We all know how much more popular the illustrative method of
exposition is found, than that in which bare processes of reasoning
and calculation form the principal subject of discourse.

Now a truly scientific illustration is a method to enable the mind to
grasp some conception or law in one branch of science, by placing
before it a conception or a law in a different branch of science, and
directing the mind to lay hold of that mathematical form which is
common to the corresponding ideas in the two sciences, leaving out of
account for the present the difference between the physical nature of
the real phenomena.

The correctness of such an illustration depends on whether the two
systems of ideas which are compared together are really analogous in
form, or whether, in other words, the corresponding physical
quantities really belong to the same mathematical class. When this
condition is fulfilled, the illustration is not only convenient for
teaching science in a pleasant and easy manner, but the recognition of
the formal analogy between the two systems of ideas leads to a
knowledge of both, more profound than could be obtained by studying
each system separately.

There are men who, when any relation or law, however complex, is put
before them in a symbolical form, can grasp its full meaning as a
relation among abstract quantities. Such men sometimes treat with
indifference the further statement that quantities actually exist in
nature which fulfil this relation. The mental image of the concrete
reality seems rather to disturb than to assist their contemplations.
But the great majority of mankind are utterly unable, without long
training, to retain in their minds the unembodied symbols of the pure