Commentary Aristotle Physics
This being given, it occurs that the first B, by its motion, arrives at the last A, and likewise the first C, by its motion, arrives at the first A, opposite the last. Also, at the same time, when this is at the last B, as though crossing at the same moment, that is, next to all the Bs, which are moved by one another. When this has been done, it is evident that this first C has passed all the As, but B has passed by only half. Since, therefore, B and C are equal in velocity, and an equal velocity passes by a smaller magnitude in less time, it follows that the time in which B travels to the last A is half the time in which C arrives at the first A opposite; but each of them (C and B) occupies an equal time in passing each A, that is, in equal times pass each section of A.
This being supposed—that the time in which B arrives at the last A is half the time in which C arrives at the first A opposite—it must be further considered how Zeno wished to conclude that this half time is equal to its double. For from the supposition that the time of the motion of C is double the time of the motion of B, it is supposed that, in the first half of the time, B was still and C moved, and thus C arrived in that half of the time at the middle of the space, where B was; and then B began to move to one part and C to another. When B arrived at the last A, it had to pass all of C, because, at the same time, the first B and the first C are at contrary ultimates, one at the first A and the other at the last, and as he said, C is next to each B in the same amount of time as it takes to pass each one of the As. This is so, because B and C each pass one A in the same interval of time. Thus, it seems that, if B covers a time equal to that in which it passes C, then C passes B and A in an equal interval of time. Therefore, the interval in which C passes all of B is equal to the time in which it passed all of A. The time in which C passed all of B is equal to the time in which C or B passed the middle of A, as was said. But it was proved that the time in which B passed the middle of A is half the time in which C reached all of A. Therefore, it follows that the half is equal to the double, which is impossible.
This is the argument of Zeno. But he falls into the error previously mentioned: he assumes that C, in the same interval of time, crosses B moving in a counter direction and A resting, which is false, as was said above.
870. Then, at “nor, in reference to” (240a19), he gives the argument by which Zeno rejected change between contradictories. For he said that whatever is being changed is in neither of the extremities while it is being changed, for while it is in the terminus from which, it is not yet being changed, and while it is in the terminus to which, it has already been changed. Therefore, if something is being changed from one contradictory to another, as from non-white to white, it follows that, while it is being changed, it is neither white nor non-white—which is impossible.
Now, although this strange conclusion would follow for those who posit that an indivisible can be moved, yet for us who posit that whatever is being moved is divisible, nothing impossible follows. For even though it is not totally in one or the other of the extremes, it is not for that reason neither white nor non-white: one part could be white and the other non-white. For a thing is called white not only when all of it is white but also when very many or its main parts are white—that is, the parts to which it is more proper to be such—because it is one thing not to be something at all and another not to be entirely something, for example, white or non-white.
And what has been said of white and non-white is to be understood of unqualified being or non-being and of all things that are contradictorily opposed, as hot and non-hot and so on. For what is being changed must always be in one of the opposites, because it is described in terms of whichever opposite predominates in it. But it does not follow that it is always as a whole in neither of the extremities, as Zeno supposed.
Now, it should be known that this answer is sufficient to refute Zeno’s argument, and that is what Aristotle’s main intention is. But the truth of this matter will be more fully given in book 8. For it is not true in all cases that part is altered or generated after part, but sometimes the whole comes all at once, as was said above. In that case, it is not this answer that would apply, but the one in book 8.
871. Then, at “again, in the case” (240a29), he refutes the argument by which Zeno rejected spherical motion. For he said that it is not possible for anything to be moved circularly or spherically or in such a way that certain things are moved within themselves, that is, the motion is confined within the space occupied by the mobile. And he proved this with the following argument. Anything that is, in its entirety and in respect of its parts, in one and the same place for a period of time is not in motion, but at rest. But all the above-mentioned fulfill these conditions, even when they are apparently in motion. Therefore, they are at once in motion and at rest—which is impossible.
The Philosopher attacks this argument on two points.
He first attacks the statement that the parts of the moving sphere are in the same place for some time. For Zeno was speaking of the place of the whole, and it is true that, while the sphere is in motion, no part passes out of the place of the sphere, but Aristotle speaks of the particular place of each part, according as a part has a place. For it was said in book 4 that the parts of a continuum are in place potentially. But it is evident in spherical motion that a part does change its particular place, although it does not leave the place of the whole, because where one part was, another part succeeds.
Second, he attacks the statement that the whole remains in the same place for some time. Against this, Aristotle says that even the whole is changing its place. For in order that two places be not the same, it is not required that one of them be entirely outside the other, but sometimes the second place is partly joined to part of the first and partly divided from it, as is clear in things moved in a straight line. For let a body of one cubit be moved from place AB to place BC—both places being one cubit each. While the mobile is being moved from one place to the other, it must partly desert one place and enter the other; for example, it could leave the portion AD of AB and enter the portion BE of BC. Therefore, it is clear that the place DE is distinct from AB, although not entirely, but only partly separated from it.
But, if it were assumed that that part of the mobile that entered the second place reentered part of the place deserted, there would be two places, yet in no way separated. They would differ not really, but only in account, that is, in the sense that the beginning of the place might be successively called by different letters each time the mobile reentered it—namely, where the beginning of the mobile is, that is, some spot in the mobile that is taken as a beginning. Thus, there would be two places conceptually, but one and the same in reality.
This is how we must understand what Aristotle says here: it is not the same revolution when it is taken as beginning at A and as beginning at B and as beginning at C or any other mark, unless you insist that it is the same revolution as to subject, as in the case of “musical man” and “man,” since one happens to the other. Hence it is clear that the mobile is always being moved from one circular place to another and is not at rest, as Zeno tried to prove. And it is the same with the sphere and everything else whose motion is confined within the space it occupies, as in the case of a potter’s wheel and a (rotating) pillar or anything of that sort.