nishing 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.
