But even this, what does it mean? The impressions that have come to us from these objects have followed paths absolutely different, the optic nerve for the object _A_, the acoustic nerve for the object _B_.

They have nothing in common from the qualitative point of view. The representations we are able to make of these two objects are absolutely heterogeneous, irreducible one to the other. Only I know that to reach the object _A_ I have just to extend the right arm in a certain way; even when I abstain from doing it, I represent to myself the muscular sensations and other a.n.a.logous sensations which would accompany this extension, and this representation is a.s.sociated with that of the object _A_.

Now, I likewise know I can reach the object _B_ by extending my right arm in the same manner, an extension accompanied by the same train of muscular sensations. And when I say these two objects occupy the same place, I mean nothing more.

I also know I could have reached the object _A_ by another appropriate motion of the left arm and I represent to myself the muscular sensations which would have accompanied this movement; and by this same motion of the left arm, accompanied by the same sensations, I likewise could have reached the object _B_.

And that is very important, since thus I can defend myself against dangers menacing me from the object _A_ or the object _B_. With each of the blows we can be hit, nature has a.s.sociated one or more parries which permit of our guarding ourselves. The same parry may respond to several strokes; and so it is, for instance, that the same motion of the right arm would have allowed us to guard at the instant [alpha] against the object _A_ and at the instant [beta] against the object _B_. Just so, the same stroke can be parried in several ways, and we have said, for instance, the object _A_ could be reached indifferently either by a certain movement of the right arm or by a certain movement of the left arm.

All these parries have nothing in common except warding off the same blow, and this it is, and nothing else, which is meant when we say they are movements terminating at the same point of s.p.a.ce. Just so, these objects, of which we say they occupy the same point of s.p.a.ce, have nothing in common, except that the same parry guards against them.

Or, if you choose, imagine innumerable telegraph wires, some centripetal, others centrifugal. The centripetal wires warn us of accidents happening without; the centrifugal wires carry the reparation.

Connections are so established that when a centripetal wire is traversed by a current this acts on a relay and so starts a current in one of the centrifugal wires, and things are so arranged that several centripetal wires may act on the same centrifugal wire if the same remedy suits several ills, and that a centripetal wire may agitate different centrifugal wires, either simultaneously or in lieu one of the other when the same ill may be cured by several remedies.

It is this complex system of a.s.sociations, it is this table of distribution, so to speak, which is all our geometry or, if you wish, all in our geometry that is instinctive. What we call our intuition of the straight line or of distance is the consciousness we have of these a.s.sociations and of their imperious character.

And it is easy to understand whence comes this imperious character itself. An a.s.sociation will seem to us by so much the more indestructible as it is more ancient. But these a.s.sociations are not, for the most part, conquests of the individual, since their trace is seen in the new-born babe: they are conquests of the race. Natural selection had to bring about these conquests by so much the more quickly as they were the more necessary.

On this account, those of which we speak must have been of the earliest in date, since without them the defense of the organism would have been impossible. From the time when the cellules were no longer merely juxtaposed, but were called upon to give mutual aid, it was needful that a mechanism organize a.n.a.logous to what we have described, so that this aid miss not its way, but forestall the peril.

When a frog is decapitated, and a drop of acid is placed on a point of its skin, it seeks to wipe off the acid with the nearest foot, and, if this foot be amputated, it sweeps it off with the foot of the opposite side. There we have the double parry of which I have just spoken, allowing the combating of an ill by a second remedy, if the first fails.

And it is this multiplicity of parries, and the resulting coordination, which is s.p.a.ce.

We see to what depths of the unconscious we must descend to find the first traces of these spatial a.s.sociations, since only the inferior parts of the nervous system are involved. Why be astonished then at the resistance we oppose to every attempt made to dissociate what so long has been a.s.sociated? Now, it is just this resistance that we call the evidence for the geometric truths; this evidence is nothing but the repugnance we feel toward breaking with very old habits which have always proved good.

III

The s.p.a.ce so created is only a little s.p.a.ce extending no farther than my arm can reach; the intervention of the memory is necessary to push back its limits. There are points which will remain out of my reach, whatever effort I make to stretch forth my hand; if I were fastened to the ground like a hydra polyp, for instance, which can only extend its tentacles, all these points would be outside of s.p.a.ce, since the sensations we could experience from the action of bodies there situated, would be a.s.sociated with the idea of no movement allowing us to reach them, of no appropriate parry. These sensations would not seem to us to have any spatial character and we should not seek to localize them.

But we are not fixed to the ground like the lower animals; we can, if the enemy be too far away, advance toward him first and extend the hand when we are sufficiently near. This is still a parry, but a parry at long range. On the other hand, it is a complex parry, and into the representation we make of it enter the representation of the muscular sensations caused by the movements of the legs, that of the muscular sensations caused by the final movement of the arm, that of the sensations of the semicircular ca.n.a.ls, etc. We must, besides, represent to ourselves, not a complex of simultaneous sensations, but a complex of successive sensations, following each other in a determinate order, and this is why I have just said the intervention of memory was necessary.

Notice moreover that, to reach the same point, I may approach nearer the mark to be attained, so as to have to stretch my arm less. What more? It is not one, it is a thousand parries I can oppose to the same danger.

All these parries are made of sensations which may have nothing in common and yet we regard them as defining the same point of s.p.a.ce, since they may respond to the same danger and are all a.s.sociated with the notion of this danger. It is the potentiality of warding off the same stroke which makes the unity of these different parries, as it is the possibility of being parried in the same way which makes the unity of the strokes so different in kind, which may menace us from the same point of s.p.a.ce. It is this double unity which makes the individuality of each point of s.p.a.ce, and, in the notion of point, there is nothing else.

The s.p.a.ce before considered, which might be called _restricted s.p.a.ce_, was referred to coordinate axes bound to my body; these axes were fixed, since my body did not move and only my members were displaced. What are the axes to which we naturally refer the _extended s.p.a.ce_? that is to say the new s.p.a.ce just defined. We define a point by the sequence of movements to be made to reach it, starting from a certain initial position of the body. The axes are therefore fixed to this initial position of the body.

But the position I call initial may be arbitrarily chosen among all the positions my body has successively occupied; if the memory more or less unconscious of these successive positions is necessary for the genesis of the notion of s.p.a.ce, this memory may go back more or less far into the past. Thence results in the definition itself of s.p.a.ce a certain indetermination, and it is precisely this indetermination which const.i.tutes its relativity.

There is no absolute s.p.a.ce, there is only s.p.a.ce relative to a certain initial position of the body. For a conscious being fixed to the ground like the lower animals, and consequently knowing only restricted s.p.a.ce, s.p.a.ce would still be relative (since it would have reference to his body), but this being would not be conscious of this relativity, because the axes of reference for this restricted s.p.a.ce would be unchanging!

Doubtless the rock to which this being would be fettered would not be motionless, since it would be carried along in the movement of our planet; for us consequently these axes would change at each instant; but for him they would be changeless. We have the faculty of referring our extended s.p.a.ce now to the position _A_ of our body, considered as initial, again to the position _B_, which it had some moments afterward, and which we are free to regard in its turn as initial; we make therefore at each instant unconscious transformations of coordinates.

This faculty would be lacking in our imaginary being, and from not having traveled, he would think s.p.a.ce absolute. At every instant, his system of axes would be imposed upon him; this system would have to change greatly in reality, but for him it would be always the same, since it would be always the _only_ system. Quite otherwise is it with us, who at each instant have many systems between which we may choose at will, on condition of going back by memory more or less far into the past.

This is not all; restricted s.p.a.ce would not be h.o.m.ogeneous; the different points of this s.p.a.ce could not be regarded as equivalent, since some could be reached only at the cost of the greatest efforts, while others could be easily attained. On the contrary, our extended s.p.a.ce seems to us h.o.m.ogeneous, and we say all its points are equivalent.

What does that mean?

If we start from a certain place _A_, we can, from this position, make certain movements, _M_, characterized by a certain complex of muscular sensations. But, starting from another position, _B_, we make movements _M'_ characterized by the same muscular sensations. Let _a_, then, be the situation of a certain point of the body, the end of the index finger of the right hand for example, in the initial position _A_, and _b_ the situation of this same index when, starting from this position _A_, we have made the motions _M_. Afterwards, let _a'_ be the situation of this index in the position _B_, and _b'_ its situation when, starting from the position _B_, we have made the motions _M'_.

Well, I am accustomed to say that the points of s.p.a.ce _a_ and _b_ are related to each other just as the points _a'_ and _b'_, and this simply means that the two series of movements _M_ and _M'_ are accompanied by the same muscular sensations. And as I am conscious that, in pa.s.sing from the position _A_ to the position _B_, my body has remained capable of the same movements, I know there is a point of s.p.a.ce related to the point _a'_ just as any point _b_ is to the point _a_, so that the two points _a_ and _a'_ are equivalent. This is what is called the h.o.m.ogeneity of s.p.a.ce. And, at the same time, this is why s.p.a.ce is relative, since its properties remain the same whether it be referred to the axes _A_ or to the axes _B_. So that the relativity of s.p.a.ce and its h.o.m.ogeneity are one sole and same thing.

Now, if I wish to pa.s.s to the great s.p.a.ce, which no longer serves only for me, but where I may lodge the universe, I get there by an act of imagination. I imagine how a giant would feel who could reach the planets in a few steps; or, if you choose, what I myself should feel in presence of a miniature world where these planets were replaced by little b.a.l.l.s, while on one of these little b.a.l.l.s moved a liliputian I should call myself. But this act of imagination would be impossible for me had I not previously constructed my restricted s.p.a.ce and my extended s.p.a.ce for my own use.

IV

Why now have all these s.p.a.ces three dimensions? Go back to the "table of distribution" of which we have spoken. We have on the one side the list of the different possible dangers; designate them by _A1_, _A2_, etc.; and, on the other side, the list of the different remedies which I shall call in the same way _B1_, _B2_, etc. We have then connections between the contact studs or push b.u.t.tons of the first list and those of the second, so that when, for instance, the announcer of danger _A3_ functions, it will put or may put in action the relay corresponding to the parry _B4_.

As I have spoken above of centripetal or centrifugal wires, I fear lest one see in all this, not a simple comparison, but a description of the nervous system. Such is not my thought, and that for several reasons: first I should not permit myself to put forth an opinion on the structure of the nervous system which I do not know, while those who have studied it speak only circ.u.mspectly; again because, despite my incompetence, I well know this scheme would be too simplistic; and finally because on my list of parries, some would figure very complex, which might even, in the case of extended s.p.a.ce, as we have seen above, consist of many steps followed by a movement of the arm. It is not a question then of physical connection between two real conductors but of psychologic a.s.sociation between two series of sensations.

If _A1_ and _A2_ for instance are both a.s.sociated with the parry _B1_, and if _A1_ is likewise a.s.sociated with the parry _B2_, it will generally happen that _A2_ and _B2_ will also themselves be a.s.sociated.

If this fundamental law were not generally true, there would exist only an immense confusion and there would be nothing resembling a conception of s.p.a.ce or a geometry. How in fact have we defined a point of s.p.a.ce. We have done it in two ways: it is on the one hand the aggregate of announcers _A_ in connection with the same parry _B_; it is on the other hand the aggregate of parries _B_ in connection with the same announcer _A_. If our law was not true, we should say _A1_ and _A2_ correspond to the same point since they are both in connection with _B1_; but we should likewise say they do not correspond to the same point, since _A1_ would be in connection with _B2_ and the same would not be true of _A2_.

This would be a contradiction.

But, from another side, if the law were rigorously and always true, s.p.a.ce would be very different from what it is. We should have categories strongly contrasted between which would be portioned out on the one hand the announcers _A_, on the other hand the parries _B_; these categories would be excessively numerous, but they would be entirely separated one from another. s.p.a.ce would be composed of points very numerous, but discrete; it would be _discontinuous_. There would be no reason for ranging these points in one order rather than another, nor consequently for attributing to s.p.a.ce three dimensions.

But it is not so; permit me to resume for a moment the language of those who already know geometry; this is quite proper since this is the language best understood by those I wish to make understand me.

When I desire to parry the stroke, I seek to attain the point whence comes this blow, but it suffices that I approach quite near. Then the parry _B1_ may answer for _A1_ and for _A2_, if the point which corresponds to _B1_ is sufficiently near both to that corresponding to _A1_ and to that corresponding to _A2_. But it may happen that the point corresponding to another parry _B2_ may be sufficiently near to the point corresponding to A1 and not sufficiently near the point corresponding to _A2_; so that the parry _B2_ may answer for _A1_ without answering for _A2_. For one who does not yet know geometry, this translates itself simply by a derogation of the law stated above. And then things will happen thus:

Two parries _B1_ and _B2_ will be a.s.sociated with the same warning _A1_ and with a large number of warnings which we shall range in the same category as _A1_ and which we shall make correspond to the same point of s.p.a.ce. But we may find warnings _A2_ which will be a.s.sociated with _B2_ without being a.s.sociated with _B1_, and which in compensation will be a.s.sociated with _B3_, which _B3_ was not a.s.sociated with _A1_, and so forth, so that we may write the series

_B1_, _A1_, _B2_, _A2_, _B3_, _A3_, _B4_, _A4_,

where each term is a.s.sociated with the following and the preceding, but not with the terms several places away.

Needless to add that each of the terms of these series is not isolated, but forms part of a very numerous category of other warnings or of other parries which have the same connections as it, and which may be regarded as belonging to the same point of s.p.a.ce.

The fundamental law, though admitting of exceptions, remains therefore almost always true. Only, in consequence of these exceptions, these categories, in place of being entirely separated, encroach partially one upon another and mutually penetrate in a certain measure, so that s.p.a.ce becomes continuous.

On the other hand, the order in which these categories are to be ranged is no longer arbitrary, and if we refer to the preceding series, we see it is necessary to put _B2_ between _A1_ and _A2_ and consequently between _B1_ and _B3_ and that we could not for instance put it between _B3_ and _B4_.

There is therefore an order in which are naturally arranged our categories which correspond to the points of s.p.a.ce, and experience teaches us that this order presents itself under the form of a table of triple entry, and this is why s.p.a.ce has three dimensions.

V

So the characteristic property of s.p.a.ce, that of having three dimensions, is only a property of our table of distribution, an internal property of the human intelligence, so to speak. It would suffice to destroy certain of these connections, that is to say of the a.s.sociations of ideas to give a different table of distribution, and that might be enough for s.p.a.ce to acquire a fourth dimension.

Some persons will be astonished at such a result. The external world, they will think, should count for something. If the number of dimensions comes from the way we are made, there might be thinking beings living in our world, but who might be made differently from us and who would believe s.p.a.ce has more or less than three dimensions. Has not M. de Cyon said that the j.a.panese mice, having only two pair of semicircular ca.n.a.ls, believe that s.p.a.ce is two-dimensional? And then this thinking being, if he is capable of constructing a physics, would he not make a physics of two or of four dimensions, and which in a sense would still be the same as ours, since it would be the description of the same world in another language?

It seems in fact that it would be possible to translate our physics into the language of geometry of four dimensions; to attempt this translation would be to take great pains for little profit, and I shall confine myself to citing the mechanics of Hertz where we have something a.n.a.logous. However, it seems that the translation would always be less simple than the text, and that it would always have the air of a translation, that the language of three dimensions seems the better fitted to the description of our world, although this description can be rigorously made in another idiom. Besides, our table of distribution was not made at random. There is connection between the warning _A1_ and the parry _B1_, this is an internal property of our intelligence; but why this connection? It is because the parry _B1_ affords means effectively to guard against the danger _A1_; and this is a fact exterior to us, this is a property of the exterior world. Our table of distribution is therefore only the translation of an aggregate of exterior facts; if it has three dimensions, this is because it has adapted itself to a world having certain properties; and the chief of these properties is that there exist natural solids whose displacements follow sensibly the laws we call laws of motion of rigid solids. If therefore the language of three dimensions is that which permits us most easily to describe our world, we should not be astonished; this language is copied from our table of distribution; and it is in order to be able to live in this world that this table has been established.

I have said we could conceive, living in our world, thinking beings whose table of distribution would be four-dimensional and who consequently would think in hypers.p.a.ce. It is not certain however that such beings, admitting they were born there, could live there and defend themselves against the thousand dangers by which they would there be a.s.sailed.

VI

A few remarks to end with. There is a striking contrast between the roughness of this primitive geometry, reducible to what I call a table of distribution, and the infinite precision of the geometers' geometry.

And yet this is born of that; but not of that alone; it must be made fecund by the faculty we have of constructing mathematical concepts, such as that of group, for instance; it was needful to seek among the pure concepts that which best adapts itself to this rough s.p.a.ce whose genesis I have sought to explain and which is common to us and the higher animals.

The evidence for certain geometric postulates, we have said, is only our repugnance to renouncing very old habits. But these postulates are infinitely precise, while these habits have something about them essentially pliant. When we wish to think, we need postulates infinitely precise, since this is the only way to avoid contradiction; but among all the possible systems of postulates, there are some we dislike to choose because they are not sufficiently in accord with our habits; however pliant, however elastic they may be, these have a limit of elasticity.