Again — If things in themselves were many, they would be both finite and infinite in number. Finite, because they are as many as they are, neither more nor less: and every number is a finite number. Infinite, because being essentially separate, discontinuous, units, each must be kept apart from the rest by an intervening unit; and this again by something else intervening. Suppose a multitude A, B, C, D, &c. A and B would be continuous unless they were kept apart by some intervening unit Z. But A and Z would then be continuous unless they were kept apart by something else — Y: and so on ad infinitum: otherwise the essential discontinuousness could not be maintained.[18]
[18] See the argument cited by Simplikius in the words of the Zenonian treatise, in Preller, Hist. Philos. Græc. ex font. context. p. 101, sect. 156.
By these two arguments,[19] drawn from the hypothesis which affirmed perpetual divisibility and denied any Continuum, Zeno showed that such Entia multa discontinua would have contradictory attributes: they would be both infinitely great and infinitely small — they would be both finite and infinite in number. This he advanced as a reductio ad absurdum against the hypothesis.
[19] Simplikius ad Aristot. Physic. f. 30. καὶ οὔτω μὲν τὸ κατὰ τὸ πλῆθος ἄπειρον ἐκ τῆς διχοτομίας ἔδειξε, τὸ δὲ κατὰ τὸ μέγεθος πρότερον κατὰ τὴν αὐτὴν ἐπιχείρησιν. Compare Zeller, Phil. d. Griech. i. p. 427.
Each thing must exist in its own place — Grain of millet not sonorous.
Again — If existing things be many and discontinuous, each of these must exist in a place of its own. Nothing can exist except in some place. But the place is itself an existing something: each place must therefore have a place of its own to exist in: the second place must have a third place to exist in and so forth ad infinitum.[20] We have here a farther reductio ad impossibile of the original hypothesis: for that hypothesis denies the continuity of space, and represents space as a multitude of discontinuous portions or places.
[20] Aristotel. Physic. iv. 1, p. 209, a. 22; iv. 3, p. 210, b. 23.
Aristotle here observes that the Zenonian argument respecting place is easy to be refuted; and he proceeds to give the refutation. But his refutation is altogether unsatisfactory. Those who despise these Zenonian arguments as sophisms, ought to look at the way in which they were answered, at or near the time.
Eudêmus ap. Simplik. ad Aristot. Physic. f. 131. ἄξιον γὰρ πᾶν τῶν ὄντων ποῦ εἶναι· εἰ δὲ ὁ τόπος τῶν ὄντων, ποῦ ἂν εἴη;
Another argument of Zeno is to the following effect:—“Does a grain of millet, when dropped upon the floor, make sound? No. — Does a bushel of millet make sound under the same circumstances? Yes. — Is there not a determinate proportion between the bushel and the grain? There is. — There must therefore be the same proportion between the sonorousness of the two. If one grain be not sonorous, neither can ten thousand grains be so.”[21]