Let us take an example in algebra. Its calculus is commonly used to investigate, not only the relations of quantity generally, but geometrical facts in particular. Now it is at once too wide and too narrow for such a purpose, fitting on to the doctrine of lines and angles with a bad fit, as the coat of a short and stout man might serve the needs of one who was tall and slim. Certainly it works well for geometrical purposes up to a certain point, as when it enables us to dispense with the cumbrous method of proof in questions of ratio and proportion, which is adopted in the fifth book of Euclid; but what are we to make of the fourth power of _a_, when it is to be translated into geometrical language? If from this algebraical expression we determined that space admitted of four dimensions, we should be enunciating a mystery, because we should be applying to space a notion which belongs to quantity. In this case algebra is in excess of geometrical truth. Now let us take an instance in which it falls short of geometry,—What is the meaning of the square root of _minus a_? Here the mystery is on the side of algebra; and, in accordance with the principle which I am illustrating, it has sometimes been considered as an abortive effort to express, what is really beyond the capacity of algebraical notation, the direction and position of lines in the third dimension of space, as well as their length upon a plane. When the calculus is urged on by the inevitable course of the working to do what it cannot do, it stops short as if in resistance, and protests by an absurdity.