With regard to the peculiar case of mathematics, which is widely applicable because formal and as exact as formal, it seems enough to say that while mathematics has very properly become the ideal of all knowledge, not excluding such sciences as psychology and sociology, the final value, the peculiar applicability of mathematics, lies in its character as a general attitude or method. It is not strictly a science, but the ideal method of science. Doctrinally, that is, as to any specific intellectual content, there can hardly be said to be any pure mathematics, any final body of formula absolutely exact and fully applicable. Has not doctrinal mathematics had a history? Has it now no promise of future changes? But whatever has a history--can this be quite "pure"? Have even those axioms, which once upon a time you and I learned to respect for their self-evidence, been free from the criticism and revision of the mathematical experts? Then, too, taking any particular formula from so-called applied mathematics, such as that simple but altogether typical one of the lever, what do we find? An equation is said to exist between the product of the weight by its distance from the fulcrum, and that of the power by its distance from the same point, but in application this formula can never be fully exemplified. The fulcrum never is a point. The perfectly homogeneous lever, so [p.216] necessary to the equation, is unattainable, if not also unthinkable. There can never be complete absence of friction, nor perfectly ideal suspension of the weight or application of the power. And the necessary atmospheric disturbances, even in a "vacuum," to say nothing of the difficulties of absolute measurements, are not less fatal. Only as method, therefore, which really means as procedure according to standards of strictest accuracy and of highest logical consistency, or as closest, most constant loyalty to a spirit of truth, not as doctrine, can mathematics be said to be freely applicable. Mathematics seems to me to be at the very heart of the working hypothesis. Its tests of accuracy are such as forever save science from anything like doctrinal dogmatism. Historically there is much significance in the fact that our doubter, Descartes, was almost the inventor of the Analytic Geometry, and that this and the Calculus, which came afterwards, and which we owe chiefly to Leibnitz and Newton, comprise rather a methodological than a doctrinal mathematics. With their invention and development the application of mathematics to material facts, or it would be better to say to the investigation of material facts, took tremendous strides. So Descartes, who doubted mathematics only because it was not satisfying doctrinally, regained in this case, as in that of his God or his material world, not exactly what he had lost. Alike in mathematics and theology he lost doctrine and creed; he won method and life. And, to return, with reference to the relation of mathematics to the free