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αὐτῶν τῶν ἀριθμῶν: ‘numbers themselves,’ e.g. 1, 2, 3, 4 etc., in other words individual mathematical numbers and nothing more. αὐτῶν means ‘by themselves,’ ‘alone,’ i.e. with nothing αἰσθητόν about them, such as is present in the ὁρατὰ ἢ ἁπτὰ σώματα ἔχοντας ἀριθμούς (=Aristotle's αἰσθητικοὶ or σωματικοὶ ἀριθμοί: v. Bonitz Ind. Arist. s. v. ἀριθμός), e.g. one man, two men etc. These mathematical numbers are not Ideas, but (like τὰ μαθηματικά generally) a half-way house between sensibles and Ideas, and for this reason valuable as a προπαιδεία to Dialectic: cf. 526 A note and see on VI 510 D and App. I. For αὐτῶν in this sense cf. αὐτὸ τὸ ἕν in E, αὐτῇ τῇ νοήσει 526 B and ἀριθμῶν αὐτῶν ἀλλ᾽ οὐ σώματα ἐχόντων [Epin.] 990 C. δεινούς. The word δύο, which was originally written after δεινούς (see cr. n.) in Α and Π, is probably due to a marginal adscript on the words ἐάν τις αὐτὸ τὸ ἓν ἐπιχειρῇ—τέμνειν. Burnet neatly conjectures δεινοὺς αὖ, but αὖ is inappropriate here.
The Republic of Plato. James Adam. 1902. Cambridge. Cambridge University Press.
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