# Popular Science Monthly/Volume 69/December 1906/The Value of Science: The Notion of Displacement IV

 THE VALUE OF SCIENCE
By M. H. POINCARE

MEMBER OF THE INSTITUTE OF FRANCE

5. The Notion of Displacement

I HAVE shown in 'Science and Hypothesis' the preponderant role played by the movements of our body in the genesis of the notion of space. For a being completely immovable there would be neither space nor geometry; in vain would exterior objects be displaced about him, the variations which these displacements would make in his impressions would not be attributed by this being to changes of position, but to simple changes of state; this being would have no means of distinguishing these two sorts of changes, and this distinction, fundamental for us, would have no meaning for him.

The movements that we impress upon our members have as effect the varying of the impressions produced on our senses by external objects; other causes may likewise make them vary; but we are led to distinguish the changes produced by our own motions and we easily discriminate them for two reasons: (1) because they are voluntary; (2) because they are accompanied by muscular sensations.

So we naturally divide the changes that our impressions may undergo into two categories to which perhaps I have given an inappropriate designation: (1) the internal changes, which are voluntary and accompanied by muscular sensations; (2) the external changes, having the opposite characteristics.

We then observe that among the external changes are some which can be corrected, thanks to an internal change which brings everything back to the primitive state; others can not be corrected in this way (it is thus that when an exterior object is displaced, we may then by changing our own position replace ourselves as regards this object in the same relative position as before, so as to reestablish the original aggregate of impressions; if this object was not displaced, but changed its state, that is impossible). Thence comes a new distinction among external changes: those which may be so corrected we call changes of position; and the others, changes of state.

Think, for example, of a sphere with one hemisphere blue and the other red; it first presents to us the blue hemisphere, then it so revolves as to present the red hemisphere. Now think of a spherical vase containing a blue liquid which becomes red in consequence of a chemical reaction. In both cases the sensation of red has replaced that of blue; our senses have experienced the same impressions which have succeeded each other in the same order, and yet these two changes are regarded by us as very different; the first is a displacement, the second a change of state. Why? Because in the first case it is sufficient for me to go around the sphere to place myself opposite the red hemisphere and reestablish the original red sensation.

Still more; if the two hemispheres, in place of being red and blue, had been yellow and green, how should I have interpreted the revolution of the sphere? Before, the red succeeded the blue, now the green succeeds the yellow; and yet I say that the two spheres have undergone the same revolution, that each has turned about its axis; yet I can not say that the green is to yellow as the red is to blue; how then am I led to decide that the two spheres have undergone the same displacement? Evidently because, in one case as in the other, I am able to reestablish the original sensation by going around the sphere, by making the same movements, and I know that I have made the same movements because I have felt the same muscular sensations; to know it, I do not need, therefore, to know geometry in advance and to represent to myself the movements of my body in geometric space.

Another example: An object is displaced before my eye; its image was first formed at the center of the retina; then it is formed at the border; the old sensation was carried to me by a nerve fiber ending at the center of the retina; the new sensation is carried to me by another nerve fiber starting from the border of the retina; these two sensations are qualitatively different; otherwise, how could I distinguish them?

Why then am I led to decide that these two sensations, qualitatively different, represent the same image, which has been displaced? It is because I can follow the object with the eye and by a displacement of the eye, voluntary and accompanied by muscular sensations, bring back the image to the center of the retina and reestablish the primitive sensation.

I suppose that the image of a red object has gone from the center A to the border B of the retina, then that the image of a blue object goes in its turn from the center A to the border B of the retina; I shall decide that these two objects have undergone the same displacement. Why? Because in both cases I shall have been able to reestablish the primitive sensation, and that to do it I shall have had to execute the same movement of the eye, and I shall know that my eye has executed the same movement because I shall have felt the same muscular sensations.

If I could not move my eye, should I have any reason to suppose that the sensation of red at the center of the retina is to the sensation of red at the border of the retina as that of blue at the center is to that of blue at the border? I should only have four sensations qualitatively different, and if I were asked if they are connected by the proportion I have just stated, the question would seem to me ridiculous, just as if I were asked if there is an analogous proportion between an auditory sensation, a tactile sensation and an olfactory sensation.

Let us now consider the internal changes, that is, those which are produced by the voluntary movements of our body and which are accompanied by muscular changes. They give rise to the two following observations, analogous to those we have just made on the subject of external changes.

1. I may suppose that my body has moved from one point to another but that the same attitude is retained; all the parts of the body have therefore retained or resumed the same relative situation, although their absolute situation in space may have varied. I may suppose that not only has the position of my body changed, but that its attitude is no longer the same, that, for instance, my arms which before were folded are now stretched out.

I should therefore distinguish the simple changes of position without change of attitude, and the changes of attitude. Both would appear to me under form of muscular sensations. How then am I led to distinguish them? It is that the first may serve to correct an external change, and that the others can not, or at least can only give an imperfect correction.

This fact I proceed to explain as I would explain it to some one who already knew geometry, but it need not thence be concluded that it is necessary already to know geometry to make this distinction; before knowing geometry I ascertain the fact (experimentally, so to speak), without being able to explain it. But merely to make the distinction between the two kinds of change, I do not need to explain the fact, it suffices me to ascertain it.

However that may be, the explanation is easy. Suppose that an exterior object is displaced; if we wish the different parts of our body to resume with regard to this object their initial relative position, it is necessary that these different parts should have resumed likewise their initial relative position with regard to one another. Only the internal changes which satisfy this latter condition will be capable of correcting the external change produced by the displacement of that object. If, therefore, the relative position of my eye with regard to my finger has changed, I shall still be able to replace the eye in its initial relative situation with regard to the object and reestablish thus the primitive visual sensations, but then the relative position of the finger with regard to the object will have changed and the tactile sensations will not be reestablished.

2. We ascertain likewise that the same external change may be corrected by two internal changes corresponding to different muscular sensations. Here again I can ascertain this without knowing geometry: and I have no need of anything else; but I proceed to give the explanation of the fact employing geometrical language. To go from the position ${\displaystyle A}$ to the position ${\displaystyle B}$ I may take several routes. To the first of these routes will correspond a series ${\displaystyle S}$ of muscular sensations; to a second route will correspond another series ${\displaystyle S''}$ of muscular sensations which generally will be completely different, since other muscles will be used.

How am I led to regard these two series ${\displaystyle S}$ and ${\displaystyle S''}$ as corresponding to the same displacement ${\displaystyle AB}$? It is because these two series are capable of correcting the same external change. Apart from that, they have nothing in common.

Let us now consider two external changes: ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, which shall be, for instance, the rotation of a sphere half blue, half red, and that of a sphere half yellow, half green; these two changes have nothing in common, since the one is for us the passing of blue into red and the other the passing of yellow into green. Consider, on the other hand, two series of internal changes ${\displaystyle S}$ and ${\displaystyle S''}$; like the others, they will have nothing in common. And yet I say that ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ correspond to the same displacement, and that ${\displaystyle S}$ and ${\displaystyle S''}$ correspond also to the same displacement. Why? Simply because ${\displaystyle S}$ can correct ${\displaystyle \beta }$ as well as ${\displaystyle \alpha }$ and because ${\displaystyle \alpha }$ can be corrected by ${\displaystyle S''}$ as well as by ${\displaystyle S}$. And then a question suggests itself: If I have ascertained that ${\displaystyle S}$ corrects ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ and that ${\displaystyle S''}$ corrects ${\displaystyle \alpha }$, am I certain that ${\displaystyle S''}$ likewise corrects ${\displaystyle \beta }$? Experiment alone can teach us whether this law is verified. If it were not verified, at least approximately, there would be no geometry, there would be no space, because we should have no more interest in classifying the internal and external changes as I have just done, and, for instance, in distinguishing changes of state from changes of position.

It is interesting to see what has been the rôle of experience in all this. It has shown me that a certain law is approximately verified. It has not told me wherefore space is, and that it satisfies the condition in question. I knew in fact, before all experience, that space satisfied this condition or that it would not be; nor have I any right to say that experience told me that geometry is possible; I very well see that geometry is possible, since it does not imply contradiction; experience only tells me that geometry is useful.

6. Visual Space

Although motor impressions have had, as I have just explained, an altogether preponderant influence in the genesis of the notion of space, which never would have taken birth without them, it will not be without interest to examine also the role of visual impressions and to investigate how many dimensions 'visual space' has, and for that purpose to apply to these impressions the definition of § 3.

A first difficulty presents itself: consider a red color sensation affecting a certain point of the retina; and on the other hand a blue color sensation affecting the same point of the retina. It is necessary that we have some means of recognizing that these two sensations, qualitatively different, have something in common. Now, according to the considerations expounded in the preceding paragraph, we have been able to recognize this only by the movements of the eye and the observations to which they have given rise. If the eye were immovable, or if we were unconscious of its movements, we should not have been able to recognize that these two sensations, of different quality, had something in common; we should not have been able to disengage from them what gives them a geometric character. The visual sensations, without the muscular sensations, would have nothing geometric, so that it may be said there is no pure visual space.

To do away with this difficulty, consider only sensations of the same nature, red sensations for instance, differing one from another only as regards the point of the retina that they affect. It is clear that I have no reason for making such an arbitrary choice among all the possible visual sensations, for the purpose of uniting in the same class all the sensations of the same color, whatever may be the point of the retina affected. I should never have dreamt of it, had I not before learned, by the means we have just seen, to distinguish changes of state from changes of position, that is, if my eye were immovable. Two sensations of the same color affecting two different parts of the retina would have appeared to me as qualitatively distinct, just as two sensations of different color.

In restricting myself to red sensations, I therefore impose upon myself an artificial limitation and I neglect systematically one whole side of the question; but it is only by this artifice that I am able to analyze visual space without mingling any motor sensation.

Imagine a line traced on the retina and dividing in two its surface; and set apart the red sensations affecting a point of this line, or those differing from them too little to be distinguished from them. The aggregate of these sensations will form a sort of cut that I shall call ${\displaystyle C,}$ and it is clear that this cut suffices to divide the manifold of possible red sensations, and that if I take two red sensations affecting two points situated on one side and the other of the line, I can not pass from one of these sensations to the other in a continuous way without passing at a certain moment through a sensation belonging to the cut.

If, therefore, the cut has n dimensions, the total manifold of my red sensations, or, if you wish, the whole visual space, will have ${\displaystyle n+1}$.

Now, I distinguish the red sensations affecting a point of the cut ${\displaystyle C}$. The assemblage of these sensations will form a new cut ${\displaystyle C'}$. It is clear that this will divide the cut ${\displaystyle C,}$ always giving to the word divide the same meaning.

If, therefore, the cut ${\displaystyle C'}$ has ${\displaystyle n}$ dimensions, the cut ${\displaystyle C}$ will have ${\displaystyle n+1}$ and the whole of visual space ${\displaystyle n+2}$.

If all the red sensations affecting the same point of the retina were regarded as identical, the cut ${\displaystyle C'}$ reducing to a single element would have 0 dimension, and visual space would have 2.

And yet most often it is said that the eye gives us the sense of a third dimension, and enables us in a certain measure to recognize the distance of objects. When we seek to analyze this feeling, we ascertain that it reduces either to the consciousness of the convergence of the eyes, or to that of the effort of accommodation which the ciliary muscle makes to focus the image.

Two red sensations affecting the same point of the retina will therefore be regarded as identical only if they are accompanied by the same sensation of convergence and also by the same sensation of effort of accommodation or at least by sensations of convergence and accommodation so slightly different as to be indistinguishable.

On this account the cut ${\displaystyle C'}$ is itself a continuum and the cut ${\displaystyle C}$ has more than one dimension.

But it happens precisely that experience teaches us that when two visual sensations are accompanied by the same sensation of convergence, they are likewise accompanied by the same sensation of accommodation. If then we form a new cut ${\displaystyle C''}$ with all those of the sensations of the cut ${\displaystyle C'}$, which are accompanied by a certain sensation of convergence, in accordance with the preceding law they will all be indistinguishable and may be regarded as identical. Therefore ${\displaystyle C''}$ will not be a continuum and will have 0 dimension; and as ${\displaystyle C''}$ divides ${\displaystyle C'}$ it will thence result that ${\displaystyle C'}$ has one, ${\displaystyle C}$ two and the whole visual space three dimensions.

But would it be the same if experience had taught us the contrary and if a certain sensation of convergence were not always accompanied by the same sensation of accommodation? In this case two sensations affecting the same point of the retina and accompanied by the same sense of convergence, two sensations which consequently would both appertain to the cut ${\displaystyle C''}$ could nevertheless be distinguished since they would be accompanied by two different sensations of accommodation. Therefore ${\displaystyle C''}$ would be in its turn a continuum and would have one dimension (at least); then ${\displaystyle C'}$ would have two, ${\displaystyle C}$ three and the whole visual space would have four dimensions.

Will it then be said that it is experience which teaches us that space has three dimensions, since it is in setting out from an experimental law that we have come to attribute three to it? But we have therein performed, so to speak, only an experiment in physiology; and as also it would suffice to fit over the eyes glasses of suitable construction to put an end to the accord between the feelings of convergence and of accommodation, are we to say that putting on spectacles is enough to make space have four dimensions and that the optician who constructed them has given one more dimension to space? Evidently not; all we can say is that experience has taught us that it is convenient to attribute three dimensions to space.

But visual space is only one part of space, and in even the notion of this space there is something artificial, as I have explained at the beginning. The real space is motor space and this it is that we shall examine in the following chapter.

Chapter IV. Space and its Three Dimensions

§ 1. The Group of Displacements

Let us sum up briefly the results obtained. We proposed to investigate what was meant in saying that space has three dimensions and we have asked first what is a physical continuum and when it may be said to have n dimensions. If we consider different systems of impressions and compare them with one another, we often recognize that two of these systems of impressions are indistinguishable (which is ordinarily expressed in saying that they are too close to one another, and that our senses are too crude, for us to distinguish them) and we ascertain besides that two of these systems can sometimes be discriminated from one another though indistinguishable from a third system. In that case we say the manifold of these systems of impressions forms a physical continuum ${\displaystyle C}$. And each of these systems is called an element of the continuum ${\displaystyle C}$.

How many dimensions has this continuum? Take first two elements ${\displaystyle A}$ and ${\displaystyle B}$ of ${\displaystyle C,}$ and suppose there exists a series ${\displaystyle \Sigma }$ of elements, all belonging to the continuum ${\displaystyle C,}$ of such a sort that ${\displaystyle A}$ and ${\displaystyle B}$ are the two extreme terms of this series and that each term of the series is indistinguishable from the preceding. If such a series ${\displaystyle \Sigma }$ can be found, we say that ${\displaystyle A}$ and ${\displaystyle B}$ are joined to one another; and if any two elements of ${\displaystyle C}$ are joined to one another, we say that ${\displaystyle C}$ is all of one piece.

Now take on the continuum ${\displaystyle C}$ a certain number of elements in a way altogether arbitrary. The aggregate of these elements will be called a cut. Among the various series ${\displaystyle \Sigma }$ which join ${\displaystyle A}$ to ${\displaystyle B}$, we shall distinguish those of which an element is indistinguishable from one of the elements of the cut (we shall say that these are they which cut the cut) and those of which all the elements are distinguishable from all those of the cut. If all the series ${\displaystyle \Sigma }$ which join ${\displaystyle A}$ to ${\displaystyle B}$ cut the cut, we shall say that ${\displaystyle A}$ and ${\displaystyle B}$ are separated by the cut, and that the cut divides ${\displaystyle C}$. If we can not find on ${\displaystyle C}$ two elements which are separated by the cut, we shall say that the cut does not divide ${\displaystyle C}$.

These definitions laid down, if the continuum ${\displaystyle C}$ can be divided by cuts which do not themselves form a continuum, this continuum ${\displaystyle C}$ has only one dimension; in the contrary case it has several. If a cut forming a continuum of 1 dimension suffices to divide ${\displaystyle C,C}$ will have 2 dimensions; if a cut forming a continuum of 2 dimensions suffices, ${\displaystyle C}$ will have 3 dimensions, etc. Thanks to these definitions, we can always recognize how many dimensions any physical continuum has. It only remains to find a physical continuum which is, so to speak, equivalent to space, of such a sort that to every point of space corresponds an element of this continuum, and that to points of space very near one another correspond indistinguishable elements. Space will have then as many dimensions as this continuum.

The intermediation of this physical continuum, capable of representation, is indispensable; because we can not represent space to ourselves, and that for a multitude of reasons. Space is a mathematical continuum, it is infinite, and we can represent to ourselves only physical continua and finite objects. The different elements of space, which we call points, are all alike, and, to apply our definition, it is necessary that we know how to distinguish the elements from one another, at least if they are not too close. Finally absolute space is nonsense, and it is necessary for us to begin by referring space to a system of axes invariably bound to our body (which we must always suppose put back in the initial attitude).

Then I have sought to form with our visual sensations a physical continuum equivalent to space; that certainly is easy and this example is particularly appropriate for the discussion of the number of dimensions; this discussion has enabled us to see in what measure it is allowable to say that 'visual space' has three dimensions. Only this solution is incomplete and artificial. I have explained why, and it is not on visual space, but on motor space that it is necessary to bring our efforts to bear. I have then recalled what is the origin of the distinction we make between changes of position and changes of state. Among the changes which occur in our impressions, we distinguish, first the internal changes, voluntary and accompanied by muscular sensations, and the external changes, having opposite characteristics. We ascertain that it may happen that an external change may be corrected by an internal change which reestablishes the primitive sensations. The external changes capable of being corrected by an internal change are called changes of position, those not capable of it are called changes of state. The internal changes capable of correcting an external change are called displacements of the whole body; the others are called changes of attitude.

Now let ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ be two external changes, ${\displaystyle \alpha '}$ and ${\displaystyle \beta '}$ two internal changes. Suppose that ${\displaystyle \alpha }$ may be corrected either by ${\displaystyle \alpha '}$ or by ${\displaystyle \beta '}$, and that ${\displaystyle \alpha '}$ can correct either ${\displaystyle \alpha }$ or ${\displaystyle \beta }$; experience tells us then that ${\displaystyle \beta '}$ can likewise correct ${\displaystyle \beta }$. In this case we say that ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ correspond to the same displacement and also that ${\displaystyle \alpha '}$ and ${\displaystyle \beta '}$ correspond to the same displacement. That postulated, we can imagine a physical continuum which we shall call the continuum or group of displacements and which we shall define in the following manner. The elements of this continuum shall be the internal changes capable of correcting an external change. Two of these internal changes ${\displaystyle \alpha '}$ and ${\displaystyle \beta '}$ shall be regarded as indistinguishable: (1) if they are so naturally, that is, if they are too close to one another; (2) if ${\displaystyle \alpha '}$ is capable of correcting the same external change as a third internal change naturally indistinguishable from ${\displaystyle \beta '}$. In this second case, they will be, so to speak, indistinguishable by convention, I mean by agreeing to disregard circumstances which might distinguish them.

Our continuum is now entirely defined, since we know its elements and have fixed under what conditions they may be regarded as indistinguishable. We thus have all that is necessary to apply our definition and determine how many dimensions this continuum has. We shall recognize that it has six. The continuum of displacements is, therefore, not equivalent to space, since the number of dimensions is not the same; it is only related to space. Now how do we know that this continuum of displacements has six dimensions? We know it by experience.

It would be easy to describe the experiments by which we could arrive at this result. It would be seen that in this continuum cuts can be made which divide it and which are continua; that these cuts themselves can be divided by other cuts of the second order which yet are continua, and that this would stop only after cuts of the sixth order which would no longer be continua. From our definitions that would mean that the group of displacements has six dimensions.

That would be easy, I have said, but that would be rather long; and would it not be a little superficial? This group of displacements, we have seen, is related to space, and space could be deduced from it, but it is not equivalent to space, since it has not the same number of dimensions; and when we shall have shown how the notion of this continuum can be formed and how that of space may be deduced from it, it might always be asked why space of three dimensions is much more familiar to us than this continuum of six dimensions, and consequently doubted whether it was by this detour that the notion of space was formed in the human mind.

§ 2. Identity of Two Points

What is a point? How do we know whether two points of space are identical or different? Or, in other words, when I say: The object ${\displaystyle A}$ occupied at the instant ${\displaystyle \alpha ''}$ the point which the object ${\displaystyle B}$ occupies at the instant ${\displaystyle \beta }$, what does that mean?

Such is the problem we set ourselves in the preceding chapter, § 4. As I have explained it, it is not a question of comparing the positions of the objects ${\displaystyle A}$ and ${\displaystyle B}$ in absolute space; the question then would manifestly have no meaning. It is a question of comparing the positions of these two objects with regard to axes invariably bound to my body, supposing always this body replaced in the same attitude.

I suppose that between the instants ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ I have moved neither my body nor my eye, as I know from my muscular sense. Nor have I moved either my head, my arm or my hand. I ascertain that at the instant a impressions that I attributed to the object ${\displaystyle A}$ were transmitted to me, some by one of the fibers of my optic nerve, the others by one of the sensitive tactile nerves of my finger; I ascertain that at the instant ${\displaystyle \beta }$ other impressions which I attribute to the object ${\displaystyle B}$ are transmitted to me, some by this same fiber of the optic nerve, the others by this same tactile nerve.

Here I must pause for an explanation; how am I told that this impression which I attribute to ${\displaystyle A}$, and that which I attribute to ${\displaystyle B}$, impressions which are qualitatively different, are transmitted to me by the same nerve? Must we suppose, to take for example the visual sensations, that ${\displaystyle A}$ produces two simultaneous sensations, a sensation purely luminous ${\displaystyle a}$ and a colored sensation ${\displaystyle a'}$, that ${\displaystyle B}$ produces in the same way simultaneously a luminous sensation ${\displaystyle b}$ and a colored sensation ${\displaystyle b'}$, that if these different sensations are transmitted to me by the same retinal fiber, ${\displaystyle a}$ is identical with ${\displaystyle b}$, but that in general the colored sensations ${\displaystyle a'}$ and ${\displaystyle b'}$ produced by different bodies are different? In that case it would be the identity of the sensation ${\displaystyle a}$ which accompanies ${\displaystyle a'}$ with the sensation ${\displaystyle b}$ which accompanies ${\displaystyle b'}$, which would tell that all these sensations are transmitted to me by the same fiber.

However it may be with this hypothesis and although I am led to prefer to it others considerably more complicated, it is certain that we are told in some way that there is something in common between these sensations ${\displaystyle a+a'}$ and ${\displaystyle b+b'}$, without which we should have no means of recognizing that the object ${\displaystyle B}$ has taken the place of the object ${\displaystyle A}$.

Therefore I do not further insist and I recall the hypothesis I have just made: I suppose that I have ascertained that the impressions which I attribute to ${\displaystyle B}$ are transmitted to me at the instant ${\displaystyle \beta }$ by the same fibers, optic as well as tactile, which, at the instant ${\displaystyle \alpha }$, had transmitted to me the impressions that I attributed to ${\displaystyle A}$. If it is so, we shall not hesitate to declare that the point occupied by ${\displaystyle B}$ at the instant ${\displaystyle \beta }$ is identical with the point occupied by ${\displaystyle A}$ at the instant ${\displaystyle \alpha }$.

I have just enunciated two conditions for these points being identical; one is relative to sight, the other to touch. Let us consider them separately. The first is necessary, but is not sufficient. The second is at once necessary and sufficient. A person knowing geometry could easily explain this in the following manner: Let 0 be the point of the retina where is formed at the instant ${\displaystyle a}$ the image of the body ${\displaystyle A;}$ let ${\displaystyle M}$ be the point of space occupied at the instant ${\displaystyle a}$ by this body ${\displaystyle A;}$ let ${\displaystyle M}$ be the point of space occupied at the instant ${\displaystyle \beta }$ by the body ${\displaystyle B}$. For this body ${\displaystyle B}$ to form its image in 0, it is not necessary that the points ${\displaystyle M}$ and ${\displaystyle M'}$ coincide; since vision acts at a distance, it suffices for the three points 0 ${\displaystyle MM'}$ to be in a straight line. This condition that the two objects form their image on is therefore necessary, but not sufficient for the points ${\displaystyle M}$ and ${\displaystyle M'}$ to coincide. Let now ${\displaystyle P}$ be the point occupied by my finger and where it remains, since it does not budge. As touch does not act at a distance, if the body ${\displaystyle A}$ touches my finger at the instant ${\displaystyle \alpha }$, it is because ${\displaystyle M}$ and ${\displaystyle P}$ coincide; if ${\displaystyle B}$ touches my finger at the instant ${\displaystyle \beta }$, it is because ${\displaystyle M'}$ and ${\displaystyle P}$ coincide. Therefore ${\displaystyle M}$ and ${\displaystyle M'}$ coincide. Thus this condition that if A touches my finger at the instant ${\displaystyle \alpha }$, ${\displaystyle B}$ touches it at the instant ${\displaystyle \beta }$, is at once necessary and sufficient for ${\displaystyle M}$ and ${\displaystyle M'}$ to coincide.

But we who, as yet, do not know geometry can not reason thus; all that we can do is to ascertain experimentally that the first condition relative to sight may be fulfilled without the second, which is relative to touch, but that the second can not be fulfilled without the first.

Suppose experience had taught us the contrary, as might well be; this hypothesis contains nothing absurd. Suppose, therefore, that we had ascertained experimentally that the condition relative to touch may be fulfilled without that of sight being fulfilled, and that, on the contrary, that of sight can not be fulfilled without that of touch being also. It is clear that if this were so we should conclude that it is touch which may be exercised at a distance, and that sight does not operate at a distance.

But this is not all; up to this time I have supposed that to determine the place of an object, I have made use only of my eye and a single finger; but I could just as well have employed other means, for example, all my other fingers.

I suppose that my first finger receives at the instant a a tactile impression which I attribute to the object ${\displaystyle A}$. I make a series of movements, corresponding to a series ${\displaystyle S}$ of muscular sensations. After these movements, at the instant ${\displaystyle \alpha }$, my ${\displaystyle second}$ finger receives a tactile impression that I attribute likewise to ${\displaystyle A}$. Afterwards, at the instant ${\displaystyle \beta }$, without my having budged, as my muscular sense tells me, this same second finger transmits to me anew a tactile impression which I attribute this time to the object ${\displaystyle B;}$ I then make a series of movements, corresponding to a series ${\displaystyle S'}$ of muscular sensations. I know that this series ${\displaystyle S'}$ is the inverse of the series ${\displaystyle S}$ and corresponds to contrary movements. I know this because many previous experiences have shown me that if I made successively the two series of movements corresponding to ${\displaystyle S}$ and to ${\displaystyle S'}$, the primitive impressions would be reestablished, in other words, that the two series mutually compensate. That settled, should I expect that at the instant ${\displaystyle \beta '}$, when the second series of movements is ended, my first finger would feel a tactile impression attributable to the object ${\displaystyle B}$?

To answer this question, those already knowing geometry would reason as follows: There are chances that the object ${\displaystyle A}$ has not budged, between the instants ${\displaystyle \alpha }$ and ${\displaystyle \alpha '}$, nor the object ${\displaystyle B}$ between the instants ${\displaystyle \beta }$ and ${\displaystyle \beta '}$; assume this. At the instant ${\displaystyle \alpha }$, the object ${\displaystyle A}$ occupied a certain point ${\displaystyle M}$ of space. Now at this instant it touched my first finger, and as touch does not operate at a distance, my first finger was likewise at the point ${\displaystyle M}$. I afterward made the series S of movements and at the end of this series, at the instant ${\displaystyle \alpha '}$, I ascertained that the object ${\displaystyle A}$ touched my second finger. I thence conclude that this second finger was then at ${\displaystyle M}$, that is, that the movements ${\displaystyle S}$ had the result of bringing the second finger to the place of the first. At the instant ${\displaystyle \beta }$ the object ${\displaystyle B}$ has come in contact with my second finger: as I have not budged, this second finger has remained at ${\displaystyle M}$; therefore the object ${\displaystyle B}$ has come to ${\displaystyle M}$; by hypothesis it does not budge up to the instant ${\displaystyle \beta '}$. But between the instants ${\displaystyle \beta }$ and ${\displaystyle \beta '}$ I have made the movements ${\displaystyle S'}$; as these movements are the inverse of the movements ${\displaystyle S}$, they must have for effect bringing the first finger in the place of the second. At the instant ${\displaystyle \beta '}$ this first finger will, therefore, be at ${\displaystyle M}$; and as the object ${\displaystyle B}$, is likewise at ${\displaystyle M}$, this object B will touch my first finger. To the question put, the answer should, therefore, be yes.

We who do not yet know geometry can not reason thus; but we ascertain that this anticipation is ordinarily realized; and we can always explain the exceptions by saying that the object ${\displaystyle A}$ has moved between the instants ${\displaystyle \alpha }$ and ${\displaystyle \alpha '}$, or the object ${\displaystyle B}$ between the instants ${\displaystyle \beta }$ and ${\displaystyle \beta '}$.

But could not experience have given a contrary result? Would this contrary result have been absurd in itself? Evidently not. What should we have done then if experience had given this contrary result? Would all geometry thus have become impossible? Not the least in the world. We should have contented ourselves with concluding that touch can operate at a distance.

When I say, touch does not operate at a distance, but sight operates at a distance, this assertion has only one meaning, which is as follows: To recognize whether ${\displaystyle B}$ occupies at the instant ${\displaystyle \beta }$ the point occupied by ${\displaystyle A}$ at the instant ${\displaystyle \alpha '}$, I can use a multitude of different criteria. In one my eye intervenes, in another my first finger, in another my second finger, etc. Well, it is sufficient for the criterion relative to one of my fingers to be satisfied in order that all the others should be satisfied, but it is not sufficient that the criterion relative to the eye should be. This is the sense of my assertion, I content myself with affirming an experimental fact which is ordinarily verified.

At the end of the preceding chapter we analyzed visual space; we saw that to engender this space it is necessary to bring in the retinal sensations, the sensation of convergence and the sensation of accommodation; that if these last two were not always in accord, visual space would have four dimensions in place of three; we also saw that if we brought in only the retinal sensations, we should obtain 'simple visual space,' of only two dimensions. On the other hand, consider tactile space, limiting ourselves to the sensations of a single finger, that is in sum the assemblage of positions this finger can occupy. This tactile space that we shall analyze in the following section and which consequently I ask permission not to consider further for the moment, this tactile space, I say, has three dimensions. Why has space properly so called as many dimensions as tactile space and more than simple visual space? It is because touch does not operate at a distance, while vision does operate at a distance. These two assertions have the same meaning and we have just seen what this is.

Now I return to a point over which I passed rapidly in order not to interrupt the discussion. How do we know that the impressions made on our retina by ${\displaystyle A}$ at the instant ${\displaystyle \alpha }$ and ${\displaystyle B}$ at the instant are transmitted by the same retinal fiber, although these impressions are qualitatively different? I have suggested a simple hypothesis, while adding that other hypotheses, decidedly more complex, would seem to me more probably true. Here then are these hypotheses, of which I have already said a word. How do we know that the impressions produced by the red object ${\displaystyle A}$ at the instant ${\displaystyle \alpha }$, and by the blue object ${\displaystyle B}$ at the ${\displaystyle \beta }$, if these two objects have been imaged on the same point of the retina, have something in common? The simple hypothesis above made may be rejected and we may suppose that these two impressions, qualitatively different, are transmitted by two different though contiguous nervous fibers. What means have I then of knowing that these fibers are contiguous? It is probable that we should have none, if the eye were immovable. It is the movements of the eye which have told us that there is the same relation between the sensation of blue at the point ${\displaystyle A}$ and the sensation of blue at the point ${\displaystyle B}$ of the retina as between the sensation of red at the point ${\displaystyle A}$ and the sensation of red at the point ${\displaystyle B}$. They have shown us, in fact, that the same movements, corresponding to the same muscular sensations, carry us from the first to the second, or from the third to the fourth. I do not emphasize these considerations, which belong, as one sees, to the question of local signs raised by Lotze.