LECTURE 12 (240 b 8-241 a 26)

THAT WHICH IS WITHOUT QUANTITATIVE PARTS CAN BE MOVED ONLY ACCIDENTALLY

             872. After the Philosopher has answered the arguments of Zeno, who denied motion, he here intends to show that that which is without parts is not moved. In doing this he refutes the opinion of Democritus, who held that atoms are in themselves mobile.

             Concerning this he makes two points. First he sets forth his intention, and secondly he proves his position, where he says, 'For suppose that . . .' (240 b 21).

             He says, therefore, first that, granting what was proven above, we must say that that which is without parts cannot be moved, except perhaps per accidens. For example, a point is moved per accidens in a whole body or in any other magnitude, like a line or a surface, in which there is a point.

             873. To be moved by the motion of another happens in two ways. The first is when that which is moved by the motion of another is not a part of that other. For example, that which is in a boat is moved by the motion of the boat, or whiteness is moved by the motion of a body of which it is not a part. The second is when a part is moved by the motion of the whole.

             Since 'being without parts' has many meanings, as also does 'being with parts', he explains what he means by 'being without parts'. He says, '. . . by "that which is without parts" I mean that which is quantitatively indivisible . . .' (240 b 13). 'Being without parts' can also refer to 'being without parts in respect to species', as when we say that fire or water is without parts because it cannot be resolved into many bodies of diverse species. But nothing prevents that which is without parts in this sense from being moved. Therefore he intends to deny motion of that which is quantitatively without parts.

             874. He has said that a part is moved by the motion of the whole. But someone might say that a part is not moved in any way. Therefore he adds that some motions of the parts, insofar as there are parts, are diverse from the motion of the whole, insofar as there is a motion of the whole.

             This difference can best be understood in spherical motion. For the velocity of the parts which are moved near the centre is not the same as the velocity of the parts which are near the exterior surface of the sphere. Nor is it the same as the velocity of the whole. It is as though there is not one but diverse motions. For it is clear that that which crosses a greater magnitude in an equal time is faster. And while the sphere is being moved, it is clear that an exterior part of the sphere crosses a greater circle than an interior part. Hence the velocity of the exterior part is greater than the velocity of the interior part. However, the velocity of the whole is the same as the velocity of the interior and the exterior part.

             However, this diversity of motions must be understood according to the way in which motion pertains to the parts of a continuum, that is, potentially. The motion of the whole and of the parts is one motion in act. But the motions of the parts are potentially diverse from each other and from the motion of the whole. And thus, when it is said that a part is moved per accidens by the motion of the whole, this means that that which is per se in potency is so moved per accidens. This is not motion per accidens in the sense in which accidents or forms are said to be moved per accidens.

             875. Having made this distinction in regard to that which is moved, he explains his intention. He says that that which is quantitatively without parts can be moved per accidens by the motion of a body. However, it is not moved as a part, because no magnitude is composed of indivisibles, as was shown. Rather it is moved in the way that something is moved by the motion of another of which it is not a part, as a man sitting in a boat is moved by the motion of the boat. But that which is without parts is not moved per se.

             He has already proven this above, not as his main point, but incidentally. Hence he here explains the truth more fully beyond the argument given above. He adds arguments which effectively prove his position.

             876. Next where he says, 'For suppose that . . .' (240 b 21), he proves his position with three arguments. The first is as follows. Let that which is without parts be moved from A B to B C. In this argument it makes no difference whether A B and B C are two magnitudes or two places, as in local motion and increase and decrease, or two qualities, as in alteration, or two contradictory opposites, as in generation and corruption. And let E D be the time in which the thing is changed from the one terminus to the other primarily, that is, not by reason [ratio] of a part. In this time that which is changed must be either in A B, the terminus from which, or in B C, the terminus to which, or partly in the one terminus and partly in the other. For whatever is changed must be in one of these three states, as was said above. One cannot grant the third member; namely, that it is in both termini according to its diverse parts. For then it would be composed of parts, and it was granted that it is without parts. And likewise one cannot grant the second member; namely, that it is in B C or the terminus to which. For when it is in the terminus to which, it has already been changed, as is clear from the above. And it was granted that in this time the body is being changed. It follows, therefore, that during the whole time in which the indivisible is being changed, it is in A B or the terminus from which. And from this it follows that it is at rest. For being at rest is nothing else than being in one and the same condition through some whole time. Since in every time there is a prior and a posterior because time is divisible, then whatever is in one and the same condition through some time is the same now as it was before. And this is being at rest. But it is impossible for a thing to be at rest while it is being changed. Therefore, it follows that that which is without parts is not moved or changed in any way.

             There could be a motion of an indivisible thing only if time were composed of 'nows'. For in the 'now' there is always a 'has been moved' or 'has been changed'. And since the 'has been moved' as such is not being moved, it follows that in the 'now' nothing is being moved, but rather has been moved. Hence, if time were composed of 'nows', it could be granted that the indivisible is moved in time. For it could be granted that in each of the 'nows' from which time is composed, the indivisible is in one condition. And in the whole time, that is, in all the 'nows', it is in many conditions. And thus it would be moved in the whole time, but not in any one 'now'.

             But it is impossible for time to be composed of 'nows', as was shown above. For it was shown that time is not composed of 'nows', and that a line is not composed of points, and that motion is not composed of impulses [momenta]. By impulses we mean states of 'having been changed'. And he who says that the indivisible is moved, or that motion is composed of indivisibles, is saying that time is composed of 'nows', or that magnitude is composed of points, which is impossible. Therefore that which is without parts cannot be moved.

             877. He gives the second argument where he says, 'Again, it may be shown . . .' (241 a 7). He says that from what follows it can be shown that neither a point nor any other indivisible can be moved. This is a special argument dealing with local motion. Whatever is moved in respect to place cannot cross a magnitude greater than itself before it crosses a magnitude equal to or less than itself. Rather the mobile body always crosses a magnitude equal to or less than itself before it crosses a magnitude greater than itself. Hence, if this is so, then a point, if it is moved, will clearly cross a magnitude less than or equal to itself before it crosses a magnitude greater than itself. But it is impossible for a point to cross something less than itself, because it is indivisible. Hence it follows that it will cross something equal to itself. And thus it is necessary that this point number all the points in the line. For since the point is moved by a motion equal to the line because it is moved through the whole line, it follows that the point measures the whole line. And this is done by numbering all the points. Therefore, it follows that the line is composed of points. And since this is impossible, the indivisible cannot be moved.

             878. He gives the third argument where he says, 'Again, since motion . . .' (241 a 15). The argument is as follows.

             Whatever is moved is moved in time. And nothing is moved in the 'now', as was proven above. Moreover, it was shown above that all time is divisible. Therefore, in whatever time something is moved, there will be a smaller time in which some smaller mobile body is moved. For, given the same velocity, it is clear that a smaller mobile body crosses some designated distance in less time than a larger body. For example, a part is moved in less time than the whole, as is clear from the above. Therefore, if a point is moved, there will be some time which is less than the time in which the point is moved. But this is impossible, for it would follow that in that smaller time something smaller than a point is moved. And thus the indivisible would be divisible into something smaller, just as time is divided into time. The indivisible could be moved only if it were possible for something to be moved in the indivisible 'now'. For just as there could be nothing smaller than the 'now' in which a body is moved, likewise there could not be some smaller mobile body.

             And thus it is clear that motion occurring in the 'now' and the indivisible being moved are the same thing. But it is impossible for motion to occur in the 'now'. Therefore the indivisible cannot be moved.