Where the empirical bases of a science are continually in the foreground, as in physics or chemistry, the foregoing formulation of procedure is intelligible and acceptable to most men. Mathematics seem, however, to stand peculiarly apart. Many, with Descartes, have delighted in them "on account of the certitude and evidence of their reasonings" and recognized their contribution to the advancement of mechanical arts. But since the days of Kant even this value has become a problem, and many a young philosophic student has the question laid before him as to why it is that mathematics, "a purely conceptual science," can tell us anything about the character of a world which is, apparently at least, free from the idiosyncrasies of individual mind. It may be that mathematics began in empirical practice, such philosophers admit, but they add that, somehow, in its later career, it has escaped its lowly origin. Now it moves in the higher circles of postulated relations and arbitrarily defined entities to which its humble progenitors and relatives are denied the entree. Parvenus, however, usually bear with them the mark of history, and in the case of this one, at least, we may hope that the history will be sufficient to drag it from the affectations of its newly acquired set and reinstate it in its proper place in the workaday world. For the sake of this hope, we shall take the risk of being tedious by citing certain striking moments of mathematical progress; and then we shall try to interpret its genuine status in the world of working truths. I BEGINNINGS OF ARITHMETIC AND GEOMETRY