CHAPTER I.

as regards its geometrical relations, such an impression may be substituted for the body itself, without altering the reasonings respecting it. As to the physical nature of this indefinite space, we are spontaneously led to represent it to ourselves, as being entirely analogous to the actual medium in which we live; so that if this medium was liquid instead of gaseous, our geometrical space would undoubtedly be conceived as liquid also. This circumstance is, moreover, only very secondary, the essential object of such a conception being only to make us view extension separately from the bodies which manifest it to us. We can easily understand in advance the importance of this fundamental image, since it permits us to study geometrical phenomena in themselves, abstraction being made of all the other phenomena which constantly accompany them in real bodies, without, however, exerting any influence over them. The regular establishment of this general abstraction must be regarded as the first step which has been made in the rational study of geometry, which would have been impossible if it had been necessary to consider, together with the form and the magnitude of bodies, all their other physical properties. The use of such an hypothesis, which is perhaps the most ancient philosophical conception created by the human mind, has now become so familiar to us, that we have difficulty in exactly estimating its importance, by trying to appreciate the consequences which would result from its suppression.

Different Kinds of Extension. The second preliminary geometrical conception which we have to examine is that of the different kinds of extension, designated by the words volume, surface, line, and even point, and of which the ordinary explanation is so unsatisfactory.

Although it is evidently impossible to conceive any extension absolutely deprived of any one of the three fundamental dimensions, it is no less incontestable that, in a great number of occasions, even of immediate utility, geometrical questions depend on only two dimensions, considered separately from the third, or on a single dimension, considered separately from the two others. Again, independently of this direct motive, the study of extension with a single dimension, and afterwards with two, clearly presents itself as an indispensable preliminary for facilitating the study of complete bodies of three dimensions, the immediate theory of which would be too complicated. Such are the two general motives which oblige geometers to consider separately extension with regard to one or to two dimensions, as well as relatively to all three together.

The general notions of surface and of line have been formed by the human mind, in order that it may be able to think, in a permanent manner, of extension in two directions, or in one only. The hyperbolical expressions habitually employed by geometers to define these notions tend to convey false ideas of them; but, examined in themselves, they have no other object than to permit us to reason with facility respecting these two kinds of extension, making complete abstraction of that which ought not to be taken into consideration. Now for this it is sufficient to conceive the dimension which we wish to eliminate as becoming gradually smaller and smaller, the two others remaining the same, until it arrives at such a degree of tenuity that it can no longer fix the attention. It is thus that we naturally acquire the real idea of a surface, and, by a second analogous operation, the idea of a line, by repeating for breadth what we had at first done for thickness. Finally, if we again repeat the same operation, we arrive at the idea of a point, or of an extension considered only with reference to its place, abstraction being made of all magnitude, and designed consequently to determine positions.

Surfaces evidently have, moreover, the general property of exactly circumscribing volumes; and in the same way, lines, in their turn, circumscribe surfaces and are limited by points. But this consideration, to which too much importance is often given, is only a secondary one.