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ὅ τι νόησις -- εἰκασίαν. That is to say, in the Simile of the Line (see Fig. i on p. 65), (1) CB : AC :: EB : DC and (2) CB : AC :: CE : AD. We have already seen that CE : EB :: AD : DC (VI 509 D note); therefore componendo CE + EB : EB :: AD + DC : DC i.e. CB : EB :: AC : DC ; therefore alternando CB : AC :: EB : DC. This proves (1), and (2) is proved as follows. Since CE : EB :: AD : DC, therefore invertendo EB : CE :: DC : AD; hence componendo EB + CE : CE :: DC + AD : AD i.e. CB : CE :: AC : AD; therefore alternando CB : AC :: CE : AD. I owe this proof to the kindness of a mathematical friend. τὴν δ᾽ ἐφ᾽ οἷς κτλ. Liebhold (who also conjectured καὶ ἔτι νόησις for καὶ ὅ τι νόησις) makes the extraordinary suggestion τὴν δ᾽ ἐφ᾽ οἶς ταῦτ᾽ ἂν διέχῃ ἀναλογίαν καὶ διαίρεσιν ἑκατέρου (Philol. 1876 p. 372). The text is of course quite sound and=τὴν δὲ <τούτων> ἐφ᾽ οἷς ταῦτά <ἐστιν> ἀναλογίαν κτλ.: cf. VI 511 E. I cannot agree with Shorey when he says (Idea of Good etc. p. 235) that Plato “avoids drawing out the proportion εἴδη: objects of διάνοια=σκευαστά etc.: εἰκόνες, because he is aware that the second member is a blank and the fourth is largely fantastic.” Both of these assertions are in my opinion quite wrong, and if they were true, Plato would have refrained from drawing out the proportions between the faculties themselves for exactly the same reasons. See App. I. As it is, we should take Plato at his word. He may well decline to enter on the tedious and unprofitable task of expounding and illustrating in detail the proportions which may be conjectured to obtain between the different objects of our intellectual powers. It would for example lead to no useful result if we tried to establish a proportion between a particular ε<*>δος, one of the five μαθήματα, a particular object of πίστις, and a particular object of εἰκασία. Such attempts would certainly involve us in an endless amount of talk, and would hardly result in anything but a series of barren and pedantic formulae and subdivisions. ἢ ὅσων. See cr. n. ὅσων is read by a large majority of MSS, and the confusion of ο and ω is common: see Introd. § 5. The construction (as Schneider points out) is ἢ ὅσων λόγων οἱ παρεληλυθότες λόγοι ἡμᾶς ἐνέπλησαν: cf. (with Schneider) παρὰ δόξαν τοῖς νῦν δοκουμένοις VI 490 A. Madvig's ὅσοι has little probability, although it avoids a certain awkwardness.
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