utterly different.

He is thus led to recognize a classification of quantities on a new
principle, according to which the physical nature of the quantity is
subordinated to its mathematical form. This is the point of view
which is characteristic of the mathematician; but it stands second to
the physical aspect in order of time, because the human mind, in order
to conceive of different kinds of quantities, must have them presented
to it by nature.

I do not here refer to the fact that all quantities, as such, are
subject to the rules of arithmetic and algebra, and are therefore
capable of being submitted to those dry calculations which represent,
to so many minds, their only idea of mathematics.

The human mind is seldom satisfied, and is certainly never exercising
its highest functions, when it is doing the work of a calculating
machine. What the man of science, whether he is a mathematician or a
physical inquirer, aims at is, to acquire and develope clear ideas of
the things he deals with. For this purpose he is willing to enter on
long calculations, and to be for a season a calculating machine, if he
can only at last make his ideas clearer.

But if he finds that clear ideas are not to be obtained by means of
processes the steps of which he is sure to forget before he has
reached the conclusion, it is much better that he should turn to
another method, and try to understand the subject by means of
well-chosen illustrations derived from subjects with which he is more
familiar.