the perfectly just, which is the fifth, that we may know which is the happier, and then we shall be able to determine whether the argument of Thrasymachus or our own is the more convincing. And as before we began with the State and went on to the individual, so now, beginning with timocracy, let us go on to the timocratical man, and then proceed to the other forms of government, and the individuals who answer to them. But how did timocracy arise out of the perfect State? Plainly, like all changes of government, from division in the rulers. But whence came division? ‘Sing, heavenly Muses,’ as Homer says;—let them condescend to answer us, as if we were children, to whom they put on a solemn face in jest. ‘And what will they say?’ They will say that human things are fated to decay, and even the perfect State will not escape from this law of destiny, when ‘the wheel comes full circle’ in a period short or long. Plants or animals have times of fertility and sterility, which the intelligence of rulers because alloyed by sense will not enable them to ascertain, and children will be born out of season. For whereas divine creations are in a perfect cycle or number, the human creation is in a number which declines from perfection, and has four terms and three intervals of numbers, increasing, waning, assimilating, dissimilating, and yet perfectly commensurate with each other. The base of the number with a fourth added (or which is 3:4), multiplied by five and cubed, gives two harmonies:—the first a square number, which is a hundred times the base (or a hundred times a hundred); the second, an oblong, being a hundred squares of the rational diameter of a figure the side of which is five, subtracting one from each square or two perfect squares from all, and adding a hundred cubes of three. This entire number is geometrical and contains the rule or law of generation. When this law is neglected marriages will be unpropitious; the inferior offspring who are then born will in time become the rulers; the State will decline, and education fall into decay; gymnastic will be preferred to music, and the gold and silver and brass and iron will form a chaotic mass—thus division will arise. Such is the Muses’ answer to our question. ‘And a true answer, of course:—but what more have they to say?’ They say that the two races, the iron and brass, and the silver and gold, will draw the State different ways;—the one will take to trade and moneymaking, and the others, having the true riches and not caring for money, will resist them: the contest will end in a compromise; they will agree to have private property, and will enslave their fellow-citizens who were once their friends and nurturers. But they will retain their warlike character, and will be chiefly occupied in fighting and exercising rule. Thus arises timocracy, which is intermediate between aristocracy and oligarchy.