their multiplication, and the ratios they bear to one another.

For when you add one hundred to another hundred, the total is two hundred. If you divide that hundred by ten, each share is ten; and if you then multiply that ten by ten, the resulting product is one hundred.

Multiplication stands opposite division: to multiply whole numbers is simply to repeat the units of one number as many times as there are units in the other, whereas division distributes the units of one number across the units of the other. Thus, if the product obtained by multiplication is divided by one of its two factors, the other factor appears; and if the quotient obtained by division is multiplied by the divisor, the dividend re-emerges. The dividend therefore corresponds to the product, and each of the two factors corresponds, in its own way, to both the dividend and the divisor. Proportion gathers all of this together: the ratio of one factor to the product is like the ratio of one to the other factor, and the ratio of the product to one factor is like that of the other factor to one.

These matters, and others of the same sort addressed by arithmetic, are intelligible truths shared by all rational beings. There is no person who does not know something of them, for they are indispensable to both theoretical knowledge and practical life. Hence the commonplace example: “one is half of two.” There is no doubt that the propositions of arithmetic are universal, necessarily accepted, and never subject to refutation.

Digression
This formed the starting point of their philosophy, which was laid down by Pythagoras; his followers were called “the people of number,” and they imagined that abstract numbers existed outside the mind. Plato and his companions later recognized the error in that idea, but they maintained instead that abstract essences—such as absolute man or absolute horse—exist outside the intellect and are eternal and everlasting. Aristotle and his disciples then saw the flaw in that view as well, and asserted that these absolute essences exist externally only in conjunction with individual particulars; and certain later adherents of Aristotle continued along this path.