James Adam, The Republic of Plato

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γενομένῳ κτλ. This is a universally recognised principle of ancient philosophy, alluded to again by Plato in Tim. 41 A. Cf. Arist. de cael. I 12. 282^{b} 8 τὸ γὰρ γενητὸν καὶ τὸ φθαοτὸν ἀκολουθοῦσιν ἀλλήλοις and ib. 10. 279^{b} 20 ἅπαντα γὰρ τὰ γινόμενα καὶ φθειρόμενα φαίνεται. That τὸ ἀγένητον is ἄφθαρτον and τὸ ἄφθαρτον ἀγένητον was also held (cf. Phaedr. 245 D, Arist. l. c. 282^{a} 30 ff., al.): hence Plato always regards the preexistence and immortality of the soul as involving one another. γένεσις is here, as usual, σύγκρισις, and φθορά διάκρισις. The point of ἐπεὶ—ἐστιν is that the cause of decay is not contained in the ideal city itself—the city would be less than ideal if it were—but springs from a universal law of Nature, to which the city, like everything else, is necessarily subject: see on 545 C and App. I, Pt iii.

λύσις δὲ ἥδε κτλ. Here begins Plato's description of the mode of dissolution. οὐ μόνον—ἐναντίας, literally trans lated, is ‘Not only to plants within the ground, but also among animals above the ground, there cometh production or non-production of souls and bodies, as often as turnings-round join for each species’ (of animals, plants etc.) ‘circumferences of circles faring a short way for the short-lived, and the reverse for the reverse.’ See Fig. 1. Suppose the revolution starts at the fixed point A. The circumference is joined as soon as the revolving wheel reaches A again, and at that point there is φορὰ ψυχῆς τε καὶ σώματος, if the seed was sown at A and has come safely to maturity. If the seed was not sown, or, though sown, did not take root or miscarried on the way, there is ἀφορία. The phrase is only a fantastic way of saying ὅταν περίοδοι ἑκάστοις ἀποτελεσθῶσιν. The περιφορά of a short-lived species is βραχύπορος and conversely, because short-lived creatures have short periods of gestation, and long-lived creatures long (Arist. de gen. an. IV 10. 777^{a} 31 ff., al. See App. I, Pt ii § 2). Soul, viewed merely as the vital principle, is one and the same in every organic creature: hence the singular ψυχή (App. l. c.). This explanation, so far as I know, is new, the περίοδος being generally supposed to be ‘Umlaufszeit.’ Cf. App. l. c.

, B 7 γένους δὲ κτλ. The literal translation is: ‘Now of your kind’ (i.e. mankind: it is the Muses who are speaking), ‘clever though the leaders of the city be whom you educated’ (the middle of personal interest IV 421 E note), ‘none the more will they by calculation together with perception obtain’ (lit. hit the obtaining of) ‘good offspring and no offspring, but it will escape them, and the time will come when they will beget children wrongly or inopportunely’ (cf. παρὰ καιρόν 546 D). In arranging matters connected with marriages and the treatment of children etc., the rulers of our city employ both λογισμός and αἴσθησις. αἴσθησις helps them to decide what couples should be joined, what children should be reared etc.: by λογισμός they calculate what number of marriages they should permit ἵνα ὡς μάλιστα διασῴζωσι τὸν αὐτὸν ἀριθμὸν τῶν ἀνδρῶν (V 460 A) etc.: see App. I, Pt ii § 2. But however well they use these instruments, the time will come when children are begotten οὐ δέον. The fault lies not with the rulers, but with the inevitable law of Change, which is beginning to affect our city together with the rest of the Universe. ἀφορίας is said, because the rulers must if possible make illicit unions unproductive (V 461 C). Possibly the εὐ of εὐγονίας may affect ἀφορίας also: cf. 555 A note See also App. l. c.

ἔστι δὲ -- τέλειος. ‘For a divine creature, there is a period comprehended by a number which is final.’ The ‘divine creature’ is the World: it is θεῖον, because it is a God, γεννητόν, because it is created (i.e. has been brought out of chaos into order). Cf. Tim. 30 A and Proclus in Tim. 89 D. With περιλαμβάνει cf. Theaet. 148 A. The ἀριθμὸς τέλειος is the period expressing the gestation of the Universe, i.e. the time which its creation occupies. For the metaphor cf. the Orphic verses cited by Proclus in Tim. 94 B and 95 E. The number is a final or consummating number because it τελειοῖ τὴν γένεσιν: cf. Theol. Ar. p. 58 ed. Ast. Plato wisely leaves this number shrouded in silence and obscurity. See App. I, Pt ii § 3 for a full discussion of the sentence.

ἀνθρωπείῳ δὲ -- ἀπέφηναν gives us the περίοδος or period of gestation for the human creature: ‘and for a human creature the number is the first in which root-and-square increases, comprehending three distances and four limits, of elements that make like and unlike and wax and wane, render all things conversable and rational with one another.’ The construction is ἀνθρωπείῳ δὲ <γεννητῷ. ἔστιν ἀριθμὸς> ἐν ᾦ κτλ., and that is itself short for ἀνθρωπείῳ δὲ <γεννητῷ ἔστι περίοδος ἣν ἀριθμὸς περιλαμβάνει> ἐν ᾧ κτλ. The ‘first’ number is of course the first number after unity. αὐξήσεις ‘increases’ may in itself mean either ‘additions’ or ‘multiplications.’ δυνάμεναι refers to ‘roots’ (cf. Eucl. X def. 11), δυναστευόμεναι to ‘squares’ (Procl. in remp. comm. ed. Kroll II p. 36. 9—12 et al.), and ‘root-and-square increases’ means either ‘additions of roots to squares’ or ‘multiplications of roots by squares.’ τρεῖς ἀποστάσεις etc. shew that multiplications and not additions are meant. The three distances are μῆκος, πλάτος and βάθος, and the four ὅροι their attendant limits. Thus in Fig. 2 AB, BC, CD are the three ἀποστάσεις, AB μῆκος, BC πλάτος, CD βάθος, and A, B, C, D the four ὅροι. Cf. Nic. Introd. Ar. p. 116 Ast εἴ τι γὰρ στερεόν ἐστιν, τὰς τρεῖς διαστάσεις—they are called ἀποστάσεις in Theol. Ar. p. 23—πάντως ἔχει, μῆκος, πλάτος καὶ βάθος: καὶ ἔμπαλιν εἴ τι ἔχει τὰς τρεῖς διαστάσεις, ἐκεῖνο πάντως στερεόν ἐστιν, ἄλλο δ᾽ οὐδέν: also Iambl. in Nic. Introd. Ar. p. 93 Pistelli στερεὸς δέ ἐστιν ἀριθμὸς ὁ τρίτον διάστημα παρὰ τὰ ἐν ἐπιπέδοις δύο προσειληφώς, δηλονότι τετάρτου ὅρου προσγενομένου: ἐν γὰρ τέσσαρσιν ὅροις τὸ τριχῇ διαστατόν, Arist. Top. Z 5. 142^{b} 24 f. and many other passages quoted in App. I, Pt i § 1. Consequently the arithmetical meaning of αὐξήσεις—λαβοῦσαι is merely ‘root-and-square multiplications’ i.e. ‘multiplications of root by square,’ in other words cubings (thus w x w^{2}=w^{3}, x x x^{2}=x^{3}, y x y^{2}=y^{3}), or κυβικαὶ αὐξήσεις, a phrase which itself might well have been employed by Plato, except that he has an object in drawing attention to the different stages in the process of augmentation (see App. I, Pt ii § 4); and partly also he wants the Muses to preserve their character as ὑψηλολογούμεναι. The period of human gestation is therefore the first number, in which ‘cubings make everything ῥητὰ πρὸς ἄλληλα’—but cubings of what? The answer is ‘of elements which make like and unlike and wax and wane.’ These elements are the numbers 3, 4 and 5, which measure the three sides of the Pythagorean ζῳογονικὸν τρίγωνον (Procl. in remp. II p. 43. 10), the triangle which, as we are informed by many authorities— Aristotle, Plutarch, Aristides Quintilianus, Proclus and others—Plato made use of in his Number. The antecedent of ὧν in ὧν ἐπίτριτος πυθμὴν πεμπάδι συζυγείς is ὁμοιούντων τε καὶ ἀνομοιούντων καὶ αὐξόντων καὶ φθινόντων, and as ὧν ἐπίτριτος πυθμήν means ‘of which 4, 3,’ Plato himself tells us two of the numbers, and the third is also readily suggested by πεμπάδι. 3, 4 and 5 are said to ‘make like,’ because, as we shall see, in the latter part of the Number, where the triangle fulfils its office as a κοσμικὸν τρίγωνον (Proclus l.c. II p. 45. 23), they produce the ‘harmony’ 3600^{2}, and square numbers are ὅμοιοι (Iambl. l.c. p. 82): they ‘make unlike’ because they produce the ‘harmony’ 4800 x 2700, and oblong numbers are ἀνόμοιοι (ib.): they are said to wax and wane in a figurative sense—to wax in the first harmony, which represents in a certain sense the waxing of the Universe, and to wane in the second, which represents its wane. As the elements out of which the Universe is formed, they may be said to grow with its growth, and decline with its decline. The words have also a further meaning as a description of 3, 4, 5 regarded as the ἀρχαί of everything which exists: see App. I, Pt ii § 5. Now the first number in which cubings of 3, 4 and 5 are present is 3^{3} + 4^{3} + 5^{3}=216, and Aristides Quintilianus, in the passage where he refers to Plato's number, speaking of the Pythagorean triangle, remarks ἀλλ᾽ εἰ καὶ τῶν πλευρῶν ἑκάστην κατὰ βάθος αὐξήσαιμεν (βάθος γὰρ ἡ σώματος φύσις) ποιήσαιμεν ἂν τὸν διακόσια δεκαέξ, ἰσάριθμον σύνεγγυς τῷ τῶν ἑπταμήνων (p. 151 Meibom). Aristotle also in Pol. E 12. 1316^{a} 5—8, according to Schneider's interpretation of his words, which I believe to be right, informs us that the whole number of this section is 216: see App. I, Pt iii. On πάντα— ἄλληλα see next note. In App. I, Pt ii § 4 I have fully treated of the meaning of all these calculations. The different mathematical terms are discussed in detail ib. Pt i § 1. My explanation of this passage is, as far as I can discover, new, except as regards αὐξήσεις—δυναστευόμεναι. Some other views are mentioned in App. I, Pt i § 1 ad fin. notes

πάντα προσήγορα κτλ. Cf. Philol. Fr. 13 Mullach πάντα γνωστὰ καὶ ποτάγορα ἀλλήλοις—ἀπεργάζεται. The Pythagoreans asserted that the embryo develops according to the proportions of the ἁρμονία or musical scale. The first stage is complete in 6 days, the second in 8, and 8 : 6 is ‘the fourth’ (διὰ τεσσάρων). The third stage (making flesh) takes 9 days, and 9 : 6 is ‘the fifth’ (διὰ πέντε). In the next 12 days the body is formed: and 12 : 6 is the octave (διὰ πασῶν). Total 6 + 8 + 9 + 12=35, and 35 is a ἁρμονία (Plut. de anim. gen. in Tim. 1017 F). Now 216=(6 x 35) + 6, so that 216 contains 6 ἁρμονίαι together with 6 times ἡ πάντων ἀρχή i.e. the unit (Excerpt. ex Nicom. in v. Jan's Mus. Script. Gr. p. 279), or if you like together with the marriage number 6. For the evidence and further details see App. I, Pt ii § 4.