James Adam, The Republic of Plato

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ἅ γε μείζω κτλ. is an important principle with Plato, who does not believe in any royal road to learning: cf. 530 C and VI 503 E. In antiquity, while algebra was still unknown, ἀριθμητική must have taxed the powers of thought far more than now, and been, from the Platonic point of view, all the more valuable on that account as an educative discipline. The treatment of numbers by Euclid Books VII—X will illustrate Plato's observation: see Gow Gk Math. pp. 74—85, with De Morgan's remarks there quoted.

ὡς τοῦτο . ὡς=‘quam’ instead of ἤ is found sporadically in Greek literature after comparatives: see my note on Ap. 30 B, 36 D. To say that in all such cases the comparative is equivalent to οὕτω with the positive is only to shelve the difficulty; and it is better to recognise the usage as exceptional than summarily to dismiss it as a barbarism (with Thompson on Gorg. 492 E). J. and C. after οὐδὲ πολλά supply ἃ πόνον οὕτω μέγαν παρέχεται, but the ellipse is too difficult, especially as οὐδὲ πολλά is only a kind of afterthought to or elaboration of οὐ ῥᾳδίως.

526C - 527C Next in order comes Plane Geometry. On its practical uses we need not dilate; the important question is whether it tends to turn the soul towards Being. A mere tiro in Geometry knows that it is not a practical art, in spite of such terms as ‘squaring’ etc., which the poverty of language compels it to employ. The object of geometrical knowledge is ever-existent Being. For this reason we shall prescribe the study of Geometry, a subject which is moreover practically useful and an excellent educational propaedeutic.

ff. On the subject of this section consult Blass and Cantor referred to on 524 D, Rothlauf l.c. pp. 50—69, and App. II to this book. The great importance attached by Plato and his school to geometry and kindred studies is attested from many sources: see for example Philoponus in Arist. de an. I 3 (Comment. in Arist. p. 117. 26 ὁ Πλάτων οὗ καὶ πρὸ τῆς διατριβῆς ἐπεγέγραπτο Ἀγεωμέτρητος μὴ εἰσίτω, Tzetzes Chil. VIII 973 μηδεὶς ἀγεωμέτρητος εἰσίτω μου τὴν στέγην, Proclus in Euclid. pp. 29 f. Friedlein Πλάτων καθαρτικὴν τῆς ψυχῆς καὶ ἀναγωγὸν τὴν μαθηματικὴν εἶναι σαφῶς ἀποφαίνεται, τὴν ἀχλὺν ἀφαιροῦσαν τοῦ νοεροῦ τῆς διανοίας φωτὸς κτλ., and D. L. IV 10 πρὸς δὲ τὸν μήτε μουσικὴν μήτε γεωμετρίαν μήτε ἀστρονομίαν μεμαθηκότα, βουλόμενον δὲ παρ᾽ αὐτὸν (Ξενοκράτη) φοιτᾶν: Πορεύου, ἔφη: λαβὰς γὰρ οὐκ ἔχεις φιλοσοφίας. Among Plato's companions or pupils in the Academy, Eudoxus and Menaechmus rendered the most conspicuous services to mathematical science (see Allman Greek Geometry from Thales to Euclid pp. 129—179), and Euclid himself, according to Proclus (in Euclid. p. 68), was τῇ προαιρέσει Πλατωνικὸς καὶ τῇ φιλοσοφία ταύτῃ οἰκεῖος. That ὁ θεὸς ἀεὶ γεωμετρεῖ was a characteristic and profound saying of Plato's (Plut. Conv. Disp. VIII 2. 718 C ff.), on the meaning of which see App. I.

τὸ ἐχόμενον τούτου. If γεωμετρία i.e. ἡ τοῦ ἐπιπέδου (plane surfaces) πραγματεία (528 D) concerns itself with δευτέρα αὔξη, and Stereometry with τρίτη αὔξη, we may infer that ἀριθμητική deals with the πρώτη αὔξη, i.e. presumably the line, which, according to the Pythagoreans, is a collection of points (cf. Laws 894 A and Rothlauf l.c. p. 51). And in point of fact the line represented number among the Pythagoreans exactly as the point is the geometrical symbol for the unit: cf. IX 587 D note Hence ἐχόμενον τούτου: we take the δευτέρα αὔξη after the first. See also App. II to this Book, and App. I to Book VIII Part I § 2.

ἢ γεωμετρίαν κτλ. The sequence —Geometry after ἀριθμητική—was probably a usual one with teachers, even in Plato's time: see Grasberger Erziehung u. Unterricht II p. 340 and cf. App. II.

ὅσον μὲν κτλ. is exactly the attitude of the historical Socrates, as Krohn (Pl. St. p. 376) and others have pointed out, comparing Xen. Mem. IV 7. 2 ff. Practical necessities of this kind probably originated the science (Gow Gk Math. pp. 134 ff.) and gave it its name γεωμετρία. The name μαθήματα (or μαθηματικά) in the special sense of Mathematics owes its origin, no doubt, to the position occupied by mathematical studies in Plato's μαθήματα: but the usage itself is not found till Aristotle (Rothlauf l.c. p. 18), although it is clear from [Epin.] 990 D, that some Platonists resented the γελοῖον ὄνομα γεωμετρίαν. Glauco represents the practical point of view throughout: cf. 527 D.