'It is not easy to estimate the loss of the tyrant, except perhaps in this way,' says Plato.

So we might say, that although the life of a good man is not to be compared to that of a bad man, yet you may measure the difference between them by valuing one minute of the one at an hour of the other ('One day in thy courts is better than a thousand'), or you might say that 'there is an infinite difference.' But this is not so much as saying, in homely phrase, 'They are a thousand miles asunder.' And accordingly Plato finds the natural vehicle of his thoughts in a progression of numbers; this arithmetical formula he draws out with the utmost seriousness, and both here and in the number of generation seems to find an additional proof of the truth of his speculation in forming the number into a geometrical figure; just as persons in our own day are apt to fancy that a statement is verified when it has been only thrown into an abstract form.

In speaking of the number 729 as proper to human life, he probably intended to intimate that one year of the tyrannical = 12 hours of the royal life.
The simple observation that the comparison of two similar solids is effected by the comparison of the cubes of their sides, is the mathematical groundwork of this fanciful expression.

There is some difficulty in explaining the steps by which the number 729 is obtained; the oligarch is removed in the third degree from the royal and aristocratical, and the tyrant in the third degree from the oligarchical; but we have to arrange the terms as the sides of a square and to count the oligarch twice over, thus reckoning them not as = 5 but as = 9.

The square of 9 is passed lightly over as only a step towards the cube.
3.