It has been said that to appreciate what virtue and morals mean, men must live virtuous and moral lives. It is equally true, that a knowledge of the objects of science is not to be attained by any scheme of definitions, however carefully contrived. He who would know what geometry is, must venture boldly into its depths and learn to think and feel as a geometer.
It seems to be expected of every pilgrim up the slopes of the mathematical Parnassus, that he will at some point or other of his journey sit down and invent a definite integral or two towards the increase of the common stock.
As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention.