James Adam, The Republic of Plato

ὅσῳ ἀηδέστερον κτλ. Plato loved to play with mathematics, and in the following passage he endeavours to give an arithmetical expression to the pleasures of justice and injustice. His motive in introducing this “hedonistic calculus,” as Bosanquet calls it, has been much discussed. The following considerations deserve attention. (1) On artistic grounds, now that the argument has been concluded, it is not inappropriate that Justice and Injustice, represented by the King and the Tyrant, should as it were be weighed in the balance. The importance of ἀριθμεῖν, μετρεῖν, ἱστάναι is constantly present to Plato's mind. See on X 602 D. (2) The Pythagoreans were in the habit of expressing virtues and other immaterial ideas in terms of numbers (see Zeller^{5} I pp. 389 ff.), and there is reason to suppose that the number 729 played a part in a Pythagorean calendar (588 A note). Some of the terms employed by Plato, such as τρίτη αὔξη, are also in all probability of Pythagorean origin. See App. I to Book VIII pp. 279 ff. (3) The arithmetical method of calculation enables Plato to set forth in a very striking and dramatic way his own dissent from the popular estimate of the tyrant's happiness (Schneider). (4) When all is said we must allow that some of the steps are arbitrary, and that Plato's main object is to reach the significant number 729, so as to indicate that the king has more pleasure than the tyrant every day and every night of his life. There is of course an element of playfulness in the episode, and we need not suppose that Plato set any particular store by his calculations: but neither ought we on the other hand to dismiss the whole reckoning as a meaningless and foolish jest. See also on VIII 545 C.

τριῶν ἡδονῶν κτλ. The three pleasures are those of the king, the timocrat, and the oligarch. The first variety is genuine, the second and third spurious: but the tyrant has ‘crossed the line into the region beyond the spurious,’ i.e. his pleasures represent a still lower depth (see 571 B ff.), being in fact only εἴδωλα twice removed of the oligarch's spurious pleasures (587 C). Schleiermacher made τῶν νόθων depend partitively on τὸ ἐπέκεινα (“so ist der Tyrann auf die jenseitige der unächten hinübergestiegen”). This view is linguistically defensible (cf. Phaed. 112 B), and even attractive at first sight; but τῶν νόθων must be interpreted by δυοῖν νόθαιν, and δυοῖν νόθαιν certainly does not include the tyrant's species of pleasure. The feminine dual ending -ᾳιν (instead of the commoner -οιν) is “magis elatioris quam vulgaris sermonis” (Roeper de dual. us. Pl. p. 6). Cf. IV 422 E note