Part 11 (1/2)
BOOK II.
GEOMETRY.
CHAPTER I.
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 a.n.a.lysis, 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 cla.s.sification 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 const.i.tute 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 s.p.a.ce._ The first is that of _s.p.a.ce_. 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, as regards its geometrical relations, such an _impression_ may be subst.i.tuted for the body itself, without altering the reasonings respecting it. As to the physical nature of this indefinite _s.p.a.ce_, we are spontaneously led to represent it to ourselves, as being entirely a.n.a.logous to the actual medium in which we live; so that if this medium was liquid instead of gaseous, our geometrical _s.p.a.ce_ would undoubtedly be conceived as liquid also. This circ.u.mstance 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.[13]
[Footnote 13: Lacroix has justly criticised the expression of _solid_, commonly used by geometers to designate a _volume_. It is certain, in fact, that when we wish to consider separately a certain portion of indefinite s.p.a.ce, conceived as gaseous, we mentally solidify its exterior envelope, so that a _line_ and a _surface_ are habitually, to our minds, just as _solid_ as a _volume_. It may also be remarked that most generally, in order that bodies may penetrate one another with more facility, we are obliged to imagine the interior of the _volumes_ to be hollow, which renders still more sensible the impropriety of the word _solid_.]
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 a.n.a.logous 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 circ.u.mscribing volumes; and in the same way, _lines_, in their turn, circ.u.mscribe _surfaces_ and are limited by _points_. But this consideration, to which too much importance is often given, is only a secondary one.
Surfaces and lines are, then, in reality, always conceived with three dimensions; it would be, in fact, impossible to represent to one's self a surface otherwise than as an extremely thin plate, and a line otherwise than as an infinitely fine thread. It is even plain that the degree of tenuity attributed by each individual to the dimensions of which he wishes to make abstraction is not constantly identical, for it must depend on the degree of subtilty of his habitual geometrical observations. This want of uniformity has, besides, no real inconvenience, since it is sufficient, in order that the ideas of surface and of line should satisfy the essential condition of their destination, for each one to represent to himself the dimensions which are to be neglected as being smaller than all those whose magnitude his daily experience gives him occasion to appreciate.
We hence see how devoid of all meaning are the fantastic discussions of metaphysicians upon the foundations of geometry. It should also be remarked that these primordial ideas are habitually presented by geometers in an unphilosophical manner, since, for example, they explain the notions of the different sorts of extent in an order absolutely the inverse of their natural dependence, which often produces the most serious inconveniences in elementary instruction.
THE FINAL OBJECT OF GEOMETRY.
These preliminaries being established, we can proceed directly to the general definition of geometry, continuing to conceive this science as having for its final object the _measurement_ of extension.