ἄναγε ‐‐ εἰς τοὐπίσω
: ‘fall back then’: cf.
Ar.
Birds 383
ἄναγ’ ἐπὶ σκέλος, with Blaydes' note. The metaphor is not naval (as Ast and Stallbaum hold), but military, nor is
ἀνάγειν (
ναῦν) even in naval language ‘inhibere,’ but ‘put out to sea,’ as in Hdt. VII 100, VIII 76 et al. and occasionally in Attic (for
ἀνάγεσθαι). Cf.
ἀνεχώρησας 528 D.
οὐκ ὀρθῶς
: see on
τὸ ἐχόμενον τούτου 526 C. The subjects ought to follow each other in the order of their complexity: see App. II. Plato's error was of course deliberately “contrived to emphasize the principle which it violated” (Bosanquet), and also, it may be added, to enable him to call especial attention to the study of Stereometry, on which he laid very great stress (527 D note).
ἤδη
should be taken with
ὄν (‘already in revolution’), not (as D. and V.) with
λαβόντες.
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.
εἰ δὲ πόλις κτλ.
is perhaps the earliest demand in literature for the State-encouragement—we might almost say the State-endowment—of pure science (cf. Krohn
Pl. St. p. 169). Plato implies that in his city this claim will be fully satisfied; and the Platonic Utopia is in fact “la revendication du pouvoir pour la science” (Tannery l.c. p. 521).
ξυνεπιστατοῖ κτλ.
: ‘should cooperate with the superintendent’ etc. not (as Jowett) ‘become the director of these studies’: for a special
ἐπιστάτης—Plato has just said—is needed in any case. Plato's picture of the
odium stereometricum, if the phrase may be allowed, is evidently drawn from life. He seems to speak as if he had himself an
ἐπιστάτης ready, and wished to secure for him public support in order that students might be willing to work under him. Now although
ὡς νῦν ἔχει belongs, strictly speaking, to the following clause, the words
may, so far as the Greek is concerned, be connected with
ἔπειτα καὶ γενομένου, and will then be equivalent to
ὡς νῦν ἐγένετο ἐπιστάτης. I think it not impossible that Plato intended his readers to suspect him of this further meaning. If there is anything in this conjecture, to whom does Plato allude? Not, surely, to himself, although some have suspected the philosopher of blowing his own trumpet in a somewhat similar passage of the
Phaedo (78 A): see Lutoslawski's
Plato's Logic pp. 263 f. We are told by Plutarch
de genio Socratis 7. 579 C that Plato referred the Delian deputation to Eudoxus, telling them that the problem was
οὔ τοι φαῦλον οὐδ’ ἀμβλὺ διανοίας ὁρώσης, ἄκρως δὲ τὰς γραμμὰς ἠσκημένης ἔργον εἶναι· τοῦτο μὲν οὖν Εὔδοξον αὐτοῖς τὸν Κνίδιον ἢ τὸν Κυζικηνὸν Ἑλικῶνα συντελέσειν κτλ. Now we know that Eudoxus not only himself achieved a solution of the Delian problem (Sturm l.c. pp. 32—37), but was also, in the fullest sense of the term, ‘the founder of scientific Stereometry’ (Günther in Müller's
Handbuch V 1, p. 30), and did more for the subject than any of Plato's disciples (Cantor l.c. pp. 208—210). For these reasons I think it not unlikely that Plato has Eudoxus in his mind. Eudoxus and his pupils seem to have been living and working in the Academy along with the followers of Plato sometime between Plato's second and third visits to Sicily (368 B.C. and 361 B.C.: see Allman
Gk Geometry etc. p. 178), and it is a pleasing and I hope pardonable conjecture—I do not claim that it is more—to suppose that Plato avails himself of this opportunity to pay a graceful compliment to his fellow-workers. See also on line 19 below and
Introd. § 4.
ἐντίμως ἄγουσα
. The phrase is illustrated by Lobeck
Phryn. p. 419.
ὑπὸ δὲ κτλ.
ὑπὸ depends on
ἀτιμαζόμενα καὶ κολουόμενα. There is a sense in which the students also
ἀτιμάζουσι καὶ κολούουσι a subject, which they
ἀσθενῶς ζητοῦσιν (B above).
κολουόμενα is in harmony with
αὐξάνεται—though cut short, the study still grows or advances. For other views on this sentence see App. VIII.
λόγον κτλ.
The
ζητοῦντες are the
ζητητικοί of B—not, I think, Plato's pupils, but men who cannot explain the
true utility of stereometry (as described in 527 D, E), and are unwilling to throw their whole hearts into a ‘useless’ study.
βίᾳ ‐‐ αὐξάνεται
. Blass (l.c. p. 22) observes that in these words “sine dubio mathematici ex schola Platonis profecti intelligendi sunt.” It is just conceivable —though of course no stress should be laid on the conjecture—that
ὑπὸ χάριτος conceals some complimentary allusion to a particular person. If so, Eudoxus may be intended (see above on 528 C). There is, it is true, a tradition that Plato and Eudoxus had not always been on the best of terms (Allman
Gk Geom. pp. 128 f.), but during the visit of Eudoxus to Athens between 368 and 361 B.C., they appear to have worked harmoniously and even cordially together (ib. pp. 133, 178). See also 530 A note But we have no evidence to shew that Eudoxus bore the sobriquet of
χάρις, though his character and personality (see Arist.
Eth. Nic. X 2. 1172^{b} 15 ff.), and even perhaps his name, deserved such a compliment. I think Plato means merely ‘through elegance,’ i.e. through the inherent elegance of the subject: cf.
τό γε ἐπίχαρι καὶ διαφερόντως ἔχει. The use of
ὑπό is as in
ὑπὸ δέους φωνὴν ἔρρηξε and the like: see KühnerGerth
Gr. Gr. II 1, p. 523. Badham's
ἐπιχάριτα for
ὑπὸ χάριτος is an unlucky venture. Dr Jackson suggests that
ὑπὸ χάριτος may perhaps mean ‘by grace, favour,’ ‘on sufferance’: but Glauco's reply appears to me against this view.
οὐδὲν ‐‐ φανῆναι
: ‘be brought to light,’ ‘discovered,’ ‘solved’: cf. X 602 D and
ηὑρῆσθαι and
ἐκφανῆ above. Unless Badham, Madvig, and Baiter had entirely mistaken the meaning of
φανῆναι, they could scarcely have conjectured or approved of
τοιαῦτα in place of
αὐτά. Plato's language seems to point to some exceptional activity in connexion with the study of stereometrical problems, such as may have been occasioned by the application from Delos (527 D note), and to encourage his pupils to hope for success at no distant date.
ἀλλά μοι κτλ.
The recapitulation is intended to emphasize once more the principle regulating Plato's sequence of subjects (528 A note and App. II).
σπεύδων ‐‐ βραδύνω
: a proverbial saying, like our ‘more haste, less speed’: cf. (with Stallbaum)
Pol. 264 B
. If we
σπεύδομεν ταχέως, we are apt
σπεύδοντες βραδύνειν; hence the proverb
σπεῦδε βραδέως ‘Eile mit Weile.’ See Jebb on
Soph.
Ant. 231
.
ὅτι τῇ ζητήσει κτλ.
: not “quia ita est comparata, ut de ea quaerere ridiculum sit” (Stallbaum), but ‘quia ridicule tractatur’: cf. (with Schneider) 529 E.
ᾗ σὺ μετέρχει
: lit. ‘in respect of that, in respect of which you pursue it,’ i.e. ‘in the way in which you pursue it,’ no longer for its practical uses, as I did before (527 D), but because it leads the soul ‘on high,’ and from things here yonder (“from the things of this world to the next,” say D. and V., quite wrongly). The object of
ἐπαινῶ is not
ᾗ σὺ μετέρχει, but astronomy. Glauco has assimilated the phraseology of Socrates without its meaning. ‘On high’ and ‘yonder’ mean to Glauco the material heavens, not the
νοητὸς τόπος: and he thinks the soul looks upwards if the bodily eye is turned aloft! The essence of Glauco's error consists in materializing the spiritual; and Plato here warns us against a danger which is responsible for countless errors, not only in Platonic criticism, but in every department of human thought and dogma. See also on 529 B, C.
