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
mathematician, so that, if science is ever to become popular, and yet
remain scientific, it must be by a profound study and a copious
application of those principles of the mathematical classification of
quantities which, as we have seen, lie at the root of every truly
scientific illustration.
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