James Adam, The Republic of Plato

τριπλασίου ἄρα κτλ. The distance of the tyrant from true pleasure is measured first ἀριθμῷ, i.e. “numero seu secundum longitudinem, numerus enim omnis quatenus monadibus constat, lineae instar habendus” (Schneider III p. LXXXXV. See also for ἀριθμῷ VII 526 C note and cf. the expression γραμμικὸς ἀριθμὸς in

Nicom. Introd. Ar. p. 117 Ast. Relatively to themselves, we reckoned the oligarch, democrat, and tyrant, as 1, 2, 3; but we have since found that the distance of the oligarch from true pleasure is in reality 3 times I: hence that of the tyrant must be 3 times 3, as in the line AB. We should doubtless regard the intervening numbers (4, 5, and 7, 8) as indicating different stages in the gradual degeneration of the oligarch into the democrat (559 D ff.) and the democrat into the tyrant or tyrannical man (572 D ff.). It might seem more natural to make the distance of the tyrant from true pleasure 5 and not 9 (King 1, Timocrat 2, Oligarch 3, Democrat 4, Tyrant 5); but (as Schneider reminds us) the pleasures of the Democrat and Tyrant lie beyond the two spurious pleasures, so that the modulus of progression may reasonably be increased. Plato's chief object is however to reach the number 729, and he could not do so except by making a fresh departure with the oligarch.

ἐπίπεδον ἄρα κτλ. The number 9 is ἐπίπεδος, because=3 x 3: εἰσὶ δὲ τῶν ἀριθμῶν οἱ μὲν ἐπίπεδοι, ὅσοι ὑπὸ δύο ἀριθμῶν πολλαπλασιάζονται, οἷον μήκους καὶ πλάτους: τούτων δὲ οἱ μὲν τρίγωνοι, οἱ δὲ τετράγωνοι κτλ. (Theo. Smyrn. p. 31 Hiller. Cf. Gow Gk Math. p. 69 and Müller in Hermes 1870 p. 394 note 1). This explanation, which so far agrees with that of the Scholiast, is adopted by the English translators and editors; but Schneider (l.c. and on p. 313 of his translation) holds that ἐπίπεδον κτλ. invites us to raise to the second power not 3, but 9—the number which we are presently expected to raise to the third power. The inferential ἄρα seems to me in favour of the Scholiast's view, as well as κατὰ τὸν τοῦ μήκους ἀριθμόν (cf. κατὰ τὸ μῆκος in Theo p. 31 al.), an expression which corresponds to ἀριθμῷ in the previous sentence while at the same time preparing us for κατὰ δὲ δύναμιν καὶ τρίτην αὔξην in the next. The whole sentence is, I believe, only a way of saying that, if the tyrant is 3 x 3 degrees distant from true pleasure, his εἴδωλον of pleasure may be represented by 9. The use of the mathematical term ἐπίπεδον has a playful effect, both in itself and also because it sounds wilful and eccentric to express a number of one ‘increase’ (τὸν τοῦ μήκους ἀριθμόν) in terms of two. δύναμις= δευτέρα αὔξη: cf. Tim 54 B and Cantor Gesch. d. Mathem. p. 178. The first increase (viz. of the unit or point) was ὁ τοῦ μήκους ἀριθμός, i.e. in this case (1 x 9=) 9: by the second-and-third increases (on the same scale) we obtain 9 x 9 (second increase or δύναμις) x 9 (third increase or τρίτη αὔξη)=729. See App. I to Book VIII p. 279. Schneider's erroneous idea that the squaring of 9 has already been alluded to in ἐπίπεδον—εἴη leads him to take δύναμιν as merely ‘power’ and τρίτην αὔξην as “per epexegesin ad δύναμιν additum.” What motive induced Plato to cube the distance? Was it something purely fanciful, e.g. “in order to gauge the depth of the tyrant's misery” (J. and C.), or because the king and the tyrant are themselves solid creatures (cf. Arist. Quint. de Mus. III p. 89. 35 Jahn)? I think not. He probably intended to suggest that “the degradation proceeds by increasingly wide intervals” (Bosanquet), but the actual calculations are inspired by a desire to reach the total 729. See on 588 A.

μεταστρέψας: ‘conversely’ (“umgekehrt” Schneider).