it must increase the unity and coherence of his personality by promoting the subordination of all his lower interests to his interests in the ideal world. May I illustrate what I mean by an example from a narrow field—the teaching of geometry. In teaching geometry we must, by exercises in drawing and measuring, and by a course of theory, enable a boy to gain a mastery over the simple geometrical properties of the bodies in his environment. And I would add that this mastery must be thorough. But our teaching must bear a direct relation to the boy's present and after life. I refer not only to the study of geometry as a direct preparation for such callings as that of an engineer, but also to the utility of geometry as an aid to the appreciation of the scientific thought common to educated men and to its practical use in daily life. "There is no reason," says Dr. Nunn, "why mathematics should not be taught systematically as a means necessary for appreciating the main results of human industry and invention, including those embodied in the mechanism of commerce and the financial machinery of civic and national life." And, lastly, we have the ideal purpose of learning geometry which must permeate all our teaching of it. This purpose has been described as follows by one of the most suggestive thinkers of our day. "For the health of the moral life," says Bertrand Russell, "for ennobling the tone of an age or nation, the austerer virtues have a strange power, exceeding the power of those not informed and purified by thought. Of these austere virtues the love of truth is the chief, and in mathematics, more than
