LECTURE 2 (231 b 19-232 a 18)

IF MAGNITUDE IS COMPOSED OF INDIVISIBLE PARTS, THEN SO IS MOTION. BUT THIS IS IMPOSSIBLE

             758. The arguments given above are rather clear in regard to lines and other continuous quantities which have position and in which contact is properly found. He wishes here to show that the same reasoning [ratio] applies to magnitude, time, and motion.

             This discussion is divided into two parts. First he states his intention. Secondly, where he says, 'This may be made . . .' (231 b 20), he proves his position.

             He says, therefore, first that for the same reasons [ratio] magnitude and time and motion are either all composed of indivisible parts and divided into indivisible parts, or else none of them are. For whatever is granted for one of these necessarily applies also to the others.

             759. Next where he says, 'This may be made . . .' (231 b 20), he proves his position. He does this first in regard to magnitude and motion, and secondly in regard to time and magnitude, where he says, 'And if length and motion . . .' (232 a 19).

             Concerning the first part he makes three points. First he states his position. Secondly, where he says, '. . . e.g. if the magnitude . . .' (231 b 23), he gives an example. Thirdly, where he says, 'Therefore, since where . . .' (231 b 25), he proves his position.

             His position is as follows. If magnitude is composed of indivisible parts, then the motion which crosses a magnitude will be composed of indivisible motions equal in number to the indivisible parts from which the magnitude is composed.

             760. He gives the following example. Let there be a line A B C which is composed of three indivisible parts, A, B, and C. Let Z be a mobile object which is moved in the space of the line A B C; and let the motion of Z be D E F. If the parts of the space or line are indivisible, then the designated parts of the motion must also be indivisible.

             Next where he says, 'Therefore, since where . . .' (231 b 25), he proves his position.

             Concerning this he makes three points. First he sets forth certain things which are necessary to prove his position. Secondly, where he says, '. . . and, as we saw . . .' (231 b 30), he proves that if magnitude is composed of points, then motion is not composed of motions but of minute impulses [momenta]. Thirdly, where he says, '. . . and will take place . . .' (232 a 8), he shows that motion cannot be composed of minute impulses.

             761. He makes two preliminary points. The first is that in respect to every part of a given motion there must be something which is moved. And conversely, if something is moved, it is necessary that some motion be present to it. And if this is true, then the mobile object Z must be moved through A, which is part of the whole magnitude, by that part of the motion which is D. And in regard to B, another part of the magnitude, it is moved by another part of the motion which is E. And in regard to C, the third part of the magnitude, it is moved by the third part of the motion which is F. Thus the individual parts of the motion correspond to the individual parts of the magnitude.

             He gives the second point where he says, 'Now a thing that is . . .' (231 b 28). He says that that which is being moved from one terminus to another cannot simultaneously be 'being moved' and 'having been moved'. For it is being moved when it is being moved. For example, if someone is walking to Thebes, it is impossible for him simultaneously to be walking to Thebes and to have walked to Thebes.

             These two points are stated as though they are per se evident. The fact that something must be moved in order for motion to be present is also clear in all accidents and forms. For in order that a thing be white, it must have whiteness. And conversely, if whiteness be present, it is necessary that something be white. Furthermore, it is apparent from the succession in motion that a thing cannot simultaneously be 'being moved' and 'having been moved'. For it is impossible for two parts of time to exist simultaneously, as was said in Book IV. Thus is it impossible for 'being moved' to be simultaneous with 'having been moved', which is the terminus of the motion.

             762. Next where he says, '. . . and, as we saw . . .' (231 b 30), he proves his position from the foregoing.

             When some part of motion is present, it is necessary that something is moved. And when something is moved, motion must be present. Therefore, if the mobile object Z is moved in respect to A, an indivisible part of the magnitude, it is necessary that some motion, D, be present to Z. Hence, Z either is simultaneously 'being moved' and 'has been moved' through A, or it is not. If these are not simultaneous, that is, the object is 'being moved' before it 'has been moved', then it follows that A is divisible. For while Z was being moved, it was not at rest in A, because the state of rest precedes the motion. Nor had it traversed the whole of A, for it was not yet moved through A. (Nothing is ever being moved through a space through which it has already gone.) Rather the object must be in an intermediate state. Therefore, when the object is being moved through A, it has already crossed part of A and another part still remains. And thus it follows that A is divisible, which is contrary to what was assumed.

             If the object is being moved through A and has been moved through A simultaneously, then it follows that while it is coming it has already arrived. It will have come and will have been moved to where it is being moved. This is contrary to the second preliminary point.

             Thus it is clear that nothing can be moved in respect to an indivisible magnitude. For otherwise it must be that either 'being moved' and 'having been moved' are simultaneous, or else the magnitude is divided.

             Let us grant that nothing can be moved through the indivisible magnitude A. Then, if one says that the mobile object is moved through the whole magnitude A B C, and if the whole motion by which it is moved is D E F, and if nothing is being moved in the indivisible magnitude A, but already has been moved, then it follows that motion is not composed of motions but of minute impulses. This is true because of the following. That part of the motion which is D corresponds to that part of the magnitude which is A. Now if D were a motion, it would be necessary for it to be a motion through A, because the mobile object is moved by the motion which is present to it. But it has been proven that, in respect to the indivisible magnitude A, the object is not being moved, but only has been moved when it has crossed this indivisible magnitude. Therefore it follows that D is not a motion but a minute impulse, by which it is called a motion, just as 'to be moved' is called a motion. And a minute impulse is related to motion in the same way that an indivisible point is related to a line. The same argument applies to the other parts of the motion and the magnitude. Therefore it necessarily follows that if magnitude is composed of indivisible parts, then motion is composed of indivisible minute impulses. This is what he intended to prove.

             763. But it is impossible for motion to be composed of minute impulses, just as it is impossible for a line to be composed of points. Hence where he next says, '. . . and will take place . . .' (232 a 8), he shows that this is impossible in three ways.

             The first is as follows. If motion is composed of minute impulses, and if magnitude is composed of indivisible parts, so that the object is not being moved but has been moved through an indivisible part of the magnitude, then it follows that the object has been moved even though it was not previously moved. It was granted that in respect to an indivisible magnitude the object has been moved, but is not being moved. For the object cannot be moved in an indivisible magnitude. Hence it follows that the object has crossed a magnitude without at some time being in a state of crossing it. This is impossible, just as it is impossible for something to be past without having ever been present.

             764. But since this impossibility can be admitted by one who says that motion is composed of minute impulses, he brings out a second impossibility where he says, 'Since, then, everything . . .' (232 a 12). The argument is as follows.

             Everything which is naturally in motion or at rest must be either in motion or at rest. But while the mobile object is in A, it is not being moved. And likewise while it is in B and in C, it is not being moved. Hence, while it is in A and in B and in C it is at rest. Therefore it follows the object is simultaneously and continuously in motion and at rest.

             That this follows he proves in this way. It was granted that the object is moved through the whole magnitude A B C. Further, it was granted that in respect to each part of the magnitude the object is at rest. But that which is at rest through each part is at rest through the whole. Therefore, it follows that the object is at rest through the whole magnitude. And thus it follows that the object is continuously at rest and in motion through the whole magnitude. This is altogether impossible.

             765. He gives the third impossibility where he says, 'Moreover, if the indivisibles . . .' (232 a 15). The argument is as follows.

             It was shown that if magnitude is composed of indivisible parts, then so is motion. Hence, these indivisible parts of motion, D, E, and F, are either so constituted that each of them is a motion, or they are not. Now if each of these is a motion, then since each of them corresponds to an indivisible part of the magnitude in which the object is not being moved but has been moved, it follows that the mobile object is not moved by the present motion, but is at rest. This is contrary to the first supposition. And if they are not motions, it follows that motion is composed of non-motions. This seems impossible, just as it is impossible for a line to be composed of non-lines.