Commentary Aristotle Physics
863. In regard to the first, he lists four reasons. Zeno used four arguments against motion that have caused difficulty for many of those who tried to answer them. The first is this. If anything is being moved through a certain space, it must reach the middle before it reaches the end. But, since the first half is divisible, half of it must be first traversed and so on indefinitely, since a magnitude can be divided to infinity. Infinites, however, cannot be traversed in finite time. Therefore, nothing can be moved.
Therefore, Aristotle says that he has already answered this argument (in the beginning of this book 6), when he proved that time is divided to infinity in the same way as a magnitude is. This answer is directed less to the immediate question than to one who asks whether infinites can be traversed in finite time, as he will say in book 8, where he answers this argument by showing that a mobile does not use the infinites that exist in a magnitude as though they were actually existing, but only as existing potentially. For a thing in motion uses a point in space as actually existing when it uses it as a beginning and as an end, and it is then that the mobile must be at rest, as will be explained in book 8. But, if it had to traverse infinites that were actually existing, then it would never reach the end.
864. The second argument is given at “the second is” (239b14), and he says that they called this one the Achilles, as though it were invincible and unanswerable. The argument was this. If anything is being moved, it follows that a slower thing, if it started earlier, will never be caught by anything moving most rapidly. And it was proved in the following way. If a slower object began to be moved for some time before a very swift one, then, in that time, it has traversed some distance. Therefore, before the very swift one in pursuit could reach the slower, which is still running, it must leave the place first left by the pursued and reach the place that the pursued reached during the time the pursuer was not in motion. But the very fast pursuer must traverse this space in some time, during which the slower has meanwhile traversed a certain space, and so on forever. Therefore, the slower must always hold a lead—that is, is always some distance ahead of the most swift pursuer—and so the swifter will never catch the slower. But this is unacceptable. Therefore, it is better to say that nothing is moved.
865. In solving this argument, he says that it is the same as the first, which proceeded by dividing the distance into two halves and then continually halving one part of the remainder.
But the difference between them is that, in the second, the given magnitude of space is not divided into halves, but according to the difference between the swift and the slower motion.
For in the first period of time, in which only the slower was in motion, there is a greater magnitude involved; in the second period (in which the faster traversed the distance covered by the slower between its start and the start of the faster), which is a shorter time period, a smaller magnitude was traversed by the slower, and so on forever.
Hence, the time and the magnitude are always being divided and that seems to be the reason why the slower is never caught by the swifter.
But this tends to the same thing as what was said of the division of the magnitude into halves: in both arguments, it seems that the mobile cannot reach a certain goal on account of the magnitude’s being infinitely divided, no matter how it happens to be divided—that is, whether according to halves, as happens in the first argument, or according to the excess of the faster over the slower, as in the second argument. However, in this second argument, it is further added that the very swift cannot reach the slower that it is pursuing. This tragic phraseology employs inflated language in order to excite wonder, but it does not do anything to the force of the argument.
Hence, it is clear that the solution of the two arguments is the same. For just as a false conclusion was reached in the first argument—namely, that the mobile would never reach the end of the magnitude on account of the infinite division of the magnitude—so also what the second argument tries to conclude is false, namely, that the slower will never be caught by the swifter, which is just another way of saying that a mobile will never reach its goal.
Now, it is true that, as long as the slower is ahead, it is not yet reached by the swifter. But yet it will at some time be reached, if you concede that a finite magnitude can be traversed in finite time. For the swifter pursuing mobile will traverse the whole distance by which the slower is ahead and even more, in less time than the slower was meantime moving farther ahead. Proceeding in this way, the swifter will not only catch the slower, but pass it. These, therefore, are the solutions to two of Zeno’s arguments.
866. The third argument is given at “the third is” (239b30), and he says that the third argument of Zeno was the one cited above before he began to give the arguments—namely, that an arrow in flight is always at rest. And as was said above, this seems to happen, because Zeno supposed that time is made up of instants. For unless that be granted, the syllogism fails.
867. He sets out the fourth argument at “the fourth argument” (239b33). Concerning it, he does three things: first, he sets out the argument; second, he gives the solution, at “the fallacy of” (240a1; [868]). third, he explains it by an example, at “for instance” (240a4; [869]).
First, therefore, he says that the fourth argument of Zeno proceeded from some bodies that move in a stadium such that there are two equal magnitudes that are moved in an equal manner (that is, through a space in the stadium equal to both in quantity), and this motion is in opposite directions (that is, one of the equal magnitudes is moved through that space of the stadium toward one part, and the other toward the other part, in such a way, however, that one of the mobile magnitudes begins to move from the terminus of the stadium that is equal to it, and the other begins to move from the middle of the stadium or of a space in the given stadium); both move with equal velocity. This being given, Zeno held that it would result in a half time equaling a double time. Since this is impossible, he intended to infer further that it is impossible for anything to be moved.
868. Then, at “the fallacy of” (240a1), he gives the solution. He says that Zeno was deceived in that, on the one hand, he held that the mobile is moved according to the moved magnitude and, on the other, he held that it was moved according to a resting magnitude equal to the moved magnitude. Because an equal velocity of the moved bodies is supposed, he wanted to conclude that the motion of equally swift bodies in regard to equal magnitudes, one of which is in motion and the other standing still, is done in equal times. This is seen to be false for the following reason.
When something is moved in relation to a resting magnitude, there is only one motion, but when something is moved in relation to a moving magnitude, there are two motions. If they are moving in the same direction, it takes more time, and if they are moving in opposite directions, it takes less, according to the amount of either motion. If the magnitude in relation to which something mobile is moved is moved in the same direction with an equal velocity, or even a greater velocity, the other moving body can never pass it. If the magnitude moves with less speed, it will pass by it at a certain time, but it will take more time than if it were resting. It is quite the contrary if the magnitude is moved opposite the direction of the other body. The more swiftly the magnitude moves, the less time the other body takes to pass it, because the two motions work together to pass each other.
869. Then, at “for instance” (240a4), he makes clear what he said in the latter part. Suppose that there are three magnitudes equal to each other, each designated as A, and these magnitudes are standing still, that is, not moved; thus, there might be a space of three cubits, each one of which is marked by an A. There are another three magnitudes all equal and designated as B, as there might be one moving unit of three cubits. These magnitudes begin to move from the middle of the space. There are also three other magnitudes, equal in number, size, and velocity to B, and designated as C. These begin to move from the last space, that is, from the last A.