§ 1. Definition of aim. A distinguished psychologist recently addressed a body of our American colleagues on the teaching of physics. “Your aim in the teaching of physics,” he told them, “should be—to teach physics” The advice was excellent—especially from one who is apt to be regarded as an intellectual Strong Man, full of esoteric knowledge about the development of mental muscle. Substituting “mathematics” for “physics” we may well adopt the maxim with all its implications, negative and positive.

§ 2. Negative implications. Beginning with the negative implications our amendment of Prof. Baldwin's dictum may be expanded as follows. In the first place, the central purpose in teaching mathematics is not to “train the power” of reasoning, of generalising, of “mental accuracy,” etc. The fallacies embodied in the persistent heresy of “formal training” have been repeatedly exposed, and need not detain us here. The time should soon come when an educational writer may ignore them. The chemist does not feel bound to begin his text-book by rebutting the phlogiston-theory, nor the geographer to demolish the sophistries of the flat-earthers.

So much for preliminaries. We come now to the formulation of a postulate which may be compared to Euclid's “Let it be granted that a straight line can be drawn from any point to any other point ”; a postulate, by the way, whose significance has been much underrated by later geometricians. To explain fully the closeness of the analogy would involve the discussion of the cataloguing of groups higher than the first, for which I have not space here, but I may indicate its general nature, to those familiar with the subject, thus: In “projective” geometry the existence of straight lines is assumed, and from them, by means of the Quadrilateral Construction, we obtain an unique method of determining any number of points in a straight line, with reference to three given points in it. My procedure is the exact converse of this. I assume what corresponds to an unique method of determining any number of points in a straight line, with reference to any three of them, and from that I deduce a method of drawing other straight lines, that is, if I am given a plane to draw them in.