To the Pythagoreans belongs by common consent the distinction of having raised mathematics to the level of a science. We may well feel some astonishment when we reflect that this great achievement, which implies in the achievers the spirit of sober scientific reasoning, should nevertheless have been the work of men who were also enthusiasts and mystics. But we can find it in no way strange that the complication of these disparate tendencies should have resulted in the profession of philosophical doctrines, respecting the world-significance of numbers, which to us appear fantastic to the verge of absurdity, and which not improbably wore that appearance to some few contemporary intellects, more critical if less original and creative.