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 forever, 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,[1] 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.[2] 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 100 times as great (400=4 × 100),[3] and the other a figure having one side equal to the former, but oblong,[4] consisting of 100 numbers squared upon rational diameters of a square (i. e., omitting fractions), the side of which is five (7 × 7=49 × 100=4900), each of them being less by one (than the perfect square which includes the fractions, sc. 50) or less by[5] two perfect squares of irrational diameters (of a square the side of which is five=50 + 50=
- ↑ I. e., a cyclical number, such as 6, which is equal to the sum of its divisors, 1, 2, 3, so that when the circle or time represented by 6 is completed, the lesser times or rotations represented by 1, 2, 3 are also completed.
- ↑ Probably the numbers 3, 4, 5, 6, of which the three first=the sides of the Pythagorean triangle. The terms will then be 33, 43, 53, which together=63=216.
- ↑ Or the first a square, which is 100 × 100=10000. The whole number will then be 17, 500=a square of 100 and an oblong of 100 by 75.
- ↑ Reading προμήκη δέ.
- ↑ Or, “consisting of two numbers squared upon irrational diameters,” etc.=100. For other explanations of the passage see Introduction.
