# Popular Science Monthly/Volume 70/January 1907/The Value of Science: Tactile Space V

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* THE VALUE OF SCIENCE
By M. H. POINCARÉ
MEMBER OF THE INSTITUTE OF FRANCE
§ 3. Tactile Space
THUS I know how to recognize the identity of two points, the point occupied by $\mathrm {A}$ at the instant $\alpha$ and the point occupied by $\mathrm {B}$ at the instant $\beta$ but only on one condition, namely, that I have not budged between the instants $\alpha$ and $\beta$. That does not suffice for our object. Suppose, therefore, that I have moved in any manner in the interval between these two instants, how shall I know whether the point occupied by $\mathrm {A}$ at the instant a is identical with the point occupied by $\mathrm {B}$ at the instant $\beta$? I suppose that at the instant $\alpha$, the object $\mathrm {A}$ was in contact with my first finger and that in the same way, at the instant $\beta$, the object $\mathrm {B}$ touches this first finger; but at the same time, my muscular sense has told me that in the interval my body has moved. I have considered above two series of muscular sensations $S$ and $S'$, and I have said it sometimes happens that we are led to consider two such series $S$ and $S'$ as inverse one of the other, because we have often observed that when these two series succeed one another our primitive impressions are reestablished.
If then my muscular sense tells me that I have moved between the two instants $\alpha$ and $\beta$, but so as to feel successively the two series of muscular sensations $S$ and $S'$ that I consider inverses, I shall still conclude, just as if I had not budged, that the points occupied by $\mathrm {A}$ at the instant $\alpha$ and by $\mathrm {B}$ at the instant $\beta$ are identical, if I ascertain that my first finger touches $\mathrm {A}$ at the instant $\alpha$ and $\mathrm {B}$ at the instant $\beta$.
This solution is not yet completely satisfactory, as one will see. Let us see, in fact, how many dimensions it would make us attribute to space. I wish to compare the two points occupied by $\mathrm {A}$ and $\mathrm {B}$ at the instants $\alpha$ and $\beta$, or (what amounts to the same thing since I suppose that my finger touches $\mathrm {A}$ at the instant $\alpha$ and $\mathrm {B}$ at the instant $\beta$) I wish to compare the two points occupied by my finger at the two instants $\alpha$ and $\beta$. The sole means I use for this comparison is the series $\Sigma$ of muscular sensations which have accompanied the movements of my body between these two instants. The different imaginable series $\Sigma$ form evidently a physical continuum of which the number of dimensions is very great. Let us agree, as I have done, not to consider as distinct the two series $\Sigma$ and $\Sigma +s+s'$, when $S$ and $S'$ are inverses one of the other in the sense above given to this word; in spite of this ​agreement, the aggregate of distinct series $\Sigma$ will still form a physical continuum and the number of dimensions will be less but still very great.
To each of these series $\Sigma$ corresponds a point of space; to two series $\Sigma$ and $\Sigma '$ thus correspond two points $\mathrm {M}$ and $\mathrm {M} '$. The means we have hitherto used enable us to recognize that $\mathrm {M}$ and $\mathrm {M} '$ are not distinct in two cases: (1) if $\Sigma$ is identical with $\Sigma '$ (2) if $\Sigma =\Sigma +s+s'$, $s$ and $s'$ being inverses one of the other. If in all the other cases we should regard $\mathrm {M}$ and $Mu'$ as distinct, the manifold of points would have as many dimensions as the aggregate of distinct series $\Sigma '$, that is, much more than three.
For those who already know geometry, the following explanation would be easily comprehensible. Among the imaginable series of muscular sensations, there are those which correspond to series of movements where the finger does not budge. I say that if one does not consider as distinct the series $\Sigma$ and $\Sigma +\sigma$, where the series a corresponds to movements where the finger does not budge, the aggregate of series will constitute a continuum of three dimensions, but that if one regards as distinct two series $\Sigma$ and $\Sigma '$ unless $\Sigma '$ $=$ $\Sigma +s+s'$, $s$ and $s'$ being inverses, the aggregate of series will constitute a continuum of more than three dimensions.
In fact, let there be in space a surface $\mathrm {A}$, on this surface a line $\mathrm {B}$, on this line a point $\mathrm {M}$. Let $C_{0}$ be the aggregate of all series $\Sigma$. Let $C_{1}$ be the aggregate of all the series $\Sigma$, such that at the end of corresponding movements the finger is found upon the surface A, and $C_{2}$ or $C_{3}$ the aggregate of series $\Sigma$ such that at the end the finger is found on $\mathrm {B}$, or at $\mathrm {M}$. It is clear, first that $C_{1}$ will constitute a cut which will divide $C_{0}$, that $C_{2}$ will be a cut which will divide $C_{1}$, and $C_{3}$ a cut which will divide $C_{2}$. Thence it results, in accordance with our definitions, that if $C_{3}$ is a continuum of $n$ dimensions, $C_{0}$ will be a physical continuum of $n+3$ dimensions.
Therefore, let $\Sigma$ and $\Sigma +\sigma$ a be two series forming part of $C_{3}$; for both, at the end of the movements, the finger is found at M; thence results that at the beginning and at the end of the series σ, the finger is at the same point M. This series a is therefore one of those which correspond to movements where the finger does not budge. If $\Sigma$ and $\Sigma +\sigma$ are not regarded as distinct, all the series of $C_{3}$ blend into one; therefore $C_{3}$ will have dimension, and $C_{0}$ will have 3, as I wished to prove. If, on the contrary, I do not regard $\Sigma$ and $\Sigma +\sigma$ as blending (unless $\sigma =s+s'$, $s$ and $s'$ being inverses), it is clear that $C_{3}$ will contain a great number of series of distinct sensations; because, without the finger budging, the body may take a multitude of different attitudes. Then $C_{3}$ will form a continuum and $C_{0}$ will have more than three dimensions, and this also I wished to prove.
​We who do not yet know geometry can not reason in this way; we can only verify. But then a question arises; how, before knowing geometry, have we been led to distinguish from the others these series a-where the finger does not budge? It is, in fact, only after having made this distinction that we could be led to regard $\Sigma$ and $\Sigma +\sigma$ as identical, and it is on this condition alone, as we have just seen, that we can arrive at space of three dimensions.
We are led to distinguish the series $\sigma$, because it often happens that when we have executed the movements which correspond to these series o-of muscular sensations, the tactile sensations which are transmitted to us by the nerve of the finger that we have called the first finger, persist and are not altered by these movements. Experience alone tells us that and it alone could tell us.
If we have distinguished the series of muscular sensations $s+s'$ formed by the union of two inverse series, it is because they preserve the totality of our impressions; if now we distinguish the series $\sigma$, it is because they preserve certain of our impressions. (When I say that a series of muscular sensations $s$ 'preserves' one of our impressions $A$, I mean that we ascertain that if we feel the impression $A$, then the muscular sensations $s$, we still feel the impression $A$ after these sensations $s$.)
I have said above it often happens that the series $\sigma$ do not alter the tactile impressions felt by our first finger; I said often, I did not say $always.$ This it is that we express in our ordinary language by saying that the tactile impressions would not be altered if the finger has not moved, on the condition that neither has the object $A$, which was in contact with this finger, moved. Before knowing geometry, we could not give this explanation; all we could do is to ascertain that the impression often persists, but not always.
But that the impression often continues is enough to make the series o-appear remarkable to us, to lead us to put in the same class the series $\Sigma$ and $\Sigma +\sigma$, and hence not regard them as distinct. Under these conditions we have seen that they will engender a physical continuum of three dimensions.
Behold then a space of three dimensions engendered by my first finger. Each of my fingers will create one like it. It remains to consider how we are led to regard them as identical with visual space, as identical with geometric space.
But one reflection before going further; according to the foregoing, we know the points of space, or more generally the final situation of our body, only by the series of muscular sensations revealing to us the movements which have carried us from a certain initial situation to this final situation. But it is clear that this final situation will depend, on the one hand, upon these movements and, on the other hand, upon the initial situation from which we set out. Now these movements are ​revealed to us by our muscular sensations; but nothing tells us the initial situation; nothing can distinguish it for us from all the other possible situations. This puts well in evidence the essential relativity of space.
§ 4. Identity of the Different Spaces
We are therefore led to compare the two continua $C$ and $C'$ engendered, for instance, one by my first finger $D$, the other by my second finger $D'$. These two physical continua both have three dimensions. To each element of the continuum $C$, or, if you prefer, to each point of the first tactile space, corresponds a series of muscular sensations $\Sigma$, which carry me from a certain initial situation to a certain final situation. Moreover, the same point of this first space will correspond to $\Sigma$ and to $\Sigma +\sigma$, if $\sigma$ is a series of which we know that it does not make the finger $D$ move.
Similarly to each element of the continuum $C'$, or to each point of the second tactile space, corresponds a series of sensations $\Sigma '$, and the same point will correspond to $\Sigma '$ and to $\Sigma '+\sigma '$, if $\sigma '$ is a series which does not make the finger $D'$ move.
What makes us distinguish the various series designated $\sigma$ from those called $\sigma '$ is that the first do not alter the tactile impressions felt by the finger $D$ and the second preserve those the finger $D'$ feels.
Now see what we ascertain: in the beginning my finger $D'$ feels a sensation $A'$; I make movements which produce muscular sensations $S$; my finger $D$ feels the impression $A$; I make movements which produce a series of sensations $\sigma$; my finger $D$ continues to feel the impression $A$, since this is the characteristic property of the series $\sigma$; I then make movements which produce the series $S'$ of muscular sensations, inverse to $S$ in the sense above given to this word. I ascertain then that my finger $D'$ feels anew the impression $A'$. (It is of course understood that $S$ has been suitably chosen.)
This means that the series $s+\sigma +s'$, preserving the tactile impressions of the finger $D'$, is one of the series I have called $\sigma '$. Inversely, if one takes any series $\sigma '$, $s'+\sigma '+s$ will be one of the series that we call $\sigma$.
Thus if $s$ is suitably chosen, $s+\sigma +s'$ will be a series $\sigma '$, and by making $\sigma$ vary in all possible ways, we shall obtain all the possible series $\sigma '$.
Not yet knowing geometry, we limit ourselves to verifying all that, but here is how those who know geometry would explain the fact. In the beginning my finger $D'$ is at the point $M$, in contact with the object $a$, which makes it feel the impression $A'$. I make the movements corresponding to the series $S$; I have said that this series should be suitably ​chosen, I should so make this choice that these movements carry the finger $D$ to the point originally occupied by the finger $D'$, that is, to the point $M$; this finger $D$ will thus be in contact with the object $a$, which will make it feel the impression $A$.
I then make the movements corresponding to the series $\sigma$; in these movements, by hypothesis, the position of the finger $D$ does not change, this finger therefore remains in contact with the object $a$ and continues to feel the impression $A$. Finally I make the movements corresponding to the series $S'$. As $S'$ is inverse to $S$, these movements carry the finger $D'$ to the point previously occupied by the finger $D$, that is, to the point $M$. If, as may be supposed, the object a has not budged, this finger $D'$ will be in contact with this object and will feel anew the impression $A'$. . . . $Q.E.D.$
Let us see the consequences. I consider a series of muscular sensations $\Sigma$. To this series will correspond a point $M$ of the first tactile space. Now take again the two series $s$ and $s'$, inverses of one another, of which we have just spoken. To the series $s+\Sigma +s'$ will correspond a point $N$ of the second tactile space, since to any series of muscular sensations corresponds, as we have said, a point, whether in the first space or in the second.
I am going to consider the two points $N$ and $M$, thus defined, as corresponding. What authorizes me so to do? For this correspondence to be admissible, it is necessary that if two points $M$ and $M'$, corresponding in the first space to two series $\Sigma$ and $\Sigma '$, are identical, so also are the two corresponding points of the second space $N$ and $N'$, that is the two points which correspond to the two series $s+\Sigma +s'$ and $s+\Sigma '+s'$. Now we shall see that this condition is fulfilled.
First a remark. As $S$ and $S'$ are inverses of one another, we shall have $S+S'=0$, and consequently $S+S'+\Sigma =\Sigma +S+S'=\Sigma$, or again $\Sigma +S+S'+\Sigma '=\Sigma +\Sigma '$; but it does not follow that we have $S+\Sigma +S'=\Sigma$; because, though we have used the addition sign to represent the succession of our sensations, it is clear that the order of this succession is not indifferent: we can not, therefore, as in ordinary addition, invert the order of the terms; to use abridged language, our operations are associative, but not commutative.
That fixed, in order that $\Sigma$ and $\Sigma '$ should correspond to the same point $M=M'$ of the first space, it is necessary and sufficient for us to have $\Sigma '=\Sigma +\sigma$. We shall then have: $S+\Sigma '+\Sigma '=S+\Sigma +\sigma +S'=8+\Sigma +S'+S+\sigma +S'$.
But we have just ascertained that $S+\sigma +8'$ was one of the series $\sigma$. We shall therefore have: $S+\Sigma '+S'=S+\Sigma +S'+\sigma '$, which means that the series $S+\Sigma '+S'$ and $S+\Sigma +S'$ correspond to the same point $N=N'$ of the second space. $Q.E.D.$
Our two spaces therefore correspond point for point; they can be ​'transformed' one into the other; they are isomorphic. How are we led to conclude thence that they are identical?
Consider the two series $\sigma$ and $S+\sigma +S'=\sigma '$. I have said that often, but not always, the series $o-$ preserves the tactile impression $A$ felt by the finger $D$; and similarly it often happens, but not always, that the series $\sigma '$ preserves the tactile impression $A'$ felt by the ringer $D'$. Now I ascertain that it happens very often (that is, much more often than what I have just called 'often') that when the series $\sigma$ has preserved the impression $A$ of the finger $D$, the series $\sigma '$ preserves at the same time the impression $A'$ of the finger $D'$; and, inversely, that if the first impression is altered, the second is likewise. That happens very often, but not always.
We interpret this experimental fact by saying that the unknown object a which gives the impression $A$ to the finger $D$ is identical with the unknown object $a'$ which gives the impression $A'$ to the finger $D'$. And in fact when the first object moves, which the disappearance of the impression $A$ tells us, the second likewise moves, since the impression $A'$ disappears likewise. When the first object remains motionless, the second remains motionless. If these two objects are identical, as the first is at the point $M$ of the first space and the second at the point $N$ of the second space, these two points are identical. This is how we are led to regard these two spaces as identical; or better this is wbat we mean when we say that they are identical.
What we have just said of the identity of the two tactile spaces makes unnecessary our discussing the question of the identity of tactile space and visual space, which could be treated in the same way.
§ 5. Space and Empiricism
It seems that I am about to be led to conclusions in conformity with empiristic ideas. I have, in fact, sought to put in evidence the role of experience and to analyze the experimental facts which intervene in the genesis of space of three dimensions. But whatever may be the importance of these facts, there is one thing we must not forget and to which besides I have more than once called attention. These experimental facts are often verified but not always. That evidently does not mean that space has often three dimensions, but not always.
I know well that it is easy to save oneself and that, if the facts do not verify, it will be easily explained by saying that the exterior objects have moved. If experience succeeds, we say that it teaches us about space; if it does not succeed, we hie to exterior objects which we accuse of having moved; in other words, if it does not succeed, it is given a fillip.
These fillips are legitimate; I do not refuse to admit them; but they suffice to tell us that the properties of space are not experimental truths, properly so called. If we had wished to verify other laws, we ​could have succeeded also, by giving other analogous fillips. Should we not always have been able to justify these fillips by the same reasons? One could at most have said to us: 'Your fillips are doubtless legitimate, but you abuse them; why move the exterior objects so often?'
To sum up, experience does not prove to us that space has three dimensions; it only proves to us that it is convenient to attribute three to it, because thus the number of fillips is reduced to a minimum.
I will add that experience brings us into contact only with representative space, which is a physical continuum, never with geometric space, which is a mathematical continuum. At the very most it would appear to tell us that it is convenient to give to geometric space three dimensions, so that it may have as many as representative space.
The empiric question may be put under another form. Is it impossible to conceive physical phenomena, the mechanical phenomena for example, otherwise than in space of three dimensions? We should thus have an objective experimental proof, so to speak, independent of our physiology, of our modes of representation.
But it is not so; I shall not here discuss the question completely, I shall confine myself to recalling the striking example given us by the mechanics of Hertz. You know that the great physicist did not believe in the existence of forces, properly so called; he supposed that visible material points are subjected to certain invisible bonds which join them to other invisible points and that it is the effect of these invisible bonds that we attribute to forces.
But that is only a part of his ideas. Suppose a system formed of $n$ material points, visible or not; that will give in all $3n$ coordinates; let us regard them as the coordinates of a single point in space of $3n$ dimensions. This single point would be constrained to remain upon a surface (of any number of dimensions $<3n)$ in virtue of the bonds of which we have just spoken; to go on this surface from one point to another, it would always take the shortest way; this would be the single principle which would sum up all mechanics.
Whatever should be thought of this hypothesis, whether we be allured by its simplicity, or repelled by its artificial character, the simple fact that Hertz was able to conceive it, and to regard it as more convenient than our habitual hypotheses, suffices to prove that our ordinary ideas, and, in particular, the three dimensions of space, are in no wise imposed upon mechanics with an invincible force.
§ 6. Mind and Space
Experience, therefore, has played only a single role, it has served as occasion. But this rôle was none the less very important; and I have thought it necessary to give it prominence. This rôle would have been useless if there existed an a priori form imposing itself upon our sensitivity, and which was space of three dimensions.
​Does this form exist, or, if you choose, can we represent to ourselves space of more than three dimensions? And first what does this question mean? In the true sense of the word, it is clear that we can not represent to ourselves space of four, nor space of three, dimensions; we can not first represent them to ourselves empty, and no more can we represent to ourselves an object either in space of four, or in space of three, dimensions: (1) Because these spaces are both infinite and we can not represent to ourselves a figure in space, that is, the part in the whole, without representing the whole, and that is impossible, because it is infinite; (2) because these spaces are both mathematical continua and we can represent to ourselves only the physical continuum; (3) because these spaces are both homogeneous, and the frames in which we enclose our sensations, being limited, can not be homogeneous.
Thus the question put can only be understood in another manner; is it possible to imagine that, the results of the experiences related above having been different, we might have been led to attribute to space more than three dimensions; to imagine, for instance, that the sensation of accommodation might not be constantly in accord with the sensation of convergence of the eyes; or indeed that the experiences of which we have spoken in paragraph 2 and of which we express the result by saying 'that touch does not operate at a distance,' might have led us to an inverse conclusion.
And then evidently yes that is possible. From the moment one imagines an experience, one imagines just by that the two contrary results it may give. That is possible, but that is difficult, because we have to overcome a multitude of associations of ideas, which are the fruit of a long personal experience and of the still longer experience of the race. Is it these associations (or at least those of them that we have inherited from our ancestors), which constitute this a priori form of which it is said that we have pure intuition? Then I do not see why one should declare it refractory to analysis and should deny me the right of investigating its origin.
When it is said that our sensations are 'extended' only one thing can be meant, that is that they are always associated with the idea of certain muscular sensations, corresponding to the movements which enable us to reach the object which causes them, which enable us, in other words, to defend ourselves against it. And it is just because this association is useful for the defense of the organism, that it is so old in the history of the species and that it seems to us indestructible. Nevertheless, it is only an association and we can conceive that it may be broken; so that we may not say that sensation can not enter consciousness without entering in space, but that in fact it does not enter consciousness without entering in space, which means, without being entangled in this association.
No more can I understand one's saying that the idea of time is ​logically subsequent to space, since we can represent it to ourselves only under the form of a straight line; as well say that time is logically subsequent to the cultivation of the prairies, since it is usually represented armed with a scythe. That one can not represent to himself simultaneously the different parts of time, goes without saying, since the essential character of these parts is precisely not to be simultaneous. That does not mean that we have not the intuition of time. So far as that goes, no more should we have that of space, because neither can we represent it, in the proper sense of the word, for the reasons 1 have mentioned. What we represent to ourselves under the name of straight is a crude image which as ill resembles the geometric straight as it does time itself.
Why has it been said that every attempt to give a fourth dimension to space always carries this one back to one of the other three? It is easy to understand. Consider our muscular sensations and the 'series' they may form. In consequence of numerous experiences, the ideas of these series are associated together in a very complex woof, our series are classed. Allow me, for convenience of language, to express my thought in a way altogether crude and even inexact by saying that our series of muscular sensations are classed in three classes corresponding to the three dimensions of space. Of course this classification is much more complicated than that, but that will suffice to make my reasoning understood. If I wish to imagine a fourth dimension, I shall suppose another series of muscular sensations, making part of a fourth class. But as all my muscular sensations have already been classed in one of the three preexistent classes, I can only represent to myself a series belonging to one of these three classes, so that my fourth dimension is carried back to one of the other three.
What does that prove? This: that it would have been necessary first to destroy the old classification and replace it by a new one in which the series of muscular sensations should have been distributed into four classes. The difficulty would have disappeared.
It is presented sometimes under a more striking form. Suppose I am enclosed in a chamber between the six impassable boundaries formed by the four walls, the floor and the ceiling; it will be impossible for me to get out and to imagine my getting out. Pardon, can you not imagine that the door opens, or that two of these walls separate? But of course, you answer, one must suppose that these walls remain immovable. Yes, but it is evident that I have the right to move; and then the walls that we suppose absolutely at rest will be in motion with regard to me. Yes, but such a relative motion can not be anything; when objects are at rest, their relative motion with regard to any axes is that of a rigid solid; now, the apparent motions that you imagine are not in conformity with the laws of motion of a rigid solid. ​Yes, but it is experience which has taught us the laws of motion of a rigid solid; nothing would prevent our imagining them different. To sum up, for me to imagine that I get out of my prison, I have only to imagine that the walls seem to open, when I move.
I believe, therefore, that if by space is understood a mathematical continuum of three dimensions, were it otherwise amorphous, it is the mind which constructs it, but it does not construct it out of nothing; it needs materials and models. These materials, like these models, preexist within it. But there is not a single model which is imposed upon it; it has choice; it may choose, for instance, between space of four and space of three dimensions. What then is the rôle of experience? It gives the indications following which the choice is made.
Another thing: whence does space get its quantitative character? It comes from the rôle which the series of muscular sensations play in its genesis. These are series which may repeat themselves, and it is from their repetition that number comes; it is because they can repeat themselves indefinitely that space is infinite. And finally we have seen, at the end of section 3, that it is also because of this that space is relative. So it is repetition which has given to space its essential characteristics; now, repetition supposes time; this is enough to tell that time is logically anterior to space.
§ 7. Rôle of the Semicircular Canals
I have not hitherto spoken of the role of certain organs to which the physiologists attribute with reason a capital importance, I mean the semicircular canals. Numerous experiments have sufficiently shown that these canals are necessary to our sense of orientation; but the physiologists are not entirely in accord; two opposing theories have been proposed, that of Mach-Delage and that of M. de Cyon.
M. de Cyon is a physiologist who has made his name illustrious by important discoveries on the innervation of the heart; I can not, however agree with his ideas on the question before us. Not being a physiologist, I hesitate to criticize the experiments he has directed against the adverse theory of Mach-Delage; it seems to me, however, that they are not convincing, because in many of them the total pressure was made to vary in one of the canals, while, physiologically, what varies is the difference between the pressures on the two extremities of the canal; in others the organs were subjected to profound lesions, which must alter their functions.
Besides, this is not important; the experiments, if they were irreproachable, might be convincing against the old theory. They would not be convincing for the new theory. In fact, if I have rightly understood the theory, my explaining it will be enough for one to understand that it is impossible to conceive of an experiment confirming it.
​The three pairs of canals would have as sole function to tell us that space has three dimensions. Japanese mice have only two pairs of canals; they believe, it would seem, that space has only two dimensions, and they manifest this opinion in the strongest way; they put themselves in a circle, and, so ordered, they spin rapidly around. The lampreys, having only one pair of canals, believe that space has only one dimension, but their manifestations are less turbulent.
It is evident that such a theory is inadmissible. The sense-organs are designed to tell us of changes which happen in the exterior world. We could not understand why the Creator should have given us organs destined to cry without cease: Remember that space has three dimensions, since the number of these three dimensions is not subject to change.
We must, therefore, come back to the thory of Mach-Delage. What the nerves of the canals can tell us is the difference of pressure on the two extremities of the same canal, and thereby: (1) the direction of the vertical with regard to three axes rigidly bound to the head; (2) the three components of the acceleration of translation of the center of gravity of the head; (3) the centrifugal forces developed by the rotation of the head; (4) the acceleration of the motion of rotation of the head.
It follows from the experiments of M. Delage that it is this last indication which is much the most important; doubtless because the nerves are less sensible to the difference of pressure itself than to the brusque variations of this difference. The first three indications may thus be neglected.
Knowing the acceleration of the motion of rotation of the head at each instant, we deduce from it, by an unconscious integration, the final orientation of the head, referred to a certain initial orientation taken as origin. The circular canals contribute, therefore, to inform us of the movements that we have executed, and that on the same ground as the muscular sensations. When, therefore, above we speak of the series $S$ or of the series $\Sigma$, we should say, not that these were series of muscular sensations alone, but that they were series at the same time of muscular sensations due to the semicircular canals. Apart from this addition, we should have nothing to change in what precedes.
$S$
$\Sigma$
NOBEL MEDALS
The gold medals conferred in connection with the Nobel prizes are here shown. Above is the medal in physics and in chemistry. The obverse of the medals in medicine and in literature is the same; the reverse of each of these medals is shown beneath. At the bottom is the medal for the promotion of peace.
* ↑ In place of saying that we refer space to axes rigidly bound to our body, perhaps it would be better to say, in conformity to what precedes, that we refer it to axes rigidly bound to the initial situation of our body.
* ↑ Image must be fuzzy due to copyright protection.