This part of mathematical science is independent of the nature of the objects which it examines, and only bears upon the numerical relations which they present. Consequently, it may happen that the same relations may exist among a great number of different phenomena. Notwithstanding their extreme diversity these phenomena will be considered by the mathematician as presenting a single analytical question, which can be solved once for all. “Thus, for instance, the same law which reigns between space and time when we examine the vertical fall of a body _in vacuo_, is found again for other phenomena which present no analogy with the former nor among themselves; for it also expresses the relation between the area of a sphere and the length of its diameters; it equally determines the decrease in intensity of light or of heat by reason of the distance of the objects lighted and heated, etc.”[104] We have no general method which serves indifferently for establishing the equations of any natural phenomena whatever: we need special methods for the several classes of geometrical, optical, mechanical phenomena, etc. But, whatever may be these phenomena, once the equation is established, the method for solving it is uniform. In this sense, abstract mathematics is really an “_organon_.” Geometry and mechanics, on the contrary, should be regarded as real natural sciences, resting as the others do upon observation. But, adds Comte, these two sciences present this peculiarity, that in the present state of the human mind, they are already used, and will continue to be used as methods far more than as direct doctrine. In this way mathematics is in fact “instrumental,” not only in abstract parts, but also in its relatively concrete parts. It is entirely used as a “tool” by the more complicated sciences, such as astronomy and physics. It is truly the real logic of our age.