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545C - 547C How does Timarchy arise out of Aristocracy? We may lay it down as a universal rule that constitutional change is originated by dissension within the governing class. Socrates invokes the Muses to tell ‘how first sedition entered.’ Like everything else, our perfect city is subject to Nature's universal law, that whatever is created perishes. Out of the elements of the number which expresses the shortest period of gestation in the human kind, Socrates builds up a ‘geometrical number,’ which he calls ‘the lord of better and worse births.’ When, through ignorance of these, couples are united inopportunely, as one day they will be, a degenerate race of offspring arises. The best of these in due course become rulers; but the mixture of races—golden, silver, copper, iron—waxes greater, and sedition is the result. The contending parties finally, by means of a compromise, effect the transition to Timarchy—a form of commonwealth standing midway between Aristocracy and Oligarchy.

ff. I have discussed the famous ‘Number of Plato’ at length in Appendix I, and must refer the reader to that Appendix for a fuller justification of views which considerations of space preclude me from defending totis viribus throughout the notes. The connexion of the episode with the argument of the Republic may be expressed as follows. In accordance with the form of a historical narrative which he employs throughout these two books, Plato invites us to conceive of his perfect city as having actually existed long ago, just as in the Timaeus (23 C ff.) and Critias (109 B ff.) the Platonic Utopia appears as prehistoric Athens. In making this demand upon the imagination of his countrymen, Plato could count upon the support to be derived from the prevalence of the view that mankind had degenerated from an age of innocence and bliss in the far-distant past: see the references in my edition of the Protagoras p. xxiii and Rohde Griech. Roman pp. 216 ff. What, then, was the originating cause of degeneration? Plato finds the cause, not in anything peculiar to the Ideal city, but in a law which prevails throughout the whole of Nature—the law that everything created is doomed to decay. There cannot be any ἴδιος μεταβολή (to quote the phrase of Aristotle Pol. E 12. 1316^{a} 12) of a perfect City; for a city which carries within itself the seeds of decay is not perfect, but imperfect. In the sequel Plato first describes the manner in which degeneration begins to take effect (οὐ μόνονδέον 546 A, B), and afterwards proceeds to construct a Number which is the expression of that law of inevitable degeneration to which the Universe and all its parts are subject. The substance of what he has to say on the first head is that a psychologically inferior offspring gradually makes its appearance because children are sometimes begotten inopportunely. It is noteworthy that here, as everywhere in Books VIII and IX, the decline of the constitution or soul of the State (543 A note) is traced to the decline of the soul of the individual. In the words ἀνθρωπείῳ δὲτριάδος (546 B, C), Plato, copying the method of the Pythagoreans, and closely following their calculations, at all events in the first part of the reckoning, attempts to give an arithmetical expression to the Law of Change in that which he calls the γεωμετρικὸς ἀριθμός. According to the view which I have endeavoured to establish in Appendix I, the arithmetic, in which each of the factors and processes involved was full of significance to ancient speculators on the theory of numbers, may be thus expressed in modern arithmetical notation: (1) 3^{3} + 4^{3} + 5^{3}=216. (2) (3 x 4 x 5)^{4}=12,960,000 =3600^{2}=4800 x 2700.

The first number, 216, is the shortest period of gestation in the human race expressed in days. In the second equation, the number 12,960,000 expresses, also in days, the duration of a Great Year in the life of the Universe. Expressed in years, the number is 36,000, if we count, as Plato here does, 360 days in the year. The two ‘harmonies,’ 3600^{2} and 4800 x 2700, are the two cycles described in the Politicus, each of which is a Great Year. In the first ὁμοιότης prevails, in the second ἀνομοιότης: the World ‘waxes’ in the first, and ‘wanes’ in the second, without, however, suffering dissolution. In what sense the whole number 36,000 years, which astronomers sometimes called the Platonicus annus in the middle ages, is at once the numerical Cause of Change, and the ‘lord of better and worse births,’ is pointed out in App. I, Pt ii § 7, and also in the notes on 546 C. How far Plato attached a serious value to his Number and the calculations from which he derives it, I have briefly discussed at the end of App. I, Pt ii. Here it must suffice to say that the episode, like many other passages in Plato, is half-serious, and half-playful. The setting of the whole is mythical, for it is only for literary and artistic purposes that Plato pictures his ideal city as historically true: and the meaning of the latter part of the Number is deciphered by the aid of one of Plato's myths. Moreover, the style of the whole passage, though extraordinarily rhythmical and highly-wrought, acquires a touch of fantastic humour from the bewildering parade of mathematical terms, at some of which even Plato's own contemporaries would probably have smiled. On its serious side, the Number affords an interesting example of the application of Number and Mathematics to explain the life of the Universe and Man; and, as I have said in the Appendix, finds its fittest apology in the saying θεὸς ἀεὶ γεωμετρεῖ. It is of some importance in the history of philosophy because of its connexion with Pythagorean embryology and physics, and its employment by the Neoplatonists to justify the wildest astrological vagaries. The extreme difficulty of the Greek has made the Platonic Number a favourite hunting-ground of successive generations of scholars, and the works which have been written on the subject, a few of which are mentioned in the Appendix, are very numerous.

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