In order to know the magnitudes which do not allow of direct measurement, we must evidently connect them with others which are capable of being immediately determined, and according to which we succeed in discovering the former, by means of the relations which exist between them and the latter. “Such is the precise object of mathematical science in its entirety.”[102] We see immediately how extremely vast it is. If we must insert a large number of intermediaries between the quantities which we desire to know, and those which we can measure immediately, the operations may become very complicated. Fundamentally, according to Comte, there is no question, whatever it may be, which cannot be finally conceived as consisting in determining one quantity by another, and consequently which does not depend ultimately upon mathematics. It will be said that we must take into account not only the quantity, but also the quality of the phenomena. This objection, decisive in the eyes of Aristotle, who could not conceive that we could legitimately [Greek: metaballein] [Greek: eis allo genos], no longer holds good for modern thinkers. Since Descartes’ time, they have seen analysis applied to geometrical, mechanical and physical phenomena. There is no absurdity in conceiving that what has been done for these phenomena is possible for the others. We must be able to represent every relation between any phenomena whatever by an equation, allowing for the difficulty of finding this equation and of solving it.[103] As a matter of fact, we are quickly stopped by the complexity of the data. In the present state of the human mind there are only two great categories of phenomena of which we regularly know the equations: these are geometry and mechanics. This being established, the whole of mathematical science is divided into two parts: abstract and concrete mathematics. The one studies the laws of geometrical and mechanical phenomena. The other is constituted by the _calculus_, which, if we take this word in its largest sense, applies to the most sublime combinations of transcendent analysis, as well as to the simplest numerical operations. It is purely “instrumental.” Fundamentally, it is nothing else than an “immense admirable extension of natural logic to a certain order of deductions.”