(b) The second proof, which is also the presupposition of continuity and the manifestation of division, is called “Achilles, the Swift.” The ancients loved to clothe difficulties in sensuous representations. Of two bodies moving in one direction, one of which is in front and the other following at a fixed distance and moving quicker than the first, we know that the second will overtake the first. But Zeno says, “The slower can never be overtaken by the quicker.” And he proves it thus: “The second one requires a certain space of time to reach the place from which the one pursued started at the beginning of the given period.” Thus during the time in which the second reached the point where the first was, the latter went over a new space which the second has again to pass through in a part of this period; and in this way it goes into infinity.
| c | d | e | f | g |
| B | A |
B, for instance, traverses two miles (c d) in an hour, A in the same time, one mile (d e); if they are two miles (c d) removed from one another, B has in one hour come to where A was at the beginning of the hour. While B, in the next half hour, goes over the distance crossed by A of one mile (d e), A has got half a mile (e f) further, and so on into infinity. Quicker motion does not help the second body at all in passing over the interval of space by which he is behind: the time which he requires, the slower body always has at its avail in order to accomplish some, although an ever shorter advance; and this, because of the continual division, never quite disappears.
Aristotle, in speaking of this, puts it shortly thus. “This proof asserts the same endless divisibility, but it is untrue, for the quick will overtake the slow body if the limits to be traversed be granted to it.” This answer is correct and contains all that can be said; that is, there are in this representation two periods of time and two distances, which are separated from one another, i.e. they are limited in relation to one another; when, on the contrary, we admit that time and space are continuous, so that two periods of time or points of space are related to one another as continuous, they are, while being two, not two, but identical. In ordinary language we solve the matter in the easiest way, for we say: “Because the second is quicker, it covers a greater distance in the same time as the slow; it can therefore come to the place from which the first started and get further still.” After B, at the end of the first hour, arrives at d and A at e, A in one and the same period, that is, in the second hour, goes over the distance e g, and B the distance d g. But this period of time which should be one, is divisible into that in which B accomplishes d e and that in which B passes through e g. A has a start of the first, by which it gets over the distance e f, so that A is at f at the same period as B is at e. The limitation which, according to Aristotle, is to be overcome, which must be penetrated, is thus that of time; since it is continuous, it must, for the solution of the difficulty, be said that what is divisible into two spaces of time is to be conceived of as one, in which B gets from d to e and from e to g, while A passes over the distance e g. In motion two periods, as well as two points in space, are indeed one.
If we wish to make motion clear to ourselves, we say that the body is in one place and then it goes to another; because it moves, it is no longer in the first, but yet not in the second; were it in either it would be at rest. Where then is it? If we say that it is between both, this is to convey nothing at all, for were it between both, it would be in a place, and this presents the same difficulty. But movement means to be in this place and not to be in it, and thus to be in both alike; this is the continuity of space and time which first makes motion possible. Zeno, in the deduction made by him, brought both these points into forcible opposition. The discretion of space and time we also uphold, but there must also be granted to them the overstepping of limits, i.e. the exhibition of limits as not being, or as being divided periods of time, which are also not divided. In our ordinary ideas we find the same determinations as those on which the dialectic of Zeno rests; we arrive at saying, though unwillingly, that in one period two distances of space are traversed, but we do not say that the quicker comprehends two moments of time in one; for that we fix a definite space. But in order that the slower may lose its precedence, it must be said that it loses its advantage of a moment of time, and indirectly the moment of space.
Zeno makes limit, division, the moment of discretion in space and time, the only element which is enforced in the whole of his conclusions, and hence results the contradiction. The difficulty is to overcome thought, for what makes the difficulty is always thought alone, since it keeps apart the moments of an object which in their separation are really united. It brought about the Fall, for man ate of the tree of the knowledge of good and evil; but it also remedies these evils.
(c) The third form, according to Aristotle, is as follows:—Zeno says: “The flying arrow rests, and for the reason that what is in motion is always in the self-same Now and the self-same Here, in the indistinguishable;” it is here and here and here. It can be said of the arrow that it is always the same, for it is always in the same space and the same time; it does not get beyond its space, does not take in another, that is, a greater or smaller space. That, however, is what we call rest and not motion. In the Here and Now, the becoming “other” is abrogated, limitation indeed being established, but only as moment; since in the Here and Now as such, there is no difference, continuity is here made to prevail against the mere belief in diversity. Each place is a different place, and thus the same; true, objective difference does not come forth in these sensuous relations, but in the spiritual.
This is also apparent in mechanics; of two bodies the question as to which moves presents itself before us. It requires more than two places—three at least—to determine which of them moves. But it is correct to say this, that motion is plainly relative; whether in absolute space the eye, for instance, rests, or whether it moves, is all the same. Or, according to a proposition brought forward by Newton, if two bodies move round one another in a circle, it may be asked whether the one rests or both move. Newton tries to decide this by means of an external circumstance, the strain on the string. When I walk on a ship in a direction opposed to the motion of the ship, this is in relation to the ship, motion, and in relation to all else, rest.
In both the first proofs, continuity in progression has the predominance; there is no absolute limit, but an overstepping of all limits. Here the opposite is established; absolute limitation, the interruption of continuity, without however passing into something else; while discretion is presupposed, continuity is maintained. Aristotle says of this proof: “It arises from the fact that it is taken for granted that time consists of the Now; for if this is not conceded, the conclusions will not follow.”
(d) “The fourth proof,” Aristotle continues, “is derived from similar bodies which move in opposite directions in the space beside a similar body, and with equal velocity, one from one end of the space, the other from the middle. It necessarily results from this that half the time is equal to the double of it. The fallacy rests in this, that Zeno supposes that what is beside the moving body, and what is beside the body at rest, move through an equal distance in equal time with equal velocity, which, however, is untrue.”