by Aristotle
3 · In consequence of what has been said, in part by way of assumption and in part by way of demonstration, it is clear that not every body possesses either lightness or heaviness. We must explain in what sense we are using the words [20] ‘heavy’ and ‘light’, sufficiently, at least, for our present purposes: we can examine the terms more precisely later, when we come to consider their essential nature. 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. The heaviest thing will be that which sinks to the bottom of all things that move downward, and the [25] lightest that which rises to the surface of everything that moves upward. 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 water light relatively to earth. But the body which moves in a circle cannot possibly [30] possess heaviness or lightness. For neither naturally nor unnaturally can it move either towards or away from the centre. 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. Suppose, then, that the movement is unnatural. In that [270a1] 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. For we decided that of contrary movements, if the one is unnatural to anything, the other will be natural to it. But since the natural movement of the whole and of its [5] 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 mean that it could move by its own nature either from or towards the centre); and, secondly, that it cannot possibly move in the way of locomotion by being dragged upwards or pulled downwards. For neither [10] 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.
It is equally reasonable to assume that this body will be ungenerated and indestructible and exempt from increase and alteration, since everything that comes [15] to be comes into being from a contrary and some substrate, and passes away likewise in a substrate by the action of a contrary into a contrary, as we explained in our opening discussions.1 Now the motions of contraries are contrary. If then this body can have no contrary, because there can be no contrary motion to the circular, [20] nature seems justly to have exempted from contraries the body which was to be ungenerated and indestructible. For it is on contraries that generation and destruction depend. 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 [25] which this body can have been generated. And if it is exempt from increase and destruction, the same reasoning leads us to suppose that it is also unalterable. 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. [30] But all natural bodies which change their properties we see to be subject to increase and diminution. 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. 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.
[270b1] 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. Our theory seems to [5] confirm the phenomena and to be confirmed by them. 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 immortal and regard any [10] other supposition as impossible. If then there is, as there certainly is, anything divine, what we have just said about the primary bodily substance was well said. The mere evidence of the senses is enough to convince us of this, at least with human certainty. 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 [15] heaven or in any of its proper parts. The name, too, of that body seems to have been handed down right to our own day from our distant ancestors who conceived of it in the fashion which we have been expressing. The same ideas, one must believe, recur in men’s minds not once or twice but again and again. And so, implying that the [20] primary body is something else beyond earth, fire, air, and water, they gave the highest place the name of aether, derived from the fact that it ‘runs always’2 for an eternity of time. Anaxagoras, however, misuses this name, taking aether as equivalent to fire [25].
It is also clear from what has been said why the number of what we call simple bodies cannot be greater than it is. 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 [30] the centre.
4 · That there is no other form of motion 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. For concave and convex are not only regarded as opposed to one another, but they are also coupled together and treated as a unity in [271a1] opposition to the straight. 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. 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. Secondly, it may [5] be thought that the same reasoning which holds good of the rectilinear path applies also to 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. For that is limited, while the circular paths which pass through the same points are infinite in number. Even [10] if we are confined to the single semicircle and the opposition is between movement from C to D and from D to C along that semicircle, the case is no better. 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. It is no more satisfactory to construct a circle and treat motion along one semicircle as contrary to motion along the other. For example, taking a whole circle, motion from [15] E to F on the semicircle G may be opposed to motion from F to E on the semicircle H. But even supposing these are contraries, it in no way follows that the motions on the whole circle are contraries. Nor again can motion along the circle from A to B be regarded as the contrary of motion from A to C; for the motion goes from the [20] same point towards the same point, and contrary motion was distinguished as motion from a contrary to its contrary. And even if one circular motion is the contrary of another, one of the two would be pointless; for that which moves in a circle, at whatever point it begins, must necessarily pass through all the contrary [25] places alike. (By contrarieties of place I mean up and down, back and front, and right and left.) But contrarieties of movements correspond to those of places. For if the two motions were equal, there would be no movement, and if one of the two were [30] preponderant, the other would not occur. So that if both bodies were there, one of them, inasmuch as it would not be moving with its own movement, would be pointless, in the sense in which a shoe is pointless when it is not worn. But God and nature create nothing that is pointless.
[271b1] 5 · 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 impos
sibility? The decision of this question, either way, is not [5] unimportant, but rather all important, to our search for the truth. It is this problem which practically always has been and may be expected to be the source of the differences of those who have written about nature as a whole, since the least initial deviation from the truth is multiplied later a thousandfold. Admit, for instance, the [10] existence of a minimum magnitude, and you will find that the minimum which you have introduced causes the greatest truths of mathematics to totter. 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. Now the infinite possesses this power of principles, and indeed in the sphere of quantity possesses it in the highest degree; so [15] that it is in no way absurd or unreasonable that the assumption that an infinite body exists should be of peculiar moment to our inquiry. The infinite, then, we must now discuss, opening the whole matter from the beginning.
Every body is necessarily to be classed either as simple or as composite; the infinite body, therefore, will be either simple or composite. But it is clear, further, [20] 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 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. Let us [25] try the primary body first, and then go on to consider the others.
The body which moves in a circle must necessarily be finite in every respect, for the following reasons. If the body so moving is infinite, the radii drawn from the [30] centre will be infinite. But the space between infinite radii is infinite—by the space between the lines I mean the area outside which no magnitude which is in contact with the lines can be found. This, I say, will be infinite; for in the case of finite radii it is always finite; and again one can always go on to take more than the given [272a1] quantity, so that just as we say that number is infinite, because there is no greatest, the same argument 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 [5] necessarily infinite: circular motion therefore is an impossibility. Yet we see that the heavens revolve in a circle, and by argument also we have determined that there is something to which circular movement belongs.
Again, if from a finite time a finite time be subtracted, what remains must be finite and have a beginning. And if the time of a journey has a beginning, there must [10] be a beginning also of the movement, and consequently also of the distance traversed. This applies universally. Take a line, ACE, infinite in one direction, E, and another line, BB, infinite in both directions. Let ACE describe a circle, revolving upon A as centre. In its circular movement it will cut BB for a certain [15] finite time; 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. 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 [20].
That the infinite cannot move away 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. Now if the two were moving in contrary directions, they would pass clear of one [25] 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. This, however, is clear: that it is impossible to traverse an infinite line in a finite time. Infinite time, then, would be required. (This we demonstrated above in the discussion of [30] movement.)3 And it makes no difference whether a finite is passing by an infinite or an infinite by a finite. For when A is passing B, then B overlaps A, and it makes no [272b1] difference whether B is moved or unmoved, except that, if both move, they pass clear of one another more quickly. 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. One has only to imagine the movement to be slow [5] where both move and much faster where one is stationary. 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. If, therefore, the time which the finite moving line takes to pass the other is infinite, [10] then necessarily the time occupied by the motion of the infinite past the finite is also infinite. For the infinite to move at all is thus absolutely impossible; since the very smallest movement must take an infinity of time. Moreover the heavens certainly revolve, and they complete their circular orbit in a finite time; so that they pass [15] round the whole extent of any line within their orbit, such as the finite line AB. The revolving body, therefore, cannot be infinite.
Again, as a line which has a limit4 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. E.g. if it is a square or a circle or a sphere, it cannot be infinite, any more than a foot-long line [20] can. There is then no such thing as an infinite sphere 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 infinite circle being itself an impossibility, there can be no circular motion of an infinite body.
Again, take a centre C, an infinite line, AB, another infinite line at right angles [25] to it, E, and a moving radius, CD. CD will never cease contact with E, but the position will always be something like CE, CD cutting E at F. The infinite line, therefore, does not complete the circle.
Again, if the heaven is infinite and moves in a circle, we shall have to admit [30] 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. It results that when the infinite body has completed its revolution, it has traversed an infinite equal to itself in a [273a1] finite time. But that we know to be impossible.
It can also be shown, conversely, that if the time of revolution is finite, the distance traversed must also be finite; but the distance traversed was equal to itself; therefore, it is itself finite.
[5] We have now shown that the body which moves in a circle is not endless or infinite, but has its limit.
6 · 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. But if one of a pair of [10] 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. 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. Further, if up [15] and down are determinate, the intermediate place is also necessarily determinate. For, if it is indeterminate, there will be infinite motion; and that we have already shown to be an impossibility.5 The middle region then is determinate, and consequently any body which either is in it, or might be in it, is determinate. But the [20] bodies which move up and down may be in it, since the one moves naturally away from the centre and the other towards it.
From this 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. For the supposed infinite body would have to be infinite [25] in weight. (The same argument applies to lightness; for if there is infinite weight, there is infinite lightness, if the
rising body is infinite.) This is proved as follows. Assume the weight to be finite, and take an infinite body, AB, of the weight C. [30] Subtract from the infinite body a finite mass, BD, the weight of which shall be E. E then is less than C, since it is the weight of a lesser mass. Suppose then that the smaller goes into the greater a certain number of times, and take BF bearing the [273b1] same proportion to BD which the greater weight bears to the smaller. For you may subtract as much as you please from an infinite. If now the masses are proportionate to the weights, and the lesser weight is that of the lesser mass, the greater must be [5] that of the greater. The weights, therefore, of the finite and of the infinite body are equal. 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, further, the weight of unequal masses will be [10] the same, since the infinite and the finite cannot be equal. It does not matter whether the weights are commensurable or not. If they are incommensurable the same reasoning holds. For instance, suppose E multiplied by three is rather more than C: the weight of three masses of the full size of BD will be greater than C. We thus arrive at the same impossibility as before. Again we may assume weights [15] which are commensurate; for it makes no difference whether we begin with the weight or with the mass. For example, assume the weight E to be commensurate with C, 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 [20] another. (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. Nor again does it make any difference to our demonstration whether the total mass has its weight equally or unequally distributed. For it must always be possible to take from the infinite mass bodies of equal weight to BD by diminishing or [25] increasing the size of the section to the necessary extent.