"Now although Euclid makes no mention of incommensurable quantities in his fifth book, he was well aware of their existence; and therefore, to render his theorems on proportion general, he had to take care that this class of quantities should be comprehended in his reasonings. But proportion limited to numbers, or to the symbols for numbers, would necessarily exclude incommensurables; he therefore had to proceed quite independently of arithmetic, and to secure to his propositions such a universality that each theorem should rigorously apply, whether the quantities or magnitudes spoken of be measurable, or beyond the powers of numerical representation. He has executed his difficult task with consummate ability; for as Dr. Barrow remarks, "there is nothing, in the whole body of the Elements, of a more subtile invention,—nothing more solidly established, and more accurately handled, than the doctrine of proportionals.""