e
arrests some fundamental concretion in discourse, and formulates the
first principles of logic. Mastering such definitions, sinking into the
dry life of such forms, he may spin out and develop indefinitely, in the
freedom of his irresponsible logic, their implications and congruous
extensions, opening by his demonstration a depth of knowledge which we
should otherwise never have discovered in ourselves. But if the geometer
had a fanatical zeal and forbade us to consider space and the triangles
it contains otherwise than as his own ideal science considers them:
forbade us, for instance, to inquire how we came to perceive those
triangles or that space; what organs and senses conspired in furnishing
the idea of them; what material objects show that character, and how
they came to offer themselves to our observation--then surely the
geometer would qualify his service with a distinct injury and while he
opened our eyes to one fascinating vista would tend to blind them to
others no less tempting and beautiful. For the naturalist and
psychologist have also their rights and can tell us things well worth
knowing; nor will any theory they may possibly propose concerning the
origin of spatial ideas and their material embodiments ever invalidate
the demonstrations of geometry. These, in their hypothetical sphere, are
perfectly autonomous and self-generating, and their applicability to
experience will hold so long as the initial images they are applied to
continue to abound in perception.

If we awoke to-morrow in a world containing nothing but music, geometry
would indeed lose its relevance to our future experience; but it would
keep its ideal cogency, and become again a living language if any
spatial objects should ever reappear in sense.

The history of such reappearances--natural history--is meantime a good
subject for observation and experiment. Chronicler and critic can always
approach experience with a method complementary to the deductive methods
pursued in mathematics and logic: instead of developing the import of a
definition, he can investigate its origin and describe its relation to
other disparate phenomena. The mathematician develops the import of
given ideas; the psychologist investigates their origin and describes
their relation to the rest of human experience. So the prophet develops
the import of his trance, and the theologian the import of the prophecy:
which prevents not the historian from coming later and sho