The general proposition in the theory of commensurables is to determine the conditions under which lines, surfaces, or solidities, connected with prescribed figures or forms, may have their ratios expressible by integer numbers.

The attention of geometers must have been drawn to this subject by the contemplation of incommensurable lines: the altitude of an equilateral trigon is incommensurable with the base; the diagonal of a square incommensurable with the side, and so on. And, when the two sides of a right angle are expressed by two numbers, the hypotenuse is, in the great multitude of cases, incommensurable with the sides: thus, if the sides be 5 and 7 inches respectively, the length of the subtense cannot be accurately expressed either in integers or fractions. However, when the sides are 3 and 4 units, the hypotenuse is exactly 5 of the same units.