Plato Republic 8.546

ὧν ἐπίτριτος πυθμὴν κτλ. In ἀνθρωπείῳ δὲ—ἀπέφηναν the Pythagorean triangle was employed to construct the period of gestation for the microcosm or man: here it is used to construct two periods in the lifetime of the macrocosm or Universe, for that is what Plato means by the two ‘harmonies.’ The translation is: ‘of which, 4, 3 married with 5, yields two harmonies when thrice increased, the one equal an equal number of times, so many times 100, the other of equal length one way, but oblong:—on the one side, of 100 squares of rational diameters of five diminished by one each, or if of irrational diameters, by two: on the other of one hundred cubes of three.’ The antecedent of ὧν is ὁμοιούντων τε καὶ ἀνομοιούντων καὶ αὐξόντων καὶ φθινόντων, which I have already interpreted as the numbers 3, 4, 5. Of these numbers ( ὧν) the ἐπίτριτος πυθμήν i.e. 3, 4 (cf. Theo Smyrn. p. 80 ed. Hiller, Proclus l. c. II p. 37 ὁ ἐπίτριτος πυθμὴν γ/ καὶ δ/), is ‘married’ or ‘coupled’ with 5. That is to say, 3, 4, and 5 are multiplied together: whence we get 3 x 4 x 5=60. ‘Thrice increased’ is ‘three times multiplied by itself’; and 60 thrice increased is therefore 60 x 60 x 60 x 60. This sum, which is 12,960,000, yields two harmonies. One of the two harmonies is ‘equal an equal number of times, viz. so many times 100,’ in other words, it is a square (cf. Theaet. 147 E ) each of whose sides is a certain number of times 100 (for τοσαυτάκις cf. τοσοῦτον in Alc. I 108 E), viz. of course 36 times 100, for 60 x 60 x 60 x 60=3600^{2}. See Fig. 4. The other harmony which 60 x 60 x 60 x 60 yields is a rectangle (with προμήκη cf. Theaet. 148 A ), one of whose sides is one hundred cubes of 3, i.e. 2700, and the other the number which Plato describes in ἑκατὸν μὲν—δυοῖν. What is that number? ἀριθμοὶ ἀπό means (numerical) ‘squares of’ (cf. Procl. l. c. II p. 38. 9 et al.): the side in question is therefore ‘100 squares of’—what? Of the rational diameter of 5 etc. Now the ‘rational diameter of 5’ is the nearest rational number to the real diameter of a square whose side is 5 (Theo l. c. pp. 43 ff. and other authorities). The real diameter of a square whose side is 5 is √50. See Fig. 5. AC^{2}=5^{2} + 5^{2}=50 (by Pythagoras' famous εὕρημα Eucl. I 47): therefore AC=√50. And the nearest rational number to √50 is 7: for √49=7. Consequently 7 is ‘the rational diameter’ of 5. And 100 squares of 7=100 x 49=4900. But we are told to diminish the 100 squares by 1 each. Do so: 4900 - (1 x 100)=4800. This side is therefore 4800. The words ἀρρήτων δὲ δυοῖν give us an alternative way of reaching the number 4800. The construction is <ἀπὸ> ἀρρήτων δὲ <διαμέτρων δεομένων> δυοῖν <ἑκάστων> = (or of 100) ‘squares of irrational diameters of 5, wanting 2 each.’ Now the irrational diameter of 5 is √50. Square this and it becomes 50. 100 squares of 50= 5000. Subtract 2 from each square and you have 5000 - (2 x 100)=4800. The two sides of the oblong are therefore 4800 and 2700 (‘one hundred cubes of three’). The area is 4800 x 2700= 12,960,000 which is 60 x 60 x 60 x 60. See Fig. 6. Thus the arithmetical meaning of this part of Plato's Number may be expressed by us as follows:

συζυγείς . The metaphor is from marriage, and marriage, among the Pythagoreans, was usually expressed by multiplication. Thus 6, which is the product of the first male number 3 and the first female number 2, was called by them marriage. συζυγεῖσα also means ‘multiplied with’ in Proclus l. c. II p. 544 (App. I, Pt i § 2).

δύο ἁρμονίας . The square and oblong may be regarded as ἁρμονίαι because in them, as in the number 216 above, all things are προσήγορα καὶ ῥητὰ πρὸς ἄλληλα. Thus 12,960,000=(35 + 1) x 360,000, so that, as 35 is a ἁρμονία, 12,960,000 contains the portentous number of 360,000 ἁρμονίαι plus (1 x 360,000=) 360,000, each ἁρμονία thus having added to it, as before, the unit which is ἡ πάντων ἀρχή. The analogy between the Microcosm and the Macrocosm is thus preserved : see on πἁντα—ῥητά 546 B above. So much for the arithmetical meaning of the term ἁρμονίας. In App. I, Pt ii § 5 I have given my reasons for connecting the two ἁρμονίαι with the myth of the Politicus. In that myth we are told how two cycles of equal and vast duration invariably succeed one another in the life of the Universe, a progressive and a retrogressive cycle. These two cycles are two Great Years, in the first of which ὁμοιότης prevails and the Universe is fresh and strong, while in the second, in which we are living now, ἀνομοιότης begins to assert itself and the Universe flags and wanes. Cf. 547 A note Here the first ἁρμονία, which is a square and therefore ὅμοιον, represents the progressive cycle, the cycle of όμοιότης, and the second ἁρμονία, which is an oblong, and therefore ἀνόμοιον (see above on 546 B line 12), stands for the retrogressive cycle, the cycle of ἀνομοιότης. If this identification is, as I believe, correct, each ἁρμονία represents a Great Year. The area or number of each harmony, according to Plato, is 12,960,000, and as Plato elsewhere says that the Great Year is measured τῷ τοῦ ταὐτοῦ καὶ ὁμοίως ἰόντος κύκλῳ ( Tim. 39 D ), i.e. by the diurnal revolutions of the heavens, we may take this number as denoting days. Converted into years, on the astronomical calculation of 360 days to the year, followed by Plato here and elsewhere, the number becomes 36,000 years, which was known in Ptolemaic astronomy as the magnus Platonicus annus. For the evidence on all these points, see App. I, Pt ii §§ 5, 6.

ξύμπας δὲ οὗτος κτλ. ‘This whole number, a number measuring the earth, is lord of better and worse births.’ On its arithmetical side, γεωμετρικός means only that the number is reached by means of γεωμετρία and expressed in geometrical figures: but I have no doubt that Plato meant the word to bear another and profounder meaning, suitable to the real import of the two harmonies whereof this is the number. The number is τῷ ὄντι γεωμετρικός, for it measures an aeon of the Universe, of which the Earth is part (cf. 555 A note and VI 511 B note): and indeed it is artistically right that the meaning of the two harmonies should be summed up at the climax of the whole in a single pregnant word. How do good and bad births depend upon this number? Because in the early days of our era, when God had but lately left the world, and ἀνομοιότης and ἀνωμαλία were young, Nature produced better children than οἶοι νῦν βροτοί εἰσιν. Plato in fact invites us to think of his city as having existed soon after the change to the aeon in which we now live, just as throughout Book VIII and part of IX the Ideal City is figured in the past. For more on this subject see App. I, Pt ii §§ 5—7. I know not what others will think, but to me it seems that the extraordinary range and elevation of its central ideas make the Platonic number worthy even of a writer who is full of ‘thoughts that wander through eternity.’ The connexion between the Human Child and the Divine, the Microcosm and the Macrocosm, has played no small part in the history of human thought, and the story of a Great Year, with the hope which it affords of the ἀποκατάστασις of all things ( Acts 3. 21), has been and is, in its religious setting, the solace and support of many a ‘human child.’

παρὰ καιρόν : ‘inopportunely,’ ‘improperly’: cf. Pol. 277 A and οὐ δέον 546 B. The phrase does not, as I once thought, imply that Nature has appointed certain periodic times or seasons in the life of men and women when their union will produce good offspring, but refers to unions of wrong couples, superabundance of marriages, and the like: cf. V 459 E ff. The notion that the number of the Great Year is to be used by the rulers as a means of determining at what time unions should take place, derives no support from the Greek, and ought not to be entertained. In point of fact, the number is not a nuptial but a secular number, being γεωμετρικὸς ὡς ἀληθῶς. The expression ‘nuptial number’ is not applied to it either by Plato or by Aristotle, and it is only in later writers that we meet with ὁ τοῦ λεγομένου γάμου τόπος (Nicom. Introd. Ar. p. 144 Ast), γαμήλιον διάγραμμα (Plut. de Is. et Os. 373 F) and γαμικὸς ἀριθμός (Iambl. in Nic. Ar. p. 82. 21 Pistelli).

καταστήσονται . The active καταστήσουσι, found in v and two other MSS of little moment, is read by Hermann. Some may prefer it because καταστήσονται is passive just below: but Plato is careless about matters of this kind, and it is better to follow the best MSS.

ἡμῶν κτλ. : ‘us they will first begin to neglect when they are Guardians’ (i. e. after they have come εἰς τὰς τῶν πατέρων δυνάμεις), ‘setting too little store by music first, and second by gymnastic.’ Political decay is constantly associated by Plato with neglect of ‘Music’: see on IV 424 C. In place of δεύτερον δὲ τὰ γυμναστικῆς, which is in all MSS, Baiter adopts Madvig's conjecture δεύτερά τε γυμναστικῆς. At first sight ὅθεν ἀμουσότεροι—νέοι would seem to favour such an alteration, as well as the fact that in the city which comes next in order Gymnastic is more esteemed than Music (548 C). But πρῶτον after ἡμῶν supports the MS tradition, and the decline of the ideal city, which, as we have seen, arises from inevitable organic deterioration, shews itself in a general lowering of vital energy, rather than in the exaltation of any one pursuit at the expense of another. In the Spartan city Gymnastic ranks higher than Music, because Music has fallen from the high position which she formerly occupied, and not because Gymnastic stands higher than before.

ὅθεν ‐‐ νέοι : ‘and so our children will forget us.’ The Muses are speaking, and the children of Plato's Muse may well be called the Muses' children. This is the force of ἡμῖν, which is the reading of A, Ξ and some other MSS: Π and others have ὑμῖν. Schneider says ‘ Μούσαις—ἄμουσον γίγνεσθαι nullo modo tolerabile est.’ That is true, only ἡμῖν does not go with ἀμουσότεροι, but is an ethic dative, and seems to me at least to be full of a strange beauty and pathos. As true Gymnastic educates the soul and not the body (III 410 C ff.), the neglect of Gymnastic in the ideal city itself contributes to ἀμουσία.

τὰ παρ’ ὑμῖν γένη . See III 415 A ff.

ἀνομοιότης καὶ ἀνωμαλία ἀνάρμοστος . We have already seen that as the second scroll of the World's life unfolds itself, ἀνομοιότης, ἀνωμαλία, and ἀναρμοστία, with their attendant retinue of sedition, strife and war, make their appearance and wax more and more aggressive, until at last, in the words of the Politicus, careful lest the world χειμασθεὶς ὑπὸ ταραχῆς διαλυθεὶς εἰς τὸν τῆς ἀνομοιότητος ἄπειρον ὄντα τόπον δύῃ, God takes the helm again and κοσμεῖ τε καὶ ἐπανορθῶν ἀθάνατον αὐτὸν καὶ ἀγήρων ἀπεργάζεται ( Pol. 273 D f.). See on 546 C and App. I, Pt ii § 5. The same insidious enemies, not from any fault of the rulers, but because the part must neces sarily suffer with the whole, fasten both on the perfect individual and on the perfect State, and the fall of men and cities, which Plato describes in VIII and IX, is one long record of the triumphal progress of ἀνομο<*>ότης, until at last she sits enthroned in the soul and city of the tyrant. The Platonic number is thus the setting in which Plato's ‘Philosophy of History’ is framed.

ταύτης τοι γενεᾶς . From Homer Il. VI 211 al. ταύτης τοι γενεῆς τε καὶ αἵματος εὔχομαι εἶναι. Plato means of course ‘Such, as we must say, is the pedigree of Sedition, wheresoever she arises.’ D. and V. are wholly wrong when they translate: “so that we may positively assert that the rise of such a generation will invariably be marked by divisions.”

εἱλκέτην κτλ. The logical object of εἱλκέτην and ἠγέτην (in line 13) is τὴν πολιτείαν. For the omission of the object with ἕλκω and ἄγω cf. 560 B, X 604 A, 604 D et al. It is only another way of expressing oneself to say that the verbs are practically intransitive. The sedition which arises is not between rulers and ruled, but between the rulers among themselves, as is clear from 545 D and elsewhere: hence τὸ μὲν σιδηροῦν καὶ χαλκοῦν does not refer to the farmers and artisans, who probably possessed οἰκίαι from the first (III 417 A note), but to the section of the rulers who have become σιδηροῦν and χαλκοῦν by the intermixture of different breeds. Cf. the oracle foretelling the destruction of the city ὅταν αὐτὴν ὁ σίδηρος ἢ ὁ χαλκὸς φυλάξῃ (III 415 C). γῆς κτῆσιν κτλ. means ‘the possession of land and a private dwelling-place as well as of gold and silver,’ all of which were forbidden to Plato's rulers. ἔγκτησις γῆς καὶ οἰκίας was a familiar expression to the Greeks, and one of the well-recognised privileges of μέτοικοι at Athens (Gilbert Gr. Staatsalt. II p. 295). On the reading χρυσοῦ see cr. n. It is usual to read χρυσίου, but ἀργύρου immediately following favours χρυσοῦ, whose authority is not much inferior to that of χρυσίου. Cf. χρυσόν τε καὶ ἄργυρον 548 A. Herwerden, retaining χρυσίου, would write ἀργυρίου instead of ἀργύρου with some MSS of little value: but the reading printed above has much more MS support. His further proposal ἐπὶ χρηματισμὸν χρυσίου τε καὶ ἀργυρίου καὶ γῆς κτῆσιν καὶ οἰκίας will not find favour among scholars.

φύσει ‐‐ ψυχάς . They are not rich in worldly possessions, but they have the true riches—the riches of the soul. Cf. the prayer of Socrates in Phaedr. 279 C πλούσιον δὲ νομίζοιμι τὸν σοφόν, and see also on III 416 E. The usual view, which makes τὰς ψυχάς depend on ἠγέτην (Schneider, J. and C., D. and V., etc.) is surely wrong. Jowett from his translation appears to have caught the meaning.

εἰς μέσον ὡμολόγησαν κτλ. The change is effected, as in the case of the corresponding man (550 B), by a peaceful compromise. In the later stages of political decay, when ἀνομοιότης has gathered strength, revolution is attended by civil war (557 A), and the tyrant wades through bloodshed to his throne (565 E ff.).

περιοίκους τε καὶ οἰκέτας . We meet with περίοικοι not only in Sparta, of which city Plato is chiefly thinking, but also in Crete, Thessaly and Argos: see Gilbert Gr. Staatsalt. II pp. 16, 74, 220. In each of these States there was also an inferior grade, in Sparta the Helots, in Crete the vοικέες of the Gortynian inscription, sometimes also spoken of as οἰκέται, in Thessaly the πενέσται, and in Argos the γυμνῆτες or γυμνήσιοι (Gilbert l.c.). It is clear, I think, that in οἰκέτας Plato is thinking of this lowest order. The Spartan Helots had to perform the duties of domestic servants, as appears from Plut. Lyc. et Num. comp. 2. 4 ἦν ἡ περὶ τὰ χρήματα κατασκευὴ δεδομένη δούλοις καὶ Εἵλωσιν, ὥσπερ ἡ περὶ τὸ δεῖπνον καὶ ὄψον διακονία.

φυλακῆς αὐτῶν : the duty of watching and guarding— sensu inimico— the περίοικοι and οἰκέται. The institutions and history of Sparta are a sufficient commentary on the phrase.

οὐκοῦν ‐‐ πολιτεία . Cf. 547 C note

γεωργιῶν ‐‐ χρηματισμοῦ : as in Sparta: see Xen. Rep. Lac. 7. 1—2 and other authorities in HermannThumser l.c. p. 182 notes

ξυσσίτια . Cf. Hermann-Thumser l.c. pp. 185—191. In Crete, the ξυσσίτια were maintained at the expense of the State; in Sparta, by the contributions of the ξύσσιτοι. The former arrangement of course prevailed in the ideal city (III 416 E), and as timarchy copies the ideal city in regard to ξυσσίτια ( τὴν προτέραν μιμήσεται below), we may suppose that in the timarchical polity, as conceived by Plato, the Cretan method was observed. Cf. Arist. Pol. Β 9. 1271^{a} 28 ff. and see also on 551 A, B.

τῷ δέ γε φοβεῖσθαι κτλ. The Spartans were notorious for their dislike and distrust of knowledge and intellectual cultivation: see Hipp. Mai. 285 B ff. and Arist. Pol. Β 9. 1271^{b} 1 ff., with Susemihl and Hicks' notes. For the anacoluthon τῷ δέ γε—τὰ πολλά—ἕξει (548 A) cf. (with Schneider) Laws 931 C, 949 A.

κεκτημένη . See cr. n. Bekker's silence is often untrustworthy, and as he omitted Α and Ξ as well as v in the list of MSS which read κεκτημένην, it is doubtful if even v has the nominative here. κεκτημένην is however very awkward from its position between τὴν προτέραν μιμήσεται and πολεμοῦσα, and the error is so easy and common that I agree with Bekker and others in rejecting the accusative. See Introd. § 5. In illustration of what Plato says we may contrast Pausanias for example and Lysander with Brasidas and Callicratidas. The former were σοφοί but far from ἁπλοῖ: the latter θυμοειδεῖς and ἁπλούστεροι.

τοὺς τοιούτους : i.q. τοὺς σοφούς. With μεικτούς cf. 547 A.

ἁπλουστέρους = ‘more singleminded.’ Since they distrust σοφοί because their σοφοί are not ἁπλοῖ, it is natural enough that they should seek ἁπλούστεροι, and find them in θυμοειδεῖς, whose single all-engrossing idea is war (548 A) and φιλονικίαι (548 C). The text would hardly have been suspected if critics had grasped the meaning of ἁπλοῦς (see on I 351 A, II 370 B and IV 434 C). As it is, there is a host of superfluous conjectures: ἀλλοκωτέρους, αὐστηρούς, αὐθαδεστέρους, ποικιλωτέρους (Ast), πολλαπλουστέρους (Stallbaum), ἀγχινουστέρους (Müller), ὑποαμουσοτέρους (Herwerden), ἀμουσοτέρους (Herwerden, Apelt, Richards). It should be remembered that the Spartans prided themselves upon τὸ ἁπλοῦν in the ordinary sense of the term.