The Basic Works of Aristotle (Modern Library Classics) (56 page)

2
     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 greater magnitude in an equal time,
(25)
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. Now since of two things that which changes sooner is quicker,
(30)
in the time FG, in which A has changed from C to D, B will not yet have arrived at D but will be short of it: so that in an equal time the quicker will pass over a greater magnitude. More than this, it will pass over a greater magnitude in less time: for in the time in which A has arrived at D, B being the slower has arrived, let us say, at E.
[232b]
Then since A has occupied the whole time FG in arriving at D, it will have arrived at H in less time than this, say FJ. Now the magnitude CH that A has passed over is greater than the magnitude CE, and the time FJ is less than the whole time FG: so that the quicker will pass over a greater magnitude in less time.
(5)
And from this it is also clear that the
quicker will pass over an equal magnitude in less time than the slower. For since it passes over the greater magnitude in less time than the slower, and (regarded by itself) passes over KL the greater in more time than KN the lesser, the time PQ in which it passes over KL will be more than the time PR in which it passes over KN: so that,
(10)
the time PQ being less than the time PV in which the slower passes over KN, the time PR will also be less than the time PV: for it is less than the time PQ, and that which is less than something else that is less than a thing is also itself less than that thing. Hence it follows that the quicker will traverse an equal magnitude in less time than the slower. 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,
(15)
and since, whereas a thing is slower if its motion occupies more time and of equal velocity if its motion occupies 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. 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.
(20)

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 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,
(25)
it follows necessarily that time is continuous. 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 CD in the time FG.
(30)
Now it is clear that the quicker will traverse the same magnitude in less time than this: let us say in the time FH. Again, since the quicker has passed over the whole CD in the time FH, the slower will in the same time pass over CJ, say, which is less than CD.
[233a]
And since B, the slower, has passed over CJ in the time FH, the quicker will pass over it in less time: so that the time FH will again be divided. And if this is divided the magnitude CJ will also be divided just as CD was: and again, if the magnitude is divided, the time will also be divided. And we can carry on this process for ever,
(5)
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 slower will divide the length. If, then, this alternation always holds good, and at
every turn involves a division,
(10)
it is evident that all time must be continuous. 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.

Moreover, the current popular arguments make it plain that, if time is continuous, magnitude is continuous also,
(15)
inasmuch as a thing passes over half a given magnitude in half the time taken to cover the whole: in fact without qualification it passes over 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. 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,
(20)
magnitude is also infinite in both respects.

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. 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.
(25)
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 by the passage over the infinite is not a finite but an infinite time,
(30)
and the contact with the infinites is made by means of moments not finite but infinite in number.

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. This may be shown as follows. Let AB be a finite magnitude, and let us suppose that it is traversed in infinite time C,
(35)
and let a finite period CD of the time be taken.
[233b]
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. (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,
(5)
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 BE measures the whole of which it is a part,
(10)
and if an equal magnitude is passed over in an equal time, then it follows that the time like the magnitude 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, the time occupied in traversing this part must be finite, the limit in 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. It is evident, then, from what has been said that neither a line nor a surface nor in fact anything continuous can be indivisible.
(15)

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 period of time and in an equal time the quicker passes over a greater length,
(20)
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 as that traversed by the slower, and that the respective magnitudes are divided, that of the quicker, the magnitude ABCD, into three indivisibles, and that of the slower into the two indivisibles EF, FG. Then the time may also be divided into three indivisibles,
(25)
for an equal magnitude will be passed over in an equal time. Suppose then that it is thus divided into JK, KL, LM. Again, since in the same time the slower has been carried over EF, FG, 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.
³
(30)
It is evident, therefore, that nothing continuous is without parts.

3
     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,
⁴
but in its proper and primary sense; in which sense it is inherent in all time.
(35)
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,
⁵
a limit of both.
[234a]
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.
(5)

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 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.
(10)
But if the intermediate thing is time, it will be divisible: for all time has been shown
⁶
to be divisible. Thus on this assumption the present is divisible. 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. 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.
⁷
(15)
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. 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.
(20)
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. It is clear, then, from what has been said that time contains something indivisible, and this is what we call a present.

We will now show that nothing can be in motion in a present.
(25)
For if this is possible, there can be both quicker and slower motion in the present. Suppose then that in the present M the quicker has traversed the distance AB. That being so, the slower will in the same present traverse a distance less than AB, say AC. But since the slower will have occupied the whole present in traversing AC,
(30)
the quicker will occupy less than this in traversing it. 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.

Nor can anything be at rest in a present: for, as we were saying,
⁸
that 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 motion in a present, it is clear that nothing can be at rest in a present either.

Moreover, inasmuch as it is the same present that belongs to both the times,
⁹
and it is possible for a thing to be in motion throughout one time and to be at rest throughout the other,
(35)
and that which is in motion or at rest for the whole of a time will be 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 at the same time be at rest and in motion: for both the times have the same extremity, viz.
[234b]
the present.

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,
(5)
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.

4
     Further, everything that changes must be divisible.
(10)
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 and in part in an unvarying condition is not in a state of change); it follows, therefore,
(15)
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. (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,
(20)
therefore, that everything that changes must be divisible.

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. if the whole AC is in motion, there will be a motion of AB and a motion of BC. That being so, let DE be the motion of the part AB and EF the motion of the part BC.
(25)
Then the whole DF must be the motion of AC: for DF must constitute the motion of AC inasmuch as DE and EF severally constitute the motions of each of its parts. But the motion of a thing can never be constituted by
the motion or something else: consequently the whole motion is the motion of the whole magnitude.

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 DE, EF is the motion of one of the parts AB, BC)
(30)
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 of DF are the motions of AB, BC and of nothing else: for, as we saw,
¹⁰
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 each of the parts may be subtracted from it: and these motions will be equal to DE,
(35)
EF respectively: for the motion of that which is one must be one.
[235a]
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 JI, this will be a motion of nothing: for it can be the 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.
(5)
And the same result follows if the division of HI reveals a surplus on the side of the motions of the parts. 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 the parts: and it must be applicable to everything that is divisible into parts.

Motion is also susceptible of another kind of division,
(10)
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,
(15)
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 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.
(20)
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. 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.

In the same way the being-in-motion will also be divisible.
(25)
For let C be the 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 to infinity. Moreover by setting out successively 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 if it were some other being-in-motion that corresponded to the whole motion,
(30)
there would be more than one being-in-motion corresponding to the same motion), the argument being the same as that whereby we showed
¹¹
that the motion of a thing is divisible into the motions of the parts of the thing: for if we take separately the being-in-motion corresponding to each of the two motions, we shall see that the whole being-in-motion 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 accidentally divisible because that which changes is so): for the division of one term will involve the division of all.
(35)
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 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.
[235b]
That divisibility does so we have already
¹²
shown; that infinity does so will be made clear in what follows.
¹³
(5)