few primitive ideas, which are represented by symbols, and used according to certain given postulates, it becomes possible to bring the whole body of mathematics within a single treatment. The development of this pure mathematics, which comes to be a logic of the mathematical sciences, has been made possible by such a generalization of number theory and theories of the elements of space and time that the rigor of mathematical reasoning is secured, while the physical scientist is left the widest freedom in the choice and construction of concepts and imagery for his hypotheses. The only compulsion is a logical compulsion. The metaphysical compulsion has disappeared from mathematics and the sciences whose techniques it provides. It was just this compulsion which confined ancient science. Euclidian geometry defined the limits of mathematics. Even mechanics was cultivated largely as a geometrical field. The metaphysical doctrine according to which physical objects had their own places and their own motions determined the limits within which astronomical speculations could be carried on. Within these limits Greek mathematical genius achieved marvelous results. The achievements of any period will be limited by two variables: the type of problem against which science formulates its methods, and the materials which analysis puts at the scientist's disposal in attacking the problems. The technical problems of the trisection of an angle and the duplication of a cube are illustrations of the problems which characterize a geometrical doctrine that was finding its technique. There appears also the method of analysis of the problem into simpler problems, the assumption of the truth of the conclusion to be proved and the process of arguing from this to a known truth. The more fundamental problem which appears first as the squaring of the circle, which becomes that of the determination of the relation of the circle to its diameter and development of the method of exhaustion, leads up to the sphere, the regular polyhedra, to conic sections and the beginnings of trigonometry. Number was not freed from the relations of geometrical magnitudes, though Archimedes could conceive of a number greater or smaller than any assignable magnitude. With the method of exhaustion, with the conceptions of number found in writings of Archimedes and others, with the beginnings of spherical geometry and trigonometry, and with the slow growth of algebra finding its highest expression in that last flaring up of Greek mathematical creation, the work of Diophantes; there were present all the conceptions which were necessary for attack upon the problems of velocities and changing velocities, and the development of the method of analysis which has been the revolutionary tool of Europe since the Renaissance. But the problems of a relation between the time and space of a motion that should change just as a motion, without reference to the essence of the