The Basic Works of Aristotle (Modern Library Classics) (59 page)

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 above
²⁹
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 above
³⁰
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.

¹
v. 3.

²
Which is
ex hypothesi
impossible (231
^b
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
222
^a
12.

⁶
Chapter 2.

⁷
i. e. it will not be a
point
of division but merely something intermediate between past and future.

⁸
226
^b
12 sqq.

⁹
viz. past and future.

¹⁰
223
^b
1 sqq.

¹¹
234
^b
24 sqq., especially 234
^b
34 sqq.

¹²
234
^b
10–20.

¹³
Chapter 7.

¹⁴
sc.
BC will have more right than AC to be regarded as that in which the change has been completed.

¹⁵
234
^b
10 sqq.

¹⁶
235
^b
33. The ‘primary time’ is the irreducible minimum: thus the very terms of the definition make it clear that a thing must be changing in the
whole
of the ‘primary time’ in which it changes.

¹⁷
235
^b
6 sqq.

¹⁸
231
^b
6 sqq.

¹⁹
i. e. you may begin by cutting off half the line, then half of what remains, and so on, the part cut off thus continuously increasing and the part remaining continually decreasing.

²⁰
Ch. 6.

²¹
238
^b
31 sqq.

²²
226
^b
12 sqq.

²³
sc.
time.

²⁴
i. e. a space only just large enough to contain it, not a larger space of which only part is occupied.

²⁵
233
^a
13 sqq.

²⁶
viz. the first argument given above, ll. 11–14.

²⁷
234
^b
10 sqq.

²⁸
239
^a
27.

²⁹
231
^b
18 sqq.

³⁰
232
^b
23 sqq.

1
     Everything that is in motion must be moved by something.
(25)
For if it has not the source of its motion in itself it is evident that it is moved by something other than itself, for there must be something else that moves it. If on the other hand it has the source of its motion in itself, let AB be taken to represent that which is in motion essentially of itself and not in virtue of the fact that something belonging to it is in motion. Now in the first place to assume that AB,
(30)
because it is in motion as a whole and is not moved by anything external to itself, is therefore moved by itself—this is just as if, supposing that JK is moving KL and is also itself in motion, we were to deny that JL is moved by anything on the ground that it is not evident which is the part that is moving it and which the part that is moved. In the second place that which is in motion without being moved by anything does not necessarily cease from its motion because something else is at rest, but a thing must be moved by something if the fact of something else having ceased from its motion causes it to be at rest.
[242a]
Thus, if this is accepted, everything that is in motion must be moved by something.
(5)
For AB, which has been taken to represent that which is in motion, must be divisible, since everything that is in motion is divisible. Let it be divided, then, at C. Now if CB is not in motion, then AB will not be in motion: for if it is, it is clear that AC would be in motion while BC is at rest,
(10)
and thus AB cannot be in motion essentially and primarily. But
ex hypothesi
AB is in motion essentially and primarily. Therefore if CB is not in motion AB will be at rest. But we have agreed that that which is at rest if something else is not in motion must be moved by something. Consequently, everything that is in motion must be moved by something: for that which is in motion will always be divisible,
(15)
and if a part of it is not in motion the whole must be at rest.

Since everything that is in motion must be moved by something, let us take the case in which a thing is in locomotion and is moved by something that is itself in motion, and that again is moved by something else that is in motion, and that by something else,
(20)
and so on continually: then the series cannot go on to infinity, but there must be some first movent. For let us suppose that this is not so
and take the series to be infinite. Let A then be moved by B, B by C, C by D, and so on, each member of the series being moved by that which comes next to it. Then since
ex hypothesi
the movent while causing motion is also itself in motion, and the motion of the moved and the motion of the movent must proceed simultaneously (for the movent is causing motion and the moved is being moved simultaneously) it is evident that the respective motions of A,
(25)
B, C, and each of the other moved movents are simultaneous. Let us take the motion of each separately and let E be the motion of A, F of B, and G and H respectively the motions of C and D: for though they are all moved severally one by another, yet we may still take the motion of each as numerically one, since every motion is from something to something and is not infinite in respect of its extreme points.
(30)
By a motion that is numerically one I mean a motion that proceeds from something numerically one and the same to something numerically one and the same in a period of time numerically one and the same: for a motion may be the same generically, specifically,
(35)
or numerically: it is generically the same if it belongs to the same category, e. g. substance or quality: it is specifically the same if it proceeds from something specifically the same to something specifically the same, e. g. from white to black or from good to bad, which is not of a kind specifically distinct: it is numerically the same if it proceeds from something numerically one to something numerically one in the same period of time, e. g. from a particular white to a particular black, or from a particular place to a particular place, in a particular period of time: for if the period of time were not one and the same, the motion would no longer be numerically one though it would still be specifically one.
[242b]
We have dealt with this question above.
¹
(4)
Now let us further take the time in which A has completed its motion,
(8)
and let it be represented by J. Then since the motion of A is finite the time will also be finite. But since the movents and the things moved are infinite, the motion EFGH, i. e. the motion that is composed of all the individual motions,
(15)
must be infinite. For the motions of A, B, and the others may be equal, or the motions of the others may be greater: but assuming what is conceivable, we find that whether they are equal or some are greater, in both cases the whole motion is infinite. And since the motion of A and that of each of the others are simultaneous, the whole motion must occupy the same time as the motion of A: but the time occupied by the motion of A is finite: consequently the motion will be infinite in a finite time, which is impossible.

It might be thought that what we set out to prove has thus been shown,
(20)
but our argument so far does not prove it, because it does not yet prove that anything impossible results from the contrary supposition: for in a finite time there may be an infinite motion, though not of one thing, but of many: and in the case that we are considering this is so: for each thing accomplishes its own motion, and there is no impossibility in many things being in motion simultaneously. But if (as we see to be universally the case) that which primarily is moved locally and corporeally must be either in contact with or continuous with that which moves it,
(25)
the things moved and the movents must be continuous or in contact with one another, so that together they all form a single unity: whether this unity is finite or infinite makes no difference to our present argument; for in any case since the things in motion are infinite in number the whole motion will be infinite, if, as is theoretically possible, each motion is either equal to or greater than that which follows it in the series: for we shall take as actual that which is theoretically possible. If,
(30)
then, A, B, C, D form an infinite magnitude that passes through the motion EFGH in the finite time J, this involves the conclusion that an infinite motion is passed through in a finite time: and whether the magnitude in question is finite or infinite this is in either case impossible. Therefore the series must come to an end, and there must be a first movent and a first moved: for the fact that this impossibility results only from the assumption of a particular case is immaterial, since the case assumed is theoretically possible, and the assumption of a theoretically possible case ought not to give rise to any impossible result.
[243a]