Those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a great deal about them; if they do not expressly mention them, but prove attributes which are their results or definitions, it is not true that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.
Now the soul of man is divided into two parts, one of which has a rational principle in itself, and the other, not having a rational principle in itself, is able to obey such a principle. And we call a man in any way good because he has the virtues of these two parts.
Xenophanes states that the fire in Lipara once failed for sixteen years, but returned in the seventeenth year. They say that the lava-stream in Etna is neither flaming nor continuous, but returns only after an interval of many years.
Moral virtue is ... a mean between two vices, that of excess and that of defect, and ... it is no small task to hit the mean in each case, as it is not, for example, any chance comer, but only the geometer, who can find the center of a given circle.