by Aristotle
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. [233b1] Let AB be a finite magnitude, and an infinite time C, and let a finite period CD of the time be taken. Now in this period the thing will pass over a certain segment of the magnitude: let BE be the segment that it has thus passed over. (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 [5] over in an equal time, and BE measures the whole magnitude, the whole time 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. Moreover, if it is the case that infinite time is not occupied in passing over every magnitude, but it is possible to pass over some magnitude, say BE, in a finite time, and if this measures [10] the whole, and if an equal magnitude is passed over in an equal time, then it follows that the time too is finite. 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, this must be finite, the limit in one direction being given. The same demonstration will also show the falsity of the assumption [15] that infinite length can be traversed in a finite time. It is evident, then, from what has been said that neither a line nor a surface nor in fact anything continuous can be indivisible.
This conclusion follows not only from the present argument but from the consideration that the opposite assumption implies the divisibility of the indivisible. For since the distinction of quicker and slower may apply to motions occupying any [20] 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 passed over by the slower; for their respective velocities may stand to one another in this proportion. Suppose, then, that the quicker has in the same time been carried over a length one and a half times as great, and that the respective magnitudes are divided, that of the quicker into three indivisibles, AB, BC, CD, and that of the [25] slower into two, EF, FG. 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. Again, since in the same time the slower has been carried over EZ, ZH, the time may also be divided into two. Thus the indivisible will [30] be divisible, and that which has no parts will be passed over not in an indivisible but in a greater time. It is evident, therefore, that nothing continuous is without parts.
3 · Necessarily, too, the now—the now so-called not derivatively but in its own right and primarily—is indivisible and is inherent in all time. For the now is an extremity of the past (no part of the [234a1] future being on this side of it), and again of the future (no part of the past being on that side of it): it is, we maintain, a limit of both. And if it is proved that it is of this character and one and the same, it will at once be evident also that it is indivisible.
Now the now that is the extremity of both times must be one and the same; for [5] 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 apart from the other, there will be time between them, because everything continuous is such that there is something between its limits described by the same name as itself. But if the intermediate thing is time, it will be divisible; for all [10] time has been shown to be divisible. Thus on this assumption the now is divisible. But if the now 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. Also the now will be a now not in its own right but derivatively, for [15] the division will not be a division in its own right. Furthermore, there will be a part of the now that is past and a part that is future, and it will not always be the same part that is past or future. Nor, then, will the now be the same; for the time may be divided at many points. If, therefore, the now cannot possibly have these characteristics, it follows that it must be the same now that belongs to each of the two times. [20] But if it is the same, it is evident that it is also indivisible; for if it is divisible it will be involved in the same implications as before. It is clear, then, from what has been said that time contains something indivisible, and this is what we call the now.
We will now show that nothing can be in motion in a now. For if this is possible, there can be both quicker and slower motion. Suppose then that in the now N the [25] quicker has traversed the distance AB. That being so, the slower will in the same now have traversed a distance less than AB, say AC. But since the slower will have occupied the whole now in traversing AC, the quicker will occupy less than this in traversing it. Thus we shall have a division of the now, whereas we found it to be [30] indivisible. It is impossible, therefore, for anything to be in motion in a now.
Nor can anything be at rest; for we assert that, that only can be at rest which is of such a nature to be in motion but is not in motion when, where, or as it would naturally be so; since, therefore, nothing is of such a nature as to be in motion in a now, it is clear that nothing can be at rest either.
Moreover, inasmuch as it is the same now 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 [234b1] throughout the other, and that which is in motion or at rest for the whole of a time will be in motion or at rest in any part of it in which it is of such a nature as to be in motion or at rest: it will follow that the same thing can at the same time be at rest and in motion; for both the times have the same extremity, viz. the now.
Again, we say that a thing is at rest if its condition in whole and in part is [5] uniform now and before; but the now contains no before; consequently, there can be no rest in it.
It follows then that the motion of that which is in motion and the rest of that which is at rest must occupy time.
[10] 4 · Further, everything that changes must be divisible. For since every change is from something to something, and when a thing is at the point to which it was changing it is no longer changing, and when both it itself and all its parts are at the point from which it was changing it is not39 changing (for that which is in whole and in part in an unvarying condition is not in a state of change); it follows, [15] therefore, that part of that which is changing must be at the starting-point and part at the goal; for it cannot be in both or in neither. (Here by ‘goal of change’ I mean that which comes first in the process of change: e.g. in a process of change from white the goal in question will be grey, not black; for it is not necessary that that which is changing should be at either of the extremes.) It is evident, therefore, that [20] everything that changes must be divisible.
Now motion is divisible in two ways—in virtue of the time that it occupies, according to the motions of the parts of that which is in motion: e.g. if the whole AC is in motion, there will be a motion of AB and a motion of BC. Let DE be the motion [25] of the part AB and EF the motion of the part BC. Then the whole DF must be the motion of AC; for it must constitute its motion inasmuch as they severally constitute the motions of each of its parts. 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.
[30] Again, since every motion is a motion of something, and the whole motion DF is not the motion of either of the parts (for each of the parts is the motion of one of the parts) or of anything else (for, the whole motion being the motion of a whole, the parts of the motion are the motions of the parts of that whole; and the parts are the motions of AB, BC 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 motion will be the motion of the magnitude ABC.
/> Again, if there is a motion of the whole other than DF, say HI, the motion of [235a1] each of the parts may be subtracted from it; and these motions will be equal to DE, EF; for the motion of that which is one must be one. So if the whole motion HI may be divided into the motions of the parts, HI will be equal to DF; if on the other hand there is any remainder, say KI, this will be a motion of nothing; for it can be the [5] motion neither of the whole nor of the parts (as the 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 HI reveals a surplus. Consequently, if this is impossible, the whole motion must be the same as and equal to DF.
This then is what is meant by the division of motion according to the motions of [10] the parts; and it must be applicable to everything that is divisible into parts.
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. 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 [15] being-in-motion, the thing that is in motion, and the sphere of the motion must all be susceptible of the same divisions (though spheres of motion are not all divisible in a like manner: thus place is essentially, quality accidentally divisible). For suppose that A is the time occupied by the motion B. 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 [20] again to occupy a further subdivision of the time, and so on always. Similarly, if the motion is divisible, the time too will be divisible; 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.
In the same way the being-in-motion will also be divisible. For let C be the [25] whole being-in-motion. 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 always. Moreover by setting out the being-in-motion corresponding to each of the two motions DC (say) and CE, we may argue that the whole being-in-motion will correspond to the whole motion (for [30] if something else did, 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 the being-in-motion corresponding to each of the two motions, we shall see that the whole is continuous.
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 [35] accidentally divisible because that which changes is so); for the division of one term will involve the 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. And we now see that in most cases [235b1] 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 the thing that changes. That divisibility does so we have already shown; that infinity does so will be made clear in what [5] follows.
5 · 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. 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 [10] of it. 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.
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 [15] in being; for everything must either be or not be. It is evident, then, that in contradictory change that which has changed must be in that to which it has changed. And if this is true in this kind of change, it will be true in all other kinds as well; for what holds good in the case of one will hold good likewise in the case of the rest.
Moreover, if we take each kind of change separately, the truth of our conclusion will be equally evident, on the ground that that which has changed must [20] be somewhere or in something. 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. If, then, that which has changed to B is in something other than B, say C, it must again be changing from C to B; for B was not assumed to be contiguous, and change is continuous. Thus we have the result that the thing that [25] 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. So it is evident likewise 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 [30] most obvious in 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.
Now the time primarily in which that which has changed has changed must be indivisible, where by ‘primary’ I mean a thing’s being such-and-such not because [35] some part of it is such-and-such. For let AC be divisible, and let it be divided at B. If then it has changed in AB or again in BC, AC cannot be the primary thing in which it has changed. If, on the other hand, it has been changing in both AB and BC (for it [236a1] must either have changed or be changing in each of them), it must have been changing in the whole too; but our assumption was that it had changed in that. The same argument applies if we suppose that it changes in one part and has changed in the other; for then we shall have something prior to what is primary. So that in [5] which a thing has changed must be indivisible. It is also evident, therefore, 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.
But there are two ways of talking about that primarily in which something has changed. On the one hand it may mean the primary time at which the change is completed—the moment when it is correct to say ‘it has changed’; on the other hand it may mean the primary time at which it began to change. Now the primary time [10] that has reference to the end of the change is something really existent; for a change may 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. But that which has reference to the beginning is not existent at all; for there is no such thing as a beginning of change, [15] nor any primary time at which it was changing. For suppose that AD is such a primary time. Then it cannot be indivisible; for, if it were, the nows will be consecutive. Again, if the changing thing is at rest in the whole time CA (for we may suppose that it is at rest), it is at rest in A also; so if AC is without parts, it will [20] simultaneously be at rest and have changed; for it is at rest in A and has changed in D. 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; if, on the other hand, it is changing in both parts, it is likewise changing in the whole; and if, again,40 it has changed in one of the two parts, the whole is not the primary time in which it has changed: it must therefore have changed in every part). It is evident, [25] then, that there is no primary time in which it has changed; for the divisions are infinite.
So, too, of that which has changed there is no primary part that has changed. For suppose that of DE the primary part that has changed is DF (everything that changes having been shown to be divisible); and let HI b
e the time in which DF has [30] changed. If, then, in the whole time DF has changed, in half the time there will be a part that has changed, less than and prior to DF; and again there will be another part prior to this, and yet another, and so on always. 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. [35]
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. For in a [236b1] process of change we may distinguish three terms—that which changes, that in which it changes, and that to which it changes, e.g. the man, the time, and the pallor. Of these the man and the time are divisible; but with the pallor it is otherwise [5] (though they are all divisible accidentally; for that of which the pallor or any other quality is an accident is divisible). For things which are divisible in their own right and not accidentally have no primary part. Take the case of magnitudes: let AB [10] be a magnitude, and suppose that it has moved from B to a primary C. Then if BC is taken to be indivisible, two things without parts will have to be contiguous; if on the other hand it is taken to be divisible, there will be something prior to C to which the magnitude has changed, and something else again prior to that, and so on to always, because the process of division never gives out. Thus there can be no primary thing [15] to which a thing has changed. 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 indivisible in its own right.