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δευτέραν αὔξην κτλ. It is better (with Schneider) to translate αὔξη by ‘increase’ than by ‘dimension’; for αὔξη always implies something increased, and in the phrases δευτέρα αὔξη etc. this ‘something’ is the point. Among the Pythagoreans, who probably originated these expressions, the line was regarded as an αὔξη of the point, the plane of the line, the solid of the plane. See App. II.

κύβων αὔξην: ‘cubic increase,’ i.e. the increase which belongs to, or results in, cubes, with perhaps also a play on a different sense of κύβων αὔξην, ‘how to increase cubes,’ as in the famous ‘Delian problem’ of the διπλασιασμὸς κύβου (so also Tannery l. c. X p. 525). See on 527 D. But as cubes are not the only solid bodies, Plato adds τὸ βάθους μετέχον. By Aristotle's time the name στερεομετρία had been invented to designate the science as a whole (An. Post. II 13. 78^{b} 38).

ταῦτά γε -- ηὑρῆσθαι. Plato does not of course mean to say that the study of Stereometry had not yet been invented, for the subject had already in one form or another engaged the attention of the Py thagoreans, Anaxagoras and Democritus (Blass l.c. p. 21, Tannery l.c. X p. 524), not to speak of Hippocrates of Chios, who had concerned himself in the fifth century B.C. with the question of the duplication of the cube (Allman Gk Geometry etc. pp. 84 ff.). He only means that its problems had not yet been ‘discovered’ (ηὑρῆσθαι as in Pythagoras' ηὕρηκα) or solved. When and by whom the ‘Delian problem’ in particular was definitively solved to the satisfaction of the Academy, is not quite clear. The tradition which ascribes a solution of it to Plato himself is beset with grave difficulties, as Blass (l.c. pp. 21—30) and others have pointed out (see especially Cantor l.c. pp. 194—202 and Sturm Das Delische Problem pp. 49 ff.). It is however universally allowed that the principle involved —the finding of “two mean proportionals between one straight line and another twice as long” (Gow Gk Math. p. 169) —was first stated by Hippocrates of Chios and well known to Plato, at all events when he wrote the Timaeus (32 A ff.: see also Häbler Ueber zwei Stellen in Platons Timaeus etc. pp. 1—17). We may perhaps infer from οὔπω ηὑρῆσθαι that Plato did not think a final solution of this as of other stereometrical problems had yet been reached: there is at all events nothing in the Republic to justify the curious statement of Diogenes Laertius that (Ἀρχύτας) πρῶτος κύβου διπλασιας μὸν εὗρεν, ὥς φησι Πλάτων ἐν πολιτείᾳ (VIII 83), although it is probably true that Archytas was the first to offer a solution of the famous difficulty (see Sturm l.c. pp. 22—32). In D. L. l.c. Cobet reads πρῶτος κύβον εὗρεν κτλ., whether on his own responsibility, or on MS authority, he does not tell us. See also on 527 D, 528 C.

ὅτι τε κτλ. In Laws 819 E ff. Plato reproaches the Greeks for their ignorance of and indifference to stereometrical questions.

ἐντίμως ἔχει: ‘holds in honour,’ as in VIII 548 A. The expression usually means ‘is honoured’ (Xen. An. II 1. 7): hence ἄγει for ἔχει is proposed by Herwerden, who compares 528 C, 538 E. But the error is not an easy one in such a MS as A, and it is safer to keep ἔχει and take the phrase as=ἐν τιμῇ ἔχει (cf. ἐν ἀτιμίῃ ἔχει Hdt. III 3, ἐν εὐνοίᾳ ἔχειν [Dem.] 284. 11, and Jebb on Soph. Ant. 639) as ἄγειν ἐντίμως=ἄγειν ἐν τιμῇ (538 E).

ὡς νῦν ἔχει belongs no doubt to the following clauses (IV 419 A note): but see also on 528 C.

μεγαλοφρονούμενοι is condemned as un-Attic by Cobet (V. L.^{2} pp. 232, 531); but μεγαλόφρων, μεγαλοφροσύνη are Attic, and Xenophon uses μεγαλοφρονεῖν. μεγαλαυχούμενοι (Cobet's emendation) would mean ‘vaunting’: cf. III 395 D.

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  • Commentary references from this page (2):
    • Sophocles, Antigone, 639
    • Xenophon, Anabasis, 2.1.7
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