Geometry is the first portion of concrete mathematics. Undoubtedly the facts with which it deals are more connected among themselves than the facts studied by the other sciences, and this allows us easily to deduce some of these facts once the others are given. But there is a certain number of primary phenomena which, not being established by any reasoning, can only be founded upon observation, and which stand as the basis of all geometrical deductions.[107] Although very small, this part of observation is indispensable because it is the initial one, and never can quite vanish. In this way, metaphysical discussions upon the origin of geometrical definitions and space are set aside. Comte here adopts d’Alembert’s opinion. The latter had said: “The true principles of the sciences are simple recognised facts, which do not suppose any others, and which consequently can neither be explained nor questioned: in geometry they are the properties of extension as apprehended by sense. Upon the nature of extension there are notions common to all men, a common point at which all sects are united as it were in spite of themselves, common and simple principles from which unawares they all start. The philosopher will seize upon these common primitive notions to make them the basis of the geometrical truths.”[108] Extension is a property of bodies. But, instead of considering this extension in the bodies themselves, we consider it in an indefinite milieu which appears to us to contain all the bodies, of the universe and which we call space. Let us think, for instance, of the impression left by a body in a fluid in which it might be immersed. From the geometrical point of view this impression can quite conveniently be substituted to the body itself. Thus, by a very simple abstraction, we divest matter of all its sensible properties, only to contemplate in a certain manner its phantom, according to d’Alembert’s expression. From that moment we can study not only the geometrical forms realised in nature, but also all those which can be imagined. Geometry assumes a “rational” character. Similarly, it is by a simple abstraction of the mind that geometry regards lines as having no thickness, and surfaces as being without depth. It suffices to conceive the dimension to be diminished as becoming gradually smaller and smaller until it reaches such a degree of thinness that it can no longer fix the attention. It is thus that we naturally acquire the “real idea” of surface, then of the line, and then of the point. There is therefore no necessity to appeal to the _a priori_.