[Footnote 2: Pol. v. 12, § 8:--'He only says that nothing is abiding, but that all things change in a certain cycle; and that the origin of the change is a base of numbers which are in the ratio of 4 : 3; and this when combined with a figure of five gives two harmonies; he means when the number of this figure becomes solid.'] Plato begins by speaking of a perfect or cyclical number (cp. Tim. 39 D), i.e. a number in which the sum of the divisors equals the whole; this is the divine or perfect number in which all lesser cycles or revolutions are complete. He also speaks of a human or imperfect number, having four terms and three intervals of numbers which are related to one another in certain proportions; these he converts into figures, and finds in them when they have been raised to the third power certain elements of number, which give two 'harmonies,' the one square, the other oblong; but he does not say that the square number answers to the divine, or the oblong number to the human cycle; nor is any intimation given that the first or divine number represents the period of the world, the second the period of the state, or of the human race as Zeller supposes; nor is the divine number afterwards mentioned (cp. Arist.). The second is the number of generations or births, and presides over them in the same mysterious manner in which the stars preside over them, or in which, according to the Pythagoreans, opportunity, justice, marriage, are represented by some number or figure. This is probably the number 216. The explanation given in the text supposes the two harmonies to make up the number 8000. This explanation derives a certain plausibility from the circumstance that 8000 is the ancient number of the Spartan citizens (Herod. vii. 34), and would be what Plato might have called 'a number which nearly concerns the population of a city' (588 A); the mysterious disappearance of the Spartan population may possibly have suggested to him the first cause of his decline of States. The lesser or square 'harmony,' of 400, might be a symbol of the guardians,--the larger or oblong 'harmony,' of the people, and the numbers 3, 4, 5 might refer respectively to the three orders in the State or parts of the soul, the four virtues, the five forms of government. The harmony of the musical scale, which is elsewhere used as a symbol of the harmony of the state (Rep. iv. 443 D), is also indicated. For the numbers 3, 4, 5, which represent the sides of the Pythagorean triangle, also denote the intervals of the scale.