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αὕτη ἐπιστήμη κτλ.: ‘the nature of this science is in precise contradiction to’ etc. πᾶν τοὐναντίον is adverbial, and ἔχει intransitive. In what follows Plato is not (as Stallbaum strangely supposes) censuring contemporary geometricians, “sed vitium quoddam ipsius scientiae” (Schneider, who refers to VI 510 B—511 A), for even those ὅσοι καὶ σμικρὰ γεωμετρίας ἔμπειροι agree with him. It is however probable on other grounds that Plato himself was afraid lest his pupils should depend too much on geometrical figures and instruments; and Plutarch tells us that he reproved Eudoxus and others for this very fault: ἀπόλλυσθαι γὰρ οὕτω καὶ διαφθείρεσθαι τὸ γεωμετρίας ἀγαθὸν αὖθις ἐπὶ τὰ αἰσθητὰ παλινδρομούσης, καὶ μὴ φερομένης ἄνω μηδὲ ἀντιλαμβανομένης τῶν ἀϊδίων καὶ ἀσωμάτων εἰκόνων, πρὸς οἷσπερ ὢν θεὸς ἀεὶ θεός ἐστι (Conv. Disp. VIII 2. 718 F: cf. also Vit. Marc. 14. 5—7. The anecdote refers in particular to stereometrical instruments, but its moral applies equally to the study of plane geometry). See Rothlauf l.c. pp. 66 ff. and Gow Gk Math. p. 181.

ἀναγκαίως ‘in beggarly fashion’ (“nothgedrungenen Weise” Prantl). They are ‘hard up’ for words to express their meaning, and must consequently use language which does not: cf. VI 510 D. J. and C.'s explanation “with merely practical needs in view” is quite wrong; nor does the word appear to me to contain “a facetious allusion to geometrical necessity” (in the sense of V 458 D), as they assert. Madvig's ἀκάκως for ἀναγκαίως is a singularly unhappy suggestion.

παρατείνειν = ‘to apply,’ e.g. a parallelogram to a given line: cf. Men. 87 A (παρὰ τὴν δοθεῖσανγραμμὴν παρατείναντα κτλ.) with E. S. Thompson's note. Blass takes the same view (de Pl. math. p. 19). Euclid's expression for this is παραβάλλειν (X 18 ff. and passim: cf. Cantor l.c. p. 145). The ordinary translation ‘produce’ (as e.g. a line) has no authority, and Euclid's word for ‘producing’ a line is ἐκβάλλειν or προσεκβάλλειν (I 5, 17 and passim. So also in Aristotle: see the Ind. Arist. s.v.).

προστιθέναι: ‘to add,’ sc. one μέγεθος to another in any geometrical construction: cf. Nicom. Introd. Ar. II 12. 2. Euclid uses συντιθέναι, σύγκειμαι (X 16 and passim) of such an operation.

φθεγγόμενοι is said “non sine eleganti quadam irrisione” (Stallbaum): cf. VI 505 C note

τὸ δέ: ‘whereas’: cf. I 340 D note

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