James Adam, The Republic of Plato

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καὶ ὁ Γλαύκων κτλ. “Glauco exclaimed, very comically, ‘Save us all, what an amazing transcendence!’” It is Glauco's προθυμία which is γελοῖον: see 506 D. ὑπερβολῆς is not ‘exaggeration’ (Jowett), but refers to ὑπερέχοντος: cf. ἀμήχανον κάλλος λέγεις 509 A. A ὑπερβολή which transcends existence may well be called δαιμονία (‘supernatural,’ ‘miraculous’).

εἰ μή τι, ἀλλὰ κτλ. Stephanus proposed ἄλλο for ἀλλά (as in 501 E), but cf. Men. 86 E εἰ μή τι οὖν, ἀλλὰ σμικρόν γέ μοι τῆς ἀρχῆς χάλασον.

509C - 511E Socrates, at Glauco's request, now proceeds to expound the similitude more fully. Let us take a line, and divide it into two unequal parts, to represent the objects of Sight and the objects of Thought respectively. If we further subdivide each part in the ratio of the original sections, we shall have four segments, representing, in order of clearness, (1) Images and the like, (2) so-called real things, (3) the objects of that intellectual method which descends from assumptions to a conclusion, using sensible objects as images or illustrations, (4) the objects of that intellectual method, which ascends from assumptions to an unassumed first principle, without making use of any sensible illustrations whatsoever, and thereafter descends to a conclusion. The third section represents the subjects investigated by the so-called ‘Arts’ or mathematical sciences; the fourth is the sphere of Dialectic. The corresponding mental states are called by Socrates εἰκασία, πίστις, διάνοια, and νόησις. Each of these is clear or sure exactly in proportion as its objects are true.