necessary relations, the discoveries of physics have revealed to the
mathematician new forms of quantities which he could never have
imagined for himself.

Of the methods by which the mathematician may make his labours most
useful to the student of nature, that which I think is at present most
important is the systematic classification of quantities.

The quantities which we study in mathematics and physics may be
classified in two different ways.

The student who wishes to master any particular science must make
himself familiar with the various kinds of quantities which belong to
that science. When he understands all the relations between these
quantities, he regards them as forming a connected system, and he
classes the whole system of quantities together as belonging to that
particular science. This classification is the most natural from a
physical point of view, and it is generally the first in order of
time.

But when the student has become acquainted with several different
sciences, he finds that the mathematical processes and trains of
reasoning in one science resemble those in another so much that his
knowledge of the one science may be made a most useful help in the
study of the other.

When he examines into the reason of this, he finds that in the two
sciences he has been dealing with systems of quantities, in which the
mathematical forms of the relations of the quantities are the same in
both systems, though the physical nature of the quantities may be