Now, Euclidean geometry is, and will remain, the most convenient:

1 Because it is the simplest; and it is so not only in consequence of our mental habits, or of I know not what direct intuition that we may have of Euclidean s.p.a.ce; it is the simplest in itself, just as a polynomial of the first degree is simpler than one of the second; the formulas of spherical trigonometry are more complicated than those of plane trigonometry, and they would still appear so to an a.n.a.lyst ignorant of their geometric signification.

2 Because it accords sufficiently well with the properties of natural solids, those bodies which our hands and our eyes compare and with which we make our instruments of measure.

CHAPTER IV

s.p.a.cE AND GEOMETRY

Let us begin by a little paradox.

Beings with minds like ours, and having the same senses as we, but without previous education, would receive from a suitably chosen external world impressions such that they would be led to construct a geometry other than that of Euclid and to localize the phenomena of that external world in a non-Euclidean s.p.a.ce, or even in a s.p.a.ce of four dimensions.

As for us, whose education has been accomplished by our actual world, if we were suddenly transported into this new world, we should have no difficulty in referring its phenomena to our Euclidean s.p.a.ce.

Conversely, if these beings were transported into our environment, they would be led to relate our phenomena to non-Euclidean s.p.a.ce.

Nay more; with a little effort we likewise could do it. A person who should devote his existence to it might perhaps attain to a realization of the fourth dimension.

GEOMETRIC s.p.a.cE AND PERCEPTUAL s.p.a.cE.--It is often said the images of external objects are localized in s.p.a.ce, even that they can not be formed except on this condition. It is also said that this s.p.a.ce, which serves thus as a ready prepared _frame_ for our sensations and our representations, is identical with that of the geometers, of which it possesses all the properties.

To all the good minds who think thus, the preceding statement must have appeared quite extraordinary. But let us see whether they are not subject to an illusion that a more profound a.n.a.lysis would dissipate.

What, first of all, are the properties of s.p.a.ce, properly so called? I mean of that s.p.a.ce which is the object of geometry and which I shall call _geometric s.p.a.ce_.

The following are some of the most essential:

1 It is continuous;

2 It is infinite;

3 It has three dimensions;

4 It is h.o.m.ogeneous, that is to say, all its points are identical one with another;

5 It is isotropic, that is to say, all the straights which pa.s.s through the same point are identical one with another.

Compare it now to the frame of our representations and our sensations, which I may call _perceptual s.p.a.ce_.

VISUAL s.p.a.cE.--Consider first a purely visual impression, due to an image formed on the bottom of the retina.

A cursory a.n.a.lysis shows us this image as continuous, but as possessing only two dimensions; this already distinguishes from geometric s.p.a.ce what we may call _pure visual s.p.a.ce_.

Besides, this image is enclosed in a limited frame.

Finally, there is another difference not less important: _this pure visual s.p.a.ce is not h.o.m.ogeneous_. All the points of the retina, aside from the images which may there be formed, do not play the same role.

The yellow spot can in no way be regarded as identical with a point on the border of the retina. In fact, not only does the same object produce there much more vivid impressions, but in every _limited_ frame the point occupying the center of the frame will never appear as equivalent to a point near one of the borders.

No doubt a more profound a.n.a.lysis would show us that this continuity of visual s.p.a.ce and its two dimensions are only an illusion; it would separate it therefore still more from geometric s.p.a.ce, but we shall not dwell on this remark.

Sight, however, enables us to judge of distances and consequently to perceive a third dimension. But every one knows that this perception of the third dimension reduces itself to the sensation of the effort at accommodation it is necessary to make, and to that of the convergence which must be given to the two eyes, to perceive an object distinctly.

These are muscular sensations altogether different from the visual sensations which have given us the notion of the first two dimensions.

The third dimension therefore will not appear to us as playing the same role as the other two. What may be called _complete visual s.p.a.ce_ is therefore not an isotropic s.p.a.ce.

It has, it is true, precisely three dimensions, which means that the elements of our visual sensations (those at least which combine to form the notion of extension) will be completely defined when three of them are known; to use the language of mathematics, they will be functions of three independent variables.

But examine the matter a little more closely. The third dimension is revealed to us in two different ways: by the effort of accommodation and by the convergence of the eyes.

No doubt these two indications are always concordant, there is a constant relation between them, or, in mathematical terms, the two variables which measure these two muscular sensations do not appear to us as independent; or again, to avoid an appeal to mathematical notions already rather refined, we may go back to the language of the preceding chapter and enunciate the same fact as follows: If two sensations of convergence, _A_ and _B_, are indistinguishable, the two sensations of accommodation, _A'_ and _B'_, which respectively accompany them, will be equally indistinguishable.

But here we have, so to speak, an experimental fact; _a priori_ nothing prevents our supposing the contrary, and if the contrary takes place, if these two muscular sensations vary independently of one another, we shall have to take account of one more independent variable, and 'complete visual s.p.a.ce' will appear to us as a physical continuum of four dimensions.

We have here even, I will add, a fact of _external_ experience. Nothing prevents our supposing that a being with a mind like ours, having the same sense organs that we have, may be placed in a world where light would only reach him after having traversed reflecting media of complicated form. The two indications which serve us in judging distances would cease to be connected by a constant relation. A being who should achieve in such a world the education of his senses would no doubt attribute four dimensions to complete visual s.p.a.ce.

TACTILE s.p.a.cE AND MOTOR s.p.a.cE.--'Tactile s.p.a.ce' is still more complicated than visual s.p.a.ce and farther removed from geometric s.p.a.ce.

It is superfluous to repeat for touch the discussion I have given for sight.

But apart from the data of sight and touch, there are other sensations which contribute as much and more than they to the genesis of the notion of s.p.a.ce. These are known to every one; they accompany all our movements, and are usually called muscular sensations.

The corresponding frame const.i.tutes what may be called _motor s.p.a.ce_.

Each muscle gives rise to a special sensation capable of augmenting or of diminishing, so that the totality of our muscular sensations will depend upon as many variables as we have muscles. From this point of view, _motor s.p.a.ce would have as many dimensions as we have muscles_.

I know it will be said that if the muscular sensations contribute to form the notion of s.p.a.ce, it is because we have the sense of the _direction_ of each movement and that it makes an integrant part of the sensation. If this were so, if a muscular sensation could not arise except accompanied by this geometric sense of direction, geometric s.p.a.ce would indeed be a form imposed upon our sensibility.

But I perceive nothing at all of this when I a.n.a.lyze my sensations.

What I do see is that the sensations which correspond to movements in the same direction are connected in my mind by a mere _a.s.sociation of ideas_. It is to this a.s.sociation that what we call 'the sense of direction' is reducible. This feeling therefore can not be found in a single sensation.

This a.s.sociation is extremely complex, for the contraction of the same muscle may correspond, according to the position of the limbs, to movements of very different direction.

Besides, it is evidently acquired; it is, like all a.s.sociations of ideas, the result of a _habit_; this habit itself results from very numerous _experiences_; without any doubt, if the education of our senses had been accomplished in a different environment, where we should have been subjected to different impressions, contrary habits would have arisen and our muscular sensations would have been a.s.sociated according to other laws.

CHARACTERISTICS OF PERCEPTUAL s.p.a.cE.--Thus perceptual s.p.a.ce, under its triple form, visual, tactile and motor, is essentially different from geometric s.p.a.ce.

It is neither h.o.m.ogeneous, nor isotropic; one can not even say that it has three dimensions.

It is often said that we 'project' into geometric s.p.a.ce the objects of our external perception; that we 'localize' them.

Has this a meaning, and if so what?

Does it mean that we _represent_ to ourselves external objects in geometric s.p.a.ce?

Our representations are only the reproduction of our sensations; they can therefore be ranged only in the same frame as these, that is to say, in perceptual s.p.a.ce.