sake of the goddess who is the common patron and parent and educator of both our cities.
six. if abc be a having ab not greater than ac, a line ag, drawn from a to any point g in bc, is less than ac. for the angle acb xviii. is not greater than abc; but agc xvi. is greater than abc; therefore agc is greater than acg.
number of movements, while another cannot attain health itself, but only running or reduction of flesh, so that one or other of these is for such a being the end.
for it alone is embraced by a single surface, while rectilinear solids have several. the sphere is among solids what the circle is among plane figures.
a limit is a thing which contains; and this motion, being perfect, contains those imperfect motions which have a limit and a goal, having itself no beginning or end, but unceasing through the infinity of time, and of other movements, to some the cause of
but when you take it in this way, using the one point as two, a pause is necessary, if the same point is to be the beginning and the end. the 'now' on the other hand, since the body carried is moving, is always different.
more sensible is a rod of stone wall that bounds an honest man's field than a hundred-gated thebes that has wandered farther from the true end of life.
locomotion: it cannot continue to infinity.
but when he goes on to speak of the essence which is compounded out of both, the track becomes fainter and we can only follow him with hesitating steps.
then again, considering that the bone would be too brittle and inflexible, and when heated and again cooled would soon mortify and destroy the seed within-having this in view, he contrived the sinews and the flesh, that so binding all the members together
fifty nine. divide a given semicircle into two parts by a perpendicular to the diameter, so that the radii of the circles inscribed in them may have a given ratio.
for that without which nothing else can exist, while it can exist without the others, must needs be first; for place does not pass out of existence when the things in it are annihilated.
eight. the four angular points of a cyclic quadrilateral determine four triangles whose orthocentres (the intersections of their perpendiculars) form an equal quadrilateral.
assuming, therefore, that 'to touch' is-as we have defined it in a previous work'-'to have the extremes together', only those things will touch one another which, being separate magnitudes and possessing position, have their extremes 'together'.
therefore that which turns back in traversing a rectilinear course must in so doing come to a stand. consequently there cannot be a continuous rectilinear motion that is eternal.
thirty one. the inscribed and escribed circles of any triangle are all touched by its nine-points circle.
again, when a small body of fire is contained in a larger body of air or water or earth, and both are moving, and the fire struggling is overcome and broken up, then two volumes of fire form one volume of air; and when air is overcome and cut up into
in like manner gk is equal to one hundred, and fg is equal to b (const.) hence the three sides of the triangle kfg are respectively equal to the three lines a, b, one hundred.
suppose that a is the time occupied by the infinite power in the performance of an act of heating or pushing, and that ab is the time occupied by a finite power in the performance of the same act: then by adding to the latter another finite power and
wherefore it lives and does not differ from a living being, but is fixed and rooted in the same spot, having no power of self-motion.
if, then, it is moved in virtue of some part of it being moved by that part itself, it is this part that will be the primary self-movent, since, if this part is separated from the whole, the part will still move itself, but the whole will do so no longer.
nineteen. if r denotes the radius of the circle circumscribed about a triangle abc, r, r, r, r the radii of its inscribed and escribed circles, , , the perpendiculars from its circumcentre on the sides; , , the segments of these perpendiculars between the
wherefore of all modes of purifying and re-uniting the body the best is gymnastic; the next best is a surging motion, as in sailing or any other mode of conveyance which is not fatiguing; the third sort of motion may be of use in a case of extreme
dem.-one. since the lines ab, ef intersect, the angle agh is equal to egb xv. ; but egb is equal to ghd (hyp.); therefore agh is equal to ghd, and they are alternate angles. hence xxvii. ab is parallel to four hundred.
again, the tripod fell 'of itself', because, though when it fell it stood on its feet so as to serve for a seat, it did not fall for the sake of that.
into q), a new contrariety, other than those mentioned, will belong to fire and q: for it has been assumed that q is not the same as any of the four, e w a and f. let k, then, belong to f and y to q.
again, make the angle boc indefinitely small, and make boc equal to it; the triangles boc, boc are similar. proceeding in this way, we see that the circles can be divided into the same number of similar elementary triangles.
as a matter of fact, this word 'duration' possessed a divine significance for the ancients, for the fulfilment which includes the period of life of any creature, outside of which no natural development can fall, has been called its duration.
anyhow if these functions belong to some of these things and not to others, they should explain the difference, instead of speaking in quite general terms as they do. again, combustion of a body produces fire, and fire is a sphere or a pyramid.
it will not, however, move in a mass to a single point on the circumference-the only possible result on the lines of the indifference theory-but rather each corresponding portion of fire to the corresponding part of the extremity, each fourth part, for
secondly, we see that a thing that neither is in motion nor contains any motion within itself can be set in motion; e.g.
thus wishing to preserve the entire seed, he enclosed it in a stone-like casing, inserting joints, and using in the formation of them the power of the other or diverse as an intermediate nature, that they might have motion and flexure.
(eight) in the strictest sense of all, as a thing is 'in' a vessel, and generally 'in' place.
again, even on their own presuppositions it does not seem as if the clements would be infinite in number.
thus, if a be the side and b the diagonal of a square, the ratio of a : b is
hence, too, what effects motion as a physical agent can be moved: when a thing of this kind causes motion, it is itself also moved. this, indeed, has led some people to suppose that every mover is moved.
for if the body z be as much thinner than five hundred as e exceeds h, a, if it moves through z, will traverse it in a time inverse to the speed of the movement, i.e. in a time equal to h.
the distinction must be investigated once more, for it demands some explanation.
again, since ab is bisected in o and divided unequally in e,
though there may be more). then the whole abg moves itself. but if i take away g, ab will move itself, a imparting motion and b being moved, whereas g will not move itself or in fact be moved at all.
moreover, if it is the case that infinite time is not occupied in passing over every magnitude, but it is possible to ass over some magnitude, say be, in a finite time, and if this be measures the whole of which it is a part, and if an equal magnitude is
these cannot be given by individuals to themselves; and therefore human actions, in so far as they are dependent upon them, are regarded by plato as involuntary rather than voluntary.
the freshest and acutest forms of triangles are those that are found in children, but they become more obtuse with advancing years; and when they finally wear out and fall to pieces, old age and death supervene.
for in all things, as we affirm, nature always strives after 'the better'.
now the point in the middle is potential: but this one is actual, and regarded from below it is a finishing-point, while regarded from above it is a starting-point, so that it stands in these same two respective relations to the two motions.
nine. what is the altitude of a parallelogram whose base is sixty five metres and area one thousand four hundred thirty square metres?
socrates: and do you also remember how, with a view of securing as far as we could the best breed, we said that the chief magistrates, male and female, should contrive secretly, by the use of certain lots, so to arrange the nuptial meeting, that the bad
hence the distance of i from the foot of each perpendicular, and from the middle point of each side, is one two the radius. in like manner, if oc be bisected in k, then ik one two the radius.
therefore the figure ghi is equal to a, and it is similar to bcd. hence it is the figure required.
and the fairest bond is that which makes the most complete fusion of itself and the things which it combines; and proportion is best adapted to effect such a union.
for we too hastily said that plato in the timaeus regarded all 'vices and crimes as involuntary.'
now all things rest and move naturally and by constraint. a thing moves naturally to a place in which it rests without constraint, and rests naturally in a place to which it moves without constraint.
the angles (abc, acb) at the base (bc) of an isosceles triangle are equal to one another, and if the equal sides (ab, ac) be produced, the external angles (dbc, ecb) below the base shall be equal.
it exists fully in the sense in which we say 'it is day' or 'it is the games'; and potentially as matter exists, not independently as what is finite does.
thus were created women and the female sex in general.
when exhaled through the mouth or nostrils, it leaves a vacuum which is filled up by other air finding a way in through the pores, this air being thrust out of its place by the exhalation from the mouth and nostrils.
movement as 'before' and 'after'. here, too there is a correspondence with the point; for the point also both connects and terminates the length-it is the beginning of one and the end of another.
these all apply to becoming in time, and have no meaning in relation to the eternal nature, which ever is and never was or will be; for the unchangeable is never older or younger, and when we say that he 'was' or 'will be,' we are mistaken, for these
one hundred ten. if two secants at right angles to each other, passing through one of the points of intersection of two circles, cut the circles again, and the line through their centres in the two systems of points a, b, c; a, b, c respectively, then ab
there will result instead an 'intermediate': and this 'intermediate', according as it is potentially more hot than cold or vice versa, will possess a power-of-heating that is double or triple its power-of-cooling, or otherwise related thereto in some
four. three circles touch each other externally at a, b, one hundred; the chords ab, ac of two of them are produced to meet the third again in the points five hundred and e; prove that de is a diameter of the third circle, and parallel to the line joining
and because she is composed of the same and of the other and of the essence, these three, and is divided and united in due proportion, and in her revolutions returns upon herself, the soul, when touching anything which has essence, whether dispersed in
but he who seeks after knowledge and exercises the divine part of himself in godly and immortal thoughts, attains to truth and immortality, as far as is possible to man, and also to happiness, while he is training up within him the divine principle and
this follows from the supposed juxtaposition of the worlds. for either we must refuse to admit the identical nature of the simple bodies in the various universes, or, admitting this, we must make the centre and the extremity one as suggested.
and since every change is from something to something-as the word itself (metabole) indicates, implying something 'after' (meta) something else, that is to say something earlier and something later-that which changes must change in one of four ways: from
he was the teacher of simmias and cebes, who became disciples of socrates. we have hardly any other information about him.
and we, too, who are going to discourse of the nature of the universe, how created or how existing without creation, if we be not altogether out of our wits, must invoke the aid of gods and goddesses and pray that our words may be acceptable to them and
fifty three. prove also that an equiangular circumscribed polygon is regular, and an equiangular inscribed polygon, if the number of sides be odd.
four. in the same case, find the loci of the angular points of a square described on one of the sides.
this explains why the middle of the heaven and the surface which faces us of the rotating system are held to be 'up' and 'down' in the strict and fullest sense for all men: for the one is always at rest, while the inner side of the rotating body remains
eighteen. if a transversal cut in the points a, one hundred, b three lines issuing from a point five hundred, prove that
for there are some who think it possible both for the ungenerated to be destroyed and for the generated to persist undestroyed.
it is worthy of remark that these mystical fancies are nowhere to be found in the writings of plato, although the importance of number as a form and also an instrument of thought is ever present to his mind.
the remaining part he divided into long and round figures, and to these as to anchors, fastening the mortal soul, he proceeded to make the rest of the body, first forming for both parts a covering of bone.
and unless love or strife is actually setting them in motion, the 'simple' bodies themselves have absolutely no movement or rest. but this is paradoxical: and what is more, they do in fact obviously move.
all the other thinkers, too, evidently treat the infinite as matter-that is why it is inconsistent in them to make it what contains, and not what is contained.
but when terms stand to one another as these do, f and h coincident, e and f never predicated of the same thing but one or other of everything, and g and h likewise, then e and g must needs be coincident.
the foregoing proof forms an exception to euclid's demonstrations of converse propositions, for it is direct. the following is an indirect proof:-if cb be not at right angles to ac, let four hundred be perpendicular to it. make four hundred cb. join ad.
the tendency to argue from the higher to the lower, from man to the world, has led to many errors, but has also had an elevating influence on philosophy.
one. cut a line externally in "extreme and mean ratio."
twenty. if two figures be inversely similar, how can they be changed into figures directly similar?
dem.-join bc. now since ab is parallel to four hundred, and bc intersects them, the angle abc is equal to the alternate angle dcb xxix. .
but now the sight of day and night, and the months and the revolutions of the years, have created number, and have given us a conception of time, and the power of enquiring about the nature of the universe; and from this source we have derived philosophy,
even the more recent of the ancient thinkers were in a pother lest the same thing should turn out in their hands both one and many.
that is why the using art also is in a sense directive; but it differs in that it knows the form, whereas the art which is directive as being concerned with production knows the matter.
one might also raise the question what sort of movement time is the number of. must we not say 'of any kind'?
these generally agree, and therefore may be supposed to be derived from a single original. the version is very faithful, and is a remarkable monument of cicero's skill in managing the difficult and intractable greek.
directions to one another-three of them, the sun, mercury, venus, with equal swiftness, and the remaining four-the moon, saturn, mars, jupiter, with unequal swiftness to the three and to one another, but all in due proportion.
the reason is their reciprocal transformation. for, had each of them persisted in its own place instead of being transformed by its neighbour, they would have got dissevered long ago.
draw parallels to the sides of the original triangle abc through its vertices, forming a new triangle abc described about abc; then the three perpendiculars at the middle points of the sides of abc are concurrent v. cor.
and moise is that which, being readily adaptable in shape, is not determinable by any limit of its own: while 'dry' is that which is readily determinable by its own limit, but not readily adaptable in shape.
nowhere--where is the goat-stag or the sphinx?), and because 'motion' in its most general and primary sense is change of place, which we call 'locomotion'.
the senses are not instruments, but rather passages, through which external objects strike upon the mind. the eye is the aperture through which the stream of vision passes, the ear is the aperture through which the vibrations of sound pass.
true, there never was and is not likely soon to be a nation of philosophers, nor am i certain it is desirable that there should be.
very weak, but that which is trained and exercised, very strong. wherefore we should take care that the movements of the different parts of the soul should be in due proportion.
'matter' will be identical in-potentiality as well as numerically-identical.
he was always tending to argue from what was near to what was remote, from what was known to what was unknown, from man to the universe, and back again from the universe to man.
the commentary is of little or no value, either in a philosophical or philological point of view. the writer is unable to explain particular passages in any precise manner, and he is equally incapable of grasping the whole.
we may distinguish generally between predicating b of a because it (a) is itself, and because it is something else; and particularly between place which is common and in which all bodies are, and the special place occupied primarily by each.
but when the heat changes its place, and the particles at the other exit grow warmer, the hotter air inclining in that direction and carried towards its native element, fire, pushes round the air at the other; and this being affected in the same way and
a motion from health to disease and a motion from disease to health.
now there is such a thing as locomotion, and in locomotion there is included circular movement, and everything is measured by some one thing homogeneous with it, units by a unit, horses by a horse, and similarly times by some definite time, and, as we
does it belong then to the same or to different sciences to know each severally?
iii. a right line is said to touch a circle when it meets the circle, and, being produced both ways, does not cut it; the line is called a tangent to the circle, and the point where it touches it the point of contact.
we must take this as our starting-point and try to discover-since we wish to know what time is-what exactly it has to do with movement.
but even if we consider it on its own merits the so-called vacuum will be found to be really vacuous. for as, if one puts a cube in water, an amount of water equal to the cube will be displaced; so too in air; but the effect is imperceptible to sense.
further, by 'transverse' in the universe we commonly mean, not above and below, but a direction crossing the line of the poles, which, by implication, is length: for transverse motion is motion crossing motion up and down.
plato probably did more for physical science by asserting the supremacy of mathematics than aristotle or his disciples by their collections of facts.
but we have already defined single and continuous motion to be motion of a single thing in a single period of time and operating within a sphere admitting of no further specific differentiation (for we have three things to consider, first that which is in
so in the straight line in question any one of the points lying between the two extremes is potentially a middle-point: but it is not actually so unless that which is in motion divides the line by coming to a stand at that point and beginning its motion
so, too, in the case of that which is perishing and that which has perished: for that which becomes and that which perishes must contain an element of infiniteness as an immediate consequence of the fact that they are continuous things: and so a thing
for suppose that a thing has changed from a to b in a moment.
and thus the whole mixture out of which he cut these portions was all exhausted by him.
to attain any assured knowledge about the soul is one of the most difficult things in the world. as the form of question which here presents itself, viz.
abh : cde in the duplicate ratio of ah : ce.
it is the same here with the revolving bodies: for the are intercepted by two radii will be larger in the larger circle, and hence it is not surprising that the revolution of the larger circle should take the same time as that of the smaller.
as in the republic, plato is still the enemy of the purgative treatment of physicians, which, except in extreme cases, no man of sense will ever adopt.
fourteen. if the quadrilateral abcd is not cyclic, prove that the three rectangles ab.four hundred, bc.ad, ac.bd are proportional to the three sides of a triangle which has an angle equal to the sum of a pair of opposite angles of the quadrilateral.
for the passingaway of either 'element' produces either the other or the matter.
the earth, which is our nurse, clinging (or 'circling') around the pole which is extended through the universe, he framed to be the guardian and artificer of night and day, first and eldest of gods that are in the interior of heaven.
nor is it possible for a thing to be in itself even incidentally: for two things would at the same time in the same thing.
therefore bc is greater than bh; but bh has been proved to be equal to ef; therefore bc is greater than ef.
hence the parallelogram ag is equal to one thousand one i. xxxvi. . in like manner il is equal to jf; but one thousand one is equal to il i. xliii. .
if two triangles (bac, edf) have two sides (ba, ac) of one equal respectively to two sides (ed, df) of the other, and have also the angles (a, five hundred) included by those sides equal, the triangles shall be equal in every respect-that is, their bases
bodies are either simple or compounded of such; and by simple bodies i mean those which possess a principle of movement in their own nature, such as fire and earth with their kinds, and whatever is akin to them.
virtue of the motion of the whole. (it must be remembered, however, that by 'that which is without parts' i mean that which is quantitatively indivisible (and that the case of the motion of a part is not exactly parallel): for parts have motions belonging
we have explained under what conditions 'combination', 'contact', and 'action-passion' are attributable to the things which undergo natural change.
two right lines (ab, four hundred) which intersect one another in any point (e) are coplanar, and so also are any three right lines (ec, cb, be) which form a triangle.
and if we consider the matter in yet a third wa ly we shall get this same result as follows.
now if the things that are in motion-that is to say, the things to which the motions belong essentially and not accidentally-differ specifically, then their respective motions will also differ specifically: if on the other hand they differ generically or
fifty seven. a transversal being drawn cutting the sides of a triangle, the lines from the angles of the triangle to the middle points of the segments of the transversal intercepted by those angles meet the opposite sides in collinear points.
'body' is what has extension in all directions and the infinite is what is boundlessly extended, so that the infinite body would be extended in all directions ad infinitum.
bodies, we say, which have no natural movement, have no constrained movement; and where there is no natural and no constrained movement there will be no movement at all.
of it held to be a transformation only. for, although one of these discoveries might have been supposed to be a happy guess, taken together they seem to imply a great advance and almost maturity of natural knowledge.
on the other hand, when condensed, two volumes of fire make a volume of air; and two and a half parts of air condense into one of water.
if he allowed the mechanical use of the rule and compass he could give methods of solving many problems that go beyond the limits of the "geometry of the point, line, and circle."-see notes five hundred, f at the end of this work.
six. if the square of a line four hundred, drawn from the angle one hundred of an equilateral triangle abc to a point five hundred in the side ab produced, be equal to ab ; prove that ad is cut in "extreme and mean ratio" at b.
occupy the same position for a period of time, and that therefore they will be at once at rest and in motion.
and (ii) the same consequence follows if a and b are absolutely 'other', i.e. in no respect identical. whiteness could not be affected in any way by line nor line by whiseness-except perhaps 'coincidentally', viz.
or thus: ad ac four hundred four hundred cb; bd four hundred cb. hence ad.db (four hundred cb)(four hundred cb) cd cb ; therefore ad .db cb cd .
for no element which is one and the same with itself can be changed by or change another of the same kind and in the same state. but so long as in the process of transition the weaker is fighting against the stronger, the dissolution continues.
but the race of birds was created out of innocent light-minded men, who, although their minds were directed toward heaven, imagined, in their simplicity, that the clearest demonstration of the things above was to be obtained by sight; these were
what is spoken of the unchanging or intelligible must be certain and true; but what is spoken of the created image can only be probable; being is to becoming what truth is to belief.
the heart is the house of guard in which all the veins meet, and through them reason sends her commands to the extremity of her kingdom.
senses: moreover the inanimate is unconscious of being affected by alteration, whereas the animate is conscious of it, though there is nothing to prevent the animate also being unconscious of it when the process of the alteration does not concern the
in time all things come into being and pass away; for which reason some called it the wisest of all things, but the pythagorean paron called it the most stupid, because in it we also forget; and his was the truer view.
one take then the possible case of its not moving it: then there will be a part that is moved and a part that is an unmoved movent.
the best exercise is the spontaneous motion of the body, as in gymnastics, because most akin to the motion of mind; not so good is the motion of which the source is in another, as in sailing or riding; least good when the body is at rest and the motion is
dem.-since abcde is a regular pentagon, the arc abc is two ths of the circumference; and since agh is an equilateral triangle, the arc abg is one rd of the circumference.
comes-to-be and passes-away. yet (a) it is evident that even fire is itself moved, i.e. suffers action.
prop. thirty five.-theorem. parallelograms on the same base (bc) and between the same parallels are equal.
two. the angles at the circumference, being the halves of the central angles, are therefore equal.
the mouth to the lungs;-he supposes the lung to be hollow and bloodless; the spinal marrow he conceives to be the seed of generation; he confuses the parts of the body with the states of the body-the network of fire and air is spoken of as a bodily organ;
thus it is evident that all the 'elements' will be able to be transformed out of one another; and that, in the instances we have taken, e (earth) also will contain the remaining two 'complementary factors', viz.
is best and the end of the things that lead up to it. (whether we say the 'good itself or the 'apparent good' makes no difference.)
prop. six.-problem. in a given circle (abcd) to inscribe a square.
proportion is equal to the third; therefore five. xiv. the second term ab is equal to the fourth ab, and, being parallel to it, the lines aa, bb i. xxxiii. are parallel.
similarly, we speak neither (ii) of the food as 'combining' with the body, nor (iii) of the shape as 'combining' with the wax and thus fashioning the lump.
and 'position' belongs only to those things which also have a place': for in so far as we attribute 'contact' to the mathematical things, we must also attribute 'place' to them, whether they exist in separation or in some other fashion.
which their sides are the chords; but in that year the celebrated gauss proved that if n one be a prime number, regular polygons of n one sides are inscriptable by elementary geometry.
either from oneself to something else or something else to oneself without being in contact with it: it is evident, therefore, that in all locomotion there is nothing intermediate between moved and movent.
'unequal' or 'non-existent'; nor is change either to or from these rather than to or from their opposites.
that case the same thing will be simultaneously hot and cold. both fire and air, therefore, will be something else which is the same; i.e. there will be some 'matter', other than either, common to both.
four. the distances of the vertices a, b, one hundred from the points of contact of the escribed circle which touches ab externally are s b, s a, s.
(one) there is no necessity that the place should grow with the body in it,
they argue that (i) if both the 'combined' constituents persist unaltered, they are no more 'combined' now than they were before, but are in the same condition: while (ii) if one has been destroyed, the constituents have not been 'combined'-on the
again, because the figures described on ba, ac are similar, they are in the duplicate ratio of ba : ac xx. . hence five. xi. ba : ad :: figure described on ba : figure described on ac.
if they had come in sight of this nature, all their ignorance would have been dispelled.
conversely, three is a submultiple, or measure, or part of six, nine, twelve, fifteen, c., because it is contained in each of these without a remainder; but not of ten, thirteen, seventeen, c., because in each case it leaves a remainder.
again, in plants too we find the relation of means to end, though the degree of organization is less. were there then in plants also 'olive-headed vine-progeny', like the 'man-headed ox-progeny', or not?
we speak of a soul of the universe; but more truly regarded, the universe of the timaeus is a soul, governed by mind, and holding in solution a residuum of matter or evil, which the author of the world is unable to expel, and of which plato cannot tell us
consequently the infinite magnitude will not be traversed in finite time: and it makes no difference whether the magnitude is infinite in only one direction or in both: for the same reasoning will hold good.
seven. if two polygons be directly similar, either may be turned round their centre of similitude until they become homothetic, and this may be done in two different ways.
thus the differences are reasonably distributed among the primary bodies, and the number of the latter is consonant with theory. for all who make the simple bodies 'elements' postulate either one, or two, or three, or four.
it is evident, then, that that which has become must previously have been in process of becoming, and that which is in process of becoming must previously have become: for all magnitudes and all periods of time are infinitely divisible.
data. for (one) everything is capable of acting or being acted upon, of being or not being, either for an infinite, or for a definitely limited space of time; and the infinite time is only a possible alternative because it is after a fashion defined, as a
one such source there is which he calls mind, and mind begins its work of thinking from some starting-point. so necessarily all things must have been together at a certain time, and must have begun to be moved at a certain time.
one hundred three. a line drawn through the intersection of two tangents to a circle is divided harmonically by the circle and the chord of contact.
one. if ef, gh be parallels to the adjacent sides of a parallelogram abcd, the diagonals eh, gf of two of the four picts into which they divide it and one of the diagonals of abcd are concurrent.
thirty two. the four triangles which are determined by four points, taken three by three, are such that their nine-points circles have one common point.
he would not have them too precisely distinguished. thus seems to have arisen the first dim perception of (greek) or matter, which has played so great a part in the metaphysical philosophy of aristotle and his followers.
we must admit, further, (three) that aristotle attributed to plato the doctrine of the rotation of the earth on its axis.
the whole description is figurative, as plato himself implies when he speaks of a 'fountain of fire which we compare to the network of a creel.' he really means by this what we should describe as a state of heat or temperature in the interior of the body.
this, however, is an after-stage-at present, we are only concerned with the creation of the body and soul.
thirty eight. if in the construction of the figure, proposition xlvii., ef, kg be joined,
cor. one.-the line which bisects perpendicularly one of two parallel chords of a circle bisects the other perpendicularly.
nails.' or once more, let us reflect on two serious passages in which the order of the world is supposed to find a place in the human soul and to infuse harmony into it.
thirdly, at the same moment all the b's have passed all the g's: for the first g and the first b will simultaneously reach the opposite ends of the course, since (so says zeno) the time occupied by the first g in passing each of the b's is equal to that
as i said at first, all things were originally a chaos in which there was no order or proportion. the elements of this chaos were arranged by the creator, and out of them he made the world.
i learned from my two years' experience that it would cost incredibly little trouble to obtain one's necessary food, even in this latitude; that a man may use as simple a diet as the animals, and yet retain health and strength.
uniting with the ever-flowing current, they shake the courses of the soul, stopping the revolution of the same and twisting in all sorts of ways the nature of the other, and the harmonical ratios of twos and threes and the mean terms which connect them,
fifteen. if through a, one of the points of intersection of two circles, we draw any line abc, cutting the circles again in b and one hundred, the tangents at b and one hundred intersect at a given angle.
of all other things it would be most natural to suppose that there is alteration in figures and shapes, and in acquired states and in the processes of acquiring and losing these: but as a matter of fact in neither of these two classes of things is there
therefore the animal as a whole moves itself naturally: but the body of the animal may be in motion unnaturally as well as naturally: it depends upon the kind of motion that it may chance to be suffering and the kind of element of which it is composed.
and not the substances themselves-it is a house of cards which we are pulling to pieces and putting together again (compare however laws). yet perhaps plato may regard these sides or faces as only the forms which are impressed on pre-existent matter.
and all this movement, active as well as passive, takes place in order that the body, being watered and cooled, may receive nourishment and life; for when the respiration is going in and out, and the fire, which is fast bound within, follows it, and ever
again, since the angles acb, dfe are xx. the halves of the equal angles aob, dhe, they are equal i. axiom vii. . therefore (def. x.) the segments acb, dfe are similar, and their chords ab, de have been proved equal; therefore xxiv. the segments are equal.
motion to that of the heavens. this at once leads us to expect that the body which is nearest to that first simple revolution should take the longest time to complete its circle, and that which is farthest from it the shortest, the others taking a longer
and inseparable from them. since, however, the 'elements' are seen to be more than two, the contrarieties must at the least be two.
may they not have had, like the animals, an instinct of something more than they knew?
and they, imitating him, received from him the immortal principle of the soul; and around this they proceeded to fashion a mortal body, and made it to be the vehicle of the soul, and constructed within the body a soul of another nature which was mortal,
value of what was on hand at the last date, was
further, it cannot truly be said of a thing now that it exists last year, nor could it be said last year that it exists now. it is therefore impossible for what once did not exist later to be eternal.
yet without this crude use of analogy the ancient physical philosopher would have stood still; he could not have made even 'one guess among many' without comparison.
(two) on the other hand, it is also clear that the stars do not roll. for rolling involves rotation: but the 'face', as it is called, of the moon is always seen.
and it is right to call anything that which nature intends it to be, and which belongs to it, rather than that which it is by constraint and contrary to nature. the evidence of the senses further corroborates this.
are we then to say that the all is composed of indivisible substances? some thinkers did, in point of fact, give way to both arguments.
but if a thing has for infinite time more than one capacity, another time is impossible and the times must coincide. thus if a thing which exists for infinite time is destructible, it will have the capacity of not being.
let us take this as our starting-point. all natural bodies and magnitudes we hold to be, as such, capable of locomotion; for nature, we say, is their principle of movement.
now, that which produces upward and downward movement is that which produces weight and lightness, and that which is moved is that which is potentially heavy or light, and the movement of each body to its own place is motion towards its own form.
being', it is necessary that the contradictories of both terms should be predicable of one and the same thing, and thus that, intermediate between what always is and what always is not, there should be that to which being and not-being are both possible;
cor.-every equilateral triangle is equiangular.
all the four elements entered into the original composition of the human frame; the bone was formed out of smooth earth; liquids of various kinds pass to and fro; the network of fire and air irrigates the veins.
dem.-let the triangle abc be applied to def, so that the point b will coincide with e, and the line bc with the line ef; then because bc is equal to ef, the point one hundred shall coincide with f.
always as 'such'; nor must we imply that there is any stability in any of those things which we indicate by the use of the words 'this' and 'that,' supposing ourselves to signify something thereby; for they are too volatile to be detained in any such
again, (one) some things are per se in place, namely every body which is movable either by way of locomotion or by way of increase is per se somewhere, but the heaven, as has been said, is not anywhere as a whole, nor in any place, if at least, as we must
so when what is within a thing which is moved, is moved and changes its place, as a boat on a river, what contains plays the part of a vessel rather than that of place.
now the process of repletion and evacuation is effected after the manner of the universal motion by which all kindred substances are drawn towards one another.
now since four hundred is greater than ad, it is evident that four hundred is not a common measure of ac and cb, and therefore not a common measure of ab and ac.
two. the projection on either of two intersecting planes of a normal to the other plane is perpendicular to the line of intersection of the planes.
since there must always be motion without intermission, there must necessarily be something, one thing or it may be a plurality, that first imparts motion, and this first movent must be unmoved.
prop. twelve.-problem. to describe a regular pentagon about a given circle (abcde).
take a line, ace, infinite in one direction, e, and another line, bb, infinite in both directions. let ace describe a circle, revolving upon one hundred as centre. in its movement it will cut bb continuously for a certain time.
thus arises the following progression:-moon one, sun two, venus three, mercury four, mars eight, jupiter nine, saturn twenty seven.
time is and at another time is not, that any such thing should without undergoing any process of change at one time be and at another time not be).
now if, when a is transformed into w, the 'white' persists, water will be moist and white: but if it does not persist, water will be black since change is into contraries. water, therefore, must be either white or black. let it then be the first.
the incapacities of animals, age, decay, and the like, are all unnatural, due, it seems, to the fact that the whole animal complex is made up of materials which differ in respect of their proper places, and no single part occupies its own place.
for all things go the wrong way, and having become corrupted, first they taint the blood itself, and then ceasing to give nourishment to the body they are carried along the veins in all directions, no longer preserving the order of their natural courses,
hence, for the purposes of proof, it will make no difference to them to have such an infinite instead, while its existence will be in the sphere of real magnitudes.
division cfpies of necessity a greater space. it is inevitable, too, that generation of one out of another should come to a stop, since a finite quantum cannot contain an infinity of finite quanta.
as we pointed out, where air was, water in turn comes to be, the one replacing the other; and similarly with other bodies. hence the place of a thing is neither a part nor a state of it, but is separable from it.
forcing their way, and so they leave the earth unmelted and undissolved; but particles of water, which are larger, force a passage, and dissolve and melt the earth.
one could not easily put motion and change in another genus-this is plain if we consider where some people put it; they identify motion with or 'inequality' or 'not being'; but such things are not necessarily moved, whether they are 'different' or
six. the transverse lines bk, cg are perpendicular to each other.
becoming in process of becoming. and this again was sometime in process of becoming, so that even then we should not have arrived at what was in process of simple becoming.
on the other hand, suppose (ii) they are not 'comparable in their amount' in the sense that so-much of the one yields so much of the other, but comparable in 'power of action (a pint of water, e.g.
numerically one is theoretically two. we have further the distinction between the potential and the actual.
the 'continuous' is a subdivision of the contiguous: things are called continuous when the touching limits of each become one and the same and are, as the word implies, contained in each other: continuity is impossible if these extremities are two.
but when the flesh wastes and returns into the veins there is discoloured blood as well as air in the veins, having acid and salt qualities, from which is generated every sort of phlegm and bile.
and if we take the case of quantitative change, we shall get a like result, for here too the change is in something continuous. it is evident, then, that only in qualitative motion can there be anything essentially indivisible.
the kinds of matter, then, must be as numerous as these bodies, i.e. four, but though they are four there must be a common matter of all-particularly if they pass into one another-which in each is in being different.
in consequence of what has been said, in part by way of assumption and in part by way of proof, it is clear that not every body either possesses lightness or heaviness.
two. since agh and bgh are adjacent angles, their sum is equal to two right angles xiii.
for if that which is in motion has traversed the distance kl in the primary time chrh, in half the time a thing that is in motion with equal velocity and began its motion at the same time will have traversed half the distance.
and of these two sets of things, since they cannot diverge from the necessity of their nature, it is impossible for the first not to he and impossible for the second to he.
now there is no ratio in which the void is exceeded by body, as there is no ratio of zero to a number.
xiii. points which lie on the circumference of a circle are said to be concyclic.
been compelled, by reason of the factions and troubles which he found stirring in his own country when he came home, to attend to other matters, in my opinion he would have been as famous as homer or hesiod, or any poet.
or thus: bisect the angle bac by ae ix. then the angle bea is greater than eac; but eac eab (const.); therefore the angle bea is greater than eab. hence ab is greater than be xix. . in like manner ac is greater than ec.
motion towards something else as well as the puller. we may similarly classify all the varieties of these last two, e.g. packing and combing: the former is a form of pushing together, the latter a form of pushing apart.
but still we find a trace reappearing of the teaching of anaxagoras: 'all was confusion, and then mind came and arranged things.'
one. if the opposite angles of a quadrilateral be supplemental, it is cyclic.
he is merely using a single idea, putting it as strongly as he can, and neglecting all its limitations. the first discourse is important not for any positive doctrine it contains, but as a key to the development of rousseau's mind.
in one or two cases it might not inconceivably fall out so, but to imagine it in every case alike is a mere fiction. besides, chance has no place in that which is natural, and what happens everywhere and in every case is no matter of chance.
because it is infinite and fixes itself, but owing to some other cause.
it depends either upon whether the material is or is not 'substantial', or upon whether it is more or less 'substantial', or upon whether it is more or less perceptible.
hence motions may be consecutive or successive in virtue of the time being continuous, but there can be continuity only in virtue of the motions themselves being continuous, that is when the end of each is one with the end of the other.
one has only to imagine the movement to be slow where both move and much faster where one is stationary.
dem.-let the same construction be made, we have
consequently, if o actually passed through a after being in process of passing through, the motion must be divisible: for at the time when o was passing through, it neither was at rest nor had completed its passage but was in an intermediate state: while
first, there is the immortal nature of which the brain is the seat, and which is akin to the soul of the universe. this alone thinks and knows and is the ruler of the whole.
universe eternal, so far as might be. now the nature of the ideal being was everlasting, but to bestow this attribute in its fulness upon a creature was impossible.
since every motion is continuous, a motion that is one in an unqualified sense must (since every motion is divisible) be continuous, and a continuous motion must be one.
water which is mingled with fire is called liquid because it rolls upon the earth, and soft because its bases give way.
motion of that which is one must be one) nor of anything else: for a motion that is continuous must be the motion of things that are continuous. and the same result follows if the division of oi reveals a surplus on the side of the motions of the parts.
for in the universe, which is a sphere, there is no opposition of above or below, and that which is to us above would be below to a man standing at the antipodes.
thirteen. the circumference of sectors having equal central angles are proportional to their radii. hence if a, a denote the arcs of two sectors, which subtend equal angles at the centres, and if r, r be their radii, a r a r.
one. the projection of any line on a plane is a right line.
for if one is to result from another, it will have to pass through such a vast number of contrarieties-and indeed even more than any determinate number. consequently (i) into some 'elements' transformation will never be effected-viz.
the angle acg is equal to the adjacent angle bcg, and therefore i. def. xiii. each is a right angle; but the angle acd is right (const.); therefore acd is equal to acg-a part equal to the whole-which is absurd.
thus pushing on is a form of pushing in which that which is causing motion away from itself follows up that which it pushes and continues to push it: pushing off occurs when the movent does not follow up the thing that it has moved: throwing when the
hence, since a b lies between m n- and m -one- n, c d lies between the same quantities. therefore the difference between a b and c d is less than one n; and since n may be as large as we please, the difference is nothing; therefore
let chrh be the primary time in which that which is in motion is in motion: and (as all time is divisible) let it be divided at k. now in the time chk it either is in motion or is not in motion, and the same is likewise true of the time krh.
and, again, it is intelligible that the advocates of both views, although their theories are not the same, are yet in contact with the nature of the facts. for sometimes we speak of the substratum as suffering action (e.g.
thirty eight. four concyclic points taken three by three determine four triangles, the centres of whose nine-points circles are concyclic.
five. the line of connexion of the middle points of two sides of a triangle is equal to half the third side.
when a man who is being restored to health runs or learns: and accidental change we have long ago decided to leave out of account.
twenty seven. if a variable circle touch two fixed circles, the chord of contact passes through a fixed point on the line connecting the centres of the fixed circles.
but people really mean that there is an empty interval in which there is no sensible body. they hold that everything which is in body is body and say that what has nothing in it at all is void (so what is full of air is void).
the enunciation of a problem consists of two parts, namely, the data, or things supposed to be given, and the quaesita, or things required to be done.
god, like man, is supposed to have an ideal of which plato is unable to tell us the origin. he may be said, in the language of modern philosophy, to resolve the divine mind into subject and object.
nine. the sum of the three medians of a triangle is less than its perimeter.
wherefore he made an image of eternity which is time, having an uniform motion according to number, parted into months and days and years, and also having greater divisions of past, present, and future.
this same argument may also be stated in another way as follows. every movent moves something and moves it with something, either with itself or with something else: e.g.
light. as a larger quantity of lead or of bronze is heavier than a smaller-and this holds good of all homogeneous masses, the superior weight always depending upon a numerical superiority of equal parts-in precisely the same way, they assert, lead is
dem.-since cg is parallel to the side bf of the triangle abf, ac : af :: ag : ab ii. ; but ac is the fourth part of af (const.). hence ag is the fourth part of ab five., d. . in the same manner, any other required submultiple may be cut off.
on the other hand, in motion on a circular line we shall find singleness and continuity: for here we are met by no impossible consequence: that which is in motion from a will in virtue of the same direction of energy be simultaneously in motion to a
dem.-the same construction being made, since the arc bc is equal to cg, the angle boc is equal to cog. hence the sectors boc, cog are congruent (see observation, proposition xxix., book three.); therefore they are equal.
to postulate a difference of nature in the simple bodies according as they are more or less distant from their proper places is unreasonable. for what difference can it make whether we say that a thing is this distance away or that?
indeed the reason why we do not hear, and show in our bodies none of the effects of violent force, is easily given: it is that there is no noise.
yet men have come to such a pass that they frequently starve, not for want of necessaries, but for want of luxuries; and i know a good woman who thinks that her son lost his life because he took to drinking water only.
the other and afterwards examine the nature of our hypothesis. in order, then, that the affections may follow regularly after the elements, let us presuppose the existence of body and soul.
before he learnt it. and when he is in this condition, if something does not prevent him, he actively exercises his knowledge: otherwise he would be in the contradictory state of not knowing. in regard to natural bodies also the case is similar.
lack of experience diminishes our power of taking a comprehensive view of the admitted facts.
proportionate to its greatness, cannot move at all. the time should be less in proportion as the weight is greater. but there is no proportion between the infinite and the finite: proportion can only hold between a less and a greater finite time.
for if we take a determinate part of a finite magnitude and add another part determined by the same ratio (not taking in the same amount of the original whole), and so on, we shall not traverse the given magnitude.
we must inquire therefore in what manner chance and spontaneity are present among the causes enumerated, and whether they are the same or different, and generally what chance and spontaneity are.
if then this is so, and if 'biped' is supposed to be an attribute of 'man', it must be either separable, so that 'man' might possibly not be 'biped', or the definition of 'man' must come into the definition of 'biped'-which is impossible, as the converse
the authors of our race were aware that we should be intemperate in eating and drinking, and take a good deal more than was necessary or proper, by reason of gluttony.
the centres of the escribed circles. sixteen.
dem.-since a : c :: b : c, a- c b c. hence, multiplying each by c, we get a b.
empedocles says that fire and earth and the related bodies are elementary bodies of which all things are composed; but this anaxagoras denies. his elements are the homoeomerous things, viz. flesh, bone, and the like.
let a, b, c; a, b, c be the two sets of magnitudes, and let the ratio a : b a : b, and b : c b : c; then, if a be greater than, equal to, or less than c, a will be greater than, equal to, or less than c.
each step then in the series is for the sake of the next; and generally art partly completes what nature cannot bring to a finish, and partly imitates her.
and pray what more can a reasonable man desire, in peaceful times, in ordinary noons, than a sufficient number of ears of green sweet-corn boiled, with the addition of salt?
for nothing except solids results from putting planes together: they do not even attempt to generate any quality from them.
since, however, (ii) this is false, i.e. since every body is divisible, there is no difference between 'having been divided into parts which remain in contact' and 'being divisible'.
the regularity and irregularity of nature; or of chance, the nameless or unknown cause; or of justice, symbolizing the law of compensation; are of the fates and furies, typifying the fixed order or the extraordinary convulsions of nature.
it is clear, then, that by the time that it has become sick, it must also have changed to whatever may be the other change concerned (for that it should be at rest, though logically possible, is excluded by the theory).
the elements therefore cannot be a body which is not an element, and the only remaining alternative is that they are
for some of the older philosophers thought that 'what is' must of necessity be 'one' and immovable.
decrease: of what can come to be and can pass away, coming to he and passing away: of what can be carried along, locomotion.
lastly, plato, though an idealist philosopher, is greek and not oriental in spirit and feeling. he is no mystic or ascetic; he is not seeking in vain to get rid of matter or to find absorption in the divine nature, or in the soul of the universe.
half-motion before the whole for every half-distance that we get, so that we have the result that when the whole distance is traversed we have reckoned an infinite number, which is admittedly impossible.
sol.-through a and b draw any two parallels ac and bd in opposite directions. cut off ac m, and bd n, and join four hundred; the joining line will divide ab internally at e in the ratio of m : n.
the good is the beautiful, and the beautiful is the symmetrical, and there is no greater or fairer symmetry than that of body and soul, as the contrary is the greatest of deformities.
in his treatise de natura deorum, he also refers to the timaeus, which, speaking in the person of velleius the epicurean, he severely criticises.
this tradition was sustained by the great authority of plato, and therefore the legend of the island of atlantis, though not closely connected with the voyages of the early navigators, may be truly said to have contributed indirectly to the great
the doctor has a knowledge of health and also of bile and phlegm, in which health is realized, and the builder both of the form of the house and of the matter, namely that it is bricks and beams, and so forth): if this is so, it would be the part of
it is no more satisfactory to construct a circle and treat motion 'along one semicircle as contrary to motion along the other.
be susceptible of the same divisions (though spheres of motion are not all divisible in a like manner: thus quantity is essentially, quality accidentally divisible). for suppose that a is the time occupied by the motion b.
we ought to try to make our investigation such as will render an account of place, and will not only solve the difficulties connected with it, but will also show that the attributes supposed to belong to it do really belong to it, and further will make
(two) again, if from a finite time a finite time be subtracted, what remains must be finite and have a beginning.
but further, if motion is continuous, it is one: and it is one only if the movent and the moved that constitute it are each of them one, since in the event of a thing's being moved now by one thing and now by another the whole motion will not be
wherefore the varieties of smell have no name, and they have not many, or definite and simple kinds; but they are distinguished only as painful and pleasant, the one sort irritating and disturbing the whole cavity which is situated between the head and
again, density and rarity, as instruments of generation, are equivalent to fineness and coarseness, since the fine is rare, and coarse in their use means dense.
clearly then also 'to come to be so-and-so from not-being' means 'qua not-being'.
for, in fact, whenever a thing is in motion, part of it is at the starting-point while part is at the goal to which it is changing: and consequently a motion finds its true contrary rather in another motion than in a state of rest.
finite both in respect of the quantity subtracted and of the number of times a subtraction is made.
two other points strike us in the use which the ancient philosophers made of numbers.
the same construction being made, because ab.f four hundred.e, and that f is equal to ah, and e to one hundred one, we have the parallelogram ag ck; and since these parallelograms are equiangular, the sides about their equal angles are reciprocally
this conclusion follows not only from the present argument but from the consideration that the opposite assumption implies the divisibility of the indivisible.
there is a similarity between the timaeus and the fragments of philolaus, which by some has been thought to be so great as to create a suspicion that they are derived from it.
by the finite traversing the infinite either by locomotion over it or by measuring it. therefore, since this is impossible, the infinite cannot traverse the finite.
the first of those who studied science were misled in their search for truth and the nature of things by their inexperience, which as it were thrust them into another path.
for though strife 'dissociated', it was not by strife that the 'ether' was borne upwards.
future. the later forms of such narratives contained features taken from the edda, as well as from the old and new testament; also from the tales of missionaries and the experiences of travellers and of colonists.
the atomists are committed to the view that every 'indivisible' is incapable alike of receiving a sensible property (for nothing can 'suffer action' except through the void) and of producing one-no 'indivisible' can be, e.g. either hard or cold.
and the ratio is intermediate to each pair. it is evident we may continue the series as far as we please.
dem.-since a is greater than b, dividing each by c,
twenty nine. if the position of the common point in the last question be given, the three angles of the triangle xy z are given, and conversely.
the divine nature. but of this enough for the present.
for if the whole body suffers action under these conditions, it would suffer action in the same way even if it had no pores but were just its own continuous self. moreover, how can their account of 'vision through a medium' be correct?
forty nine. the sum of the equilateral triangles described on the legs of a right-angled triangle is equal to the equilateral triangle described on the hypotenuse.
similar ratio. thus all the other bodies will result from the contraries, or rather from the 'elements', in so far as these have been 'combined': while the elements' will result from the contraries, in so far as these 'exist potentially' in a special
naturally they turned to their kindred earth, and put their forelegs to the ground, and their heads were crushed into strange oblong forms.
cor. three.-the parallelogram ac : gh :: ac : ad.six hundred.
three. if a quadrilateral be circumscribed to a circle, the sum of one pair of opposite sides is equal to the sum of the other pair.
and its newer triangles cut them up, and so the animal grows great, being nourished by a multitude of similar particles.
anything else might be in a state of rest, but there is no reason why it should not be its nature to be moved. the earth is not carried along, and would not be carried along if it were infinite, provided it is held together by the centre.
deprived of fire: and so the eye no longer sees, and we feel disposed to sleep.
it is the fulfilment of this potentiality, and by the action of that which has the power of causing motion; and the actuality of that which has the power of causing motion is not other than the actuality of the movable, for it must be the fulfilment of
formerly was, must have ceased-to-be at some time, the 'nows' too cannot be simultaneous with one another, but the prior 'now' must always have ceased-to-be.
there is no less difference between above and below or front and back in animals generally than between right and left.
(one) as the finger is 'in' the hand and generally the part 'in' the whole.
four. being given a side of a regular pentagon, construct it.
for since it passes over the greater magnitude in less time than the slower, and (regarded by itself) passes over lm the greater in more time than sixty the lesser, the time prh in which it passes over lm will be more than the time ps, which it passes
clearly, too, if there were no time, there would be no 'now', and vice versa. just as the moving body and its locomotion involve each other mutually, so too do the number of the moving body and the number of its locomotion.
they are depriving the element of its priority. generation out of the elements is, as they say, synthesis, and generation into the elements is analysis, so that the body with the finer parts must have priority in the order of nature.
sixty seven. the four sides of a cyclic quadrilateral are given; construct it.
but nothing can dissolve air, when strongly condensed, which does not reach the elements or triangles; or if not strongly condensed, then only fire can dissolve it.
this explains why it is not the primary motion that causes coming-to-be and passingaway, but the motion along the inclined circle: for this motion not only possesses the necessary continuity, but includes a duality of movements as well.
corresponding magnitudes is of their submultiple (c) (compare proposition ii.).
twelve. four circles are described on the sides of a quadrilateral as diameters. the common chord of any two on adjacent sides is parallel to the common chord of the remaining two.
the side af is equal to ag; therefore gf is double af. in like manner jf is double ef; but af is equal to ef; hence gf is equal to jf.
thirty three. for the circumscribed circle, the constant is equal to n times the square of the radius of the inscribed circle, together with one n times the square of the radius of the circumscribed circle.
these have many meanings, and though it may make no difference to the argument, yet some confusion of mind must result from treating as uniform in its use a word which has several distinct applications.
fire and earth, moreover, are extremes and purest: water and air, on the contrary are intermediates and more like blends.
it is obvious that of these terms 'in succession' is first in order of analysis: for that which touches is necessarily in succession, but not everything that is in succession touches: and so succession is a property of things prior in definition, e.g.
(two) plato is perfectly aware-and he could hardly be ignorant-that blood is a fluid in constant motion.
god gave us the faculty of sight that we might behold the order of the heavens and create a corresponding order in our own erring minds.
not to be; and also (c) in a third sense, to that which is easily destructible, to the 'easily destroyed', so to speak. of the indestructible the same account holds good.
this proof is shorter than the usual one, since it is not necessary to prove that ac, four hundred are in one right line.
natural one. in thinking of the life and moving principle of the several heavens one must regard the first as far superior to the others. such a superiority would be reasonable.
and since they thought that the 'truth lay in the appearance, and the appearances are conflicting and infinitely many, they made the 'figures' infinite in number.
so, when you have a homogeneous substance which is continuous, the parts are potentially in place: when the parts are separated, but in contact, like a heap, they are actually in place.
in so far then as it is separable from the thing, it is not the form: qua containing, it is different from the matter.
two ; but the centre is equally distant from a and b; hence the centre must be in ed; and since it must be equally distant from e and five hundred, it must be the middle point of de.
ninety nine. if two circles ten, y be so related that a triangle may be inscribed in ten and circumscribed about y , an infinite number of such triangles can be constructed.
placed in the liver the seat of divination. and herein is a proof that god has given the art of divination not to the wisdom, but to the foolishness of man.
wherefore also god in the beginning of creation made the body of the universe to consist of fire and earth. but two things cannot be rightly put together without a third; there must be some bond of union between them.
it was through failure to make this distinction that those thinkers gave the matter up, and through this error that they went so much farther astray as to suppose that nothing else comes to be or exists apart from being itself, thus doing away with all
end, and action always requires two terms, end and means. the lower animals have less variety of action than man; and plants perhaps have little action and of one kind only.
it is evident (i) that any and every part of the growing thing has increased, and that similarly in diminution every part has become smaller: also (ii) that a thing grows by the accession, and diminishes by the departure, of something.
for man remains a man and is such even when he becomes musical, whereas what is not musical or is unmusical does not continue to exist, either simply or combined with the subject.
in the first place, there is the caste of priests, which is separated from all the others; next, there are the artificers, who ply their several crafts by themselves and do not intermix; and also there is the class of shepherds and of hunters, as well as
dem.-from ac cut off ad equal to ab iii . join bd (post. i.). now since ab is equal to ad, the triangle abd is isosceles; therefore v. the angle adb is equal to abd; but the angle adb is greater than the angle acb xvi. ; therefore abd is greater than acb.
the topics which i propose briefly to reconsider are (a) the relation of the timaeus to the other dialogues of plato and to the previous philosophy; (b) the nature of god and of creation (c) the morality of the timaeus:-
of compound ratio , the enunciation is the same as if we said, "in the ratio compounded of the ratios of the sides," which is euclid's; but it is more easily understood as we have put it.
whewell and others with wasting their fine intelligences in wrong methods of enquiry; and their progress in moral and political philosophy has been sometimes contrasted with their supposed failure in physical investigations.
are we to suppose that body and magnitude come-to-be out of something which, though potentially magnitude and body, is actually incorporeal and devoid of magnitude?
the soul was first made by him-the elder to rule the younger; not in the order in which our wayward fancy has led us to describe them, but the soul first and afterwards the body.
plato found heat and air within the human frame, and the blood circulating in every part. he assumes in language almost unintelligible to us that a network of fire and air envelopes the greater part of the body.
on a given right line (ab) to construct a rectilineal figure similar to a given one (cdefg), and similarly placed as regards any side (four hundred) of the latter.
so inconsistent are the forms in which he describes the works which no tongue can utter-his language, as he himself says, partaking of his own uncertainty about the things of which he is speaking.
between the ideal and the sensible plato interposes the two natures of time and space. time is conceived by him to be only the shadow or image of eternity which ever is and never has been or will be, but is described in a figure only as past or future.
two. in the demonstration of prop. vii. the contrapositive of prop. v. occurs; show where.
might not, by being crowded and pressed and matted together, destroy sensation by reason of its hardness, and impair the memory and dull the edge of intelligence.
thirty four. construct the greatest triangle equiangular to a given one whose sides shall touch three given circles.
vain would be the labour of telling all the figures of them, moving as in dance, and their juxta-positions and approximations, and when and where and behind what other stars they appear to disappear-to tell of all this without looking at a plan of them
let agh be the greater; to each add bgh, and we have the sum of the angles agh, bgh greater than the sum of the angles bgh, ghd; but the sum of agh, bgh is two right angles; therefore the sum of bgh, ghd is less than two right angles, and therefore (axiom
so if the whole motion oi may be divided into the motions of the parts, oi will be equal to dz: if on the other hand there is any remainder, say ki, this will be a motion of nothing: for it can be the motion neither of the whole nor of the parts (as the
prop. one.-problem. on a given finite right line (ab) to construct an equilateral triangle.
again, if the possession of angles makes a body produce heat and combustion, every element produces heat, though one may do so more than another.
earth is converted into pottery when the watery part is suddenly drawn away; or if moisture remains, the earth, when fused by fire, becomes, on cooling, a stone of a black colour.
what nature is, then, and the meaning of the terms 'by nature' and 'according to nature', has been stated.
and he who would understand what he remembers to have been said, whether in a dream or when he was awake, by the prophetic and inspired nature, or would determine by reason the meaning of the apparitions which he has seen, and what indications they afford
for when a is passing b, then b overlaps a and it makes no difference whether b is moved or unmoved, except that, if both move, they pass clear of one another more quickly.
similarly, therefore, if air which is large in extent comes to have a smaller volume, or becomes greater from being smaller, it is the matter which is potentially both that comes to be each of the two.
he was finished and smooth, having neither eyes nor ears, for there was nothing without him which he could see or hear; and he had no need to carry food to his mouth, nor was there air for him to breathe; and he did not require hands, for there was
two. since the angle egb is equal to agh xv. , and ghd is equal to agh (one), egb is equal to ghd (axiom i.).
fifteen. when are two lines divided proportionally?
this is the case, for instance, with the bodies of animals and their parts and with vegetable bodies, and similarly also with those of the elements.
sol.-take any point five hundred in ab. join four hundred (post. i.), and make the angle dcf equal to the angle adc xxiii. . the line ce is parallel to ab xxvii. .
it is agreed that there are only three plane figures which can fill a space, the triangle, the square, and the hexagon, and only two solids, the pyramid and the cube.
it follows then, that there will be a process of change previous to the first.
he observed that earth, water, and air had settled down to their places, and he imagined fire or the exterior aether to have a place beyond air.
hence we must not believe the old tale which says that the world needs some atlas to keep it safe-a tale composed, it would seem, by men who, like later thinkers, conceived of all the upper bodies as earthy and endowed with weight, and therefore supported
again, if body is distinguished from body by the appropriate qualitative difference, and there is a limit to the number of differences (for the difference lies in qualities apprehended by sense, which are in fact finite in number, though this requires
but as a matter of fact that which primarily moves itself cannot contain either a single part that moves itself or a number of parts each of which moves itself.
the sound which moves swiftly is acute; that which moves slowly is grave; that which is uniform is smooth, and the opposite is harsh. loudness depends on the quantity of the sound. of the harmony of sounds i will hereafter speak.
necessarily, then, movements also will be either simple or in some sort compound-simple in the case of the simple bodies, compound in that of the composite-and in the latter case the motion will be that of the simple body which prevails in the
but alongside the all or the whole there is nothing outside the all, and for this reason all things are in the heaven; for the heaven, we may say, is the all. yet their place is not the same as the heaven.
if on the other hand (a) both are in what is moved and acted on-both the agency and the patiency (e.g.
the first expression of this was some element, rarefied by degrees into a pure abstraction, and purged from any tincture of sense.
also the classes in which the incidental attribute is included; thus 'a man' could be said to be the cause of a statue or, generally, 'a living creature'. an incidental attribute too may be more or less remote, e.g.
cor. one.-if two triangles abc, def have two sides in one proportional to two sides in the other, ab : bc :: de : ef, and the angles a, five hundred opposite one pair of homologous sides equal, the angles one hundred, f opposite the other are either equal
iv. again, is the substance of all those solids uniform, or do they fall into sets which differ from one another-as if, e.g. some of them, in their aggregated bulk, were 'fiery', others earthy'?
and of the organs they first contrived the eyes to give light, and the principle according to which they were inserted was as follows: so much of fire as would not burn, but gave a gentle light, they formed into a substance akin to the light of every-day
twelve. if the base ab of a triangle be divided in five hundred, so that mad nbd, then
two. the parallelograms about the diagonal of a square are squares.
this may be shown as follows. let ab be a finite magnitude, and let us suppose that it is traversed in infinite time g, and let a finite period gd of the time be taken.
contrary and transverse and every way diverse in relation to that which is light, heavy, below or above in an opposite place.
cor.-if ac (fig. two) be produced to e, oe joined, cutting the circle bcd in five hundred, and the line dp drawn, dp will be another tangent from p.
of perplexity. there is a similar confusion about necessity and free-will, and about the state of the soul after death. also he sometimes supposes that god is immanent in the world, sometimes that he is transcendent.
three. name the primary concepts of geometry. ans. points, lines, surfaces, and solids.
for those who hold that the void exists regard it as a sort of place or vessel which is supposed to be 'full' when it holds the bulk which it is capable of containing, 'void' when it is deprived of that-as if 'void' and 'full' and 'place' denoted the same
one of these two motions it must be. and (one) rest in whiteness is of course not contrary to rest in health.
again, suppose the body to exist in dispersion, it may be maintained none the less that the total of all these scattered particles, say, of fire, is infinite. but body we saw to be that which has extension every way.
and here, like plato (the insertion of a note in the text of an ancient writer is a literary curiosity worthy of remark), we may take occasion to correct an error.
this is indicated by the fact that rings also that have no bezel are described as 'endless', because it is always possible to take a part which is outside a given part.
hear me, then:-three tribes of mortal beings have still to be created, but if created by me they would be like gods.
that which is bound may be dissolved, but only an evil being would dissolve that which is harmonious and happy. and although you are not immortal you shall not die, for i will hold you together.
hence the sum of the two remaining angles abc, cab is one right angle; but the angle baf is right (const.); therefore the sum of the angles abc, bac is equal to baf. reject bac, which is common, and we get the angle abc equal to the angle fac.
the causes of respiration have now to be considered. the exhalation of the breath through the mouth and nostrils displaces the external air, and at the same time leaves a vacuum into which through the pores the air which is displaced enters.
(that is why a drop of wine does not 'combine' with ten thousand gallons of water: for its form is dissolved, and it is changed so as to merge in the total volume of water.) on the other hand (ii) when there is a certain equilibrium between their 'powers
seventeen. give euclid's definition of proportion.
world, but similarly to them, since in form no portion differs from another. the result is that we must either abandon our present assumption or assert that the centre and the extremity are each numerically one.
two. how many in the conclusion? name them.
if, then, the non-realization of the existence of time happens to us when we do not distinguish any change, but the soul seems to stay in one indivisible state, and when we perceive and distinguish we say time has elapsed, evidently time is not
when one of the other elements is fastened upon by fire, and is cut by the sharpness of its angles and sides, it coalesces with the fire, and then ceases to be cut by them any longer.
castor berry, oil itself, and other things of a like kind: thirdly, there is the class of substances which expand the contracted parts of the mouth, until they return to their natural state, and by reason of this property create sweetness;-these are
in other words, their coming-to-be will not be necessary.
it is evident, then, that in contradictory change that which has changed must be in that to which it has changed.
consequently the phenomenon in question occurs, because the 'irregular' coming-to-be of these things is the passing-away of other things.
it is thus, then, that in the first place the 'elements' are transformed; and that (in the second place) out of the 'elements' there come-to-be flesh and bones and the like-the hot becoming cold and the cold becoming hot when they have been brought to the
then because ad is equal to ac, the angle acd is equal to adc (v.); therefore the angle bcd is greater than the angle bdc; hence the side bd opposite to the greater angle is greater than bc opposite to the less xix. .
we have assumed, and have proved, that coming-to-be and passing-away happen to things continuously; and we assert that motion causes coming-to-be.
'damp' is that which has foreign moisture on its surface ('sodden' being that which is penetrated to its core), while 'dry' is that which has lost foreign moisture.
which they are used are far removed from one another, while sometimes there is a certain likeness between them, and sometimes again they are nearly related either generically or analogically, with the result that they seem not to be equivocal though they
cor. two.-a line cannot meet a circle in more than two points.
are no better-not that one mean to depreciate them; but every one can see that they are a tribe of imitators, and will imitate best and most easily the life in which they have been brought up; while that which is beyond the range of a man's education he
in existence either always or usually. that which is by chance, then, is by nature such as to exist at one time and not at another.
one hundred. in the same case, the circle inscribed in the triangle formed by joining the points of contact on y touches a given circle.
when, therefore, we perceive the 'now' one, and neither as before and after in a motion nor as an identity but in relation to a 'before' and an 'after', no time is thought to have elapsed, because there has been no motion either.
divides each pair of opposite sides into inversely proportional segments; is divided by each pair of opposite lines into segments which, measured from the centre, are proportional to the sides; is divided by both pairs of opposite sides into segments
yet we are frequently told that the one class of them is the reality and the other appearance; and one is often spoken of as the double or reflection of the other.
prop. nine.-problem. about a given square (abcd) to describe a circle.
motion in a present, it is clear that nothing can be at rest in a present either.
both teaching and learning, though they are two, in the learner), then, first, the actuality of each will not be present in each, and, a second absurdity, a thing will have two motions at the same time.
what initiates it must be single, unmoved, ungenerated, and incapable of 'alteration'; and if the circular movements are more than one, their initiating causes must all of them, in spite of their plurality, be in some way subordinated to a single
wherefore, the mother and receptacle of all created and visible and in any way sensible things, is not to be termed earth, or air, or fire, or water, or any of their compounds or any of the elements from which these are derived, but is an invisible and
nor, again, can a point be in succession to a point or a moment to a moment in such a way that length can be composed of points or time of moments: for things are in succession if there is nothing of their own kind intermediate between them, whereas that
again, it is easy to see that the triangles bch, fag are equiangular; therefore bc : ch :: af : ag; hence bc.ag af.ch, or ad.ag af.ch; but we have proved ab.ae af.ah. hence ad.ag ab.ae af.ac.
similarly contraries are the extreme points of processes of increase and decrease: the limit of increase is to be found in the complete magnitude proper to the peculiar nature of the thing that is increasing, while the limit of decrease is the complete
dem.-bisect ac in e x. . join be (post. i.). produce it, and from the produced part cut off ef equal to be iii . join cf.
this is why plato in the timaeus says that matter and space are the same; for the 'participant' and space are identical.
there is also another difficulty involved in the view that remaining in a particular place is contrary to motion from that place.
we must then first look at whatever others have said, and formulate the questions which require settlement in the interests of this inquiry, before we go on to state our own view of the matter.
cor. two.-the area of a triangle is equal to half the rectangle contained by its base and perpendicular.
that time should be composed of moments, in which case at any moment it would have completed a motion or a change, so that it would never be in motion, but would always have been in motion.
again, the time will be divisible similarly to the motion: for if the whole motion occupies all the time half the motion will occupy half the time, and less of the motion again will occupy less of the time.
now those who decide for a single element, which is either water or air or a body finer than water and denser than air, and proceed to generate other things out of it by use of the attributes density and rarity, all alike fail to observe the fact that
dem.-the parallelogram am is equal to ge xliii. ; but ge is equal to the triangle one hundred (const.); therefore am is equal to the triangle one hundred. again, the angle abm is equal to ebg xv.
and so in the vessel of the head, they first of all put a face in which they inserted organs to minister in all things to the providence of the soul, and they appointed this part, which has authority, to be by nature the part which is in front.
shields and spears, a custom which the goddess first taught you, and then the asiatics, and we among asiatics first received from her.
and the moisture, rising up under the sutures, watered and closed in the skin upon the crown, forming a sort of knot.
three. through a given point draw a line so that the portion intercepted by the legs of a given angle may be bisected in the point.
the thoughts of men widened to receive experience; at first they seemed to know all things as in a dream: after a while they look at them closely and hold them in their hands. they begin to arrange them in classes and to connect causes with effects.
dem.-from the construction it is evident that the figures are equiangular, and it is only required to prove that the sides about the equal angles are proportional. now because the triangle abh is equiangular to cde, ab : bh :: four hundred : de iv.
ab is equal to four hundred, and ac to bd; the angle bac to the angle bdc, and the triangle abc to the triangle bdc.
(three) that the infinite cannot move may also be shown as follows. let a be a finite line moving past the finite line, b. of necessity a will pass clear of b and b of a at the same moment; for each overlaps the other to precisely the same extent.
it would be an unworthy thing to liken it to any nature which exists as a part only; for nothing can be beautiful which is like any imperfect thing; but let us suppose the world to be the very image of that whole of which all other animals both
the world has received animals, mortal and immortal, and is fulfilled with them, and has become a visible animal containing the visible-the sensible god who is the image of the intellectual, the greatest, best, fairest, most perfect-the one only-begotten
it is in circular movement, therefore, and in cyclical coming-to-be that the 'absolutely necessary' is to be found.
in the same way the being-in-motion will also be divisible. for let g be the whole being-in-motion.
many curious and, to the early thinker, mysterious properties of them came to light when they were compared with one another.
we must therefore make a fresh start and consider the question; if a thing moves itself, in what sense and in what manner does it do so?
loss of such magnitude. locomotion, it is true, we cannot show to be finite in this way, since it is not always between contraries.
both are then, as i have said, incidental causes-both chance and spontaneity-in the sphere of things which are capable of coming to pass not necessarily, nor normally, and with reference to such of these as might come to pass for the sake of something.
everything whose motion is continuous must, on arriving at any point in the course of its locomotion, have been previously also in process of locomotion to that point, if it is not forced out of its path by anything: e.g.
he was ready to explain the phenomena of the heavens by the most trivial analogies of earth. the experiments which nature worked for him he sometimes accepted, but he never tried experiments for himself which would either prove or disprove his theories.
for time is not motion, but 'number of motion': and what is at rest, also, can be in the number of motion.
if, therefore, to be in itself is the place of the infinite, that also will be appropriate to the part. therefore it will remain in itself.
of heaven, and that the animals, as many as nature intended, might participate in number, learning arithmetic from the revolution of the same and the like.
the occasion of the rehearsal was the day of the apaturia called the registration of youth, at which our parents gave prizes for recitation. some poems of solon were recited by the boys.
two were cut down the back, along the back bone, where the skin and flesh meet, one on the right and the other on the left, having the marrow of generation between them.
but with equal swiftness; the remaining four, moon, saturn, mars, jupiter, with unequal swiftness to the former three and to one another.
space is said by plato to be the 'containing vessel or nurse of generation.'
in the first place they allow that a thing may come to be without qualification from not being, accepting on this point the statement of parmenides.
(the proof of this is given in our discussion of movement, where it is shown that no finite thing possesses infinite power, and no infinite thing finite power.) if then that which moves naturally can also move unnaturally, there will be two infinites, one
together, i.e. 'associates', homogeneous and heterogeneous things alike.
in e both one hundred and five hundred must be present, for either a or one hundred is present everywhere and therefore in e. since then a is impossible, one hundred must be present, and the same argument holds of five hundred.
he would have seen the world pervaded by number and figure, animated by a principle of motion, immanent in a principle of rest.
sensible body, alongside the so-called elements. everything can be resolved into the elements of which it is composed.
again, his account of motion is vague. for it is not an adequate explanation to say that 'love and strife set things moving, unless the very nature of love is a movement of this kind and the very nature of strife a movement of that kind.
therefore the four figures ag, one thousand one, il, jf are all equal; hence their sum is equal to four times ag; and the square co has been proved to be equal to four times ck.
if water, air, and the like are a 'matter' of which the natural bodies consist, as some thinkers in fact believe, these 'clements' must be either one, or two, or more. now they cannot all of them be one-they cannot, e.g.
since everything to which motion or rest is natural is in motion or at rest in the natural time, place, and manner, that which is coming to a stand, when it is coming to a stand, must be in motion: for if it is not in motion it must be at rest: but that
hence the sum of the squares on ac, cb exceeds the square on ab by twice ce-that is, by bc.four hundred.
and the month is accomplished when the moon has completed her orbit and overtaken the sun, and the year when the sun has completed his own orbit.
the theory resembles that about the stone being worn away by the drop of water or split by plants growing out of it: if so much has been extruded or removed by the drop, it does not follow that half the amount has previously been extruded or removed in
for alteration is movement in respect of quality; and qualitative states and dispositions, such as health and disease, do not come into being without changes of properties.
socrates: i should like, before proceeding further, to tell you how i feel about the state which we have described.
sixty nine. if four points a, b, one hundred, five hundred be collinear, find the locus of the point p at which ab and four hundred subtend equal angles.
motion is one generically according to the different categories to which it may be assigned: thus any locomotion is one generically with any other locomotion, whereas alteration is different generically from locomotion.
one. given the altitude of a triangle and the base angles, construct it.
if a right line (ab) be normal to a plane (one hundred one), any plane (de) passing through it shall be perpendicular to that plane.
this is what happens when tin is combined with bronze. for some things display a hesitating and ambiguous attitude towards one another-showing a slight tendency to combine and also an inclination to behave as 'receptive matter' and 'form' respectively.
flesh is said to have been 'altered' if, while its character and substance remain, some one of its essential properties, which was not there before, now qualifies it: on the other hand, that 'whereby' it has been 'altered' may have undergone no change,
but since pt is a tangent, and ot drawn from the centre to the point of contact, the angle otp is right xviii. . hence ot pt is equal to op .
the air has a double ingress and a double exit, through the mouth or nostrils, and through the skin.
man's gain is not another's loss, and that the stable-boy has equal cause with his master to be satisfied?
if, therefore, artificial products are for the sake of an end, so clearly also are natural products. the relation of the later to the earlier terms of the series is the same in both.
nor does it matter if the thing need not rest in the contradictory state, or if there is no state of rest as a contrary to the process of change: it may be true that the non-existent is not at rest, and that perishing is a process to the non-existent.
and there is least of definiteness, whenever either in describing the beginning or the end of the world, he has recourse to myths.
dem.-one. let the equal angles be so placed as to be vertically opposite, and that ac, four hundred may form one right line; then it may be demonstrated, as in the last proposition, that bc, ce form one right line. join bd.
four. if a line ab be divided in one hundred so that ac cb ; prove that ab bc ab.ac.
sets her free; but when there is not so much of it, and the body though wasted still holds out, the bile is itself mastered, and is either utterly banished, or is thrust through the veins into the lower or upper belly, and is driven out of the body like
but much the safest plan is to speak of them as follows:-anything which we see to be continually changing, as, for example, fire, we must not call 'this' or 'that,' but rather say that it is 'of such a nature'; nor let us speak of water as 'this'; but
the first agent being unaffected, while the last agent only acts by suffering action itself.
dem.-let ce be the common section of the planes de, one hundred one. from any point f in ce draw fg in the plane de parallel to ab i. xxxi. .
dem.-let ab, four hundred meet the plane ten in the points b, five hundred. join bd. then the lines ab, bd, four hundred are coplanar. now in the plane ten, to which ab is normal, draw de at right angles to bd. take any point e in de, and join be, ae, ad.
now, if there is no void at all, and if, as those who take this view say, there is no expansion of bodies, the impossibility of this is manifest: and if there is void and expansion, there is no accounting for the fact that the body which results from
but the series must stop somewhere, since the kinds of motion are limited; and if we say that the process is reversible, and that that which is causing alteration is in process of locomotion, we do no more than if we had said at the outset that that which
again, it may be shown in the following way that there can be no motion of a point or of any other indivisible. that which is in motion can never traverse a space greater than itself without first traversing a space equal to or less than itself.
and v., we see that if two triangles possess either condition, they also possess the other. triangles are unique in this respect. in all other rectilineal figures one of the conditions may exist without the other.
the parts of the body too must be treated in the same way-they should receive their appropriate exercise.
something else and so on throughout the series: and so we proceeded to the position that the first principle that directly causes things that are in motion to be moved is that which moves itself, and the first principle of the whole series is the unmoved.
for if four exceeds three by one, and two by more than one, and one by still more than it exceeds two, still there is no ratio by which it exceeds zero; for that which exceeds must be divisible into the excess that which is exceeded, so that will be what
for, as we affirm, a thing can only remain the same with itself, whole and sound, when the same is added to it, or subtracted from it, in the same respect and in the same manner and in due proportion; and whatever comes or goes away in violation of these
the fallacy of the reasoning lies in the assumption that a body occupies an equal time in passing with equal velocity a body that is in motion and a body of equal size that is at rest; which is false.
and now we must consider the same question in the case of becoming and perishing: how is one becoming of equal velocity with another?
for the most part the farmer gives to his cattle and hogs the grain of his own producing, and buys flour, which is at least no more wholesome, at a greater cost, at the store.
the compounds of earth and water are not soluble by water, but by fire only, and for this reason:-neither fire nor air melt masses of earth; for their particles, being smaller than the interstices in its structure, have plenty of room to move without
dem.-since ab is in the plane ten, it can be produced in it bk. i. post. ii. ; let it be produced to five hundred. then, if bc be not in ten, let any other plane passing through ad be turned round ad until it passes through the point one hundred.
therefore h must have come to a stand there: it cannot have come to be at five hundred and departed from five hundred simultaneously, for in that case it would simultaneously be there and not be there at the same moment.
now it is clear that the quicker will traverse the same magnitude in less time than this: let us say in the time zo.
there are similar differences in the air; of which the brightest part is called the aether, and the most turbid sort mist and darkness; and there are various other nameless kinds which arise from the inequality of the triangles.
for a single clod moves to the same place as the whole mass of earth, and a spark to the same place as the whole mass of fire. so that if it be weight that all possess, no body is, strictly speaking, light: and if lightness be universal, none is heavy.
as to the thinkers who explain modification of property through the movement facilitated by the pores, if this is supposed to occur notwithstanding the fact that the pores are filled, their postulate of pores is superfluous.
if then we accept both the former argument and this one, we must, to preserve both, assume a third somewhat as the substratum of the contraries, such as is spoken of by those who describe the all as one nature-water or fire or what is intermediate between
self-moved, in that it has no dependent or projecting part, as a rectilinear figure has, and is in fact as far as possible removed in shape from ambulatory bodies.
in other words, will that which is only potentially a 'this' (which only potentially is), while without the qualification 'potentially' it is not a 'this' (i.e. is not), possess, e.g. any determinate size or quality or position?
if that is impossible, and the places are finite, the whole too must be finite; for the place and the body cannot but fit each other.
when these processes take place in regular order the body is in health.
but there still remains the question whether, supposing that two alterations are of equal velocity, we ought to look for this equality in the sameness (or likeness) of the affections, or in the things altered, to see e.g.
now, when they should be implanted in bodies by necessity, and be always gaining or losing some part of their bodily substance, then in the first place it would be necessary that they should all have in them one and the same faculty of sensation, arising
nothing grows in the process; unless indeed there be something common to both things (to that which is coming-to-be and to that which passed-away), e.g. 'body', and this grows.
(the universal is more knowable in the order of explanation, the particular in the order of sense: for explanation has to do with the universal, sense with the particular.) 'the great and the small', for example, belong to the former class, 'the dense and
for (i) if all of them are uniform in substance, what is it that separated one from another?
that which is coming to be and that which was previously there, while if it is incorporeal, there must be an extra-corporeal void. but we have already shown that this is impossible.
place is thought to be something important and hard to grasp, both because the matter and the shape present themselves along with it, and because the displacement of the body that is moved takes place in a stationary container, for it seems possible that
hence a b lies between m- n and m- -one n.
separated out; for the flesh left would be less than the minimum of flesh.
but chance also and spontaneity are reckoned among causes: many things are said both to be and to come to be as a result of chance and spontaneity.
socrates: excellent, timaeus, i like your manner of approaching the subject-proceed.
for, as we are saying, none of the other philosophers made any definite statement about growth, except such as any amateur might have made.
cor.-the parallelograms af, fc, ac are, two by two, homothetic.
one. the angle between two planes is equal to the angle between two intersecting normals to these planes.
regard beauty, strength, and all the other bodily excellences and defects.
and if this is true in this kind of change, it will be true in all other kinds as well: for in this matter what holds good in the case of one will hold good likewise in the case of the rest.
with regard to coming to a standstill the question may be raised whether there is an opposite state of rest to unnatural as well as to natural motions.
the rectangle contained by the side of an inscribed square standing on the base of a triangle, and the sum of the base and altitude, is equal to twice the area of the triangle.
at the same time these two are also contrary to each other: for it would be absurd to suppose that there are contrary motions and not opposite states of rest. states of rest in contraries are opposed.
'to be not good', and so the same thing will be 'good' and 'not good', and man and horse; in fact, their view will be, not that all things are one, but that they are nothing; and that 'to be of such-and-such a quality' is the same as 'to be of
of these objections, then, the first-mentioned motion to opposites is not always the same and numerically one a correct statement; in fact, this may be said to be a necessary conclusion, provided that it is possible for the motion of that which is one and
thirty four. prove the converse of ptolemy's theorem (see xvii., ex. thirteen).
v. if a solid angle be composed of three plane angles it is called a trihedral angle; if of four, a tetrahedral angle; and if of more than four, a polyhedral angle.
(we make the same distinction in speaking of the agent: for sometimes we say that 'the man', but at other times that 'what is hot', produces heat.) now the one group of thinkers supposed that agent and patient must possess something identical, because
dem.-join ad, six hundred. since adc is a semicircle, the angle adc is right three. xxxi. . hence, since adc is a right-angled triangle, and bd a perpendicular from the right angle on the hypotenuse, bd is a mean proportional viii. cor.
of the mathematical figure were always the same with that of the thing or the natural centre.
cd ad ac i. xlvii. , and bd ad ab , we get bc ac ab bc.four hundred. therefore ab is less than bc ac by bc.four hundred.
it is plain that one must suppose as many distinct species of matter as there are bodies.
now, since time is number, the 'now' and the 'before' and the like are in time, just as 'unit' and 'odd' and 'even' are in number, i.e. in the sense that the one set belongs to number, the other to time. but things are in time as they are in number.
this great island lay over against the pillars of heracles, in extent greater than libya and asia put together, and was the passage to other islands and to a great ocean of which the mediterranean sea was only the harbour; and within the pillars the
to understand its value we must grasp its limitations; when the questions it answers seem unnaturally put, we must not conclude that they are meaningless; we must see if the answer still holds when the question is put in a more up-to-date form.
every body must necessarily be either finite or infinite, and if infinite, either of similar or of dissimilar parts. if its parts are dissimilar, they must represent either a finite or an infinite number of kinds.
the younger gods the fashioning of their mortal bodies, and desired them to furnish what was still lacking to the human soul, and having made all the suitable additions, to rule over them, and to pilot the mortal animal in the best and wisest manner which
now is the time to explain what was before obscurely said: there was an error in imagining that all the four elements might be
we have already discussed the first heaven and its parts, the moving stars within it, the matter of which these are composed and their bodily constitution, and we have also shown that they are ungenerated and indestructible.
syllables or first compounds. and let me say thus much: i will not now speak of the first principle or principles of all things, or by whatever name they are to be called, for this reason-because it is difficult to set forth my opinion according to the
but water and speech are not commensurable in respect of clearness, since that which primarily contains the attribute is different in the two cases.
now since cb is equal to four hundred, the rectangle contained by ac, cb is equal to the rectangle contained by ac, four hundred; but the rectangle contained by ac, four hundred is the figure ad.
hence the rectangle contained by a and de is the figure dh (def. v.). in like manner the rectangle contained by a and ec is the figure ek. hence we have the following identities:-
so, too, of that which has changed there is no primary part that has changed. for suppose that of ae the primary part that has changed is az (everything that changes having been shown to be divisible): and let oi be the time in which dz has changed.
for those things which are composed of the largest particles have the largest void left in their compositions, and those which are composed of the smallest particles have the least.
on the other hand (ii) hot and cold, and dry and moist, are terms, of which the first pair implies power to act and the second pair susceptibility.
he put the moon in the orbit which was nearest to the earth, the sun in that next, the morning star and mercury in the orbits which move opposite to the sun but with equal swiftness-this being the reason why they overtake and are overtaken by one another.
therefore the two sides db, bc in one are equal to the two sides ac, cb in the other; and the angle dbc in one is equal to the angle acb in the other (hyp). therefore iv.
() its reciprocal, to describe a circle about a triangle.
all of which goes to show not only that the earth is circular in shape, but also that it is a sphere of no great size: for otherwise the effect of so slight a change of place would not be quickly apparent.
one. if two figures be similar, to each point in the plane of one there will be a corresponding point in the plane of the other.
but this question depends on another set of arguments, and the truth will be made clear later. is possible for a thing to cause motion, though it is itself incapable of being moved.
greater them the angle (five hundred) contained by the sides of the other.
the liquid kind is composed of the small and unequal particles of water; and moves itself and is moved by other bodies owing to the want of uniformity and the shape of its particles; whereas the fusile kind, being formed of large and uniform particles, is
in an obtuse-angled triangle (abc), the square on the side (ab) subtending the obtuse angle exceeds the sum of the squares on the sides (bc, ca) containing the obtuse angle, by twice the rectangle contained by either of them (bc), and its continuation
for that which takes place according to nature is pleasant, but that which is contrary to nature is painful.
for as the bronze is to the statue, the wood to the bed, or the matter and the formless before receiving form to any thing which has form, so is the underlying nature to substance, i.e. the 'this' or existent.
every proposition in the fifth book is a theorem.
if then no portion of earth can move away from the centre, obviously still less can the earth as a whole so move. for it is the nature of the whole to move to the point to which the part naturally moves.
relative lightness must depend not only on an excess of void, but also an a defect of solid: for if the ratio of solid to void exceeds a certain proportion, the relative lightness will disappear.
but it is real when the point o is without the triangle-that is, when the triangle has an obtuse angle.
his cosmos would necessarily be imperfect and unequal, being the first attempt to impress form and order on the primaeval chaos of human knowledge. he would see all things as in a dream.
between air and water (coarser than air, but finer than water), or between air and fire (coarser than fire, but finer than air).
as to our own theory-we have given a general account of the causes in an earlier work,' we have now explained and distinguished the 'matter' and the 'form'.
for whatever any one of them supposed to have this character-whether one thing or more than one thing-this or these he declared to be the whole of substance, all else being its affections, states, or dispositions.
on the former view there could be an amount of earth so small as to contain less solid than a large mass of fire.
two. what proposition is this the converse of?
the words 'being,' or 'unity,' or essence,' or 'good,' became sacred to them. they did not see that they had a word only, and in one sense the most unmeaning of words.
for some time, and then begins to decrease, it is said to be a maximum at the end of the increase. thus, in the foregoing figure, pa, supposed to revolve round p and meet the circle, is a maximum.
and if there is something else that moves it, there will be two infinites, that which moves and that which is moved, differing in their form and power.
for if the infinite could traverse the finite, the finite could traverse the infinite; for it makes no difference which of the two is the thing in motion; either case involves the traversing of the infinite by the finite.
contained in it are only the transformations of the same soul of the world acting on the same matter.
and yet democritus says 'the more any indivisible exceeds, the heavier it is'-to which we must clearly add 'and the hotter it is'.
or thus: since the perpendicular must be the shortest line from o to four hundred, and oc is evidently the shortest line; therefore oc must be perpendicular to four hundred.
going to experience. thus all pleasure of this kind must be produced by sensible things: and since the presence in any one of moral defect or excellence involves the presence in him of pleasure or pain (with which moral excellence and defect are always
figure: therefore fire will be a pyramid. others, again, express no opinion on the subject of its figure, but simply regard it as the of the finest parts, which in combination will form other bodies, as the fusing of gold-dust produces solid gold.
and a yellow colour. a shoot of gold, which is so dense as to be very hard, and takes a black colour, is termed adamant.
and the converse happens when some gentle inspiration of the understanding pictures images of an opposite character, and allays the bile and bitterness by refusing to stir or touch the nature opposed to itself, but by making use of the natural sweetness
when a in the course of its locomotion comes to a stand at b and starts again towards g: but when its motion is continuous a cannot either have come to be or have ceased to be at the point b: it can only have been there at the moment of passing, its
now, of the triangles which we assumed at first, that which has two equal sides is by nature more firmly based than that which has unequal sides; and of the compound figures which are formed out of either, the plane equilateral quadrangle has necessarily
this proposition and the preceding one may be included in one enunciation, thus: "ratios compounded of equal ratios are equal."
we have now finished our examination of the heavy and the light and of the phenomena connected with them.
it is true that the point is common to both times, the earlier as well as the later, and that, while numerically one and the same, it is theoretically not so, being the finishing-point of the one and the starting-point of the other: but so far as the
same as the effect either generically-or the efficient cause of the coming-to-be of a hard thing is not a hard thing or specifically, as e.g.
i have often remarked that there are three kinds of soul located within us, having each of them motions, and i must now repeat in the fewest words possible, that one part, if remaining inactive and ceasing from its natural motion, must necessarily become
one thing then has and enjoys the ultimate good, other things attain to it, one immediately by few steps, another by many, while yet another does not even attempt to secure it but is satisfied to reach a point not far removed from that consummation.
a dream, say of all existence that it must of necessity be in some place and occupy a space, but that what is neither in heaven nor in earth has no existence.
he may fairly be called the parent of the romantic movement in art, letters and life; he affected profoundly the german romantics and goethe himself; he set the fashion of a new introspection which has permeated nineteenth century literature; he began
now it cannot be partly in each of the two: for then it would be divisible into parts. nor again can it be in bg: for then it will have completed the change, whereas the assumption is that the change is in process.
further, is astronomy different from physics or a department of it?
and the same argument shows that g is coincident with e.
if a chord of given length slide round a fixed circle-one. the locus of its middle point is a circle; two. the locus of any point fixed in the chord is a circle.
heavier than wood. for all bodies, in spite of the general opinion to the contrary, are composed of identical parts and of a single material. but this analysis says nothing of the absolutely heavy and light.
or the antediluvian athens ever had any existence except in the imagination of plato. martin is of opinion that plato would have been terrified if he could have foreseen the endless fancies to which his island of atlantis has given occasion.
thus, the ratio of two commensurable lines is the ratio of the numbers which express their lengths, measured with the same unit. and this may be extended to the case where the lines are incommensurable.
eight. through a given point draw a transversal cutting two lines given in position, so that the rectangle contained by the segments intercepted between it and the line may be given.
thus in some cases the cause is nourishment: when it is being digested animals sleep, and when it is being distributed through the system they awake and move themselves, the first principle of this motion being thus originally derived from outside.
but if there cannot be in this way a sensible body which is infinite in the full sense, evidently there can no more be a body which is potentially infinite in respect of addition, except as the inverse of the infinite by division, as we have said.
in this transcription, in order to retain the accuracy of this text, those words are rendered by spelling out each greek letter individually, such as alpha beta gamma delta... . the reader can distinguish these words by the enclosing braces .
twenty five. the orthocentres of the four triangles formed by any four lines are collinear.
for an image, since the reality, after which it is modelled, does not belong to it, and it exists ever as the fleeting shadow of some other, must be inferred to be in another (i.e.
lastly, some contraries are more primary than others, and some arise from others-for example sweet and bitter, white and black-whereas the principles must always remain principles.
there are many senses in which motion is said to be 'one': for we use the term 'one' in many senses.
suppose, no body contains it. on the line on which it is moved, its parts have place: for each is contiguous the next.
dem.-since the ratio of c : d is greater than the ratio of e : f,
plato also approaches very near to our doctrine of the primary and secondary qualities of matter.
there are other and worse diseases which are prior to these; as when the bone through the density of the flesh does not receive sufficient air, and becomes stagnant and gangrened, and crumbling away passes into the food, and the food into the flesh, and
for on our earth it is man that has the greatest variety of actions-for there are many goods that man can secure; hence his actions are various and directed to ends beyond them-while the perfectly conditioned has no need of action, since it is itself the
up to this point we have practically had most of the other writers on the subject with us, as i have said already: for all of them identify their elements, and what they call their principles, with the contraries, giving no reason indeed for the theory,
admire the hole which he made. many are concerned about the monuments of the west and the east,-to know who built them. for my part, one should like to know who in those days did not build them,-who were above such trifling.
'the hard', on the other hand, derives from the dry: for 'hard' is that which is solidified, and the solidified is dry.
thirty five. if o be the point of concurrence of the bisectors of the angles of the triangle abc, and if ao produced meet bc in five hundred, and from o, oe be drawn perpendicular to bc; prove that the angle bod is equal to the angle coe.
protrudes through nature. of this too there was a shadow in the eleatic philosophy in the realm of opinion, which, like a mist, seemed to darken the purity of truth in itself.-so far the words of plato may perhaps find an intelligible meaning.
having a power of cooling equal to that of ten pints of air); even so, they are 'comparable in their amount', though not qua 'amount' but qua iso-much power'. there is also (iii) a third possibility.
for a single thing has a single movement, and a simple thing a simple: contrary movements cannot belong to the same thing, and movement away from the centre is the contrary of movement to it.
for a thing comes to be from the privation, which in its own nature is not-being,-this not surviving as a constituent of the result.
so this reasoning also shows that when a thing is moved, if it is not moved immediately by something that moves itself, the series brings us at some time or other to a movent of this kind.
so it is necessary that all the things in time should be contained by time, just like other things also which are 'in anything', e.g. the things 'in place' by place.
changed. if, on the other hand, it is a different moment, there will be a period of time intermediate between the two: for, as we saw, moments are not consecutive.
heard; and there was no surrounding atmosphere to be breathed; nor would there have been any use of organs by the help of which he might receive his food or get rid of what he had already digested, since there was nothing which went from him or came into
sol.-in the given plane bh draw any line bc, and from a draw ad perpendicular to bc i. xii. ; then if ad be perpendicular to the plane, the thing required is done. if not, from five hundred draw de in the plane bh at right angles to bc i. xi.
penetrating according to their rarity, and thus all the elements are on their way up and down everywhere and always into their own places. hence there is a principle of inequality, and therefore of motion, in all time.
dem.-join co, and let fall perpendiculars from o on the five sides of the pentagon. now the triangles abo, cbo have the side ab equal to bc (hyp.), and bo common, and the angle abo equal to cbo (const.). hence the angle bao is equal to bco i. iv.
nor again (ii) will it be possible to say with truth, even in regard to the members of a limited sequence, that it is 'absolutely necessary' for any one of them to come-to-be. we cannot truly say, e.g.
on the other hand (ii) if, although it is not a 'this somewhat' or a substance, it is to possess some of the remaining determinations quoted above, then (as we said)' properties will be separable from substances.
he replied-there is at the head of the egyptian delta, where the river nile divides, a city and district called sais; the city was the birthplace of king amasis, and is under the protection of the goddess neith or athene.
seventy seven. in the same case the diameter of the circle circumscribed about the triangle abc ab.bc.ca divided by the area of abc.
for we maintain that one and the same 'matter' is equally, so to say, the basis of either of the two opposed things-being as it were a 'kind'; and that that which can he hot must be made hot, provided the heating agent is there, i.e. comes near.
when the lung, which is the steward of the air, is obstructed, by rheums, and in one part no air, and in another too much, enters in, then the parts which are unrefreshed by air corrode, and other parts are distorted by the excess of air; and in this
when on the other hand the body, though wasted, still holds out, then the bile is expelled, like an exile from a factious state, causing associating diarrhoeas and dysenteries and similar disorders.
to meet the objections of some inveterate cavillers, i may as well state, that if i dined out occasionally, as i always had done, and i trust shall have opportunities to do again, it was frequently to the detriment of my domestic arrangements.
to this man or that, of past, present or future good and evil, must first recover his wits.
when, then, is there a difference of species? is an attribute specifically different if the subject is different while the attribute is the same, or must the attribute itself be different as well? and how are we to define the limits of a species?
but they have never explained in detail the shapes of the various elements, except so far to allot the sphere to fire.
four. describe a circle through two given points, and intercepting a given arc on a given circle.
had a fineness equal to the punctures, was borne up by its own impulse and extended far outside the head, but being too slow to escape, was thrust back by the external air, and rolled up underneath the skin, where it took root.
the term 'immovable' we apply in the first place to that which is absolutely incapable of being moved (just as we correspondingly apply the term invisible to sound); in the second place to that which is moved with difficulty after a long time or whose
there was one more illusion to which the ancient philosophers were subject, and against which plato in his later dialogues seems to be struggling-the tendency to mere abstractions; not perceiving that pure abstraction is only negation, they thought that
nineteen. the diagonals of a lozenge are sixteen and thirty metres respectively; find the length of a side.
there is no first term)-if then everything that is in motion is moved by something, and the first movent is moved but not by anything else, it much be moved by itself.
dem.-draw be parallel to ac xxxi. . now since bc intersects the parallels be, ac, the alternate angles ebc, acb are equal xxix. . again, since ab intersects the parallels be, ac, the angle ebd is equal to bac xxix.
there was nothing behind them; they were to physical science what the poems of homer were to early greek history.
there is a story, which even you have preserved, that once upon a time paethon, the son of helios, having yoked the steeds in his father's chariot, because he was not able to drive them in the path of his father, burnt up all that was upon the earth, and
fifty. given the base of a triangle, the difference of the base angles, and the sum or difference of the sides; construct it.
for the contradictory of each will at times be true of it unless it always exists.
each thing that comes to be comes from a similar body, and there is a coming to be of all things, though not, it is true, at the same time. hence there must also be an origin of coming to be.
what makes fire burn? the fineness of the sides, the sharpness of the angles, the smallness of the particles, the quickness of the motion. moreover, the pyramid, which is the figure of fire, is more cutting than any other.
a point (p) within a circle (abc), from which more than two equal lines (pa, pb, pc, c.) can be drawn to the circumference, is the centre.
and (the first form of) the distinction, which we are investigating, results from a similar division of terms: for it makes a difference into what the changing thing changes. perhaps, e.g.
even so, this cannot be true of all such principles, since there must clearly be something that causes things that move themselves at one time to be and at another not to be.
ix. an angle in a segment is said to stand on its conjugate arc.
further, if the whole moves itself we may distinguish in it that which imparts the motion and that which is moved: so while we say that ab is moved by itself, we may also say that it is moved by a.
, because the triangle cab is isosceles; therefore the angle adc is greater than cad. hence ac is greater than four hundred i. xix.
and not only so: we also speak of the particular fluid or hot substance as being bronze, giving the material the same name as that which we use to describe the affection.
these are causes of the statue qua statue, not in virtue of anything else that it may be-only not in the same way, the one being the material cause, the other the cause whence the motion comes. some things cause each other reciprocally, e.g.
sol.-take any point a in the circumference, and join it to o, the centre of the given circle; then with a as centre, and ao as radius, describe the circle obf, intersecting the given circle in the points b, f.
even in the former class, however, there seems to be something similar, for a cold thing in a sense causes heating by turning away and retiring, just as one possessed of knowledge voluntarily makes an error when he uses his knowledge in the reverse way.
ab .bc .ca a b.b one hundred.one hundred a.
other forms of position. on these two grounds, then, they may fairly be criticized, first, for omitting the more fundamental principles, and secondly, for thinking that the two they mentioned were attributable equally to everything.
for this reason, too, place is thought to be a kind of surface, and as it were a vessel, i.e. a container of the thing.
(plato of course, if we may digress, ought to tell us why the form and the numbers are not in place, if 'what participates' is place-whether what participates is the great and the small or the matter, as he called it in writing in the timaeus.)
according to nature, so that if each of the natural bodies has not a natural movement, none of the other movements can exist); but how can there be natural movement if there is no difference throughout the void or the infinite?
hence, adding the angle abd, the sum of the angles cba, abd is equal to the sum of the three angles cbe, eba, abd. in like manner, the sum of the angles cbe, ebd is equal to the sum of the three angles cbe, eba, abd.
moreover, we say that the animal itself originates not all of its motions but its locomotion.
yet if this is so, it is a fact which deserves to be dwelt upon, and something might well have been said about it.
hence the two triangles whose base is the third side and whose vertices are the points of bisection are equal.
power, the smaller body must necessarily yield to the superior power with less reluctance than the larger; and the larger body is called heavy and said to tend downwards, and the smaller body is called light and said to tend upwards.
acquired, and would not cease from his toils and transformations until he followed the revolution of the same and the like within him, and overcame by the help of reason the turbulent and irrational mob of later accretions, made up of fire and air and
now as every animal has within him a fountain of fire, the air which is inhaled through the mouth and nostrils, on coming into contact with this, is heated; and when heated, in accordance with the law of attraction, it escapes by the way it entered toward
admit, for instance, the existence of a minimum magnitude, and you will find that the minimum which you have introduced, small as it is, causes the greatest truths of mathematics to totter.
to the breathing, air can penetrate, but not smell.
now the continuity of this movement is caused by the motion of the whole: but the approaching and retreating of the moving body are caused by the inclination.
hence these difficulties may be dismissed: but the problem immediately connected with them-whether combination is something relative to perception' must be set out and discussed.
nile might have slowly accumulated in long periods of time (hdt.). but he seems to have supposed that the course of events was recurring rather than progressive.
later i shall explain what each of them means, and also why every magnitude is divisible into magnitudes.)
illustration.-suppose a to be six inches; bd, five inches; de, four inches; ec, three inches; then bc will be twelve inches; and the rectangles will have the following values:-
it is, however, wrong to remove the foundations of a science unless you can replace them with others more convincing.
prop. six.-theorem. if two right lines (ab, four hundred) be normals to the same plane (ten), they shall be parallel to one another.
one. the squares on equal lines are equal; and, conversely, the sides of equal squares are equal.
thus, taking health as the end, there will be one thing that always possesses health, others that attain it, one by reducing flesh, another by running and thus reducing flesh, another by taking steps to enable himself to run, thus further increasing the
cor. one.-if the ratio of two sides of a triangle be given, and the angle between them, the triangle is given in species.
compounds of earth and water are unaffected by water while the water occupies the interstices in them, but begin to liquefy when fire enters into the interstices of the water.
further, if one thing alters and another accomplishes a locomotion in an equal time, we may have an alteration and a locomotion equal to one another: thus an affection will be equal to a length, which is impossible.
so that neither are there now, nor have there ever been, nor can there ever be formed more heavens than one, but this heaven of ours is one and unique and complete.
for if fire is to result from water, both the cold and the moist must pass-away: and again, both the cold and the dry must pass-away if air is to result from earth.
intersect successively in four other points e, f, g, h, forming another cyclic quadrilateral.
this series of numbers is the compound of the two pythagorean ratios, having the same intervals, though not in the same order, as the mixture which was originally divided in forming the soul of the world.
and restraining the desires when they are no longer willing of their own accord to obey the word of command issuing from the citadel.
the influence of such affinities and the chemical action of one body upon another in long periods of time have become a recognized principle of geology.
earth then has to exist; for it is earth which is at rest at the centre. (at present we may take this for granted: it shall be explained later.) but if earth must exist, so must fire.
modern educational theory; and, above all, in political thought he represents the passage from a traditional theory rooted in the middle ages to the modern philosophy of the state.
how came the poem of solon to disappear in antiquity? or why did plato, if the whole narrative was known to him, break off almost at the beginning of it?
five. if the intersections of several planes be parallel, the normals drawn to them from any point are coplanar.
let p and q be two consecutive points on the circumference. join cp, cq, pq; produce pq both ways. now since p and q are consecutive points, pq is a tangent (def. iii.).
for whatever is either of these is capable both of being and of not being for a definite time: in either case, i mean, there is a certain period of time during which the thing is and another during which it is not.
hence at once such things will act and be acted on by one another in many ways: each of them will be capable at the same time of causing alteration and of being altered.
sol.-in ab take any point five hundred, and cut off iii. ae equal to ad. join de (post. i.), and upon it, on the side remote from a, describe the equilateral triangle def i. join af. af bisects the given angle bac.
all these various uses, however, come to six in number, under each of which again the usage is twofold.
one might raise a further difficulty. what is 'that which grows'? is it that to which something is added? if, e.g. a man grows in his shin, is it the shin which is greater-but not that 'whereby' he grows, viz. not the food? then why have not both 'grown'?
twenty one. the three sides of a triangle are eight, eleven, fifteen; prove that it has an obtuse angle.
since, then, we have spoken of the primary element, of its bodily constitution, and of its freedom from destruction and generation, it remains to speak of the other two.
twenty six. the perpendiculars of a triangle are the bisectors of the angles of the triangle whose vertices are the feet of these perpendiculars.
so we may interpret the statements of heraclitus of ephesus and many others. and some subject all bodies whatever to generation, by means of the composition and separation of planes.
valentine rose and schaarschmidt, against them. but it is clear that they throw but little light upon the timaeus, and that their resemblance to it has been exaggerated.
thirty one. if from any point in the circumference of any concentric circle perpendiculars be let fall on all the sides of any regular polygon, the sum of their squares is constant.
therefore the triangle def is equiangular to abc.
moreover (c) if air is again to result out of the fire, it will do so by the conversion of the hot into its contrary: this contrary, therefore, will belong to air, and air will be a cold something: hence it is impossible for fire to be 'hot air', since in
join ea, ed, six hundred, cb; then the figure abcde is a regular pentagon.
stones and foundations take the lowest place, with earth above because it is lighter, and wood at the top of all as being the lightest.
openings, and from the lesser weels he extended cords reaching all round to the extremities of the network. all the interior of the net he made of fire, but the lesser weels and their cavity, of air.
no man, when in his wits, attains prophetic truth and inspiration; but when he receives the inspired word, either his intelligence is enthralled in sleep, or he is demented by some distemper or possession.
or thus: since the points e, f are in the plane of the parallels, the line joining these points is in that plane i. def. vi .
grote supposes, not that (greek) means 'revolving,' or that this is the sense in which aristotle understood the word, but that the rotation of the earth is necessarily implied in its adherence to the cosmical axis.
theory of any universal facilitates the understanding of its specific forms.
therefore the two sides ac, four hundred in one are equal to the two sides bc, four hundred in the other; and the angle acd is equal to the angle bcd (const.). therefore the base ad is equal to the base db iv. . hence ab is bisected in five hundred.
in good condition, while defect puts its possessor in a bad condition, to meet his proper affections.
by either. out of this he fashioned, as in a lathe, a globe made of bone, which he placed around the brain, and in this he left a narrow opening; and around the marrow of the neck and back he formed vertebrae which he placed under one another like pivots,
if, then, a move b a distance g in a time five hundred, it does not follow that e, being half of a, will in the time five hundred or in any fraction of it cause b to traverse a part of g the ratio between which and the whole of g is proportionate to that
it is clear, then, that time is 'number of movement in respect of the before and after', and is continuous since it is an attribute of what is continuous.
there is also a corresponding process of inhalation through the mouth or nostrils, and of exhalation through the pores. the inhalation through the pores appears to take place nearly at the same time as the exhalation through the mouth; and conversely.
then, if the adjacent sides ab, bc be equal, ac is a square, and the problem is solved; if not, produce ab to e, and make be equal to bc; bisect ae in f; with f as centre and fe as radius, describe the semicircle age; produce cb to meet it in g.
the veins about the nose are too narrow to admit earth and water, and too wide to detain fire and air; and for this reason no one ever perceives the smell of any of them; but smells always proceed from bodies that are damp, or putrefying, or liquefying,
as in aristotle's matter and form the connexion between them is really inseparable; for if we attempt to separate them they become devoid of content and therefore indistinguishable; there is no difference between the idea of which nothing can be
the citizens have a deity for their foundress; she is called in the egyptian tongue neith, and is asserted by them to be the same whom the hellenes call athene; they are great lovers of the athenians, and say that they are in some way related to them.
twenty. state the relations between propositions xvi., xviii., xix.
hence the rectangle ad.db, together with the square on four hundred, is equal to the whole figure cbef-that is, to the square on cb.
both belong to the mode of causation 'source of change', for either some natural or some intelligent agent is always the cause; but in this sort of causation the number of possible causes is infinite.
; and with a as centre and ae as radius, describe the circle ebf, cutting the circle abc in the points b, f. join ab. then ab is the chord required.
the philosopher himself was a child and also a man-a child in the range of his attainments, but also a great intelligence having an insight into nature, and often anticipations of the truth.
these are the second causes which god used as his ministers in fashioning the world.
for though, no doubt, plato investigated the conditions under which things come-to-be and pass-away, he confined his inquiry to these changes; and he discussed not all coming-to-be, but only that of the elements.
we need now say no more in support of the position that there is no process of change that admits of infinity or continuity except rotatory locomotion.
the following explanations will give the student clear conceptions of their meaning:-
it does not allude to the original from which it is taken; it is quite free from mysticism and neo-platonism. in length it does not exceed a fifth part of the timaeus.
two. this proposition may be proved by producing the less side.
dem.-since bd is parallel to ce, we have ii. ab : bc :: ad : de; therefore ten : y :: z : de. hence de is a fourth proportional to ten, y , z.
and (two) if the world did come into being, then, clearly, their condition must have been capable of change and not eternal: after combination therefore they will be dispersed, just as in the past after dispersion they came into combination, and this
on the other hand he doctors or fails to doctor qua doctor. but we are using words most appropriately when we say that a doctor does something or undergoes something, or becomes something from being a doctor, if he does, undergoes, or becomes qua doctor.
now, it is evident that if ma- nb is greater than unity, mc- nd is greater than unity; but if ma- nb is greater than unity, ma is greater than nb; and if mc- nd is greater than unity, mc is greater than nd.
'to be in number' means that there is a number of the thing, and that its being is measured by the number in which it is. hence if a thing is 'in time' it will be measured by time.
the more living and sensitive of the bones he enclosed in the thinnest film of flesh, and those which had the least life within them in the thickest and most solid flesh.
this must suffice for our account of the number of causes and the modes of causation.
eighteen. prove that a side and the diagonal of a square are incommensurable.
prevent there being a motion that is the same in virtue of being continuous and eternal: we shall have something to say later that will make this point clearer.
dem.-let o be the centre. join oa, ob. then the angle aob is double of the angle acb xx. , and also double of the angle adb. therefore the angle acb is equal to the angle adb.
then the rectangle ap.bp is equal to the difference of the squares of ob and op, and is therefore constant.
three. three equal lines could not be drawn from the same point to the same line.
and the elements of philosophy which entered into the conception of them: (three) the theology and physics of the timaeus, including the soul of the world, the conception of time and space, and the composition of the elements: (four) in the fourth section
v.) the multiple of the first is greater than that of the second, but the multiple of the third not greater than that of the fourth, the first has to the second a greater ratio than the third has to the fourth.
at the joints he diminished the flesh in order not to impede the flexure of the limbs, and also to avoid clogging the perceptions of the mind.
sol.-inscribe a regular pentagon abcde in the circle xi. , and also an equilateral triangle agh ii. . join cg. cg is a side of the required polygon.
now each of these belongs to all its subjects in either of two ways: namely (one) substance-the one is positive form, the other privation; (two) in quality, white and black; (three) in quantity, complete and incomplete; (four) in respect of locomotion,
if there is a single matter, as with those who compose things of triangles, nothing can be absolutely heavy or light: and if there is one matter and its contrary-the void, for instance, and the plenum-no reason can be given for the relative lightness and
cor. one.-three collinear points cannot be concyclic.
for though it separates bodies different in kind, it combines those which are the same; and the combining is essential to it, the functions of connecting and uniting being a mark of fire, while the separating is incidental.
sol.-from any point b not in the plane draw eleven. xi. bc normal to it. if this line pass through a it is the normal required. if not, from a draw ad parallel to bc i. xxxi. .
again, as air is lighter than water, so is water than earth: how then can they think that the naturally lighter substance lies below the heavier?
prop. six.-theorem. if two angles (b, one hundred) of a triangle be equal, the sides (ac, ab) opposite to them are also equal.
fashion, viz. withdrawing out of it and leaving it unchanged.
but plato delights to think of god as the author of order in his works, who, like a father, lives over again in his children, and can never have too much of good or friendship among his creatures.
for there must be a point at which what has come to be reaches completion, and also a termination of all passing away.
similarly we have a definite thing that causes alteration and a definite thing that undergoes alteration, and a certain amount, or rather degree, of alteration is completed in a certain amount of time: thus in twice as much time twice as much alteration
(for this was the difficulty that emerged in the previous argument: and it is evident that the combining constituents not only coalesce, having formerly existed in separation, but also can again be separated out from the compound.) the constituents,
a thing then which has the one kind of matter is light and always moves upward, while a thing which has the opposite matter is heavy and always moves downward.
twenty six. two parallel chords of a circle are twelve perches and sixteen perches respectively, and their distance asunder is two perches; find the length of the diameter.
twenty seven. what is the difference between the symbols denoting congruence and identity?
the smooth and the rough are severally produced by the union of evenness with compactness, and of hardness with inequality.
colour; 'musical' passes into 'not-musical'-and not into any chance thing other than musical, but into 'unmusical' or any intermediate state there may be.
(things are always in flux and decay, they say), and they go so far as to speak even of becoming and perishing as a process of alteration.
forty five. if a triangle be described about a circle, the lines from the points of contact of its sides with the circle to the opposite angular points are concurrent.
now revolution about the centre is circular motion, while the upward and downward movements are in a straight line, 'upward' meaning motion away from the centre, and 'downward' motion towards it.
before commencing it the student should read the following preliminary explanations: by their assistance it will be seen that this book, which is usually considered difficult, will be rendered not only easy, but almost intuitively evident.
hence the arc gc, which is the difference between these two arcs, is equal to two ths one rd, or -one th of the entire circumference; and therefore, if chords equal to gc i.
the sight in my opinion is the source of the greatest benefit to us, for had we never seen the stars, and the sun, and the heaven, none of the words which we have spoken about the universe would ever have been uttered.
in so far as it is in succession, it is different (which is just what its being was supposed to mean), but its substratum is an identity: for motion, as was said, goes with magnitude, and time, as we maintain, with motion.
the reasons why the primary body is eternal and not subject to increase or diminution, but unaging and unalterable and unmodified, will be clear from what has been said to any one who believes in our assumptions.
two. what theorem? ans. no theorem, only the axioms.
iii., are so related, that any one being proved directly, the other two follow by the rule of identity.
thus it is clear (i) that to describe coming-to-be and passing-away in these terms is in accordance with their fundamental assumption, and (ii) that they do in fact so describe them: nevertheless, they too must recognize 'alteration' as a fact distinct
prop. thirteen.-theorem. if two ratios are equal, and if one of them be greater than any third ratio, then the other is also greater than that third ratio.
two. if ab be produced to five hundred and e, the triangles cdf and cef are equilateral.
cause is to be found elsewhere. the void, they say, which is imprisoned in bodies, lightens them and sometimes makes the larger body the lighter. the reason is that there is more void.
if he seems to be extreme in his condemnation of medicine and to rely too much on diet and exercise, he might appeal to nearly all the best physicians of our own age in support of his opinions, who often speak to their patients of the worthlessness of
the series, however, could be traced back, as we said before, until at some time or other we arrived at the same kind of motion).
thus want of uniformity, the condition of motion, is produced.
yet by the same argument one might say it was absurd that on a large vessel the motion of mast and poop should not make a great noise, and the like might be said of the movement of the vessel itself.
or thus: describe squares on the sides. draw ae, bf, cg perpendicular to the sides; then, as in the demonstration of i. xlvii. , the rectangle bg is equal to be; ag to af, and ce to cf.
we have stated that all the 'elements' come-to-be out of any one of them; and we have explained the manner in which their mutual conversion takes place. let us nevertheless supplement our theory by the following speculations concerning them.
yet it is certain that this declamation made rousseau's name, and established his position as a great writer in parisian circles. five hundred'alembert even devoted the preface of the encyclopdia to a refutation.
the bones and hair are of the latter kind, sight and hearing of the former. ordinary affections are neither pleasant nor painful. the impressions of sight afford an example of these, and are neither violent nor sudden.
four hundred, de, ea are equal. hence the figure abcde is equilateral.
sol.-take any point five hundred on the other side of ab, and describe (post. iii.) a circle, with one hundred as centre, and four hundred as radius, meeting ab in the points f and g. bisect fg in h x. . join ch (post. i.).
plato does not look out upon the heavens and describe what he sees in them, but he builds upon the foundations of others, adding something out of the 'depths of his own self-consciousness.'
nor (two) can the infinite body be one and simple, whether it is, as some hold, a thing over and above the elements (from which they generate the elements) or is not thus qualified.
as downward movement is to a stone, will be merely accelerated by an external force, while an unnatural movement will be due to the force alone. in either case the air is as it were instrumental to the force.
the world is conceived of as a whole, and the elements are formed into and out of one another; the varieties of substances and processes are hardly known or noticed.
all of them are nearer to one another than they themselves supposed, and nearer to him than he supposed. all of them are antagonistic to sense and have an affinity to number and measure and a presentiment of ideas.
three. what is the obverse of this proposition?
for which reason the sea in those parts is impassable and impenetrable, because there is a shoal of mud in the way; and this was caused by the subsidence of the island.
four. describe a circle having its centre at a given point-one. and touching a given line; two. and touching a given circle. how many solutions of this case?
i. as to its position there is some difference of opinion. most people-all, in fact, who regard the whole heaven as finite-say it lies at the centre. but the italian philosophers known as pythagoreans take the contrary view.
again, because four hundred is parallel to bf, bc : ce :: fd : de; but fd is equal to ac, therefore bc : ce :: ac : de; hence five. xvi. bc : ac :: ce : de. therefore we have proved that ab : bc :: six hundred : ce, and that bc : ca :: ce : ed.
fourteen. in an equilateral triangle, three times the square on any side is equal to four times the square on the perpendicular to it from the opposite vertex.
which is in the same ratio to the other body as the time is to the time.
mildness and refinement of this diet, and travelling gradually down in my studies through that accidental souring of the dough which, it is supposed, taught the leavening process, and through the various fermentations thereafter, till i came to "good,
two. a square unit is the square described on a line whose length is the linear unit. thus a square inch is the square described on a line whose length is an inch; a square foot is the square described on a line whose length is a foot, c.
iv. the angle oca is equal to obp; but oca is a right angle (const.); therefore obp is a right angle, and xvi. pb touches the circle at b.
if then a body has another body outside it and containing it, it is in place, and if not, not.
dem.-ao.al ab.ak ac.ae. (six) but ao ag.ak ah.ae go.oh. (seven) hence ao.ol bg.ak ch.ae go.oh, or ao.ol eo.of io.ok go.oh. nine.
consequently the duration of the part of g which is occupied by all a in moving the whole of b, will be finite. therefore a finite thing cannot impart to anything an infinite motion.
contrary) in species. (for it is a law of nature that body is affected by body, flavour by flavour, colour by colour, and so in general what belongs to any kind by a member of the same kind-the reason being that 'contraries' are in every case within a
therefore the triangles abc, def, have the three sides of one respectively equal to the three sides of the other. hence i. viii. the angle abc is equal to def.
in all this they are not seeking for theories and causes to account for observed facts, but rather forcing their observations and trying to accommodate them to certain theories and opinions of their own.
or why, when they come into contact, do they not coalesce into one, as drops of water run together when drop touches drop (for the two cases are precisely parallel)? on the other hand (ii) if they fall into differing sets, how are these characterized?
if, then, anything is a movent while being itself moved, the series must stop somewhere and not be infinite.
hence i. xiv. bc, cg form one right line. complete the parallelogram be.
cor. two. supplies an easy demonstration of a fundamental proposition in statics.
twenty three. if three lines from the vertices of a triangle abc to any interior point o meet the opposite sides in the points a, b, one hundred; prove
also, the positive is prior to its privation (warm, for instance, to cold), and rest and heaviness stand for the privation of lightness and movement.
twenty one. what is meant by the obverse of a proposition?
prop. eight.-problem. in a given square (abcd) to inscribe a circle.
eight. inscribe a regular octagon in a given square.
series of motions. the only continuous motion, then, is that which is caused by the unmoved movent: and this motion is continuous because the movent remains always invariable, so that its relation to that which it moves remains also invariable and
propositions which are not axioms are properties of figures obtained by processes of reasoning. they are divided into theorems and problems.
he would have beheld the earth a surface only, not mirrored, however faintly, in the glass of science, but indissolubly connected with some theory of one, two, or more elements.
three. let four hundred be greater than ef, it is required to prove that og is less than oh.
now to investigate whether being is one and motionless is not a contribution to the science of nature.
eight. construct a triangle, being given a side and the two medians of the remaining sides.
water, which are called bubbles; of all these affections the cause is termed acid.
he means (five) that the idea of the world is prior to the world, just as the other ideas are prior to sensible objects; and like them may be regarded as eternal and self-existent, and also, like the idea of good, may be viewed apart from the divine mind.
but in the direction of largeness it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite.
from what we have said, then, it is clear that the weight of the infinite body cannot be finite. it must then be infinite. we have therefore only to show this to be impossible in order to prove an infinite body impossible.
as for the pyramids, there is nothing to wonder at in them so much as the fact that so many men could be found degraded enough to spend their lives constructing a tomb for some ambitious booby, whom it would have been wiser and manlier to have drowned in
again, matter is a relative term: to each form there corresponds a special matter. how far then must the physicist know the form or essence? up to a point, perhaps, as the doctor must know sinew or the smith bronze (i.e.
therefore the motion is not a single motion, since motion that is interrupted by stationariness is not single.
the causes, therefore, are these and so many in number.
again, in the case of circles and spheres and everything whose motion is confined within the space that it occupies, it is not true to say the motion can be nothing but rest, on the ground that such things in motion, themselves and their parts, will
mean that which moves upward or to the extremity, and by absolutely heavy that which moves downward or to the centre.
dem.-since the parallelograms ac, af have a common angle, they are equiangular i. xxxiv. , and all that is required to be proved is, that the sides about the equal angles are proportional.
if simple becoming was ever in process of becoming, then that which was becoming simple becoming was also in process of becoming, so that we should not yet have arrived at what was in process of simple becoming but only at what was already in process of
then five hundred is at the moment g white and not-white: for if we were right in saying that it is white during the whole time a, it is true to call it white at any moment of a, and not-white in b, and g is in both a and b.
be affected by those of them that ought to be admitted, while defect disposes its possessor to be affected by them or to be unaffected by those of them that ought to be admitted.
(b) it is used in another sense, when something which is capable of coming to be, with or without process, does not exist; such a thing is ungenerated in the sense that its generation is not a fact but a possibility.
or thus: by the first book: describe squares on the three sides. draw ae, bf, cg perpendicular to the sides of the squares. then it can be proved exactly as in the demonstration of i. xlvii. , that the rectangle bg is equal to be, ag to af, and ce to cf.
the fires from within and from without meet about the smooth and bright surface of the mirror; and because they meet in a manner contrary to the usual mode, the right and left sides of the object are transposed.
pleasure and pain are the most important of the affections common to the whole body.
thus the air is continuous with that which causes the alteration, and the body that undergoes alteration is continuous with the air. again, the colour is continuous with the light and the light with the sight.
contrarieties correlative to touch are the following: hot-cold, dry-moist, heavy-light, hard-soft, viscous-brittle, rough-smooth, coarse-fine. of these (i) heavy and light are neither active nor susceptible.
these god took and mingled them in due proportion, making as many kinds of marrow as there were hereafter to be kinds of souls.
three. draw a perpendicular to a given line to touch a given circle.
three. the internal and the external bisectors of the vertical angle of a triangle divide the base harmonically (see definition, p. one hundred ninety one).
seven. what false assumption is made in the demonstration?
he asked no questions as to how flesh or bones, or any of the other similar compound things, come-to-be; nor again did he examine the conditions under which 'alteration' or growth are attributable to things.
(one) it is not absurd that the actualization of one thing should be in another. teaching is the activity of a person who can teach, yet the operation is performed on some patient-it is not cut adrift from a subject, but is of a on b.
therefore each is both generated and destructible. therefore 'generated' and 'destructible' are coincident. now let e stand for the ungenerated, f for the generated, g for the indestructible, and h for the destructible.
something over and above the body displaced. but there is no such extension. one of the bodies which change places and are naturally capable of being in contact with the container falls in whichever it may chance to be.
if then it is possible that at any time nothing should be in motion, this must come about in one of two ways: either in the manner described by anaxagoras, who says that all things were together and at rest for an infinite period of time, and that then
particular case, building, healing, c. a similar definition will apply to each of the other kinds of motion.
further, those who divide bodies into planes and generate them out of planes seem to bear witness to the truth of this.
we must not allow, therefore, that it is white in the whole of a, but must say that it is so in all of it except the last moment g.
if time is infinite in respect of its extremities, length is also infinite in respect of its extremities: if time is infinite in respect of divisibility, length is also infinite in respect of divisibility: and if time is infinite in both respects,
now for the reason given above they cannot be continuous: and one thing can be in contact with another only if whole is in contact with whole or part with part or part with whole.
also, those mathematicians who try to calculate the size of the earth's circumference arrive at the figure four hundred,zero stades.
infinite weight is therefore impossible, and the same reasoning applies also to infinite lightness. bodies then of infinite weight and of infinite lightness are equally impossible.
at first men personify nature, then they form impressions of nature, at last they conceive 'measure' or laws of nature. they pass out of mythology into philosophy.
in like manner the triangle dbc is half the parallelogram dbcf, because the diagonal six hundred bisects it, and halves of equal things are equal (axiom vii.). therefore the triangle abc is equal to the triangle dbc.
to sum up all in a word: there are three kinds of soul located within us, and any one of them, if remaining inactive, becomes very weak; if exercised, very strong. wherefore we should duly train and exercise all three kinds.
i. if a point one hundred be taken in a line ab, the parts ac, cb are called segments, and one hundred a point of division.
because ac is equal to ce, and the angle ace is right, the angle cea is half a right angle. in like manner the angles ceb, cbe are half right angles; therefore the whole angle aef is right.
two. since the triangle abh is similar to cde, we have xix. .
it remains, then, that in the time in which it is changing, it is in ab. that being so, it will be at rest: for, as we saw, to be in the same condition for a period of time is to be at rest.
two. if any point within a triangle be joined to its angular points, the sum of the joining lines is greater than its semiperimeter.
to its like that a body moves when it moves to its own place.
what is actually hot is naturally capable of moving what is potentially hot): and similarly in the case of all other things of this kind.
underneath the sphere of the revolving body is necessarily heated by its motion, and particularly in that part where the sun is attached to it. hence warmth increases as the sun gets nearer or higher or overhead.
in the form we have given them they are freed from their usual repulsive appearance. the constructions and demonstrations are euclid's, but slightly modified.
being themselves moved; and such as are not of this kind are no longer inside the province of physics, for they cause motion not by possessing motion or a source of motion in themselves, but being themselves incapable of motion.
the end and the means towards it may come about by chance. we say, for instance, that a stranger has come by chance, paid the ransom, and gone away, when he does so as if he had come for that purpose, though it was not for that that he came.
for neither naturally nor unnaturally can it move with any other motion but its own, either itself or any part of it, since the reasoning which applies to the whole applies also to the part.
expresed universally our argument is as follows. let there be two attributes, a and b, not capable of being present in any one thing together, while either a or one hundred and either b or five hundred are capable of being present in everything.
dem.-join oa, ob, oc. the triangles bdo, cdo have the side bd equal four hundred (const.), and do common, and the angle bdo equal to the angle cdo, because each is right. hence i. iv. bo is equal to oc. in like manner ao is equal to oc.
here we see him at the beginning of the long journey which was to lead on at last to the theory of the social contract .
locomotion must be similar: for it also has its goal and starting-point--and therefore the starting-point and the goal of the natural movement must differ in form-just as the movement of coming to health does not take any direction which chance or the
the activities corresponding to these powers have not been given any name, unless it is thought that 'impetus' is such a name.
xxxii. the angle fcb is equal to the sum of the two interior angles bac, cba of the triangle abc. hence the angle acb is equal to its adjacent angle fcb, and therefore it is a right angle i. def. xiii. .
now the first alternative is impossible, and the second is fantastic: it is absurd that that which has a capacity for causing alteration should as such necessarily have a capacity, let us say, for increase.
the question as to the nature of the whole, whether it is infinite in size or limited in its total mass, is a matter for subsequent inquiry. we will now speak of those parts of the whole which are specifically distinct.
at that rate the matter of the heavy and the light, qua matter of them, would be the void; for the dense and the rare are productive of locomotion in virtue of this contrariety, and in virtue of their hardness and softness productive of passivity and
and when reason, which works with equal truth, whether she be in the circle of the diverse or of the same,-in voiceless silence holding her onward course in the sphere of the self-moved,-when reason, i say, is hovering around the sensible world, and when
twenty nine. the bases of two or more triangles having a common vertex are given, both in magnitude and position, and the sum of the areas is given; prove that the locus of the vertex is a right line.
one. if two circles touch, any line drawn through the point of contact will cut off similar segments.
this proposition may be proved as follows:
and vanishing out of place, and is apprehended by opinion and sense. there is also a third nature-that of space, which is indestructible, and is perceived by a kind of spurious reason without the help of sense.
when the change from contrary to contrary is in quantity, it is 'growth and diminution'; when it is in place, it is 'motion'; when it is in property, i.e.
something, which is at rest: but if so, we shall simply repeat the same argument; and either we shall come to an ultimate something to which rest where it is or we shall have an infinite process, which is impossible.
three. if the external angles of the triangle abc be bisected as in the annexed diagram, the three angular points o, o, o, of the triangle formed by the three bisectors will be the centres of three circles, each touching one side externally, and the other
ninety eight. inscribe in a given circle a polygon all whose sides pass through given points.
jawbones to them below the face. and they framed the mouth, having teeth and tongue and lips, with a view to the necessary and the good; for food is a necessity, and the river of speech is the best of rivers.
for in the whole range of time past, so far as our inherited records reach, no change appears to have taken place either in the whole scheme of the outermost heaven or in any of its proper parts.
for it is far more reasonable that what is should cause the coming-to-be of what is not, than that what is not should cause the being of what is.
the old man brightened up at hearing this, and said: had solon only had the leisure which was required to complete the famous legend which he brought with him from egypt he would have been as distinguished as homer and hesiod.
again, does it follow that being, if one, is motionless? why should it not move, the whole of it within itself, as parts of it do which are unities, e.g. this water? again, why is qualitative change impossible?
also the matter of magnitude and quality-being separable from these matters by definition, but not separable in place unless qualities are, in their turn, separable.
they did not understand that the content of notions is in inverse proportion to their universality-the element which is the most widely diffused is also the thinnest; or, in the language of the common logic, the greater the extension the less the
fire upward and earth downward and towards the middle of the universe, it is clear that it cannot be the void that is the condition of locomotion. what, then, will the void be the condition of?
for when the eyelids, which the gods invented for the preservation of sight, are closed, they keep in the internal fire; and the power of the fire diffuses and equalizes the inward motions; when they are equalized, there is rest, and when the rest is
if on the other hand it has the source of its motion in itself, let ab be taken to represent that which is in motion essentially of itself and not in virtue of the fact that something belonging to it is in motion.
in like manner the soul is often made vicious by the influence of bodily pain; the briny phlegm and other bitter and bilious humours wander over the body and find no exit, but are compressed within, and mingle their own vapours with the motions of the
so that to say that the things that come into being form a circle is to say that there is a circle of time; and this is to say that it is measured by the circular movement; for apart from the measure nothing else to be measured is observed; the whole is
and, secondly, the same theory which composes solids of planes clearly composes planes of lines and lines of points, so that a part of a line need not be a line.
and afterwards the causes of things which are moved by others, and which work by chance and without order. of the second or concurrent causes of sight i have already spoken, and i will now speak of the higher purpose of god in giving us eyes.
either both acute or both not acute), the triangles are similar.
then now let me explain to you the order of our entertainment; first, timaeus, who is a natural philosopher, will speak of the origin of the world, going down to the creation of man, and then i shall receive the men whom he has created, and some of whom
eighteen. explain the extended meaning of the word angle.
we will now show that the 'primary when' in which that which has changed effected the completion of its change must be indivisible, where by 'primary' i mean possessing the characteristics in question of itself and not in virtue of the possession of them
'at some time' means a time determined in relation to the first of the two types of 'now', e.g. 'at some time' troy was taken, and 'at some time' there will be a flood; for it must be determined with reference to the 'now'.
if then it did not cease-to-be in the next 'now' but in another, it would exist simultaneously with the innumerable 'nows' between the two-which is impossible.
i use the term 'absolutely', in view of the generic character of 'light' and 'heavy', in order to confine the application to bodies which do not combine lightness and heaviness.
but he does not explain the process by which surfaces become solids; and he characteristically ridicules democritus for not seeing that the worlds are finite and not infinite.
in the phrase 'moving faster' i refer to that which changes before another into the condition in question, when it moves over the same interval and with a regular movement; e.g.
if it had, there would be another body in which it sank: and if that had weight, there would be yet another which moved to the extremity and thus rose to the surface of all moving things. in fact, however, we have no evidence of such a body.
if, therefore, it can be shown that that which is pushing and that which is pushing and pulling are adjacent respectively to that which is being pushed and that which is being pulled, it will be evident that in all locomotion there is nothing intermediate
so, too, anaxagoras is right when he says that mind is impassive and unmixed, since he makes it the principle of motion: for it could cause motion in this sense only by being itself unmoved, and have supreme control only by being unmixed.
the elements are moving in a disorderly manner before the work of creation begins; and there is an eternal pattern of the world, which, like the 'idea of good,' is not the creator himself, but not separable from him.
would have to be in motion simultaneously and also to have ceased simultaneously to be in motion when the original movent ceases to move them, even if, like the magnet, it makes that which it has moved capable of being a movent.
dem.-because bd is divided into two parts in one hundred, we have
now the creation took up the whole of each of the four elements; for the creator compounded the world out of all the fire and all the water and all the air and all the earth, leaving no part of any of them nor any power of them outside.
unlike the eleatics, who relegated the world to the sphere of not-being, he admits creation to have an existence which is real and even eternal, although dependent on the will of the creator.
eleven. the angles made with the base of an isosceles triangle by perpendiculars from its extremities on the equal sides are each equal to half the vertical angle.
heights of all philosophy; and here is critias, whom every athenian knows to be no novice in the matters of which we are speaking; and as to hermocrates, i am assured by many witnesses that his genius and education qualify him to take part in any
dem.-join ac, bd. the angle abd is equal to acd, being in the same segment abcd xxi. ; and the angle dbc is equal to dac, because they are in the same segment dabc. hence the whole angle abc is equal to the sum of the two angles acd, dac.
and the ratios of their numbers, motions, and other properties, everywhere god, as far as necessity allowed or gave consent, has exactly perfected, and harmonized in due proportion.
there is no use in attempting to define or explain the first god in the platonic system, who has sometimes been thought to answer to god the father; or the world, in whom the fathers of the church seemed to recognize 'the firstborn of every creature.'
we produce a beginning of motion in ourselves spontaneously without anything having set us in motion from without.
prop. two.-problem. from a given point (a) to draw a right line equal to a given finite right line (bc).
nor again will the infinite traverse the infinite in a finite time. otherwise it would also traverse the finite, for the infinite includes the finite. we can further prove this in the same way by taking the time as our starting-point.
every one may understand the origin of diseases. they may be occasioned by the disarrangement or disproportion of the elements out of which the body is framed.
others say the earth rests upon water. this, indeed, is the oldest theory that has been preserved, and is attributed to thales of miletus.
in so far as this acceding food is potentially the double result e.g. is potentially so-much-flesh-it produces growth: for it is bound to become actually both so-much and flesh.
three. given the difference of two lines and the rectangle contained by them; find the lines.
two. if the sides bc, ca, ab of the triangle abc be denoted by a, b, c, and half their sum by s, the distances of the vertices a, b, one hundred of the triangle from the points of contact of the inscribed circle are respectively s a, s b, s c.
in equal circles (acb, dfe), angles at the centres (aob, dhe), or at the circumferences (acb, dfe), which stand on equal arcs (ab, de), are equal.
local change that is in locomotion: and so there must be something capable of being burned before there can be a process of being burned, and something capable of burning before there can be a process of burning.
forty two. through two given points describe a circle whose common chord with another given circle may be parallel to a given line, or pass through a given point.
eighteen. if there be given three parallel lines, and two fixed points a, b; then if the lines of connexion of a and b to any variable point in one of the parallels intersect the other parallels in the points one hundred and five hundred, e and f,
plato alone asserts the creation of time, saying that it had a becoming together with the universe, the universe according to him having had a becoming.
but it is difficult to conceive that the pace of each star should be exactly proportioned to the size of its circle.
as to such things as it does not contain in any way, they neither were nor are nor will be. these are those nonexistents whose opposites always are, as the incommensurability of the diagonal always is-and this will not be in time.
if there be two sets of magnitudes, which, taken two by two in direct order, have equal ratios, then the first : the last of the first set :: the first : the last of the second set ("ex aequali," or "ex aequo").
thus unqualified coming-to-be and passingaway turn out to be different according to common opinion from what they are in truth. for wind and air are in truth more real more a 'this somewhat' or a 'form'-than earth.
but i would not speak at the time, because i wanted to refresh my memory.
thus was time made in the image of the eternal nature; and it was created together with the heavens, in order that if they were dissolved, it might perish with them.
'to know or tell the origin of the other divinities is beyond us, and we must accept the traditions of the men of old time who affirm themselves to be the offspring of the gods-that is what they say-and they must surely have known their own ancestors.
contrary is nourishment to contrary: but growth is effected only by things becoming like to like. there must be alteration, then, in that there is this change from contrary to contrary.
wherefore he resolved to have a moving image of eternity, and when he set in order the heaven, he made this image eternal but moving according to number, while eternity itself rests in unity; and this image we call time.
passed over by the slower: for their respective velocities may stand to one another in this proportion.
thirty nine. place a given triangle so that its three sides shall touch three given circles.
it is evident from what has been said, then, that, if there is a void, a result follows which is the very opposite of the reason for which those who believe in a void set it up.
the modern physical philosopher is apt to dwell exclusively on the absurdities of ancient ideas about science, on the haphazard fancies and a priori assumptions of ancient teachers, on their confusion of facts and ideas, on their inconsistency and
prop. ten.-problem. to construct an isosceles triangle having each base angle double the vertical angle.
since, however, some things are-potentially while others are-actually, the constituents combined in a compound can 'be' in a sense and yet 'not-be'.
it was a subject especially congenial to the ponderous industry of certain french and swedish writers, who delighted in heaping up learning of all sorts but were incapable of using it.
one hundred eighteen. in the same case the product of the alternate segments of the sides made by the bisectors of the angles is
two. join oc, od. the two triangles bop, cop have the side ob equal to oc, and op common, and the angle bop greater than cop; therefore the base bp is greater than cp i. xxiv. . in like manner cp is greater than dp, c.
nevertheless, how this can be so remains matter for inquiry; how it comes about, i mean, that the same motive force at one time causes a thing to be in motion, and at another does not do so: for the difficulty raised by our objector really amounts to
again, since df is equal to db, df is equal to db : to each add ad , and we get ad df equal to ad db ; but ad df is equal to af ; therefore af is equal to ad db ; and we have proved af equal to ac cd . therefore ad db is equal to ac cd .
to the sphere of the same which was undivided he gave dominion, but the sphere of the other or diverse was distributed into seven unequal orbits, having intervals in ratios of twos and threes, three of either sort, and he bade the orbits move in opposite
the planets are near, so that the visual ray reaches them in its full vigour, but when it comes to the fixed stars it is quivering because of the distance and its excessive extension; and its tremor produces an appearance of movement in the star: for it
the assertion. therefore, that alteration is continuous is an extravagant calling into question of the obvious: for alteration is a change from one contrary to another. moreover, we notice that a stone becomes neither harder nor softer.
of the uses of the nerves in conveying motion and sensation, which he supposed to be communicated by the bones and veins; he was also ignorant of the distinction between veins and arteries;-the latter term he applies to the vessels which conduct air from
of rest: on the other hand rotatory motion is the only motion whose course is naturally such that it has no starting-point or finishing-point in itself but is determined from elsewhere.
nine. if ac and bk intersect in p, and through p a line be drawn parallel to bc, meeting ab in q; then cp is equal to pq.
five. make the angle poi equal pof i. xxiii. . join ip. now the triangles iop, fop have two sides io, op in one respectively equal to two sides fo, op in the other, and the angle iop equal to fop (const.); therefore i. iv. ip is equal to fp.
three. what is the difference between a chord and a secant?
so it is also impossible for a finite force to reside in an infinite magnitude.
again, because ad is parallel to ec, one of the sides of the triangle bec, bd : six hundred :: ba : ae ii. ; but ae has been proved equal to ac. therefore bd : six hundred :: ba : ac.
further, (five) place is also somewhere, not in the sense of being in a place, but as the limit is in the limited; for not everything that is is in place, but only movable body.
on the base by n times the area of the triangle.
both causes must be stated by the physicist, but especially the end; for that is the cause of the matter, not vice versa; and the end is 'that for the sake of which', and the beginning starts from the definition or essence; as in artificial products,
to a pair of radii may belong either of the two conjugate arcs into which their ends divide the circle.-newcomb.
muses as given by them with a view to irrational pleasure, which is deemed to be the purpose of it in our day, but as meant to correct any discord which may have arisen in the courses of the soul, and to be our ally in bringing her into harmony and
rectilineal figure abcd, and it has the angle e equal to the given angle ten. hence ei is a parallelogram fulfilling the required conditions.
for though it will be either flesh or bone or something else, that does not at once show that it contained these in potentiality: the further question remains, in what manner it becomes them. now anaxagoras opposes empedocles' view of the elements.
dem.-because am touches the circle at a, the angle oam is right. in like manner, the angle mbo is right; but the sum of the four angles of the quadrilateral oamb is equal to four right angles.
consequently the triangles abc, def have the sides ab, bc of one respectively equal to the sides de, ef of the other; and the contained angles abc and def equal; therefore iv. ac is equal to df, and the angle bac is equal to the angle edf.
from potential to actual substance) is coming-to-be, a change in the sphere of magnitude is growth and one in the sphere of quality is 'alteration'-both growth and 'alteration' being changes from what is-potentially to what is-actually magnitude and
this is in the truest sense the origin of creation and of the world, as we shall do well in believing on the testimony of wise men: god desired that all things should be good and nothing bad, so far as this was attainable.'
and as there are three parts in ab, and four in four hundred, we express this relation by saying that ab has to four hundred the ratio of three to four; and we denote it thus, three : four.
so, too, in its discussions concerning the divine, popular philosophy often propounds the view that whatever is divine, whatever is primary and supreme, is necessarily unchangeable. this fact confirms what we have said.
and the contraction caused by the compression thrusts the smaller particles into the interstices of the larger.
dem.-join be, ch. now since fh is a parallelogram, fg is equal to eh xxxiv. ; but bc is equal to fg (hyp.); therefore bc is equal to eh (axiom i.).
many great and wonderful deeds are recorded of your state in our histories. but one of them exceeds all the rest in greatness and valour.
an additional paradox is that the soul should consist of the 'elements', or that it should be one of them. how are the soul's 'alterations' to take place? how, e.g.
for the centre is opposed as contrary to the extremity, as that which sinks is opposed to that which rises to the surface. this also gives a reasonable ground for the duality of heavy and light in the spatial duality centre and extremity.
language, two, exercised a spell over the beginnings of physical philosophy, leading to error and sometimes to truth; for many thoughts were suggested by the double meanings of words (greek), and the accidental distinctions of words sometimes led the
again, if it is the number of planes in a body that makes one heavier than another, as the timaeus explains, clearly the line and the point will have weight. for the three cases are, as we said before, analogous.
far as it is applied to certain individual cases, he is wrong in making it of universal application.
it is clear also on other grounds that it is impossible that the destructible should not at some time be destroyed.
for the becoming of learning cannot be learning: so neither can the becoming of becoming be becoming, nor can the becoming of any process be that process.
for the study of the great writers and thinkers of the past, historical imagination is the first necessity.
for roughness is hardness mingled with irregularity, and smoothness is produced by the joint effect of uniformity and density.
for the higher intelligence of man seems to require, not only something above sense, but above knowledge, which can only be described as mind or being or truth or god or the unchangeable and eternal element, in the expression of which all predicates fail
the large bodies) are easily broken up because they collide with many other bodies. but why should indivisibility as such be the property of small, rather than of large, bodies?
five. if a transversal meet the sides of a triangle abc in the points a, b, one hundred; prove
if time is unreal, then all which is contained in time is unreal-the succession of human thoughts as well as the flux of sensations; there is no connecting link between (greek) and (greek).
like that of a substance, but consists in a process of coming to be or passing away; definite if you like at each stage, yet always different.
if two of the planes of a trihedral angle be equally inclined to the third plane, the angles contained in those planes are equal.
nor do such distinctions (up and down and right and left, c.) hold only in relation to us.
body has similar properties in a second degree, and the third body in the third degree.
moreover a conviction that there is a first unmoved something may be reached not only from the foregoing arguments, but also by considering again the principles operative in movents.
cor. two.-if two equal lines be drawn from p to either the convex or concave circumference, the diameter through p bisects the angle between them, and the parts of them intercepted by the circle are equal.
when we say that a thing changes in a particular year because it changes in a particular day. that being so, that which changes must be changing in any part of the primary time in which it changes.
this can be made clearer as follows. there are things whose constant nature it is to move away from the centre, while others move constantly towards the centre; and of these movements that which is away from the centre i call upward movement and that
or again: let o be the centre (fig. two). join oa, ob oc, od. then the four triangles aob, boc, cod, doa are each isosceles.
on this point, indeed, they were already getting into difficulties and admitted that the one was many-as if there was any difficulty about the same thing being both one and many, provided that these are not opposites; for 'one' may mean either
occupied). then, a the movement have moved b a distance g in a time five hundred, then in the same time the same force a will move one b twice the distance g, and in one d it will move one b the whole distance for g: thus the rules of proportion will be
it is, therefore, necessary for it to be of iron, it we are to have a saw and perform the operation of sawing. what is necessary then, is necessary on a hypothesis; it is not a result necessarily determined by antecedents.
but the reason of this conflict is that each group is in fact stating a part, whereas they ought to have taken a comprehensive view of the subject as a whole.
the foregoing remarks are sufficient to explain the essential nature of motion and rest, the number of kinds of change, and the different varieties of motion.
next to things that are in motion unnaturally those whose motion while natural is derived from themselves-e.g.
similarly, we have also discussed 'alteration', and explained what 'altering' is and how it differs from coming-to-be and passing-away. but we have still to investigate the so-called 'elements' of bodies.
what he says, therefore, is no explanation of 'nature'. moreover, it is this which is both 'the excellence' of each thing and its 'good': whereas he assigns the whole credit to the 'mingling'.
hence whatever is there, is of such a nature as not to occupy any place, nor does time age it; nor is there any change in any of the things which lie beyond the outermost motion; they continue through their entire duration unalterable and unmodified,
for instance (so runs the argument), let a, a...be the stationary bodies of equal size, b, b...the bodies, equal in number and in size to a, a...,originally occupying the half of the course from the starting-post to the middle of the a's, and g, g...those
every one will see that he must have looked to the eternal; for the world is the fairest of creations and he is the best of causes.
if then (one) the body, whose movement is circular, is fire or some other element, its natural motion must be the contrary of the circular motion. but a single thing has a single contrary; and upward and downward motion are the contraries of one another.
still, so much, at any rate, is clear: the things which are 'such as to impart motion', if that description be interpreted in one sense, will touch the things which are 'such as to be moved by them'-while they will not touch them, if the description be
sol.-let o (fig. one) be the centre of the given circle. join op, cutting the circumference in one hundred. with o as centre, and op as radius, describe the circle ape. erect ca at right angles to op. join oa, intersecting the circle bcd in b.
for a rectilinear figure as it revolves never continues in the same room, but where formerly was body, is now none, and where now is none, body will be in a moment because of the projection at the corners.
has as such a capacity for being built. it will have the capacity for being thus moved either immediately or through one or more links (as it will if, while everything that has a capacity for causing motion has as such a capacity for being moved by
def. vi.-if from the extremities of one right line perpendiculars be drawn to another, the intercept between their feet is called the projection of the first line on the second.
all that he did was done rationally in and by himself, and he moved in a circle turning within himself, which is the most intellectual of motions; but the other six motions were wanting to him; wherefore the universe had no feet or legs.
; hence the sides about the equal angles b and five hundred are proportional.
the statement may be false-there are similar tales about columns set up 'by the canaanites whom joshua drove out' (procop.); but even if true, it would only show that the legend, eight hundred years after the time of plato, had been transferred to egypt,
the reason is that the latter is assured to us by experiment, and is not contrasted with the certainty of ideal or mathematical knowledge.
dem.-because bc is divided into two segments in five hundred,
occupied in passing over ab will be finite: for it will be divisible into periods equal in number to the segments into which the magnitude is divisible.
astronomy, even in modern times, has made far greater progress by the high a priori road than could have been attained by any other.
two. in the same case the greater segment of the hypotenuse is equal to the least side of the triangle.
the same process of reasoning will also make clear at the same time the truth of the assumption we have made both now and at a previous stage that it is possible that there should be a motion that is continuous and eternal.
in species. in consequence of this distinction, it is evident that those things, whose 'substance'-that which is undergoing the process-is imperishable, will be numerically, as well as specifically, the same in their recurrence: for the character of the
therefore they will also move through the void with this ratio of speed. but that is impossible; for why should one move faster? (in moving through plena it must be so; for the greater divides them faster by its force.
dem.-one. let o be the centre of the circle. join oc, od. then the triangles ceo, deo have ce equal to ed (hyp.), eo common, and oc equal to od, because they are radii of the circle; hence i. viii.
this then is one principle (though not one or existent in the same sense as the 'this'), and the definition was one as we agreed; then further there is its contrary, the privation.
prop. thirty seven.-theorem. triangles (abc, dbc) on the same base (bc) and between the same parallels (ad, bc) are equal.
that there is no irregularity in the parts is obvious, since, if there were, some divergence of the stars would have taken place before now in the infinity of time, as one moved slower and another faster: but no alteration of their intervals is ever
(to illustrate my meaning: fire heats not only when in contact, but also from a distance.
rest, in parts only and by some external agency.
some parts while insusceptible in others.
thus 'sitting' is an example of a separable attribute, while 'snubness' contains the definition of 'nose', to which we attribute snubness.
hence the centres and the points a, b are in one right line; therefore ab is a diameter of each circle. hence, if ab be bisected in e, e must be the centre of each circle-that is, the circles are concentric-which v. is impossible.
for thus we define the whole-that from which nothing is wanting, as a whole man or a whole box. what is true of each particular is true of the whole as such-the whole is that of which nothing is outside.
five. describe a circle in the space acb, bounded by the line ab and the two circles.
it realized the fiction so natural to the human mind, because it answered the enquiry about the origin of the arts, that there had somewhere existed an ancient primitive civilization.
and of the other elements-the assumption by any of these of a wrong kind, or any similar irregularity, produces disorders and diseases; for when any of them is produced or changed in a manner contrary to nature, the parts which were previously cool grow
when the combining constituents have been divided into parts so small, and have been juxtaposed in such a manner, that perception fails to discriminate them one from another, have they then 'been combined or ought we to say 'no, not until any and every
house,................................ twenty eight.one thousand two hundred twelve farm one year,.......................... fourteen.seven thousand two hundred twelve food eight months,......................
thirteen. if ae be joined, the lines ae, bk, one hundred fifty, are concurrent.
fourteen (plot continued.) the tragic emotions of pity and fear should spring out of the plot itself. fifteen the element of character in tragedy. sixteen (plot continued.) recognition: its various kinds, with examples.
so that which has changed must have been changing and that which is changing must have changed, and a process of change is preceded by a completion of change and a completion by a process: and we can never take any stage and say that it is absolutely the
forty nine. describe a circle which shall bisect the circumferences of three given circles.
and if again i take another part equal to ae, that also must occupy a finite time in consequence of the same assumption.
if two triangles (abc, def) have two sides (ab, ac) of one respectively equal to two sides (de, df) of the other, but the base (bc) of one greater than the base (ef) of the other, the angle (a) contained by the sides of that which has the greater base is
again, the substance which is formed by the liquefaction of new and tender flesh when air is present, if inflated and encased in liquid so as to form bubbles, which separately are invisible owing to their small size, but when collected are of a bulk which
he was full of original thoughts, and yet liable to be imposed upon by the most obvious fallacies. he occasionally confused numbers with ideas, and atoms with numbers; his a priori notions were out of all proportion to his experience.
itself, it was indissoluble by the hand of any other than the framer.
for, since point is not 'immediately-next' to point, magnitudes are 'divisible through and through' in one sense, and yet not in another.
prop. thirty.-problem. to bisect a given arc acb.
ready to join with passion in the service of reason.
as a rule, no doubt, if a touches b, b touches a.
qualities, contains all sorts of bile and serum and phlegm.
two. let the sides given to be equal be ab and de; it is required to prove that bc is equal to ef, and ac to df. if bc be not equal to ef, suppose bg to be equal to it. join ag.
moreover 'coming to a standstill' is generally recognized to be identical or at least concomitant with the locomotion of a thing to its proper place.
(three. there is no interaction between infinites.) nor can infinite be acted upon in any way by infinite. let a and b be infinites, four hundred being the time of the action a of upon b.
dem.-produce ba to five hundred (post. ii.), and make ad equal to ac iii. . join four hundred.
dem.-let o be the centre of apb. join op. i say the centre of the smaller circle is in the line op. if not, let it be in any other position such as e. join oe, ep, and produce oe through e to meet the circles in the points one hundred, a.
but though in these respects he differs from them, he is deeply penetrated by the spirit of their philosophy; he differs from them with reluctance, and gladly recognizes the 'generous depth' of parmenides (theaet.).
one. prove that the angles at the base are equal without producing the sides. also by producing the sides through the vertex.
one. if in the second method the circle cut the line cb produced through b, prove the proposition.
clearly, then, one of the bodies will move naturally away from the centre and another towards the centre, since fire must be identical with fire, earth with earth, and so on, as the fragments of each are identical in this world.
(here by 'goal of change' i mean that which comes first in the process of change: e.g.
but the end is not due to these except as the matter, nor will it come to exist because of them.
beginning at the head and extending through the whole of the trunk.
have no power of sense-perception and of the senses themselves. for in a way even the senses undergo alteration, since the active sense is a motion through the body in the course of which the sense is affected in a certain way.
this is most obvious in the animals other than man: they make things neither by art nor after inquiry or deliberation. wherefore people discuss whether it is by intelligence or by some other faculty that these creatures work,spiders, ants, and the like.
the above, then, may be taken as sufficient to establish that there are principles and what they are and how many there are. now let us make a fresh start and proceed.
further, it is inconceivable that a movable body should be eternal, if its movement cannot be regarded as naturally eternal: and these bodies we know to possess movement. thus we see that generation is necessarily involved.
he does not, like kant, attempt to vindicate for men a freedom out of space and time; but he acknowledges him to be subject to the influence of external causes, and leaves hardly any place for freedom of the will.
unmeaningness of some of his reasons. but the timaeus still remains the greatest effort of the human mind to conceive the world as a whole which the genius of antiquity has bequeathed to us.
or itself unmoved: if, then, it is in motion, it will have to be subject to the same conditions as that which it moves, that is to say it will itself be in process of change and in being so will also have to be moved by something: so we have a series that
nature itself causes the soul to settle down and come to a state of rest for the performance of some of its functions, while for the performance of others other things do so: but in either case the result is brought about through the alteration of
the probable conclusion is as follows:-earth, when dissolved by the more penetrating element of fire, whether acting immediately or through the medium of air or water, is decomposed but not transformed.
we must no more believe that than follow empedocles when he says that the world, by being whirled round, received a movement quick enough to overpower its own downward tendency, and thus has been kept from destruction all this time.
nineteen. find the locus of a point, the sum or the difference of whose distance from two fixed lines is equal to a given length.
hither came solon, and was received with honour; and here he first learnt, by conversing with the egyptian priests, how ignorant he and his countrymen were of antiquity.
surely it does, for in any given case we have a definite thing that cause increase and a definite thing that suffers increase, and the one causes and the other suffers a certain amount of increase in a certain amount of time.
twenty four. the radii of two circles are four.twenty five and one.seventy five feet respectively, and the distance between their centres six.five feet; find the lengths of their direct and their transverse common tangents.
then ad : db :: the triangle ade : bde i. . and ae : ec :: the triangle ade : cde i. ; but ad : db :: ae : ec (hyp.). hence ade : bde :: ade : cde. therefore five. ix.
timaeus: why did the creator make the world?...he was good, and therefore not jealous, and being free from jealousy he desired that all things should be like himself. wherefore he set in order the visible world, which he found in disorder.
the fact that a thing that is in motion derives its motion from something is most evident in things that are in motion unnaturally, because in such cases it is clear that the motion is derived from something other than the thing itself.
secondly, locomotion must be primary in time: for this is the only motion possible for things.
since by v. or vi. the rectangle contained by any two lines is the square on half their sum the square on half their difference; therefore four times the rectangle contained by any two lines the square on their sum the square on their difference.
dem.-if possible, let acb, adb, be two similar segments constructed on the same side of ab. take any point five hundred in the inner one. join ad, and produce it to meet the outer one in one hundred. join bc, bd.
we must begin by distinguishing the various senses in which the term 'infinite' is used.
therefore ab ac . cor. three.-if a line be divided into any number of parts, the square on the whole is equal to the sum of the squares on all the parts, together with twice the sum of the rectangles contained by the several distinct pairs of parts.
cor. four.-if both pairs of opposite angles of a quadrilateral be equal, it is a parallelogram.
these definitions require explanation, especially def. iii., which has the fault of conveying no precise meaning-being, in fact, unintelligible.
square units. hence the area of the parallelogram is found by multiplying its length by its breadth, and this explains why we say (see def.
planets. plato seems to confuse the actual observation of the heavens with his desire to find in them mathematical perfection.
then the triangles ace, dcb are equiangular; hence ac : ce :: four hundred : cb; therefore ac.cb four hundred.ce.
more than two equal lines cannot be drawn from the given point (p) to either circumference.
we say that one thing comes to be from another thing, and one sort of thing from another sort of thing, both in the case of simple and of complex things. i mean the following.
thirteen. on what proposition is the rule for extracting the square root founded?
it is the same matter, but its being is different, as that which is receptive of disease is the same as that which is receptive of health, though in being different from it, and therefore diseasedness is different from healthiness.
it realizes its activity and continues to rise higher.
in discussing this new problem, we must begin by inquiring whether all things 'return upon themselves' in a uniform manner; or whether, on the contrary, though in some sequences what recurs is numerically the same, in other sequences it is the same only
from the given point (p) there can be drawn two equal lines to the concave or the convex circumference, both of which make equal angles with the line passing through the centre. six.
nine. what is a ratio of greater inequality?
it is part of it, the innermost part of it, which is in contact with the movable body; and for this reason the earth is in water, and this in the air, and the air in the aether, and the aether in heaven, but we cannot go on and say that the heaven is in
and not only must that which is changing have changed, but that which has changed must also previously have been changing, since everything that has changed from something to something has changed in a period of time.
five. if two right-angled s abc, abd be on the same hypotenuse ab, and the vertices one hundred and five hundred be joined, the pair of angles subtended by any side of the quadrilateral thus formed are equal.
the exposition is not clear, but this is its real meaning. and further, they say that since the atomic bodies differ in shape, and there is an infinity of shapes, there is an infinity of simple bodies.
for if it were possible, then it might actually occur, with the result, not that the body would simultaneously be actually both (indivisible and divided), but that it would be simultaneously divided at any and every point.
chance and what results from chance are appropriate to agents that are capable of good fortune and of moral action generally. therefore necessarily chance is in the sphere of moral actions.
the third objection may be thought to present more difficulty than the others, namely, that which alleges that motion arises in things in which it did not exist before, and adduces in proof the case of animate things: thus an animal is first at rest and
it represents the maturity of his thought, while the other works only illustrate his development.
sol.-take any line ab. divide it in one hundred, so that the rectangle ab.bc shall be equal to ac two. xi. . with a as centre, and ab as radius, describe the circle bde, and in it place the chord bd equal to ac i. . join ad.
and free from pain. but our creators, considering whether they should make a longer-lived race which was worse, or a shorter-lived race which was better, came to the conclusion that every one ought to prefer a shorter span of life, which was better, to a
therefore the number of two groups also-is the same number (for their number does not differ by a differentia of number), but it is not the same decad; for the things of which it is asserted differ; one group are dogs, and the other horses.
cor. one.-every triangle must have at least two acute angles.
the square described on bg will be equal to ten.
it is impossible, for instance, on a certain hypothesis that the triangle should have its angles equal to two right angles, and on another the diagonal is commensurable. but there are also things possible and impossible, false and true, absolutely.
dem.-because df bisects the chord ab and is perpendicular to it, it passes through the centre i., cor. one . in like manner ef passes through the centre.
physicist: for it is a fundamental assumption with him that motion is ultimately referable to nature herself.
the lives-of the several kinds of living things have a number by which they are distinguished: for there is an order controlling all things, and every time (i.e. every life) is measured by a period.
such an assertion would commit him to doctrines like those which plato has set forth in the timaeus.
further, it is with reason that good fortune is regarded as unstable; for chance is unstable, as none of the things which result from it can be invariable or normal.
means the presence or absence in it of more or less of the opposite quality. it is now clear, then, that there are only these three kinds of motion.
wholly in either condition will not preclude us from calling it white or not-white.
between fire and earth, the two extremes, he remarks that there are introduced, not one, but two elements, air and water, which are compared to the two mean proportionals between two cube numbers.
therefore there cannot be an alteration equal to or less than a locomotion: and consequently it is not the case that every motion is commensurable with every other.
it is what we are all inclined to do, to direct our inquiry not by the matter itself, but by the views of our opponents: and even when interrogating oneself one pushes the inquiry only to the point at which one can no longer offer any opposition.
it is also admitted that good and evil conduct are to be attributed respectively to good and evil laws and institutions.
a -- b.b - c c - d.b - c, or a- c -b d; therefore a : c :: b : d.
the containing principle may be likened to a mother, the source or spring to a father, the intermediate nature to a child; and we may also remark that the matter which receives every variety of form must be formless, like the inodorous liquids which are
it would be most natural and consequent upon what has been said that each of the stars should be composed of that substance in which their path lies, since, as we said, there is an element whose natural movement is circular.
but it is at any rate not difficult to see that place cannot be either of them. the form and the matter are not separate from the thing, whereas the place can be separated.
prop. twenty two.-theorem. the sum of the opposite angles of a quadrilateral (abcd) inscribed in a circle is two right angles.
dem.-let m and n be the multiples which a and b are of c.
it follows then that the motion of that which is in motion and the rest of that which is at rest must occupy time.
again, as there are different linear units in use, such as in this country, inches, feet, yards, miles, c., and in france, metres, and their multiples or sub-multiples, so different square units are employed.
fifty one. given the base of a triangle, the median that bisects the base, and the area; construct it.
cor. one.-if pi be produced to meet the circle again in fifty, pl is equal to pb.
of the anatomy and functions of the body he knew very little,-e.g.
ten. bisect a given triangle by a right line drawn from a given point in one of the sides.
an equal time, the quicker is neither of equal velocity nor slower, it follows that the motion of the quicker can occupy neither an equal time nor more time.
which is frothy and has a burning quality and dissolves the flesh.
astrology was the form which the lively fancy of ancient thinkers almost necessarily gave to astronomy. the observation that the lower principle, e.g. mechanics, is always seen in the higher, e.g. in the phenomena of life, further tended to perplex them.
he says, at all events, that coming-to-be and passing-away are the same as 'being altered':' yet, in common with other thinkers, he affirms that the elements are many.
six. how many parts in a triangle? ans. six; namely, three sides and three angles.
the dreams and visions, which pass through the philosopher's mind, of resemblances between different classes of substances, or between the animal and vegetable world, are put into the refiner's fire, and the dross and other elements which adhere to them
there is not a different place of the part, in which it is moved, when the whole vessel changes its place: it is always the same: for it is in the (proximate) place where they are that the air and the water (or the parts of the water) succeed each other,
considering that without them and when isolated from them, these higher things for which we look cannot be apprehended or received or in any way shared by us.
for it is evident if abc be applied to def, so that the point b shall coincide with e, and the line bc with ef, since bc is equal to ef, the point one hundred shall coincide with f; and since the angles b, one hundred are respectively equal to the angles
thing, so time is always at a beginning and at an end. and for this reason it seems to be always different; for the 'now' is not the beginning and the end of the same thing; if it were, it would be at the same time and in the same respect two opposites.
since the definition of the latter is different (for being cannot even be attributed to white, as nothing is which is not 'substance'), it follows that 'white' is not-being--and that not in the sense of a particular not-being, but in the sense that it is
the blood in order to maintain the balance of rare and dense which is necessary to its regular circulation.
"that two sides of a triangle are greater than the third" is, perhaps, self-evident; but it is not an axiom, inasmuch as it can be inferred by demonstration from other propositions; but we can give no proof of the proposition that "things which are equal
such reflections, although this is not the place in which to dwell upon them at length, lead us to take a favourable view of the speculations of the timaeus.
five. what is meant by an indirect proof?
intelligence, the 'one in many,' brighter than any promethean fire (phil.), which co-existing with them and so forming a new existence, is or becomes the intelligible world...so we may perhaps venture to paraphrase or interpret or put into other words the
six. the reciprocal of a ratio is the ratio obtained by interchanging the antecedent and the consequent. thus, four : three is the reciprocal of the ratio three : four.
existence. at the same time, the minds of men parted into the two great divisions of those who saw only a principle of motion, and of those who saw only a principle of rest, in nature and in themselves; there were born heracliteans or eleatics, as there
argument-a description which applies to the arguments both of melissus and of parmenides: their premisses are false and their conclusions do not follow.
the elements are supposed to pass into one another, but we must remember that these transformations are not the transformations of real solids, but of imaginary geometrical figures; in other words, we are composing and decomposing the faces of substances
mind worked. the circular impulse (greek) of the one philosopher answers to the circular movement (greek) of the other. but unlike anaxagoras, plato made the sun and stars living beings and not masses of earth or metal.
they say that in their statements about its generation they are doing what geometricians do when they construct their figures, not implying that the universe really had a beginning, but for didactic reasons facilitating understanding by exhibiting the
cor. one.-if a triangle and a parallelogram have equal altitudes, and if the base of the triangle be double of the base of the parallelogram, the areas are equal.
that of any of the things which enter into her; she is the natural recipient of all impressions, and is stirred and informed by them, and appears different from time to time by reason of them.
in respect of substance there is no motion, because substance has no contrary among things that are.
hence the triangle fag is similar to cad, and therefore the angle fag is equal to the angle cad. hence the line ac must pass through the point f, and therefore the parallelograms are about the same diagonal.
iii. if three quantities of the same kind be in continued proportion, the middle term is called a mean proportional between the other two.
'alteration' stand and fall together. for if the change is 'alteration', then the substratum is a single element; i.e. all things which admit of change into one another have a single matter.
seventeen. if perpendiculars be let fall on the sides of a polygon from any point, dividing each side into two segments, the sum of the squares on one set of alternate segments is equal to the sum of the squares on the remaining set.
"come-to-be" qua sharing in," to "pass-away" qua "losing," the 'form'. hence he thinks that 'assuming the truth of these theses, the forms must be causes both of coming-to-be and of passing-away'.
perhaps we may state the case thus: there is nothing to prevent each of the two parts, or at any rate one of them, that which is moved, being divisible though actually undivided, so that if it is divided it will not continue in the possession of the same
since what is 'in time' is so in the same sense as what is in number is so, a time greater than everything in time can be found.
these are the trees and plants and seeds which have been improved by cultivation and are now domesticated among us; anciently there were only the wild kinds, which are older than the cultivated.
so much for one view of the distinction between light and heavy. to others the analysis seems insufficient; and their views on the subject, though they belong to an older generation than ours, have an air of novelty.
dem.-let m denote the multiples which a, b are of a, b; then we have
dem.-erect four hundred at right angles to cb xi. , and make four hundred equal to ca iii. . join bd.
the fifth regular solid, or dodecahedron, cannot be formed by a combination of either of these triangles, but each of its faces may be regarded as composed of thirty triangles of another kind.
such times those who live upon the mountains and in dry and lofty places are more liable to destruction than those who dwell by rivers or on the seashore. and from this calamity the nile, who is our never-failing saviour, delivers and preserves us.
the triangles are equal; but the parallelogram al is double of the triangle cag xli. , because they are on the same base ag, and between the same parallels ag and one hundred fifty.
both of these views involve the same difficulties. for (one) if, on the one hand, they make the primary body an atom, the view will be open to the objections already advanced against the atomic theory.
on similar grounds, therefore, five hundred (dryness) will also belong to f.
now we are to suppose that a thing proceeds in locomotion from a to g and that at the moment of its arrival at g the continuity of its motion is unbroken and will remain so until it has arrived back at a.
since we have determined the nature of place, and void must, if it exists, be place deprived of body, and we have stated both in what sense place exists and in what sense it does not, it is plain that on this showing void does not exist, either
the motion of everything that is in process of locomotion is either rotatory or rectilinear or a compound of the two: consequently, if one of the former two is not continuous, that which is composed of them both cannot be continuous either.
if it were, the greater the quantity of fire the slower it would move, owing to the increase of weight due to the increased number of triangles.
it has suffered this change not by 'division' and composition', nor yet by 'turning' and 'intercontact' as democritus asserts; for it has passed from the liquid to the solid state without any change of 'grouping' or 'position' in the constituents of its
the story that plato had purchased three books of his writings from a relation is not worth repeating; it is only a fanciful way in which an ancient biographer dresses up the fact that there was supposed to be a resemblance between the two writers.
which subtracted from the sum of the outgoes leaves a balance of twenty five.two thousand one hundred thirty four on the one side,-this being very nearly the means with which i started, and the measure of expenses to be incurred,-and on the other, beside
born in one thousand seven hundred twelve, he issued no work of importance till one thousand seven hundred fifty; but he tells us, in the confessions, that in one thousand seven hundred forty three, when he was attached to the embassy at venice, he had
here the ratio of four : six is twenty three, and the ratio of six : nine is twenty three, so that four, six, nine are continued proportionals; but, in reality, there are four terms, for the full proportion is four : six :: six : nine.
resemble not them, but that other which included them.
and being composed of the same, the other, and the essence, these three, and also divided and bound in harmonical proportion, and revolving within herself-the soul when touching anything which has essence, whether divided or undivided, is stirred to utter
others again, with empedliocles of acragas and heraclitus of ephesus, believe that there is alternation in the destructive process, which takes now this direction, now that, and continues without end.
in which double series of numbers are two kinds of means; the one exceeds and is exceeded by equal parts of the extremes, e.g. one, four three, two; the other kind of mean is one which is equidistant from the extremes-two, four, six.
(one) the law of gravitation, according to plato, is a law, not only of the attraction of lesser bodies to larger ones, but of similar bodies to similar, having a magnetic power as well as a principle of gravitation.
this is in many ways its great strength; but where it is excessively so, the accident of time is to blame. in the eighteenth century it was, broadly speaking, safe to generalise and unsafe to particularise.
been, or what can be, counted. but if nothing but soul, or in soul reason, is qualified to count, there would not be time unless there were soul, but only that of which time is an attribute, i.e.
moreover, they act and suffer action wherever they chance to be in contact (for there they are not "one"), and they generate by being put together and becoming intertwined.
the time in question-it is impossible that in that time that which is in motion should be over against some particular thing primarily.
now in the first place to assume that ab, because it is in motion as a whole and is not moved by anything external to itself, is therefore moved by itself-this is just as if, supposing that kl is moving lm and is also itself in motion, we were to deny
the charge of premature generalization which is often urged against ancient philosophers is really an anachronism. for they can hardly be said to have generalized at all.
in existence and without beginning? or created, and had it a beginning?
xvi. a ray which turns in the sense opposite to the hands of a watch describes a positive angle, and one which turns in the same direction as the hands, a negative angle.
things are said to be in contact when their extremities are together.
motion. but since ab is therefore divisible, the thing is coming to a stand in every one of the parts of ab: for we have shown above that it is coming to a stand in every one of the parts in which it is primarily coming to a stand.
two. if two circles be concentric, all tangents to the inner from points on the outer are equal.
this last point, however, we may admit, has not yet been proved: but at any rate it is clear that if q is to be transformed in turn into yet another 'element', yet another contrariety will belong not only to q but also to f (fire).
mingling them in due proportions with one another, made the marrow out of them to be a universal seed of the whole race of mankind; and in this seed he then planted and enclosed the souls, and in the original distribution gave to the marrow as many and
for the ratios of smaller bodies may be repeated among greater bodies.
(one) what is incapable of being gone through, because it is not in its nature to be gone through (the sense in which the voice is 'invisible').
four. draw a common perpendicular to two non-coplanar lines, and show that it is the shortest distance between them.
the above-mentioned distinctions can also be made in the case of things that cause motion: some of them are capable of causing motion unnaturally (e.g. the lever is not naturally capable of moving the weight), others naturally (e.g.
dem.-join cf, cg. then the two triangles fhc, ghc have fh equal to gh (const.), and hc common; and the base cf equal to the base cg, being radii of the circle fdg (def. xxxii.). therefore the angle chf is equal to the angle chg viii.
other, and yet of all things in the world they are the most opposed and unlike.
let us explain again that there is no void existing separately, as some maintain. if each of the simple bodies has a natural locomotion, e.g.
but with regard to the assumption of 'indivisible solids', although we must not now enter upon a detailed study of its consequences, the following criticisms fall within the compass of a short digression: i.
cor. two.-two right lines cannot have a common segment.
they merely give an ingenious demonstration that air is something--by straining wine-skins and showing the resistance of the air, and by cutting it off in clepsydras.
the appropriate places of the whole and of the part are alike, e.g. of the whole earth and of a clod the appropriate place is the lower region; of fire as a whole and of a spark, the upper region.
is it that locomotion is a genus or that line is a genus?
it is the fulfilment of what is potential when it is already fully real and operates not as itself but as movable, that is motion. what i mean by 'as' is this: bronze is potentially a statue.
one hundred twenty four. from a point p in the plane of a given polygon perpendiculars are let fall on its sides; if the area of the polygon formed by joining the feet of the perpendiculars be given, the locus of p is a circle.
which this fraction expressed was in the ratio of two hundred fifty six to two hundred forty three (e.g.
hence leucippus and democritus, who say that the primary bodies are in perpetual movement in the void or infinite, may be asked to explain the manner of their motion and the kind of movement which is natural to them.
are moved by something-both those that are moved by themselves and those that are not moved by themselves (e.g.
also itself less than that thing. hence it follows that the quicker will traverse an equal magnitude in less time than the slower.
in order then that they may be mortal, and that this universe may be truly universal, do ye, according to your natures, betake yourselves to the formation of animals, imitating the power which was shown by me in creating you.
products of chance and luck. then what is the cause determining that man comes-to-be from man, that wheat (instead of an olive) comes-to-be from wheat, either invariably or generally?
substances do not allow any previous impression to remain, but begin by making the surface as even and smooth as possible.
they too conform to the principle of the same, and may be compared with the modern conception of laws of nature. they are in space, but not in time, and they are the makers of time.
for instance, although a sphere might come-to-be out of this part of a lump of wax and a pyramid out of some other part, it was nevertheless possible for either figure to have come-to-be out of either part indifferently: that is the manner of coming-to-be
(two) substance is plainly divisible into other substances, if we consider the mere nature of a definition. for instance, if 'man' is a substance, 'animal' and 'biped' must also be substances.
eight. if the middle points of any two sides of a triangle be joined, the triangle so formed with the two half sides is one-fourth of the whole.
this is indicated by the phrase 'in vain', which is used when a which is for the sake of b, does not result in b.
but water, when divided by fire or by air, on re-forming, may become one part fire and two parts air; and a single volume of air divided becomes two of fire.
this second method of coming-to-be, then, takes a longer time. but (iii) if one quality in each of two 'elements' pass-away, the transformation, though easier, is not reciprocal.
dem.-let o, o be the centres of the two fixed circles; o the centre of the variable circle; a, b the points of contact. let ab and oo meet in one hundred, and cut the fixed circles again in the points a, b respectively. join ao, ao, bo.
now, the causes being four, it is the business of the physicist to know about them all, and if he refers his problems back to all of them, he will assign the 'why' in the way proper to his science-the matter, the form, the mover, 'that for the sake of
for suppose that e is not coincident with g, then f will be, since either e or f is predictable of everything. but of that of which f is predicated h will be predicable also. h will then be coincident with g, but this we saw to be impossible.
; but abf is right (hyp.); therefore bfg is right-that is, fg is perpendicular to ce. hence every line in the plane de, drawn perpendicular to the common section of the planes de, one hundred one, is normal to the plane one hundred one. therefore eleven.
hence whatever the contrasted 'poles' of the changes may be whether fire and earth, or some other couple-the one of them will be 'a being' and the other 'a not-being'.
dem.-one. let the sides ad, df of the parallelograms ac, bf opposite to the common base bc terminate in the same point five hundred, then xxxiv. each parallelogram is double of the triangle bcd. hence they are equal to one another.
fifty two. if the diagonals ac, bd of a quadrilateral abcd intersect in e, and be bisected in the points f, g, then
what then will be the nature of its rest and of its movement, or where will they be? it will either be at home everywhere-then it will not be moved; or it will be moved everywhere-then it will not come to rest.
we may start as follows: we have three points, starting-point, middle-point, and finishing-point, of which the middle-point in virtue of the relations in which it stands severally to the other two is both a starting-point and a finishing-point, and though
for how could 'white' come from 'musical', unless 'musical' happened to be an attribute of the not-white or of the black? no, 'white' comes from 'not-white'-and not from any 'not-white', but from black or some intermediate colour.
unable to unite them in a consistent whole.
for the answer we give to this question makes the greatest difference. and again, if the primary 'reals' are indivisible magnitudes, are these bodies, as democritus and leucippus maintain? or are they planes, as is asserted in the timaeus?
other things, however, can both be and not he.) hence coming-to-be and passing-away must occur within the field of 'that which can be-and not-be'.
timaeus: all men who have any right feeling, at the beginning of any enterprise, call upon the gods; and he who is about to speak of the origin of the universe has a special need of their aid.
, and from a draw af i. xii. perpendicular to de. af is normal to the plane bh.
others, again, say that the earth, which lies at the centre, is 'rolled', and thus in motion, about the axis of the whole heaven, so it stands written in the timaeus.
therefore, since 'every' and 'all' and 'complete' do not differ from one another in respect of form, but only, if at all, in their matter and in that to which they are applied, body alone among magnitudes can be complete.
now the moved must be in motion, but it need not move anything else: the instrument of motion must both move something else and be itself in motion (for it changes together with the moved, with which it is in contact and continuous, as is clear in the
fifty four. the sum of the squares on the sides of a triangle is equal to twice the sum of the rectangles contained by each perpendicular and the portion of it comprised between the corresponding vertex and the orthocentre; also equal to r minus the sum
hence we infer generally that multiplying the terms of any fraction by two does not alter its value. in like manner it may be shown that multiplying the terms of a fraction by any whole number does not alter its value.
in the case of reckoning the halves, it is clear that this result follows: for then one point must be reckoned as two: it will be the finishing-point of the one half and the starting-point of the other, if we reckon not the one continuous whole but the
if a motion is to be regular its parts ought to fit one another.
bring up from below in their turn those who were worthy, and those among themselves who were unworthy were to take the places of those who came up?
and since b, the slower, has passed over gk in the time zo, the quicker will pass over it in less time: so that the time zo will again be divided.
but violent as were the internal and alimentary fluids, the tide became still more violent when the body came into contact with flaming fire, or the solid earth, or gliding waters, or the stormy wind; the motions produced by these impulses pass through
now we have already'proved in our course on physics that there cannot be an infinite magnitude: and we have now proved that it is impossible for a finite magnitude to have an infinite force, and also that it is impossible for a thing to be moved by a
it is necessary to class it with privation or with potentiality or with sheer actuality, yet none of these seems possible.
the extrusion of the largest atoms is a process that will in time exhaust the supply; and it is by such a process that they account for the generation of water, air, and earth from one another.
further, since to be lighter is to have fewer of these homogeneous parts and to be heavier is to have more, and air, water, and fire are composed of the same triangles, the only difference being in the number of such parts, which must therefore explain
it appears from the above estimate, that my food alone cost me in money about twenty-seven cents a week.
for we know that when a thing comes to be of such a quantity or quality or in such a relation, time, or place, a subject is always presupposed, since substance alone is not predicated of another subject, but everything else of substance.
for the nature of things is the nature which most of them possess for most of the time. thus their view brings them into the contrary position that disorder is natural, and order or system unnatural. but no natural fact can originate in chance.
he could imagine cities which had existed time out of mind (states.; laws), laws or forms of art and music which had lasted, 'not in word only, but in very truth, for ten thousand years' (laws); he was aware that natural phenomena like the delta of the
then will any predicate belonging to the remaining categories attach actually to this presupposed substance?
one. two triangles equal in area (acb, dce), which have one angle (one hundred) in one equal to one angle (one hundred) in the other, have the sides about these angles reciprocally proportional. two.
comprehend all truth. being or essence, and similar words, represented to them a supreme or divine being, in which they thought that they found the containing and continuing principle of the universe.
twenty four. how many centres of similitude can two regular polygons of n sides each have? ans. n centres, which lie on a circle.
twenty eight. classify the properties of triangles and parallelograms proved in book one.
those who make them limited in number never make them infinite in amount.
next, we must determine what are the four most beautiful figures which are unlike one another and yet sometimes capable of resolution into one another...of the two kinds of triangles the equal-sided has but one form, the unequal-sided has an infinite
again, when a few small particles, enclosed in many larger ones, are in process of decomposition and extinction, they only cease from their tendency to extinction when they consent to pass into the conquering nature, and fire becomes air and air water.
animal in motion: this is what happens when animals are asleep: though there is then no perceptive motion in them, there is some motion that causes them to wake up again.
on the same principle the fulfilment of the whole heaven, the fulfilment which includes all time and infinity, is 'duration'-a name based upon the fact that it is always-duration immortal and divine.
as if they were those very athenians whom the sacred egyptian record has recovered from oblivion, and thenceforward we will speak of them as athenians and fellow-citizens.
other objections to the view that it is not necessary to assume a third principle as a substratum may be added. (one) we do not find that the contraries constitute the substance of any thing.
one hundred fifteen. state and prove the corresponding theorems for the centres of the escribed circles.
that one of the parts into which ab is divided is equal to one of the parts into which four hundred is divided.
we will now show that nothing can be in motion in a present. for if this is possible, there can be both quicker and slower motion in the present. suppose then that in the present n the quicker has traversed the distance ab.
now, since the lines ef, bc are parallel, the triangles aef, abc are equiangular; therefore iv.
now if it exists for infinite time let this capacity be actualized; and it will be in actuality at once existent and non-existent.
there may be things not subject to generation or any kind of movement, but if so they belong to another and a higher inquiry than the study of nature.
seventeen. if the area of a rectangle be given, its perimeter is a minimum when it is a square.
(i) now we often divide terms into those which signify a 'this somewhat' and those which do not.
having established these distinctions we car now proceed to the sequel. if there are thing!
although absolutely in motion, they are relatively at rest; or we may conceive of them as resting, while the space in which they are contained, or the whole anima mundi, revolves.
two. if the medians of a triangle intersect in o, ab bc ca three(oa ob oc ).
they continued to exist for the purposes of ritual or of art; but from the sixth century onwards or even earlier there arose and gained strength in the minds of men the notion of 'one god, greatest among gods and men, who was all sight, all hearing, all
similarly, 'musical' comes to be from 'not-musical', but not from any thing other than musical, but from 'unmusical' or any intermediate state there may be.
animals unseen by reason of their smallness and without form; these again are separated and matured within; they are then finally brought out into the light, and thus the generation of animals is completed.
to each other. with this consequence there also follows another: we have a thing that is in process of change from a position in which it has not yet been: so, inasmuch as this is impossible, the thing must come to a stand at g.
let e represent half the motive power a and z half the weight b: then the ratio between the motive power and the weight in the one case is similar and proportionate to the ratio in the other, so that each force will cause the same distance to be traversed
the twenty triangular faces of an icosahedron form the faces or sides of two regular octahedrons and of a regular pyramid (twenty eight x two four); and therefore, according to plato, a particle of water when decomposed is supposed to give two particles
now it would seem that motions to contraries are at the same time motions from contraries (though their essence may not be the same; 'to health' is distinct, i mean, from 'from disease', and 'from health' from 'to disease').
all that matters is the intervention of a time: it is this that prevents the change from being continuous: so, too, in our previous instances the important thing was not the relation of contrariety but the impossibility of the two processes being present
moving in a circle along the diagonal of the same parallelogram from right to left; or, in other words, the first describing the path of the equator, the second, the path of the ecliptic.
again, because the angle bca is equal to bef, the line ca i. xxviii. is parallel to ef. in like manner, bf is parallel to four hundred; therefore the figure acdf is a parallelogram; hence ac is equal to df, and four hundred is equal to af.
one. if a were equal to five hundred, the triangles abc, def would have the two sides ab, ac of one respectively equal to the two sides de, df of the other, and the angle a contained by the two sides of one equal to the angle five hundred contained by the
again, if the motion of the heaven is the measure of all movements whatever in virtue of being alone continuous and regular and eternal, and if, in each kind, the measure is the minimum, and the minimum movement is the swiftest, then, clearly, the
these, however, are cases of addition: but it is not by addition that growing things are believed to increase. and it is far more difficult for him to account for the coming-to-be which occurs in nature.
either all things are at rest, or all things are in motion, or some things are at rest and others in motion.
now other bodies are severally light and heavy, and evidently in them the attributes are due to the difference of their uncompounded parts: that is to say, according as the one or the other happens to preponderate the bodies will be heavy and light
ac ab, and bc ab. but things which are equal to the same are equal to one another (axiom i.); therefore ac is equal to bc; therefore the three lines ab, bc, ca are equal to one another. hence the triangle abc is equilateral (def.
implies that one thing traverses an equal distance in less time than another: consequently there will be a part of a in which b will pass over a part of the circle equal to g', while g will occupy the whole of a in passing over g'.
now the whole included within the extreme circumference must be composed of all physical and sensible body, because there neither is, nor can come into being, any body outside the heaven.
it is therefore evident that there is also no place or void or time outside the heaven. for in every place body can be present; and void is said to be that in which the presence of body, though not actual, is possible; and time is the number of movement.
the view is too naive for it to be worth while to consider the impossibilities implied in it.
a divine body, and for that reason to it is given the circular body whose nature it is to move always in a circle. why, then, is not the whole body of the heaven of the same character as that part?
this would be the most intelligible way, but it is not the usual one, which is as follows:-six : nine :: ten : fifteen. in this form it is called a proportion. hence a proportion consists of two ratios which are asserted by it to be equal.
yet our eyes tell us that the heavens revolve in a circle, and by argument also we have determined that there is something to which circular movement belongs.
only the pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite.
five. the rectangle contained by two sides of a triangle is equal to the rectangle contained by the perpendicular and the diameter of the circumscribed circle. for, let ce be the diameter. join ae.
thirty six. the greater the number of sides of a regular polygon circumscribed about a given circle, the less will be its perimeter.
dem.-let o, h be the centres (see last fig.). join ao, ob, dh, he; then because the circles are equal, the angles aob, dhe at the centres, which stand on the equal arcs agb, dke, are equal xxvii. .
coming-to-be presupposes your father's, his coming-to-be does not presuppose yours)? why, on the contrary, does this coming-to-be seem to constitute a rectilinear sequence?
therefore there will be a point at which ace began for the first time to cut bb. this, however, is impossible. the infinite, then, cannot revolve in a circle; nor could the world, if it were infinite.
the fourth regular solid, or cube, is formed by the combination of four isosceles triangles into one square and of six squares into a cube.
for it is true in regard to quality, quantity, and place alike that any process of change is impossible which can have no end.
four. how many conditions are necessary to define similar triangles?
further, neither that which moves towards nor that which moves away from the centre can be infinite. for the upward and downward motions are contraries and are therefore motions towards contrary places.
take the case of magnitudes: let ab be a magnitude, and suppose that it has moved from b to a primary 'where' g.
and in like manner the body may be too much for the soul, darkening the reason, and quickening the animal desires.
plainly, too, to be in time does not mean to co-exist with time, any more than to be in motion or in place means to co-exist with motion or place.
we can distinguish, then, between the two-just as, to give another example, 'colour' and visible' are different-and clearly it is the fulfilment of what is potential as potential that is motion. so this, precisely, is motion.
twenty four. if through the point of intersection of the diagonals of a cyclic quadrilateral the minimum chord be drawn, that point will bisect the part of the chord between the opposite sides of the quadrilateral.
three. if acd, bcd be adjacent angles, any parallel to ab will meet the bisectors of these angles in points equally distant from where it meets four hundred.
for as they are formed, three of them from the triangle which has the sides unequal, the fourth from the triangle which has equal sides, three can be resolved into one another, but the fourth cannot be resolved into them nor they into it.
in order to render this identity more apparent, we have made some slight alterations in the usual proofs. the pairs of propositions thus grouped are considered in modern geometry not as distinct, but each pair is regarded as one proposition.
if we look at the question from the point of view of general theory, the same result would also appear to be indicated by the following arguments.
previously-mentioned disorders still greater.
and, to judge by what he says, the qualities are such that they can be 'attached' to things and can again be 'separated' from them, especially since strife and love are still fighting with one another for the mastery.
accomplishment of his work, but himself contriving the good in all his creations.
yet though he makes the infinites two, he does not use them.
the centre, therefore, being determinate, the upper place must also be determinate. but if these two places are determined and finite, the corresponding bodies must also be finite.
moreover, the idea of "nature" has already undergone a great development; it is no longer an empty opposition to the evils of society; it possesses a positive content.
further, everything that changes must be divisible.
of the supplement of their sum.-catalan.
seven. the circles whose diameters are the four sides of any cyclic quadrilateral intersect again in four concyclic points.
three. if two lines be at right angles, and if each bisect the other, then any point in either is equally distant from the extremities of the other.
movement is called infinite in virtue of the magnitude covered by the movement (or alteration or growth), and time because of the movement. (i use these terms for the moment.
in that which contains the attribute-that these, i mean, must not be divisible in the way in which colour is divided into kinds? thus in this respect one thing will not be commensurable with another, i.e.
again, if the earth as a whole is capable of floating upon water, that must obviously be the case with any part of it. but observation shows that this is not the case. any piece of earth goes to the bottom, the quicker the larger it is.
but neither is it open to us to say that growth or diminution occurs in the way in which e.g. air is the volume has then become greater, the change will not be growth, but a coming to-be of the one-viz.
or thus: let us denote the sides by a, b, c, and the figures by , , ; then because the figures are similar, we have xx.
world wide web presentation is copyright (one hundred) one thousand nine hundred ninety four-two thousand, daniel one hundred. stevenson, web atomics.
but neither will it in any way affect the argument if there is not a similar accession of concurrent fragments from every side.
ninety six. if the middle point of the base bc of an isosceles triangle abc be the centre of a circle touching the equal sides, prove that any variable tangent to the circle will cut the sides in points five hundred, e, such that the rectangle bd.ce will
if there be two lines (a, bc), one of which is divided into any number of parts (bd, de, ec), the rectangle contained by the two lines (a, bc), is equal to the sum of the rectangles contained by the undivided line (a) and the several parts of the divided
to say then that there is no reason why anything should not be generated and yet indestructible or ungenerated and yet destroyed, to imagine that in the one case generation and in the other case destruction occurs once for all, is to destroy part of the
the stick moves the stone and is moved by the hand, which again is moved by the man: in the man, however, we have reached a movent that is not so in virtue of being moved by something else.
now, since oe passing through the centre bisects ab, which does not pass through the centre, it is at right angles to it; therefore the angle aeo is right. in like manner the angle ceo is right.
- over one, four three, three two, - over two, eight three, three, - over four, sixteen three, six, - over eight: - over one, three two, two, - over three, nine two, six, - over nine, twenty seven two, eighteen, - over twenty seven;
these then are the reasons for which they might say that there is a void; our statement is based on the assumption that there is a single matter for contraries, hot and cold and the other natural contrarieties, and that what exists actually is produced
i saw that i could easily raise my bushel or two of rye and indian corn, for the former will grow on the poorest land, and the latter does not require the best, and grind them in a hand-mill, and so do without rice and pork; and if i must have some
what is potentially flesh or bone has not yet its own 'nature', and does not exist until it receives the form specified in the definition, which we name in defining what flesh or bone is.
squares ad, ck is equal to twice the figure ak, together with the figure gh.
fifteen. the area of a sector of a circle is equal to half the rectangle contained by the arc of the sector and the radius of the circle.
a thing, then, will be affected by time, just as we are accustomed to say that time wastes things away, and that all things grow old through time, and that there is oblivion owing to the lapse of time, but we do not say the same of getting to know or of
there will thus be a determinate time from this 'now' to that, and there was such in reference to the past event. but if there be no time which is not 'sometime', every time will be determined.
the learner becomes from one potential something another potential something: for one who possesses knowledge of a science but is not actually exercising it knows the science potentially in a sense, though not in the same sense as he knew it potentially
it is a matter of observation that fire, water, and every simple body undergo a process of analysis, which must either continue infinitely or stop somewhere. (one) suppose it infinite.
the commentary of proclus on the timaeus is a wonderful monument of the silliness and prolixity of the alexandrian age. it extends to about thirty pages of the book, and is thirty times the length of the original.
a theorem consists of two parts, the hypothesis, or that which is assumed, and the conclusion, or that which is asserted to follow therefrom. thus, in the typical theorem,
that the kinds cannot be infinite is evident, if our original presuppositions remain unchallenged.
therefore earth in motion, whether in a mass or in fragments, necessarily continues to move until it occupies the centre equally every way, the less being forced to equalize itself by the greater owing to the forward drive of the impulse.
and so, if the body which moves with a circular motion cannot admit of increase or diminution, it is reasonable to suppose that it is also unalterable.
cor. two.-if two circles touch each other at any point, they cannot have any other common point. for, since two circles cannot have more than two points common x.
dem.-the two triangles dcf, ecf have four hundred equal to ce (const.) and cf common; therefore the two sides four hundred, cf in one are respectively equal to the two sides ce, cf in the other, and the base df is equal to the base ef, being the sides of
prop. thirty one.-problem. through a given point (one hundred) to draw a right line parallel to a given right line.
the soul, interfused everywhere from the centre to the circumference of heaven, of which also she is the external envelopment, herself turning in herself, began a divine beginning of never-ceasing and rational life enduring throughout all time.
eight. the parallelogram formed by the line of connexion of the middle points of two sides of a triangle, and any pair of parallels drawn through the same points to meet the third side, is equal to half the triangle.
infinite distance. from these considerations then it is clear that the earth does not move and does not lie elsewhere than at the centre.
if, then, it is by constraint that the earth now keeps its place, the so-called 'whirling' movement by which its parts came together at the centre was also constrained.
since time is the measure of motion, it will be the measure of rest too-indirectly. for all rest is in time. for it does not follow that what is in time is moved, though what is in motion is necessarily moved.
observation.-the great difficulty which beginners find in this proposition is due to the fact that the two triangles acf, abg overlap each other.
three. of all triangles having the same base and vertical angle, the sum of the sides of an isosceles triangle is a maximum.
the arguments that may be advanced against this position are not difficult to dispose of. the chief considerations that might be thought to indicate that motion may exist though at one time it had not existed at all are the following:
(one) of the intersection of its perpendiculars;
granted, then, that they are a limited number, it is plausible to suppose them more than two. for it is difficult to see how either density should be of such a nature as to act in any way on rarity or rarity on density.
if not, produce ep, and draw cj parallel to pf; then describe the parallelogram klmn xxv.
but the motion of a thing can never be constituted by the motion of something else: consequently the whole motion is the motion of the whole magnitude.
four. draw a tangent to a given circle so that the triangle formed by it and two fixed tangents to the circle shall be-one, a maximum; two, a minimum.
compressed into the same. having made a compound of all the three, he proceeded to divide the entire mass into portions related to one another in the ratios of one, two, three, four, nine, eight, twenty seven, and proceeded to fill up the double and
whether a certain quantity of each has become white. or ought we not rather to look for it in both?
prop. eleven.-theorem. ratios that are equal to the same ratio are equal to one another.
general question we shall attain clearness when we examine the matter universally.
he looks with more favour on the pythagoreans, whose intervals of number applied to the distances of the planets reappear in the timaeus.
but (i) those thinkers are in error who postulate, beside the bodies we have mentioned, a single matter-and that corporeal and separable matter.
then neither would there be, nor could there arise, any other thing that was aquiline. similarly, suppose flesh and bones to be the matter of man, and suppose a man to be created of all flesh and all bones in indissoluble union.
it might be thought that what we set out to prove has thus been shown, but our argument so far does not prove it, because it does not yet prove that anything impossible results from the contrary supposition: for in a finite time there may be an infinite
for the supposed infinite body would have to be infinite in weight.
the character which is the ground of the predication will always remain obscure.
hence, if'unqualified not-being 'means the negation of 'being' in the sense of the primary term of the category in question, we shall have, in 'unqualified coming-to-be', a coming-to-be of a substance out of not-substance.
consequently, since these parts are finite, both in size individually and in number collectively, the whole time must also be finite: for it will be a multiple of the portion, equal to the time occupied in completing the aforesaid part multiplied by the
the same is true of the other processes of combination and separation (they will all be found to be forms of pushing apart or of pushing together), except such as are involved in the processes of becoming and perishing.
but ab is coplanar with ad, bd eleven. ii. . therefore the lines ad, bd, four hundred are coplanar; and since the angles abd, bdc are right, the line ab is parallel to four hundred i. xxviii. .
we can now understand why fire heats and the cold thing cools, and in general why the active thing assimilates to itself the patient.
who would desire better evidence than that of critias, who had heard the narrative in youth when the memory is strongest at the age of ten from his grandfather critias, an old man of ninety, who in turn had heard it from solon himself?
and the natural movement in each case is one. further, these worlds, being similar in nature to ours, must all be composed of the same bodies as it.
we call a thing white or not-white not necessarily because it is be one or the other, but cause most of its parts or the most essential parts of it are so: not being in a certain condition is different from not being wholly in that condition.
moreover if things move themselves, it would be unreasonable to suppose that in only one kind of motion is their motion derived from themselves. again, how can anything of continuous and naturally connected substance move itself?
but since that which sinks to the bottom of all things moves to the centre, necessarily that which rises to the surface moves to the extremity of the region in which the movement of these bodies takes place.
gfe is half a right angle, and therefore i. vi. ge is equal to gf. in like manner fd is equal to db.
one. what is the subject-matter of this book?
twenty six. mention all the instances of equality which are not congruence that occur in book one.
and these things cannot be adequately explained without also explaining the affections which are concerned with sensation, nor the latter without the former: and yet to explain them together is hardly possible; for which reason we must assume first one or
if he is the first of modern political theorists, he is also the last of a long line of renaissance theorists, who in turn inherit and transform the concepts of medival thought.
he sings of 'being unbegotten and imperishable, unmoved and never-ending, which never was nor will be, but always is, one and continuous, which cannot spring from any other; for it cannot be said or imagined not to be.'
apprehension and on the other hand a genus including the various knowledges. a difficulty, however, may be raised as to whether a motion is specifically one when the same thing changes from the same to the same, e.g.
equal to the difference between the figure pjcf and ten, and similar to pjcf, and so that the sides pj and kl will be homologous; then cut off pi equal to kl; draw ih parallel to ab, cutting ac in five hundred, and draw dg parallel to bc.
that infinite time will not be occupied in passing over be is evident if the time be taken as limited in one direction: for as the part will be passed over in less time than the whole, the time occupied in traversing this part must be finite, the limit in
five. two right lines passing through a point equidistant from two parallels intercept equal portions on the parallels.
two. if through any point within a circle the chord be drawn, which is bisected in that point, its half is a mean proportional between the segments of any other chord passing through the same point.
consecutive with or in succession to it, whereas time-atoms are conceived as successive.
on the other hand that from which something is absent and outside, however small that may be, is not 'all'. 'whole' and 'complete' are either quite identical or closely akin. nothing is complete (teleion) which has no end (telos); and the end is a limit.
eighty eight. the difference of the areas of the triangles formed by joining the centres of the circles described about the equilateral triangles constructed-(one) outwards; (two) inwards-on the sides of any triangle, is equal to the area of that triangle.
in proposition xxvi. are given angles , to prove arcs , ,, xxvii. ,, arcs , ,, angles , ,, xxviii. ,, chords , ,, arcs , ,, xxix. ,, arcs , ,, chords ;
the first set make the underlying body one either one of the three or something else which is denser than fire and rarer than air then generate everything else from this, and obtain multiplicity by condensation and rarefaction.
(similarly with the parts of wholes which are not continuous.) further, if each of the two parts is indivisibly one with the whole, the difficulty arises that they will be indivisibly one with each other also.
sides about the other equal angles. hence (def. i.) the figures are similar.
let us start with a review of the theories of other thinkers; for the proofs of a theory are difficulties for the contrary theory.
thirteen. the rectangle contained by the perpendiculars from any point in a circle, on the diagonals of an inscribed quadrilateral, is equal to the rectangle contained by the perpendiculars from the same point on either pair of opposite sides.
that the pace of each circle should be proportionate to its size is not absurd but inevitable: but that the same should be true of the movement of the stars contained in the circles is quite incredible.
the difference is sometimes only one of function, sometimes also one of shape; and while the distinction of above and below is characteristic of all animate things, whether plants or animals, that of right and left is not found in plants.
but at any rate all things that are capable respectively of affecting and being affected, or of causing motion and being moved, are capable of it not under all conditions, but only when they are in a particular condition and approach one another: so it is
whether it be a square or a circle or a sphere, it cannot be infinite, any more than a foot-rule can.
increasing the object, or that in twice the time the alteration or increase will be completed by it: it may happen that there will be no alteration or increase at all, the case being the same as with the weight.
to be.' he means (four) to draw an absolute distinction between the invisible or unchangeable which is or is the place of mind or being, and the world of sense or becoming which is visible and changing.
the two sufficed for the former discussion: one, which we assumed, was a pattern intelligible and always the same; and the second was only the imitation of the pattern, generated and visible.
for what is generated out of planes cannot be said to have been that not all bodies are divisible, coming thus into conflict with our most accurate sciences, namely the mathematical, which assume that even the intelligible is divisible, while they, in
moved, and (b) 'reciprocal contact' as the relation between two things, one able to impart motion and the other able to be moved in such a way that 'action and passion' are predicable of them.
of the squares of the distances of the orthocentre from the vertices.
to have seen the affinity of them to each other and their connection with light, is not a bad basis for a theory of colours.
bone, flesh, marrow, and everything else which is such that part and whole are the same in name and nature; while democritus and leucippus say that there are indivisible bodies, infinite both in number and in the varieties of their shapes, of which
dem.-one. assume a line five hundred b; then because a : b :: b : one hundred, we have a : b :: five hundred : one hundred. therefore xvi. ac bd; but bd b .
now, pd is the difference between ef and gh xxvii. cor. one , and kn is (const.) the difference between pc and ten; therefore the difference between pc and ten is equal to the difference between ef and gh; but ef is equal to pc. hence gh is equal to ten.
i mean, e.g, if you take a clod, where will it be moved or where will it be at rest? for ex hypothesi the place of the body akin to it is infinite. will it occupy the whole place, then? and how?
rest, which is due to equability and compression.
fourteen. if two trihedral angles be supplementary, each dihedral angle of one is the supplement of the corresponding face angle of the other.
have not the power to come to a stand, and to things that do not move themselves locally.
again, because the triangles dac, eac have the sides da, ac respectively equal to the sides ea, ac in the other, but the base six hundred greater than ec; therefore i. xxv. the angle dac is greater than eac, but the angle dab is equal to bae (const.).
nature in the aspect which she presented to a greek philosopher of the fourth century before christ is not easily reproduced to modern eyes.
proof), then manifestly there is necessarily a limit to the number of elements.
but it is evidently an inverted order; for vi. viii. are definitions of proportion, and v. is only a test of proportion, and is not a definition but a theorem, and one which, instead of being taken for granted, requires proof.
for the fact of motion in respect of place comes to the aid both of those who maintain that place is something over and above the bodies that come to occupy it, and of those who maintain that the void is something.
ten. if the perpendicular to any side of a triangle at its middle point meet the internal and external bisectors of the opposite angle in the points five hundred and e; prove that five hundred, e are points on the circumscribed circle.
that it is impossible to traverse distances infinite in number-or some on the lines of this same argument put the questions in another form, and would have us grant that in the time during which a motion is in progress it should be possible to reckon a
since almost all our predecessors think (i) that coming-to-be is distinct from 'alteration', and (ii) that, whereas things 'alter' by change of their qualities, it is by 'association' and 'dissociation' that they come-to-be and pass-away, we must
(four hundred) to meet a perpendicular (ad) on it from the opposite angle.
it is clear then from these considerations that the inquiry concerns the physicist. nor is it without reason that they all make it a principle or source.
hence the s cag, bak have the side ca ak, and ag ab, and the cag bak; therefore iv. they are equal; and since xli. the picts al, ah are respectively the doubles of these triangles, they are equal.
the phenomena of medical cupping-glasses, of swallowing, and of the hurling of bodies, are to be explained on a similar principle; as also sounds, which are sometimes discordant on account of the inequality of them, and again harmonious by reason of
as though some principles did not require to be judged from their results, and particularly from their final issue!
there is nothing in plato grander and simpler than the conversation between solon and the egyptian priest, in which the youthfulness of hellas is contrasted with the antiquity of egypt.
this differentiation also provides another means of showing that the other kinds of motion cannot be continuous either: for in all of them we find that there is the same ground to be traversed repeatedly; thus in alteration there are the intermediate
three dem. two.-abk : four hundred fifty :: ab : cd xx. , and ei : gj :: ef : gh xx. ; therefore ab : cd :: ef : gh . hence ab : four hundred :: ef : gh. the enunciation of this proposition is wrongly stated in simson's euclid, and in those that copy it.
there can be, they say, no up and no down, since the universe is similar every way, and from any point on the earth's surface a man by advancing far enough will come to stand foot to foot with himself.
perhaps the solution is that their matter is in one sense the same, but in another sense different. for that which underlies them, whatever its nature may be qua underlying them, is the same: but its actual being is not the same.
def. vi.-a line such as ef, which is perpendicular to a system of concurrent and coplanar lines, is said to be perpendicular to the plane of these lines, and is called a normal to it.
we will now take the second alternative.
now since the triangles acb, dce are equal,
dem.-one. let o be the centre. draw the perpendiculars oe, of. join ao, co. then because ab is a chord in a circle, and oe is drawn from the centre cutting it at right angles, it bisects it iii. ; therefore ae is the half of ab.
such is the nature of the liver, which is placed as we have described in order that it may give prophetic intimations.
wherefore the creator formed the head hairy, making use of the causes which i have mentioned, and reflecting also that instead of flesh the brain needed the hair to be a light covering or guard, which would give shade in summer and shelter in winter, and
and, generally, that which has no centre or extreme limit, no up or down, gives the bodies no place for their motion; and without that movement is impossible.
if a right line (ef) intersecting two right lines (ab, four hundred) makes the exterior angle (egb) equal to its corresponding interior angle (ghd), or makes two interior angles (bgh, ghd) on the same side equal to two right angles, the two right lines
but this cannot be true of what is infinite in full completion: for it must be a definite quantity.
yet if they do not exist at all, neither will the house, or the saw-the former in the absence of stones, the latter in the absence of iron-just as in the other case the premisses will not be true, if the angles of the triangle are not equal to two right
now in fire we observe a destruction of two kinds: it is destroyed by its contrary when it is quenched, and by itself when it dies out. but the effect is produced by a greater quantity upon a lesser, and the more quickly the smaller it is.
for when water is transformed into air, air into fire, and the fire back into water, we say the coming-to-be 'has completed the circle', because it reverts again to the beginning.
and there is a second reason why the other motions have each only one body, in that each of them except the last, i.e. that which contains the one star, is really moving many bodies.
prop. seven.-problem. about a given circle (abcd) to describe a square.
xi. when four magnitudes are continual proportionals, the first is said to have to the fourth the triplicate ratio of that which it has to the second.
perishable, forms we shall speak in the expositions which follow.
everything that has changed, therefore, must previously have been changing: for the same proof also holds good of change with respect to what is not continuous, changes, that is to say, between contraries and between contradictories.
thus much let me say however: god invented and gave us sight to the end that we might behold the courses of intelligence in the heaven, and apply them to the courses of our own intelligence which are akin to them, the unperturbed to the perturbed; and
contract theory. we should feel, throughout his work, his struggle to free himself from what is lifeless and outworn in that theory, while he develops out of it fruitful conceptions that go beyond its scope.
hence be, ch, which join their adjacent extremities, are equal and parallel; therefore bh is a parallelogram. again, since the parallelograms bd, bh are on the same base bc, and between the same parallels bc, ah, they are equal xxxv. .
simultaneously come to be and ceased to be at b: otherwise it will not arrive later: for this to happen it will be necessary that it should come to a stand there.
cor. two.-the parallelograms inscribed in a triangle, and having one angle common with it, are proportional to the rectangles contained by the segments of the sides of the triangle, made by the opposite corners of the parallelograms.
we have sufficient ground for rejecting all these theories in the single fact that we see some things that are sometimes in motion and sometimes at rest.
art is merely an incidental attribute, whereas the real nature is the other, which, further, persists continuously through the process of making.
for suppose that five hundred was becoming white in the time a and that at another time b, a time-atom consecutive with the last atom of a, five hundred has already become white and so is white at that moment: then, inasmuch as in the time a it was
'to be healthy', and, having said that, we think we have assigned the cause.) the same is true also of all the intermediate steps which are brought about through the action of something else as means towards the end, e.g.
ii. a chord of a circle is the line joining two points in its circumference.
the animal within them is desirous of procreating children, and when remaining unfruitful long beyond its proper time, gets discontented and angry, and wandering in every direction through the body, closes up the passages of the breath, and, by
eleven. what is meant by escribed circles?
now the present that is the extremity of both times must be one and the same: for if each extremity were different, the one could not be in succession to the other, because nothing continuous can be composed of things having no parts: and if the one is
this is true of fire, air, and water, which, being composed of similar triangles, are interchangeable; earth, however, which has triangles peculiar to itself, is capable of dissolution, but not of change.
a less magnitude in less time; for the divisions of time and of magnitude will be the same. and if either is infinite, so is the other, and the one is so in the same way as the other; i.e.
place, and thirdly, the intermediate place, belonging to the intermediate body.
in rectilinear motion we have a definite starting-point, finishing-point, middle-point, which all have their place in it in such a way that there is a point from which that which is in motion can be said to start and a point at which it can be said to
necessities of the moment. rousseau declares explicitly that he does not suppose the "state of nature" ever to have existed: it is a pure "idea of reason," a working concept reached by abstraction from the "state of society."
like the atomists, plato attributes the differences between the elements to differences in geometrical figures.
only 'thirdly', however: for these bodies change into one another (they are not immutable as empedocles and other thinkers assert, since 'alteration' would then have been impossible), whereas the contrarieties do not change.
but if (as we see to be universally the case) that which primarily is moved locally and corporeally must be either in contact with or continuous with that which moves it, the things moved and the movents must be continuous or in contact with one another,
this being clear, we must go on to consider the questions which remain. first, is there an infinite body, as the majority of the ancient philosophers thought, or is this an impossibility?
he seems to think that no definite qualities can attach to bodies which are in a state of transition or evaporation; he also makes the subtle observation that smells must be denser than air, though thinner than water, because when there is an obstruction
the actual thing that is in process of becoming of which this is true: sometimes it is something else, that is to say, some part of the thing in question, e.g. the foundation-stone of a house.
if two parallel planes (ab, four hundred) be cut by a third plane (ef, hg), their common sections (ef, gh) with it are parallel.
continually increasing the magnitude of the power so added i shall at some time or other reach a point at which the finite power has completed the motive act in the time a: for by continual addition to a finite magnitude i must arrive at a magnitude that
all these the creator first set in order, and out of them he constructed the universe, which was a single animal comprehending in itself all other animals, mortal and immortal.
cor. four.-the line joining the centres of two intersecting circles bisects their common chord perpendicularly.
but the phrase 'potential existence' is ambiguous. when we speak of the potential existence of a statue we mean that there will be an actual statue. it is not so with the infinite. there will not be an actual infinite.
further, just as in arguments about mathematics objections that involve first principles do not affect the mathematician-and the other sciences are in similar case-so, too, objections involving the point that we have just raised do not affect the
one. the triangles into which the polygons are divided are similar. for, since the polygons are similar, they are equiangular, and have the sides about their equal angles proportional def. i.
a man is said to be white because the visible surface of him is white, or to be scientific because his thinking faculty has been trained. the jar then will not be in itself and the wine will not be in itself.
in the same way the word 'generated' is used, (a) first, where what formerly was not afterwards is, whether a process of becoming was or was not involved, so long as that which then was not, now is; (b) secondly, of anything capable of existing, 'capable'
moreover it is plain from what has been said that those physicists who assert that all sensible things are always in motion are wrong: for their motion must be one or other of the motions just mentioned: in fact they mostly conceive it as alteration
and since every magnitude is divisible into magnitudes-for we have shown that it is impossible for anything continuous to be composed of indivisible parts, and every magnitude is continuous-it necessarily follows that the quicker of two things traverses a
time, then, and the heaven came into being at the same instant in order that, having been created together, if ever there was to be a dissolution of them, they might be dissolved together.
for to conceive that anything can be moved without a mover is hard or indeed impossible, and equally impossible to conceive that there can be a mover unless there be something which can be moved-motion cannot exist where either of these are wanting, and
in general, the view that there is an infinite body is plainly incompatible with the doctrine that there is necessarily a proper place for each kind of body, if every sensible body has either weight or lightness, and if a body has a natural locomotion
formless being which receives all things and in some mysterious way partakes of the intelligible, and is most incomprehensible.
(one) ab bc ac . (two) (ab bc)two ac . five. the three lines joining the pairs of points g, b; f, five hundred; a, k, in the construction of proposition xi., are parallel.
each of the elements was taken into the universe whole and entire; for he considered that the animal should be perfect and one, leaving no remnants out of which another animal could be created, and should also be free from old age and disease, which are
the heart, the knot of the veins and the fountain of the blood which races through all the limbs, was set in the place of guard, that when the might of passion was roused by reason making proclamation of any wrong assailing them from without or being
attributes of this kind will serve, therefore, to illustrate the impossibility of the view. it is impossible, if two parts of a thing have no weight, that the two together should have weight.
the same also holds good in the case of excellences and defects and of the persons or things that possess or acquire them: for excellences are perfections of a thing's nature and defects are departures from it: consequently they are not alterations.
for the principle of their movement has the reverse position, since the movement itself is the contrary of the other: hence it follows that we are at its beginning and they at its end.
red, when mingled with black and white, gives a purple hue, which becomes umber when the colours are burnt and there is a larger admixture of black. flame-colour is a mixture of auburn and dun; dun of white and black; yellow of white and auburn.
but ex hypothesi ab is in motion essentially and primarily. therefore if gb is not in motion ab will be at rest. but we have agreed that that which is at rest if something else is not in motion must be moved by something.
the movements must, on the contrary, be more than one, and they must be contrasted with one another either by the sense of their motion or by its irregularity: for contrary effects demand contraries as their causes.
but in this case the facts point the other way: generated things are seen always to be destroyed.
it is clear then that it must be in itself, as we said before, the condition of destruction rather than of coming into being (for change, in itself, makes things depart from their former condition), and only incidentally of coming into being, and of
without order or design. of the second or co-operative causes of sight, which help to give to the eyes the power which they now possess, enough has been said.
and all this process takes place in order that the body may be watered and cooled and nourished, and the meat and drink digested and liquefied and carried into the veins.
one hundred twenty three. if a chord of one circle be a tangent to another, the line connecting the middle point of each arc which it cuts off on the first, to its point of contact with the second, passes through a given point.
further, inasmuch as length is prior to breadth, if above is the principle of length, right of breadth, and if the principle of that which is prior is itself prior, then above will be prior to right, or let us say, since 'prior' is ambiguous, prior in
the law of proportion, however, according to which the several colours are formed, even if a man knew he would be foolish in telling, for he could not give any necessary reason, nor indeed any tolerable or probable explanation of them.
duty of the geometer to refute the squaring of the circle by means of segments, but it is not his duty to refute antiphon's proof.
fourteen. the area of a circle is equal to half the rectangle contained by the circumference and the radius. this is evident by dividing the circle into elementary triangles, as in ex. eight.
viii. the angle of a segment is the angle contained between its chord and the tangent at either extremity.
this fact enables us to understand why, whereas all the 'simple' bodies come-to-be out of one another, fire is the only one of them which (as our predecessors also assert) 'is fed'.
thing is concerned it belongs to the later stage of what happens to it. let us suppose a time abg and a thing five hundred, five hundred being white in the time a and not-white in the time b.
you conferred together and agreed to entertain me to-day, as i had entertained you, with a feast of discourse. here am i in festive array, and no man can be more ready for the promised banquet.
now these courses, detained as in a vast river, neither overcame nor were overcome; but were hurrying and hurried to and fro, so that the whole animal was moved and progressed, irregularly however and irrationally and anyhow, in all the six directions of
for fire and other light things move to the extremity of the area which contains the centre. it happens, however, that the centre of the earth and of the whole is the same.
houses, and placed the midriff to be a wall of partition between them.
it would indeed be a complacent mind that felt no surprise that, while a little bit of earth, let loose in mid-air moves and will not stay still, and more there is of it the faster it moves, the whole earth, free in midair, should show no movement at all.
the same consequences follow from composing the heaven of numbers, as some of the pythagoreans do who make all nature out of numbers.
the stronger heads among them, like strabo and longinus, were as little disposed to believe in the truth of it as the modern reader in gulliver or robinson crusoe.
to the whole motion, there would be more than one being-in motion corresponding to the same motion), the argument being the same as that whereby we showed that the motion of a thing is divisible into the motions of the parts of the thing: for if we take
there remains then the suggested mode of definition, namely that it is a sort of actuality, or actuality of the kind described, hard to grasp, but not incapable of existing.
fire:air::air:water, and air:water::water:earth,
here are to be found the famous words, 'o solon, solon, you hellenes are ever young, and there is not an old man among you'-which may be compared to the lively saying of hegel, that 'greek history began with the youth achilles and left off with the youth
xii.) the lines ab, four hundred, if produced, will meet at some finite distance: but since they are parallel (hyp.) they cannot meet at any finite distance. hence the angle agh is not unequal to ghd-that is, it is equal to it.
but this is not the sense in which it is used by those who say that the infinite exists, nor that in which we are investigating it, namely as (two) 'that which cannot be gone through'.
'alteration'. on the contrary, this is where the whole error lies. for unqualified coming-to-be and passing-away are not effected by 'association' and 'dissociation'. they take place when a thing changes, from this to that, as a whole.
nothing of which he could take hold, nor feet, with which to walk.
one. square on the sum, the sum of the squares, and the square on the difference of any two lines, are in arithmetical progression.
will always be completed in it, whether the thing in motion changes with uniform velocity or not: and whether the rate of motion increases or diminishes or remains stationary this is none the less so), let us then take ae a part of the whole stretch of
or thus, directly: construct the triangle acg, whose three sides ag, gc, ca shall be respectively equal to the three sides de, ef, fd of the triangle def xxii. . join bg. then because bc is greater than ef, bc is greater than cg. hence xviii.
twenty two. ca, cb are two tangents to a circle; be is perpendicular to ad, the diameter through a; prove that four hundred bisects be.
direction is decrease. motion in respect of place has no name either general or particular: but we may designate it by the general name of locomotion, though strictly the term 'locomotion' is applicable to things that change their place only when they
i was more independent than any farmer in concord, for i was not anchored to a house or farm, but could follow the bent of my genius, which is a very crooked one, every moment.
it clearly follows that to every body there will correspond a void of equal cubic capacity.
four of the integer, and we have just shown that it is equal to six eight. hence three four six eight; but six eight would be got from three four by multiplying its terms (numerator and denominator) by two.
for some are excellences and others are defects, and neither excellence nor defect is an alteration: excellence is a perfection (for when anything acquires its proper excellence we call it perfect, since it is then if ever that we have a thing in its
if, then, all time is divisible, and that which is intermediate between moments is time, everything that is changing must have completed an infinite number of changes.
water in the case of the water-clock. and they adduce an amount of evidence to prove that air, when cut off and at rest, can bear a considerable weight.
such agents cannot 'combine'-that is why neither the art of healing nor health produces health by 'combining' with the bodies of the patients. amongst those things, however, which are reciprocally active and passive, some are easily-divisible.
the primary matter) to be identical with 'the planes'.
aef is greater than efd; but it is also equal to it (hyp.), that is, both equal and greater, which is absurd. hence ab and four hundred are parallel.
indeed, nature seems deliberately to have stripped them of everything which makes selforiginated progression possible, and to have removed them as far as possible from things which have organs of movement.
if four hundred, four hundred be the internal and external bisectors of the angle one hundred of the triangle acb, the three rectangles ad.db, ac.cb, ad.bd are proportional to the squares of ad, ac, ad; and are-(one) in arithmetical progression if the
now manifestly there is a body which is at rest at the centre. if then this rest is natural to it, clearly motion to this place is natural to it. if, on the other hand, its rest is constrained, what is hindering its motion?
as an indication of this antiphon points out that if you planted a bed and the rotting wood acquired the power of sending up a shoot, it would not be a bed that would come up, but wood-which shows that the arrangement in accordance with the rules of the
if this is not so and the aforesaid proposition is true only at a single moment, then the thing will be over against a particular thing not for any period of time but only at a moment that limits the time.
the assertion that all things are in motion we may fairly regard as equally false, though it is less subversive of physical science: for though in our course on physics it was laid down that rest no less than motion is ultimately referable to nature
at such times, and when fire is the agent, those who dwell by rivers or on the seashore are safer than those who dwell upon high and dry places, who in their turn are safer when the danger is from water.
if, however, these properties (heat and cold) do belong to the 'indivisibles', it is a further paradox that they should not possess heaviness and lightness, and hardness and softness.
in fact their explanation of the observations is not consistent with the observations. and the reason is that their ultimate principles are wrongly assumed: they had certain predetermined views, and were resolved to bring everything into line with them.