It is as impossible for us to represent to ourselves external bodies in geometric s.p.a.ce, as it is for a painter to paint on a plane canvas objects with their three dimensions.

Perceptual s.p.a.ce is only an image of geometric s.p.a.ce, an image altered in shape by a sort of perspective, and we can represent to ourselves objects only by bringing them under the laws of this perspective.

Therefore we do not _represent_ to ourselves external bodies in geometric s.p.a.ce, but we _reason_ on these bodies as if they were situated in geometric s.p.a.ce.

When it is said then that we 'localize' such and such an object at such and such a point of s.p.a.ce, what does it mean?

_It simply means that we represent to ourselves the movements it would be necessary to make to reach that object_; and one may not say that to represent to oneself these movements, it is necessary to project the movements themselves in s.p.a.ce and that the notion of s.p.a.ce must, consequently, pre-exist.

When I say that we represent to ourselves these movements, I mean only that we represent to ourselves the muscular sensations which accompany them and which have no geometric character whatever, which consequently do not at all imply the preexistence of the notion of s.p.a.ce.

CHANGE OF STATE AND CHANGE OF POSITION.--But, it will be said, if the idea of geometric s.p.a.ce is not imposed upon our mind, and if, on the other hand, none of our sensations can furnish it, how could it have come into existence?

This is what we have now to examine, and it will take some time, but I can summarize in a few words the attempt at explanation that I am about to develop.

_None of our sensations, isolated, could have conducted us to the idea of s.p.a.ce; we are led to it only in studying the laws, according to which these sensations succeed each other._

We see first that our impressions are subject to change; but among the changes we ascertain we are soon led to make a distinction.

At one time we say that the objects which cause these impressions have changed state, at another time that they have changed position, that they have only been displaced.

Whether an object changes its state or merely its position, this is always translated for us in the same manner: _by a modification in an aggregate of impressions_.

How then could we have been led to distinguish between the two? It is easy to account for. If there has only been a change of position, we can restore the primitive aggregate of impressions by making movements which replace us opposite the mobile object in the same _relative_ situation.

We thus _correct_ the modification that happened and we reestablish the initial state by an inverse modification.

If it is a question of sight, for example, and if an object changes its place before our eye, we can 'follow it with the eye' and maintain its image on the same point of the retina by appropriate movements of the eyeball.

These movements we are conscious of because they are voluntary and because they are accompanied by muscular sensations, but that does not mean that we represent them to ourselves in geometric s.p.a.ce.

So what characterizes change of position, what distinguishes it from change of state, is that it can always be corrected in this way.

It may therefore happen that we pa.s.s from the totality of impressions _A_ to the totality _B_ in two different ways:

1 Involuntarily and without experiencing muscular sensations; this happens when it is the object which changes place;

2 Voluntarily and with muscular sensations; this happens when the object is motionless, but we move so that the object has relative motion with reference to us.

If this be so, the pa.s.sage from the totality _A_ to the totality _B_ is only a change of position.

It follows from this that sight and touch could not have given us the notion of s.p.a.ce without the aid of the 'muscular sense.'

Not only could this notion not be derived from a single sensation or even _from a series of sensations_, but what is more, an _immobile_ being could never have acquired it, since, not being able to _correct_ by his movements the effects of the changes of position of exterior objects, he would have had no reason whatever to distinguish them from changes of state. Just as little could he have acquired it if his motions had not been voluntary or were unaccompanied by any sensations.

CONDITIONS OF COMPENSATION.--How is a like compensation possible, of such sort that two changes, otherwise independent of each other, reciprocally correct each other?

A mind already familiar with geometry would reason as follows: Evidently, if there is to be compensation, the various parts of the external object, on the one hand, and the various sense organs, on the other hand, must be in the same _relative_ position after the double change. And, for that to be the case, the various parts of the external object must likewise have retained in reference to each other the same relative position, and the same must be true of the various parts of our body in regard to each other.

In other words, the external object, in the first change, must be displaced as is a rigid solid, and so must it be with the whole of our body in the second change which corrects the first.

Under these conditions, compensation may take place.

But we who as yet know nothing of geometry, since for us the notion of s.p.a.ce is not yet formed, we can not reason thus, we can not foresee _a priori_ whether compensation is possible. But experience teaches us that it sometimes happens, and it is from this experimental fact that we start to distinguish changes of state from changes of position.

SOLID BODIES AND GEOMETRY.--Among surrounding objects there are some which frequently undergo displacements susceptible of being thus corrected by a correlative movement of our own body; these are the _solid bodies_. The other objects, whose form is variable, only exceptionally undergo like displacements (change of position without change of form). When a body changes its place _and its shape_, we can no longer, by appropriate movements, bring back our sense-organs into the same _relative_ situation with regard to this body; consequently we can no longer reestablish the primitive totality of impressions.

It is only later, and as a consequence of new experiences, that we learn how to decompose the bodies of variable form into smaller elements, such that each is displaced almost in accordance with the same laws as solid bodies. Thus we distinguish 'deformations' from other changes of state; in these deformations, each element undergoes a mere change of position, which can be corrected, but the modification undergone by the aggregate is more profound and is no longer susceptible of correction by a correlative movement.

Such a notion is already very complex and must have been relatively late in appearing; moreover it could not have arisen if the observation of solid bodies had not already taught us to distinguish changes of position.

_Therefore, if there were no solid bodies in nature, there would be no geometry._

Another remark also deserves a moment's attention. Suppose a solid body to occupy successively the positions [alpha] and [beta]; in its first position, it will produce on us the totality of impressions _A_, and in its second position the totality of impressions _B_. Let there be now a second solid body, having qualities entirely different from the first, for example, a different color. Suppose it to pa.s.s from the position [alpha], where it gives us the totality of impressions _A'_, to the position [beta], where it gives the totality of impressions _B'_.

In general, the totality _A_ will have nothing in common with the totality _A'_, nor the totality _B_ with the totality _B'_. The transition from the totality _A_ to the totality _B_ and that from the totality _A'_ to the totality _B'_ are therefore two changes which _in themselves_ have in general nothing in common.

And yet we regard these two changes both as displacements and, furthermore, we consider them as the _same_ displacement. How can that be?

It is simply because they can both be corrected by the _same_ correlative movement of our body.

'Correlative movement' therefore const.i.tutes the _sole connection_ between two phenomena which otherwise we never should have dreamt of likening.

On the other hand, our body, thanks to the number of its articulations and muscles, may make a mult.i.tude of different movements; but all are not capable of 'correcting' a modification of external objects; only those will be capable of it in which our whole body, or at least all those of our sense-organs which come into play, are displaced as a whole, that is, without their relative positions varying, or in the fashion of a solid body.

To summarize:

1 We are led at first to distinguish two categories of phenomena:

Some, involuntary, unaccompanied by muscular sensations, are attributed by us to external objects; these are external changes;

Others, opposite in character and attributed by us to the movements of our own body, are internal changes;

2 We notice that certain changes of each of these categories may be corrected by a correlative change of the other category;

3 We distinguish among external changes those which have thus a correlative in the other category; these we call displacements; and just so among the internal changes, we distinguish those which have a correlative in the first category.

Thus are defined, thanks to this reciprocity, a particular cla.s.s of phenomena which we call displacements.

_The laws of these phenomena const.i.tute the object of geometry._

LAW OF h.o.m.oGENEITY.--The first of these laws is the law of h.o.m.ogeneity.

Suppose that, by an external change [alpha], we pa.s.s from the totality of impressions _A_ to the totality _B_, then that this change [alpha] is corrected by a correlative voluntary movement [beta], so that we are brought back to the totality _A_.

Suppose now that another external change [alpha]' makes us pa.s.s anew from the totality _A_ to the totality _B_.

Experience teaches us that this change [alpha]' is, like [alpha], susceptible of being corrected by a correlative voluntary movement [beta]' and that this movement [beta]' corresponds to the same muscular sensations as the movement [beta] which corrected [alpha].

This fact is usually enunciated by saying that _s.p.a.ce is h.o.m.ogeneous and isotropic_.