[792]. Arist. Met. Μ, 6. 1080 b 18 sqq., 1083 b 8 sqq.; de Caelo, Γ, 1. 300 a 16 (R. P. 76 a).
[793]. Zeller, p. 381.
[794]. We learn from Plato, Theaet. 148 b 1, that Theaitetos called surds, what Euclid calls δυνάμει σύμμετρα, by the name of δυνάμεις, while rational square roots were called μήκη. Now in Tim. 31 c 4 we find a division of numbers into ὄγκοι and δυνάμεις, which seem to mean rational and irrational quantities. Cf. also the use of ὄγκοι in Parm. 164 d. Zeno in his fourth argument about motion, which, we shall see ([§ 163]), was directed against the Pythagoreans, used ὄγκοι for points. Aetios, i. 3, 19 (R. P. 76 b), says that Ekphantos of Syracuse was the first of the Pythagoreans to say that their units were corporeal. Probably, however, “Ekphantos” was a personage in a dialogue of Herakleides (Tannery, Arch. xi. pp. 263 sqq.), and Herakleides called the monads ἄναρμοι ὄγκοι (Galen, Hist. Phil. 18; Dox. p. 610).
[795]. Zeller, p. 382.
[796]. Arist. Met. Α, 8. 990 a 22 (R. P. 81 e). I read and interpret thus: “For, seeing that, according to them, Opinion and Opportunity are in a given part of the world, and a little above or below them Injustice and Separation and Mixture,—in proof of which they allege that each of these is a number,—and seeing that it is also the case (reading συμβαίνῃ with Bonitz) that there is already in that part of the world a number of composite magnitudes (i.e. composed of the Limit and the Unlimited), because those affections (of number) are attached to their respective regions;—(seeing that they hold these two things), the question arises whether the number which we are to understand each of these things (Opinion, etc.) to be is the same as the number in the world (i.e. the cosmological number) or a different one.” I cannot doubt that these are the extended numbers which are composed (συνίσταται) of the elements of number, the limited and the unlimited, or, as Aristotle here says, the “affections of number,” the odd and the even. Zeller’s view that “celestial bodies” are meant comes near this, but the application is too narrow. Nor is it the number (πλῆθος) of those bodies that is in question, but their magnitude (μέγεθος). For other views of the passage, see Zeller, p. 391, n. 1.
[797]. Zeller, p. 404.
[798]. Ibid. pp. 467 sqq.
[799]. All this has been put in its true light by the publication of the extract from Menon’s Ἰατρικά, on which see p. 322, [n. 742].
[800]. In Aet. ii. 6, 5 (R. P. 80) the theory is ascribed to Pythagoras, which is an anachronism, as the mention of “elements” shows it must be later than Empedokles. In his extract from the same source, Achilles says οἱ Πυθαγόρειοι, which doubtless represents Theophrastos better. There is a fragment of “Philolaos” bearing on the subject (R. P. 79), where the regular solids must be meant by τὰ ἐν τᾷ σφαίρᾳ σώματα.