James Adam, The Republic of Plato

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καίτοι -- ἀρχῆς: ‘although they are intelligibles with a first principle.’ The mathematician does not ascend to an ἀρχή, and therefore does not exercise— for ἴσχειν in its original half-inchoative sense cf. IX 585 B and Kühner-Blass Gr. Gr. I 2, p. 434 note—νοῦς on his subject, but nevertheless his subject is νοητόν (as we have been told before 510 B, 511 A, C) and has an ἀρχή, viz. his ὑποθέσεις (αἷς αἱ ὑποθέσεις ἀρχαί above). καίτοι is not found elsewhere in Plato for καίπερ with a participle (Hoefer de part. Pl. p. 28) but occurs in Simonides ap. Prot. 339 C, in Axioch. 364 B and Lysias 31. 34. To write καίπερ (with Kugler de part. τοι etc. p. 18) would be rash. For other views on this difficult clause see App. XI.

καλεῖν μοι δοκεῖς. See 510 D note

ὡς -- οὖσαν . διάνοια is the most general word for a state (ἕξις) of mind or mode of thought in Greek; and the limitation here introduced is entirely Plato's own. Plato apparently attempts to fortify his innovation by etymology, hinting that the word διάνοια is by derivation that which is between (διὰ μέσου) νοῦς and δόξα. So also J. and C. Cf. εἰκασία (with allusion to εἰκόνες) in E. On δόξης see 510 A note

νόησιν is used in its strict sense of νοῦς in actual exercise, not merely the faculty of νοῦς: cf. 508 E note The exercise of νοῦς is correctly spoken of as a πάθημα ἐν τῇ ψυχῇ γιγνόμενον, but the faculty itself could hardly be thus described.