From all this it seems impossible to draw any other conclusion than that the “one” against which Zeno argued was the “one” of which a number constitute a “many,” and that is just the Pythagorean unit. [Sidenote: Space.] 162. Aristotle refers to an argument which seems to be directed against the Pythagorean doctrine of space,[888] and Simplicius quotes it in this form:[889] If there is space, it will be in something; for all that is is in something, and what is in something is in space. So space will be in space, and this goes on _ad infinitum_, therefore there is no space. R. P. 135. Footnote 888: Arist. _Phys._ Δ, 1. 209 a 23; 3. 210 b 22 (R. P. 135 a). Footnote 889: Simpl. _Phys._ p. 562, 3 (R. P. 135). The version of Eudemos is given in Simpl. _Phys._ p. 563, 26, ἀξιοῖ γὰρ πᾶν τὸ ὂν ποῦ εἷναι· εἱ δὲ ὁ τόπος τῶν ὄντων, ποῦ ἂν εἴη· οὐκοῦν ἐν ἄλλῳ τόπῳ κἀκεῖνος δὴ ἐν ἄλλῳ καὶ οὕτως εἰς τὸ πρόσω. What Zeno is really arguing against here is the attempt to distinguish space from the body that occupies it. If we insist that body must be _in_ space, then we must go on to ask what space itself is in. This is a “reinforcement” of the Parmenidean denial of the void. Possibly the argument that everything must be “in” something, or must have something beyond it, had been used against the Parmenidean theory of a finite sphere with nothing outside it. [Sidenote: Motion.] 163. Zeno’s arguments on the subject of motion have been preserved by Aristotle himself. The system of Parmenides made all motion impossible, and his successors had been driven to abandon the monistic hypothesis in order to avoid this very consequence. Zeno does not bring any fresh proofs of the impossibility of motion; all he does is to show that a pluralist theory, such as the Pythagorean, is just as unable to explain it as was that of Parmenides. Looked at in this way, Zeno’s arguments are no mere quibbles, but mark a great advance in the conception of quantity. They are as follows:— (1) You cannot get to the end of a race-course.[890] You cannot traverse an infinite number of points in a finite time. You must traverse the half of any given distance before you traverse the whole, and the half of that again before you can traverse it. This goes on _ad infinitum_, so that there are an infinite number of points in any given space, and you cannot touch an infinite number one by one in a finite time.[891] (2) Achilles will never overtake the tortoise. He must first reach the place from which the tortoise started. By that time the tortoise will have got some way ahead. Achilles must then make up that, and again the tortoise will be ahead. He is always coming nearer, but he never makes up to it.[892] Footnote 890: Arist. _Top._ Θ, 8. 160 b 8, Ζήνωνος (λόγος), ὅτι οὐκ ἐνδέχεται κινεῖσθαι οὐδὲ τὸ στάδιον διελθεῖν. Footnote 891: Arist. _Phys._ Ζ, 9. 239 b 11 (R. P. 136). Cf. Ζ, 2. 233 a 11; a 21 (R. P. 136 a). Footnote 892: