their motions and their transitions from one state to another, and of the natures inherent in them—is nobler than merely conceiving abstract magnitudes and abstract numbers. For conceiving nothing but a circle, triangle, or square—even if one were to conceive everything in Euclid—or nothing but abstract numbers yields no knowledge of anything existing in external reality. This is not the perfection of the soul. Were it not intended to provide knowledge of external things that are counted and measured, namely bodies and accidents, it would not have been regarded as a science. They made geometry a preliminary to the study of astronomy only to assist them in astronomical demonstrations or to benefit from it in developing the material world. Yet their syllogistic demonstrations establish nothing with consistent certainty, free of error, except in these mathematical subjects.
For “arithmetic,” which is the science of discrete quantity, and “geometry,” which is the science of continuous quantity, constitute certain knowledge that admits no contrary whatsoever. Examples include adding, dividing, and multiplying numbers, and determining their ratios to one another. If you add one hundred to one hundred, you know that together they make two hundred. If you divide it by ten, each receives ten; and if you multiply it by ten, the product is one hundred.
Multiplication is the inverse of division. Multiplying whole numbers means repeating the units of one number as many times as there are units in the other. If the product is divided by either number, the other factor is obtained. If the quotient is multiplied by the divisor, the dividend is obtained. Thus the dividend corresponds to the product, and each of the two factors corresponds to the dividend and the divisor, and the ratio