James Adam, The Republic of Plato

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ὁμολογουμένως=“folgerechterweise” (Cohen Pl. Ideenl. u. d. Math. p. 29) refers to the agreement between premises, intermediate steps, and conclusion: cf. VII 533 C, where ὁμολογία is used in the same way. “With perfect unanimity” (D. and V.) is incorrect and pointless.

τοῖς ὁρωμένοις εἴδεσι κτλ. They use the ‘visible kinds,’ i.e. visible squares, visible diagonals, etc., but they are thinking about mathematical squares and diagonals etc. Cf. generally Euthyd. 290 B οἱ δ᾽ αὖ γεωμέτραι καὶ οἱ ἀστρονόμοι καὶ οἱ λογιστικοί: θηρευτικοὶ γάρ εἰσι καὶ οὗτοι: οὐ γὰρ ποιοῦσι τὰ διαγράμματα ἕκαστοι τούτων ἀλλὰ τὰ ὄντα ἀνευρίσκουσιν, and VII 527 A.

ἔοικε. Visible σχήματα are imperfect copies of ‘mathematical’ σχήματα: cf. VII 526 A and App. I to Book VII.

τοῦ τετραγώνου αὐτοῦ κτλ.: ‘for that with a view to which they are discoursing is the square itself and a diagonal itself, not this which they draw’ etc. αὐτοῦ (‘by itself,’ i.e. apart from its embodiment in perceivable squares) is ambiguous, and might (so far as language is concerned) refer either to the Idea of Square (cf. v 476 A ff.) or to the Mathematical Square (cf. VII 525 D, E notes), which—see App. I to Book VII—Plato holds to be distinct from the Idea. But the ambiguity is resolved as soon as we are shewn (in 511 C ff.) how to interpret διανοούμενοι and διανοίᾳ (511 A), and we then see that Plato is here speaking of the mathematical square. The singular τοῦ τετραγώνου is generic (cf. ὁ σοφιστής for the whole class of Sophists), for there are many ‘mathematical’ squares, diagonals etc. (VII 526 A note and App. I to Book VII). It is conceivably for this reason that Plato drops the article with διαμέτρου (‘a diagonal itself’), thereby also getting a more precise antithesis to ἀλλ᾽ οὐ ταύτης, or else (if this suggestion is hypercritical) διαμέτρου is also generic. Sidgwick is, I think, mistaken when he says (J. Ph. II p. 103) that the language of this passage “in no way supports the interpolation of intermediates (Aristotle's τὰ μεταξύ) between particulars and Ideas”: for διανοούμενοι involves διάνοια, and since διάνοια is intermediate between νοῦς and δόξα (511 D), we may reasonably suppose that its objects are likewise intermediate between the higher νοητά and δοξαστά. See App. I to Book VII.

πλάττουσιν: with reference to models of geometrical figures, orreries etc., all of which belong to CD, and may themselves have shadows and likenesses in AD.

ὡς εἰκόσιν αὖ χρώμενοι. See 510 B note The anacoluthon in αὐτὰ μὲν ταῦτα—τούτοις μὲν χρώμενοι is illustrated by Engelhardt Anac. Pl. Spec. III p. 8: cf. also VII 520 D.

ζητοῦντές τε. Instead of τε, I formerly read δέ (on slight MS authority), with Ast and Stallbaum; but the corruption of δέ to τε is exceedingly improbable here. The antithetical force of the clause ζητοῦντες—ἰδεῖν is weakened by the occurrence of the words ὡς εἰκόσιν αὖ in the μέν clause. If the objects in question are used as images, the further statement that the real object of investigation is their originals (αὐτὰ ἐκεῖνα) loses its antithetical force, and becomes a sort of adjunct. Hence τε following ζητοῦντες is more appropriate than αὐτὰ δὲ ἐκεῖνα ζητοῦντες ἰδεῖν, which would be the natural way of expressing an antithesis. Cf. Laws 927 B ὀξὺ μὲν ἀκούουσι βλέπουσί τε ὀξύ (where the order is the same as here), Phaedr. 266 C and other examples cited by Hoefer de part. Pl. pp. 17 f.