The Basic Works of Aristotle (Modern Library Classics)
Page 48
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 that which is at rest cannot be coming to rest. (25) 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 slower, (30) 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 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 from itself, (35) the argument being the same as that which we used above about things in motion.20
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 stand. [239a] 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 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 motion. (5) 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 above21 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.
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, (10) 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 defined22 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 that23 in which) motion would be natural to it. Again, our use of the phrase ‘being at rest’ also implies that the previous state of a thing is still unaltered, (15) 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 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.
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, (20) and a period of time has no primary part any more than a magnitude or 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 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 the time in question—it is impossible that in that time that which is in motion should be over against some particular thing primarily.24 (25) 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 thing, itself and its parts, occupies 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, (30) 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 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. 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. [239b] (35) 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 locomotion is at rest.
9 Zeno’s reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, (5) and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.
Zeno’s arguments about motion, which cause so much disquietude to those who try to solve the problems that they present, (10) 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.25
The second is the so-called ‘Achilles’, and it amounts to this, that in a race the quickest runner can never overtake the slowest, (15) 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,26 though it differs from it in that the spaces with which we successively have to deal are not 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, (20) though the ‘Achilles’ 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. (25) 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. These then are two of his arguments.
The third is that already given above, (30) 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.
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 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. (35) This, he thinks, involves the conclusion that half a given time is equal to double that time. [240a] 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. For instance (so runs the argument), let A, (5) 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 C, C, … those 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:
First, as the B’s and the C’s pass one another, (10) the first B reaches the last C at the same moment as the first C reaches the last B. Secondly, at this moment the first C 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 C, since each of the two occupies an equal time in passing each A. Thirdly, at the same moment all the B’s have passed all the C’s: for the first C and the first B will simultaneously reach the opposite ends of the course, (15) since (so says Zeno) the time occupied by the first C in passing each of the B’s is equal to that occupied by it in passing each of the A’s, because an equal time is occupied by both the first B and the first C in passing all the A’s. This is the argument, but it presupposed the aforesaid fallacious assumption.
Nor in reference to contradictory change shall we find anything unanswerable in the argument that if a thing is changing from not-white, (20) say, to white, and is in neither condition, then it will be neither white nor not-white: for the fact that it is not wholly in either condition will not preclude us from calling it white or not-white. We call a thing white or not-white not necessarily because it is wholly either one or the other, but because 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. (25) 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.
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 that the motion can be nothing but rest, on the ground that such things in motion, (30) themselves and their parts, will occupy the same position for a period of time, and that therefore they will be at once at rest and in motion. 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 not be the same as the orbit as described from B or C 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. [240b] (5) 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.
10 Our next point is that that which is without parts cannot be in motion except accidentally: i. e. 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, (10) 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 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 essentially and severally to themselves distinct from the motion of the whole. (15) 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 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. (20) For suppose that it is changing from AB to BC—either from one magnitude to another, or from one form to another, or from some state to its contradictory—and let D be the primary time in which it undergoes the change. Then in the time in which it is changing it must be either in AB or in BC or partly in one and partly in the other: for this, (25) as we saw,27 is true of everything that is changing. Now it cannot be partly in each of the two: for then it would be divisible into parts. Nor again can it be in BC: for then it will have completed the change, whereas the assumption is that the change is in process. 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,28 to be in the same condition for a period of time is to be at rest. (30) 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. 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. [241a] But this we have already shown above29 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: (5) for this theory simply makes motion consist of indivisibles in exactly the same way as time is made to consist of moments or a length of points.
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. That being so, (10) it is evident that the point also must first traverse a space equal to or less than itself. 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. 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, (15) 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. 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 above30 to be divisible. Therefore, if a point is in motion, there must be a time less than that in which it has itself traversed any distance. But this is impossible, for in less time it must traverse less distance, (20) 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 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. (25)
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. Thus in contradictory changes the positive or the negative, as the case may be, is the limit, e. g. 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, (30) and consequently of every process of alteration: for alteration is always dependent upon some contraries. 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 loss of such magnitude. [241b] Locomotion, it is true, we cannot show to be finite in this way, since it is not always between contraries. 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, (5) and 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. 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. Consequently its motion is not infinite, and it will not be
in locomotion over an infinite distance, (10) for it cannot traverse such a distance.
It is evident, then, that a process of change cannot be infinite in the sense that it is not defined by limits. 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. If it is not one process, it would seem that there is nothing to prevent its being infinite in this sense; e. g. 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, (15) but it will not be one motion, 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, (20) with the single exception of rotatory locomotion.
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1 v. 3.
2 Which is ex hypothesi impossible (231b 28–30).
3 The slower will traverse EF in a greater time than the indivisible time in which the quicker traverses JK.
4 i. e. in which it means a period of time including the present proper.
5 222a 12.
6 Chapter 2.
7 i. e. it will not be a point of division but merely something intermediate between past and future.
8 226b 12 sqq.
9 viz. past and future.
10 223b 1 sqq.
11 234b 24 sqq., especially 234b 34 sqq.