μήτε ‐‐ ἀναγκάζειν κτλ.
: as in the eloquent picture of the
λογιστικόν and the
θυμοειδές, seeking for pleasures to lay at the feet of the
φιλοχρήματον VIII 553 C, D. Cf. also infra 590 B. For
μήτε—τε see IV 430 B note
ὁ δέ
: viz.
ὁ βασιλεύς. It is unnecessary to insert
βασιλεύς in the text, as Baiter wishes to do: cf. 1 349 D note
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
δορυφόροις ἡδοναῖς
. These are described in 573 D ff.
οὐδὲ
should not be taken with
εἰπεῖν. The hyperbaton is too difficult, and the meaning (“not even to express it” J. and C.) weak. We should translate ‘and it isn't very easy, either, to say’ etc., taking
οὐδέ as ‘also not,’ a usage illustrated by Riddell
Digest § 141. See on
οὐδὲ θαμίζεις κτλ. I 328 C.
ὁ τύραννος
. We might expect
ὁ τυραννικός, but throughout the whole of this comparison Plato is content to take the tyrant—who is, we remember,
τυραννικώτατος 575 D—as the type of the tyrannical, and the king as the type of the kingly or aristocratic man. See above on 587 B and also below 587 E note
οὐκοῦν καὶ ἡδονῆς κτλ.
If the tyrant is third from the oligarch, his pleasure will also be, in respect of truth, third from the oligarch's, i.e. will be an image of an image (
τρίτῳ εἰδώλῳ cf. X 597 E and 599 A, D) of the oligarch's pleasure.
ἐὰν εἰς ταὐτὸν κτλ.
See on IV 445 D.
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
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).
ἀμήχανον κτλ.
I do not think there can be any doubt that Schneider is right in retaining
καταπεφόρηκας, which is found in all the best and also in a majority of the inferior MSS. The word has reference to the gigantic and tempestuous numeral
ἐννεακαιεικοσικαιεπτακοσιοπλασιάκις, which Socrates has by a mighty effort of exspiration—
γηγενεῖ φυσήματι—hurled down upon his hearers' heads. Cf. generally
Theaet. 180 A
and
Ar.
Frogs 822
—825.
καταφορεῖν “contentionem vocis ad praelongum vocabulum numeri a Socrate prolati adhibendam significare potest; nec multum ab hoc abludit illa
καταφορά, quam oratoribus esse inter accusandum longiorem spiritus productionem ex Hermogene refert vetus lexicon apud Stephanum
Thes. T. IIII p. 109 C” (Schneider). See StephanusHase
Thes. s.v., and also on
καταφορικός, where the words
ἀδιόριστον τέθεικε τὸν ἀντιλέγοντα, ἵνα μετὰ ἀδικίας καταφορικῷ χρώμενος λόγῳ σφόδρα πλήξῃ τοὺς ἀκούοντας are quoted from Chrysost. X p. 386 C. The reading
καταπεφώρακας is much less dramatic and appropriate, and has very little MS authority (
Ξ^{2},
v and two other MSS, in one of which it is only a correction).
τοῦ τε δικαίου ‐‐ ἀδίκου
. These words might have been dispensed with, and accordingly Herwerden brackets them. But the point at issue, as originally raised in Book II, was the relation between Justice and Injustice, not the relation between the King and the Tyrant: and before giving his final answer, Plato does well to remind us that the King and the Tyrant are only types of Justice and Injustice. Cf.
ὁ ἀγαθὸς— ἄδικον just below.
