by Aristotle
6 · Now everything that changes changes in time, and that in two senses may be the primary time, or it may be derivative, as e.g. when we say that a thing [20] 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 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. Let TR be the primary time in which that which is in motion is in motion; and (as all [25] time is divisible) let it be divided at K. Now in the time TK it either is in motion or is not in motion, and the same is likewise true of the time TR. 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 [30] should be in motion in a time in no part of which it is in motion. If on the other hand it is in motion in only one of the two parts of the time, TR cannot be the primary time in which it is in motion; for its motion will have reference to a time other than TR. It must, then, be moving in any part of TR.
And now that this has been proved, it is evident that everything that is in motion must have been in motion before. For if that which is in motion has traversed [35] the distance KL in the primary time TR, 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 [237a1] the distance. But if the thing whose velocity is equal has traversed a certain distance in the same time, the original thing that is in motion must have traversed the same distance. Hence that which is in motion must have been in motion before.
[5] Again, if by taking the extreme now of the time—for it is the now that defines the time, and time is that which is intermediate between nows—we are enabled to say that motion has taken place in the whole time TR or in fact in any period of it, 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. 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 nows. If, then, all time is divisible, and that which [10] is intermediate between nows is time, everything that is changing must have completed an infinite number of changes.
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 now, it follows that it must have [15] changed at every now in the time: consequently, since the nows are infinite in number, everything that is changing must have completed an infinite number of changes.
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. For suppose [20] that a thing has changed from A to B in a now. Now the now 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 which has changed, when it has changed, is not in that from which it has changed. If, on the other hand, it is a different now, there will be a period of time intermediate between the two; for, as we [25] saw, nows are not consecutive. 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 always: consequently it must have previously been changing.
Moreover, the truth of what has been said is more evident in the case of magnitude, because the magnitude over which what is changing is continuous. For [30] suppose that a thing has changed from C to D. Then if CD is indivisible, two things without parts will be consecutive. But since this is impossible, that which is intermediate between them must be a magnitude and divisible into an infinite number of segments: consequently, it has previously been changing to those segments. Everything that has changed, therefore, must previously have been changing; for the same demonstration also holds good of change with respect to [237b1] what is not continuous, changes, that is to say, between contraries and between contradictories. In such cases we have only to take the time in which a thing has changed and again apply the same reasoning. So that which has changed must have been changing and that which is changing must have changed, and a process of [5] change is preceded by a completion of change and a completion by a process; and we can never take any first stage. The cause of this is that no two things without parts can be contiguous; for the division is infinite, as in the case of lines which are increasing and decreasing.
So it is evident also that that which has become must previously have been [10] becoming, and that which is becoming must previously have become, everything (that is) that is divisible and continuous; though it is not always the actual thing that is 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. 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 since [15] they are continuous things; and so a thing cannot be becoming without having become or have become without having been becoming. So, too, in the case of perishing and having perished: perishing must be preceded by having perished, and having perished by perishing. It is evident, then, that that which has become must previously have been becoming, and that which is becoming must previously have [20] become; for all magnitudes and all periods of time are always divisible. Consequently, whatever a thing may be in, it is not in it primarily.
7 · 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 the same motion or a part of it is continually repeated, but that the [25] whole is occupied by the whole. 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. 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. Consequently, since these parts are finite, both in size individually and in number [30] collectively, the whole time must also be finite; for it will be a multiple equal to the time occupied in completing the part multiplied by the number of the parts.
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 infinite time, and let CD be the infinite time. Now if one part of the [238a1] 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 will always be completed in it, whether it changes with uniform velocity or not; and whether the [5] motion increases or diminishes or remains stationary this is none the less so), let us then take AE a part of the interval AB which shall be a measure of AB. Now this occupies a certain period of the infinite time: it cannot itself occupy an infinite time, [10] for that is occupied by the whole AB. And if again I take another part equal to AE, that also must occupy a finite time in consequence of the same assumption. And if I go on taking parts in this way, since 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 of things finite [15] in multitude or in magnitude, which, whether they are equal or unequal, are none the less limited in magnitude), and the finite interval is measured by the quantities AE: consequently the motion AB must be accomplished in a finite time. (It is the same with coming to rest.) And so it is impossible for one and the same thing to be always in process of becoming or of perishing.
[20] The same reasoning 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. 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 the whole occupies all the time. 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, [25] whether equal or unequal to the part originally taken. 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, the infinite magnitude will not be exhausted, since the process of subtraction is finite both in respect of the quantity subtracted and of the number of times a subtraction is made. Consequently the infinite magnitude will not be [30] traversed in a 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.
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 [35] several part likewise, so that in the whole time it will traverse a finite magnitude.
And since a finite magnitude will not traverse an infinite in a finite time, it is [238b1] clear that neither will an infinite traverse a finite. 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 [5] the finite. For when the infinite magnitude A is in motion a part of it, say CD, will occupy the finite B, and then another, and then another, and so on to always. 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 [10] the motion of the infinite over the finite to occur in any way other than 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.
Nor again will the infinite traverse the infinite in a finite time. Otherwise it [15] 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.
Since, then, 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 does it make whether we take [20] the motion or the magnitude to be infinite? If either of the two is infinite, the other must be so too; for all locomotion is in place.
8 · 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 [25] that which is at rest cannot be coming to rest. 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 must occupy a period of time.
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 [30] slower, the same conclusion follows.
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. 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 which is coming to a stand will not be coming to a stand. If on the other hand it is coming to a stand in only one of the two parts, the whole cannot be the primary time in which it is coming to a stand; for it is coming to a stand in this derivatively, as we said before in the case of things in [35] motion.
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 [239a1] coming to a stand, there being no primary stage either of being in motion or of coming to a stand. For let AB be the primary time in which a thing is coming to a stand. Now AB cannot be without parts; for there cannot be motion in that which is without parts, because a moving thing would have moved for a part of it, and that which is coming to a stand has been shown to be in motion. But since AB is divisible, [5] the thing is coming to a stand in every one of its parts; 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. 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 [10].
Nor again can there be a primary time at which a thing at rest was resting; for it cannot have been resting in that which has no parts, because there cannot be motion in that which is indivisible, and that in which rest takes place is the same as that in which motion takes place (for we said that rest occurs if a thing which naturally moves is not moving when and at a time in which motion would be natural to it). Again, we say that a thing rests when it is now in the same state as it was in [15] earlier, judging rest not by any one point but by at least two: consequently that in which a thing is at rest cannot be 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.
[20] So there can be no primary time; and the reason is that rest and motion are always in time, and there is no primary time—nor magnitude nor in fact anything continuous; for everything continuous is divisible into an infinite number of parts.
And since everything that is in motion is in motion in time and changes from something to something, in the time in which in its own right (i.e. not merely in a [25] part of the time) something moves, it is impossible that that which is in motion should be over against some particular thing primarily. 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 now after another it can be said with truth that a thing, itself and its parts, occupies [30] the same space. 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 will be true to say that the thing, itself and its parts, occupies the same space. If this is not so and the aforesaid proposition is true only at a single now, 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. It is true that at any now it is always over against something; but it is not at rest; for at a now it is not possible for anything to [239b1] be either in motion or at rest. So while it is true to say that that which is in motion is at a now not in motion and is opposite some particular thing, it cannot in a period of time be at rest over against anything; for that would involve the conclusion that that which is in locomotion is at rest.
[5] 9 · 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 in a now, the flying arrow is therefore motionless. This is false; for time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles.
[10] Zeno’s arguments about motion, which cause so much trouble to those who try to answer them, are four in number. 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.41
The second is the so-called Achilles, and it amounts to this, that in
a race the [15] 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. 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 have successively to deal are not [20] divided into halves. 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 goes further in that it affirms that even the runner most famed for his speed must fail in his pursuit of the slowest), so that the solution too must be the [25] same. And the claim that that which holds a lead is never overtaken is false: it is not overtaken while it holds a lead; but it is overtaken nevertheless if it is granted that it traverses the finite distance. These then are two of his arguments.
The third is that already given above, to the effect that the flying arrow is at [30] rest, which result follows from the assumption that time is composed of moments: if this assumption is not granted, the conclusion will not follow.
The fourth argument is that concerning equal bodies which move alongside equal bodies in the stadium from opposite directions—the ones from the end of the stadium, the others from the middle—at equal speeds, in which he thinks it follows that half the time is equal to its double. The fallacy consists in requiring that a body [240a1] travelling at an equal speed travels for an equal time past a moving body and a body of the same size at rest. That is false. E.g. let the stationary equal bodies be AA; let BB be those starting from the middle of the A’s42 (equal in number and in [5] magnitude to them); and let CC be those starting from the end (equal in number and magnitude to them, and equal in speed to the B’s). Now it follows that the first B and the first C are at the end at the same time, as they are moving past one another. And it follows that the C has passed all the A’s43 and the B half; so that the [10] time is half, for each of the two is alongside each for an equal time. And at the same time it follows that the first B has passed all the C’s. For at the same time the first B and the first C will be at opposite ends,* being an equal time alongside each of the B’s as alongside each of the A’s, as he says,*44 because both are an equal time [15] alongside the A’s. That is the argument, and it rests on the stated falsity.