n 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
the best of them will be appointed by their predecessors, still they
will be unworthy to hold their fathers' places, and when they come into
power as guardians, they will soon be found to fall in taking care of
us, the Muses, first by under-valuing music; which neglect will soon
extend to gymnastic; and hence the young men of your State will be less
cultivated. In the succeeding generation rulers w