"Much of the confusion of ideas which has arisen on these subjects has been occasioned by not observing that when we say "two lines are incommensurable," the phrase is in fact elliptical, and we ought always to consider as understood, if not expressed, that "two lines, if referred to numbers, "are incommensurable." The deficiency of exact comparison in such cases is not in the geometrical relation of the quantities, but in the powers and capacities of our numerical system to express them. These observations may be necessary to enable us to appreciate better the opinions which have since been prevalent among mathematicians; more especially as regards the doctrine of proportionals."