Plato, Republic, ed. John Burnet, 7.526

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[526] 526a
τί οὖν οἴει, ὦ Γλαύκων, εἴ τις ἔροιτο αὐτούς· ὦ θαυμάσιοι, περὶ ποίων ἀριθμῶν διαλέγεσθε, ἐν οἷς τὸ ἓν οἷον ὑμεῖς ἀξιοῦτέ ἐστιν, ἴσον τε ἕκαστον πᾶν παντὶ καὶ οὐδὲ σμικρὸν διαφέρον, μόριόν τε ἔχον ἐν ἑαυτῷ οὐδέν; τί ἂν οἴει αὐτοὺς ἀποκρίνασθαι;
τοῦτο ἔγωγε, ὅτι περὶ τούτων λέγουσιν ὧν διανοηθῆναι μόνον ἐγχωρεῖ, ἄλλως δʼ οὐδαμῶς μεταχειρίζεσθαι δυνατόν.
ὁρᾷς οὖν, ἦν δʼ ἐγώ, ὦ φίλε, ὅτι τῷ ὄντι ἀναγκαῖον 526b ἡμῖν κινδυνεύει εἶναι τὸ μάθημα, ἐπειδὴ φαίνεταί γε προσαναγκάζον αὐτῇ τῇ νοήσει χρῆσθαι τὴν ψυχὴν ἐπʼ αὐτὴν τὴν ἀλήθειαν;
καὶ μὲν δή, ἔφη, σφόδρα γε ποιεῖ αὐτό.
τί δέ; τόδε ἤδη ἐπεσκέψω, ὡς οἵ τε φύσει λογιστικοὶ εἰς πάντα τὰ μαθήματα ὡς ἔπος εἰπεῖν ὀξεῖς φύονται, οἵ τε βραδεῖς, ἂν ἐν τούτῳ παιδευθῶσιν καὶ γυμνάσωνται, κἂν μηδὲν ἄλλο ὠφεληθῶσιν, ὅμως εἴς γε τὸ ὀξύτεροι αὐτοὶ αὑτῶν γίγνεσθαι πάντες ἐπιδιδόασιν;
ἔστιν, ἔφη, οὕτω. 526c
καὶ μήν, ὡς ἐγᾦμαι, ἅ γε μείζω πόνον παρέχει μανθάνοντι καὶ μελετῶντι, οὐκ ἂν ῥᾳδίως οὐδὲ πολλὰ ἂν εὕροις ὡς τοῦτο.
οὐ γὰρ οὖν.
πάντων δὴ ἕνεκα τούτων οὐκ ἀφετέον τὸ μάθημα, ἀλλʼ οἱ ἄριστοι τὰς φύσεις παιδευτέοι ἐν αὐτῷ.
σύμφημι, ἦ δʼ ὅς.
τοῦτο μὲν τοίνυν, εἶπον, ἓν ἡμῖν κείσθω· δεύτερον δὲ τὸ ἐχόμενον τούτου σκεψώμεθα ἆρά τι προσήκει ἡμῖν.
τὸ ποῖον; ἢ γεωμετρίαν, ἔφη, λέγεις;
αὐτὸ τοῦτο, ἦν δʼ ἐγώ. 526d
ὅσον μέν, ἔφη, πρὸς τὰ πολεμικὰ αὐτοῦ τείνει, δῆλον ὅτι προσήκει· πρὸς γὰρ τὰς στρατοπεδεύσεις καὶ καταλήψεις χωρίων καὶ συναγωγὰς καὶ ἐκτάσεις στρατιᾶς καὶ ὅσα δὴ ἄλλα σχηματίζουσι τὰ στρατόπεδα ἐν αὐταῖς τε ταῖς μάχαις καὶ πορείαις διαφέροι ἂν αὐτὸς αὑτοῦ γεωμετρικός τε καὶ μὴ ὤν.
ἀλλʼ οὖν δή, εἶπον, πρὸς μὲν τὰ τοιαῦτα καὶ βραχύ τι ἂν ἐξαρκοῖ γεωμετρίας τε καὶ λογισμῶν μόριον· τὸ δὲ πολὺ αὐτῆς καὶ πορρωτέρω προϊὸν σκοπεῖσθαι δεῖ εἴ τι πρὸς ἐκεῖνο 526e τείνει, πρὸς τὸ ποιεῖν κατιδεῖν ῥᾷον τὴν τοῦ ἀγαθοῦ ἰδέαν. τείνει δέ, φαμέν, πάντα αὐτόσε, ὅσα ἀναγκάζει ψυχὴν εἰς ἐκεῖνον τὸν τόπον μεταστρέφεσθαι ἐν ᾧ ἐστι τὸ εὐδαιμονέστατον τοῦ ὄντος, ὃ δεῖ αὐτὴν παντὶ τρόπῳ ἰδεῖν.
ὀρθῶς, ἔφη, λέγεις.
οὐκοῦν εἰ μὲν οὐσίαν ἀναγκάζει θεάσασθαι, προσήκει, εἰ δὲ γένεσιν, οὐ προσήκει.
φαμέν γε δή.

Republic. Plato. ed. John Burnet. Oxford. 1905.

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Πολιτεία (Greek) (Platonis Opera Tomus IV: Tetralogia VIII. Burnet, John, editor. Oxford: Clarendon Press, 1905.)

Translations (1)

Republic Perseus (Plato in Twelve Volumes, Vol. 5-6. Shorey, Paul, translator. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1935-37 (printing).) — 7.526 focus

[526] 526a “Suppose now, Glaucon, someone were to ask them, My good friends, what numbers¹ are these you are talking about, in which the one is such as you postulate, each unity equal to every other without the slightest difference and admitting no division into parts? What do you think would be their answer?” “This, I think—that they are speaking of units which can only be conceived by thought, and which it is not possible to deal with in any other way.” “You see, then, my friend,” said I, “that this branch of study really seems to be 526b indispensable for us, since it plainly compels the soul to employ pure thought with a view to truth itself.” “It most emphatically does.” “Again, have you ever noticed this, that natural reckoners are by nature quick in virtually all their studies? And the slow, if they are trained and drilled in this, even if no other benefit results, all improve and become quicker than they were²?” “It is so,” he said. 526c “And, further, as I believe, studies that demand more toil in the learning and practice than this we shall not discover easily nor find many of them.³” “You will not, in fact.” “Then, for all these reasons, we must not neglect this study, but must use it in the education of the best endowed natures.” “I agree,” he said.
“Assuming this one point to be established,” I said, “let us in the second place consider whether the study that comes next⁴ is suited to our purpose.” “What is that? Do you mean geometry,” he said. “Precisely that,” said I. “So much of it,” he said, 526d “as applies to the conduct of war⁵ is obviously suitable. For in dealing with encampments and the occupation of strong places and the bringing of troops into column and line and all the other formations of an army in actual battle and on the march, an officer who had studied geometry would be a very different person from what he would be if he had not.” “But still,” I said, “for such purposes a slight modicum⁶ of geometry and calculation would suffice. What we have to consider is 526e whether the greater and more advanced part of it tends to facilitate the apprehension of the idea of good.⁷ That tendency, we affirm, is to be found in all studies that force the soul to turn its vision round to the region where dwells the most blessed part of reality,⁸ which it is imperative that it should behold.” “You are right,” he said. “Then if it compels the soul to contemplate essence, it is suitable; if genesis,⁹ it is not.” “So we affirm.¹⁰”

Commentary

Commentary on Plato, Republic (Plato. The Republic, Vol 1-2. Adam, James, editor. Cambridge: Cambridge University Press, 1902-1903.) focus

περὶ ποίων κτλ. On the derisive ποῖος see 522 D n. Mathematical units are in every case ( ἕκαστον) equal each to each ( πᾶν παντί), and destitute of parts; whereas sensible units (e.g. one horse, one cow etc.) are not equal to each other, and are divisible. In πᾶν παντί Plato copies the formal language of mathematics: cf. ἑκατέραν ἑκατέρᾳ and the like in Euclid passim. For the sense see Phil. 56 C ff., where these two kinds of number are made the basis of a distinction between philosophical or scientific and popular or unscientific ἀριθμητική. It should be carefully noted that a plurality of mathematical units is expressly recognised both here ( ἴσον τε ἕκαστον πᾶν παντί κτλ.) and in Phil. l. c. ( μονάδα μονάδος ἑκάστης τῶν μυρίων μηδεμίαν ἄλλην ἄλλης διαφέρουσαν). This entirely confirms what Aristotle tells us, viz. that Plato placed μαφηματικά between αἰσθητά and εἴδη, τῷ τὰ μὲν πόλλ’ ἄττα ὅμοια εἶναι, τὸ δὲ εἶδος αὐτὸ ἓν ἕκαστον μόνον ( Met. A 6. 987^{b} 14 ff.). There are therefore three kinds of μονάδες in Plato's scheme—the Ideal μονάς, of which only one exists, the Mathematical and the Sensible, of each of which there are many. See on VI 510 D and App. I, where I have quoted further evidence on this subject, and endeavoured to explain the philosophical truth which is contained in the Platonic doctrine of mathematical numbers, magnitudes etc. as intermediates between the Ideas and sensibles.

ὧν κτλ. ὧν is for περὶ ὧν rather than <*> (as J. and C. hold): cf. VI 510 D οὐ περὶ τούτων διανοούμενοι, and (for the grammatical construction) III 402 A note διανοηθῆναι should be understood in the technical sense of VI 511 E.

τῷ ὄντι ἀναγκαῖον . Perhaps with a play on προσαναγκάζον (J. and C.): see on τῷ ὄντι VI 511 B.

ὀξεῖς κτλ. Plato was very emphatic on this point: see Laws 747 B and 819 C φύονται was restored by Schneider from the best MSS. Earlier editions read φαίνονται on inferior authority.

ἂν ‐‐ γυμνάσωνται κτλ. Even Isocrates admits this, although his self-styled ‘Philosophy’ was something very different from Plato's: see Antid. 265—266, especially γυμνασίαν μέντοι τῆς ψυχῆς καὶ παρασκευὴν φιλοσοφίας καλῶ τὴν διατριβὴν τὴν τοιαύτην (mathematical studies).

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.

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.

καὶ πορείαις . “Scriptum vellem καὶ ἐν πορείαις” (Stallbaum). The idiom is common enough: see KühnerBlass Gr. Gr. II 1, p. 548.

τὸ εὐδαιμονέστατον τοῦ ὄντος is cited by Stumpf (l.c. p. 95 note 3) in support of his identification of the Idea of Good with God: see on VI 505 A.

γένεσιν . 519 A note

Plato Republic 7.526

ed. John Burnet

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