GENERAL VIEW OF GEOMETRY.
Its true Nature. After the general exposition of the philosophical character of concrete mathematics, compared with that of abstract mathematics, given in the introductory chapter, it need not here be shown in a special manner that geometry must be considered as a true natural science, only much more simple, and therefore much more perfect, than any other. This necessary perfection of geometry, obtained essentially by the application of mathematical analysis, which it so eminently admits, is apt to produce erroneous views of the real nature of this fundamental science, which most minds at present conceive to be a purely logical science quite independent of observation. It is nevertheless evident, to any one who examines with attention the character of geometrical reasonings, even in the present state of abstract geometry, that, although the facts which are considered in it are much more closely united than those relating to any other science, still there always exists, with respect to every body studied by geometers, a certain number of primitive phenomena, which, since they are not established by any reasoning, must be founded on observation alone, and which form the necessary basis of all the deductions.
The scientific superiority of geometry arises from the phenomena which it considers being necessarily the most universal and the most simple of all. Not only may all the bodies of nature give rise to geometrical inquiries, as well as mechanical ones, but still farther, geometrical phenomena would still exist, even though all the parts of the universe should be considered as immovable. Geometry is then, by its nature, more general than mechanics. At the same time, its phenomena are more simple, for they are evidently independent of mechanical phenomena, while these latter are always complicated with the former. The same relations hold good in comparing geometry with abstract thermology.
For these reasons, in our classification we have made geometry the first part of concrete mathematics; that part the study of which, in addition to its own importance, serves as the indispensable basis of all the rest.
Before considering directly the philosophical study of the different orders of inquiries which constitute our present geometry, we should obtain a clear and exact idea of the general destination of that science, viewed in all its bearings. Such is the object of this chapter.
Definition. Geometry is commonly defined in a very vague and entirely improper manner, as being the science of extension. An improvement on this would be to say that geometry has for its object the measurement of extension; but such an explanation would be very insufficient, although at bottom correct, and would be far from giving any idea of the true general character of geometrical science.
To do this, I think that I should first explain two fundamental ideas, which, very simple in themselves, have been singularly obscured by the employment of metaphysical considerations.
The Idea of Space. The first is that of Space. This conception properly consists simply in this, that, instead of considering extension in the bodies themselves, we view it in an indefinite medium, which we regard as containing all the bodies of the universe. This notion is naturally suggested to us by observation, when we think of the impression which a body would leave in a fluid in which it had been placed. It is clear, in fact, that,