First, then, I said, let us enquire how timocracy (the government of honour) arises out of aristocracy (the government of the best). Clearly, all political changes originate in divisions of the actual governing power; a government which is united, however small, cannot be moved. Very true, he said. In what way, then, will our city be moved, and in what manner the two classes of auxiliaries and rulers disagree among themselves or with one another? Shall we, after the manner of Homer, pray the Muses to tell us 'how discord first arose'? Shall we imagine them in solemn mockery, to play and jest with us as if we were children, and to address us in a lofty tragic vein, making believe to be in earnest? How would they address us? After this manner:--A city which is thus constituted can hardly be shaken; but, seeing that everything which has a beginning has also an end, even a constitution such as yours will not last for ever, but will in time be dissolved. And this is the dissolution:--In plants that grow in the earth, as well as in animals that move on the earth's surface, fertility and sterility of soul and body occur when the circumferences of the circles of each are completed, which in short-lived existences pass over a short space, and in long-lived ones over a long space. But to the knowledge of human fecundity and sterility all the wisdom and education of your rulers will not attain; the laws which regulate them will not be discovered by an intelligence which is alloyed with sense, but will escape them, and they will bring children into the world when they ought not. Now that which is of divine birth has a period which is contained in a perfect number, but the period of human birth is comprehended in a number in which first increments by involution and evolution (or squared and cubed) obtaining three intervals and four terms of like and unlike, waxing and waning numbers, make all the terms commensurable and agreeable to one another. The base of these (3) with a third added (4) when combined with five (20) and raised to the third power furnishes two harmonies; the first a square which is a hundred times as great (400 = 4 X 100), and the other a figure having one side equal to the former, but oblong, consisting of a hundred numbers squared upon rational diameters of a square (i. e. omitting fractions), the side of which is five (7 X 7 = 49 X 100 = 4900), each of them being less by one (than the perfect square which includes the fractions, sc. 50) or less by two perfect squares of irrational diameters (of a square the side of which is five = 50 + 50 = 100); and a hundred cubes of three (27 X 100 = 2700 + 4900 + 400 = 8000). Now this number represents a geometrical figure which has control over the good and evil of births. For when your guardians are ignorant of the law of births, and unite bride and bridegroom out of season, the children will not be goodly or fortunate. And though only