In the eyes of philosophers, mathematics has always occupied a privileged place among the sciences. Plato located their object in an intermediate region between the world of sensible phenomena and that of intelligible realities. On the one hand mathematical objects, and in particular the geometrical figures, appeal to the imagination as sensible things; on the other hand, mathematical truths like ideas and the relations between ideas, are characterised by immutable and eternal fixity. This is why the study of mathematics is an excellent preparation for philosophy, which is the science of ideas. While still leaving to the mind the help of direct sensible perception, it accustoms it to permanent truth. During the whole of antiquity the science of mathematics, as the name indicates, was pre-eminently _the_ science. The science of physics, less sure of its object and of its method, was hardly distinguished from philosophical speculation, and lent itself with difficulty to the purely scientific form. For Plato then, and for those who followed him, mathematics has characteristics which distinguish it from the study of phenomena. In a certain measure, it partakes of the nature of science, conceived as bearing upon _what is_, upon the absolute reality which is neither subject to change nor to motion. It is true that they start from definitions and hypotheses. But, once the principles are established, they are developed _a priori_ by a succession of necessary demonstrations like the dialectics of ideas. This conception offers a mixture of metaphysical and positive elements. It implies that the object of science is reality such as it is in itself; but, at the same time, it sees in the demonstration the essential character of science. A long evolution, which culminates in Comte’s doctrine, has driven the metaphysical elements out of science while the other elements subsist in it still. Far from saying with Plato or with his successors that there is no science of the phenomenon or of that which passes away, Comte thinks on the contrary, that the only object of science is phenomenal reality so far as it is subject to laws. Science has not to search for causes or substances; it suffices for it to determine invariable relations.