 Chapter 4 of Science and Hypothesis Let us begin with a little paradox. Beings whose minds were made as ours, and with senses like ours, but without any preliminary education, might receive from a suitably chosen external world impressions, which would lead them to construct the geometry other than that of Euclid, and to localize the phenomena of this external world in a non-Euclidean space, or even in a space of four dimensions. As for us, whose education has been made by our actual world, if we were suddenly transported into this new world, we should have no difficulty in referring phenomena to our Euclidean space. Let somebody may appear on the scene someday who will devote his life to it, and be able to represent to himself the fourth dimension. Geometrical space and representative space It is often said that the images we form of external objects are localized in space, and even that they can only be formed on this condition. It is also said that this space, which thus serves as a kind of framework already prepared for our sensations and representations, is identical with the space of the geometers, having all the properties of that space. To all clear-headed men who think in this way, the preceding statement might well appear extraordinary, but it is as well to see if they are not the victims of some illusion which closer analysis may be able to dissipate. In the first place, what are the properties of space properly so-called? I mean of that space which is the object of geometry, and which I shall call geometrical space. The following are some of the more essential. First, it is continuous. Second, it is infinite. Third, it is of three dimensions. Fourth, it is homogenous, that is to say, all its points are identical, one with another. Fifth, it is isotropic. Compare this now with the framework of our representations and sensations, which I may call representative space, visual space. First of all, let us consider a purely visual impression due to an image formed on the back of the retina. A cursory analysis shows us this image as continuous, but as possessing only two dimensions, which already distinguishes purely visual from what may be called geometrical space. On the other hand, the image is enclosed within a limited framework, and there is a no less important difference. This pure visual space is not homogenous. All the points on the retina, apart from the images which may be formed, do not play the same role. The yellow spot can in no way be regarded as identical, with a point on the edge of the retina. Not only does the same object produce on it much brighter impressions, but in the whole of the limited framework, the point which occupies the center will not appear identical with a point near one of the edges. Closer analysis no doubt would show us that this continuity of visual space and its two dimensions are but an illusion. It would make visual space even more different than before from geometrical space, but we may treat this remark as incidental. However, sight enables us to appreciate distance, and therefore to perceive a third dimension. But everyone knows that this perception of the third dimension reduces to a sense of the effort of accommodation which must be made, and to a sense of the convergence of the two eyes that must take place in order to perceive an object distinctly. These are muscular sensations quite different from the visual sensations which have given us the concept of the first two dimensions. The third dimension will therefore not appear to us as playing the same role as the two others. What may be called complete visual space is not therefore an isotropic space. It has, it is true, exactly three dimensions, which means the elements of our visual sensations, those at least which concur in forming the concept of extension, will be completely defined if we know three of them, or in mathematical language there will be functions of three independent variables. But let us look at the matter a little closer. The third dimension is revealed to us in two different ways by the effort of accommodation and by the convergence of the eyes. No doubt these two indications are always in harmony. There is between them a constant relation, or in mathematical language, the two variables which measure these two muscular sensations do not appear to us as independent. Or again, to avoid and appeal to mathematical ideas which are already rather too refined, we may go back to the language of the preceding chapter and enunciate the same fact as follows. If two sensations of convergence A and B are indistinguishable, the two sensations of accommodation A prime and B prime, which accompany them respectively, will also be indistinguishable. But that is, so to speak, an experimental fact. Nothing prevents us, our priority, from assuming the contrary. If the contrary takes place, if these two muscular sensations both vary independently, we must take into account one more independent variable, and complete visual space will appear to us as a physical continuum of four dimensions. And so in this there is also a fact of external experiments. Nothing prevents us from assuming that a being with a mind like ours, with the same sense organs as ourselves, may be placed in a world in which light would only reach him after being passed through refracting media of complicated form. The two indications which enable us to appreciate distances would cease to be connected by a constant relation. A being educating his senses in such a world would no doubt attribute four dimensions to complete visual space. Tactile and Motor Space Tactile space is more complicated still than visual space, and differs even more widely from geometrical space. It is useless to repeat for the sense of touch my remarks on the sense of sight. But outside the data of sight and touch, there are other sensations which contribute as much and more than they do to the genesis of the concept of space. There are those which everybody knows, which accompany all our movements, and which we usually call muscular sensations. The corresponding framework constitutes what may be called motor space. Each muscle gives rise to a special sensation which may be increased or diminished so that the aggregate of our muscular sensations will depend upon as many variables as we have muscles. From this point of view, motor space would have as many dimensions as we have muscles. I know that it is said that if the muscular sensations contribute to form the concept of space, it is because we have the sense of the direction of each movement, and that this is an integral part of this sensation. If this were so, and if a muscular sense could not be aroused unless it were accompanied by this geometrical sense of direction, geometrical space would certainly be a form imposed upon our sensitiveness. But I do not see this at all when I analyze my sensations. What I do see is that the sensations which correspond to movements in the same direction are connected in my mind by a simple association of ideas. It is to this association that what we call the sense of direction is reduced. We cannot therefore discover this sense in a single sensation. This association is extremely complex, for the contraction of the same muscle may correspond, according to the position of the lens, to very different movements of direction. Moreover, it is evidently acquired. It is like all associations of ideas, the result of a habit. This habit itself is the result of a very large number of experiments, and no doubt if the education of our senses had taken place in a different medium, where we would have been subjected to different impressions, then contrary habits would have been acquired, and our muscular sensations would have been associated according to other laws. Characteristics of representative space Thus representative space in its triple form, visual, tactile, and motor, differs essentially from geometrical space. It is neither homogenous nor isotropic. We cannot even say that it is of three dimensions. It is often said that we project into geometrical space the objects of our external perception, that we localize them. Now, has that any meaning? And if so, what is that meaning? Does it mean that we represent to ourselves external objects in geometrical space? Our representations are only the reproduction of our sensations. They cannot therefore be arranged in the same framework. That is to say, in representative space. It is also just as impossible for us to represent to ourselves external objects in geometrical space, as it is impossible for a painter to paint on a flat surface objects with their three dimensions. Representative space is only an image of geometrical space, an image deformed by a kind of perspective, and we can only represent to ourselves objects by making them obey the laws of this perspective. Thus we do not represent to ourselves external bodies in geometrical space, but we reason about these bodies as if they were situated in geometrical space. When it is said, on the other hand, that we localize such an object in such a point of space, what does it mean? It simply means that we represent to ourselves the movements that must take place to reach that object. And it does not mean that to represent to ourselves these movements, they must be projected into space, and that the concept of space must therefore pre-exist. When I say that we represent to ourselves these movements, I only mean that we represent to ourselves the muscular sensations which accompany them, and which have no geometrical character, and which therefore in no way imply the pre-existence of the concept of space. Consciousness of state and changes of position. But it may be said, if the concept of geometrical space is not imposed upon our minds, and if on the other hand, none of our sensations can furnish us with that concept, how then did it ever come into existence? This is what we have now to examine, and it will take some time, but I can sum up in a few words the attempt at explanation which I am going to develop. None of our sensations, if isolated, could have brought us to the concept of space. We are brought to it solely by studying the laws by which those sensations succeed one another. We see at first that our impressions are subject to change, but among the changes that we ascertain, we are very soon led to make a distinction. Sometimes we say that the objects, the causes of these impressions, have changed their state. Sometimes that they have changed their position, that they have only been displaced, or there an object changes its state, or only its position, this is always translated for us in the same manner, by a modification and an aggregate of impressions. How then have we been enabled to distinguish them? If there were only a change of position, we could restore the primitive aggregate of impressions by making movements which would confront us with the movable object in the same relative situation. We thus correct the modification which was produced, and we re-establish the initial state by an inverse modification. If, for example, it were a question of the sight, and if an object be displaced before our eyes, we can follow it with the eye, and retain its image on the same point of the retina by appropriate movements of the eyeball. These movements we are conscious of because they are voluntary, and because they are accompanied by muscular sensations, but that does not mean that we can represent them to ourselves in geometrical space. So what characterizes change of position? What distinguishes it from change of state is that it can always be corrected by this means. It may therefore happen that we pass from the aggregate of impressions A to the aggregate B in two different ways. First, involuntarily, and without experiencing muscular sensations, which happens when it is the object that is displaced. Secondly, voluntarily, and with muscular sensation, which happens when the object is motionless, but when we displace ourselves in such a way that the object is relative motion with respect to us. If this be so, the translation of the aggregate A to the aggregate B is only a change of position. It follows its sight and touch could not have given us the idea of space without the help of the muscular sense. Not only could this concept not be derived from a single sensation, or even from a series of sensations, but emotionless being could never have acquired it. Because, not being able to correct by his movements the effects of the change of position of external objects, he would have had no reason to distinguish them from changes of state. Nor would he have been able to acquire it if his movements had not been voluntary, or if they were unaccompanied by any sensations whatever. Conditions of compensation. How is such a compensation possible in such a way that two changes, otherwise mutually independent, may be reciprocally corrected? A mind already familiar with geometry would reason as follows. If there is to be compensation, the different parts of the external object on the one hand and the different organs of our senses on the other must be in the same relative position after the double change. And for that to be the case, the different parts of the external body on the one hand and the different organs of our senses on the other must have the same relative position to each other after the double change, and so with the different parts of our body with respect to each other. In other words, the external object in the first change must be displaced as an invariable solid would be displaced, and it must also be so with the whole of our body in the second change, which is to correct the first. Under these conditions, compensation may be produced. But we who has yet known nothing of geometry, whose ideas of space are not yet formed, we cannot reason in this way. We cannot predict a priori if compensation is possible. But experiment shows us that it sometimes does take place, and when we start from this experimental fact, in order to distinguish changes of state from changes of position. Solid bodies and geometry. Among surrounding objects, there are some which frequently experience displacements that may be thus corrected by a correlative movement of our own body, namely solid bodies. The other objects whose form is variable only in exceptional circumstances undergo similar displacement, change of position without change of form. When the displacement of a body takes place with deformation, we can no longer by appropriate movements place the organs of our body in the same relative situation with respect to this body. We can no longer therefore reconstruct the primitive aggregate of impressions. It is only later, and after a series of new experiments, that we learn how to decompose a body of variable form into smaller elements, such that each is displaced approximately according to the same law as solid bodies. We thus distinguish deformations from other changes of state. In these deformations, each element undergoes a simple change of position which may be corrected, but the modification of the aggregate is more profound, and can no longer be corrected by a correlative movement. Such a concept is very complex even at this stage, and has been relatively slow in its appearance. It would not have been conceived at all had not the observation of solid bodies shown us beforehand how to distinguish changes of position. If then, there were no solid bodies in nature, there would be no geometry. Another remark deserves a moment's attention. Suppose a solid body to occupy successively the positions alpha and beta. In the first position, it will give us an aggregate of impressions A, and in the second position the aggregate of impressions B. Now let there be a second solid body of qualities entirely different from the first, of different color for instance. Assume at the pass from the position alpha, where it gives us the aggregate of impressions A prime to the position beta, where it gives us the aggregate of impressions B prime. In general, the aggregate A will have nothing in common with the aggregate A prime, nor will the aggregate B have anything in common with the aggregate B prime. The transition from the aggregate A to the aggregate B and that of the aggregate A prime to the aggregate B prime are therefore two changes which themselves have in general nothing in common. Yet we consider both these changes as displacements, and further we consider them the same displacement. How can this be? It is simply because they may be both corrected by the same correlative movement of our body. Correlative movement therefore constitutes the sole connection between two phenomena which otherwise we should never have dreamed of connecting. On the other hand, our body, thanks to the number of its articulations and muscles, may have a multitude of different movements, but all are not capable of correcting a modification of external objects. Those alone are capable of it, in which our whole body, or at least all those in which the organs of our senses enter into play are displaced in block, i.e. without any variation of their relative positions, as in the case of a solid body. To sum up, one, in the first place we distinguish two categories of phenomena, the first involuntary, unaccompanied by muscular sensations and attributed to external objects. They are external changes. The second of opposite character and attributed to the movements of our own body are internal changes. Two, we notice that certain changes of each in these categories may be corrected by a correlative change of the other category. Three, we distinguish among external changes those that have a correlative in the other category, which we call displacements, and in the same way we distinguish among the internal changes those which have a correlative in the first category. Thus by means of this reciprocity is defined a particular class of phenomena called displacements. The laws of these phenomena are the object of geometry. Law of homogeneity. The first of these laws is the law of homogeneity. Suppose that by an external change we pass from the aggregate of impressions A to the aggregate B, and that then this change alpha is corrected by a correlative voluntary movement beta so that we are brought back to the aggregate A. Suppose now that another external change alpha prime brings us again from the aggregate A to the aggregate B. Experiment then shows us that this change alpha prime, like the change alpha, may be corrected by a voluntary correlative movement beta prime, and that this movement beta prime corresponds to the same muscular sensations as the movement beta which corrected alpha. This fact is usually enunciated as swallows. Space is homogenous and isotropic. We may also say that a movement which is once produced may be repeated a second and a third time and so on without any variation of its properties. In the first chapter in which we discuss the nature of mathematical reasoning we saw the importance that should be attached to the possibility of repeating the same operation indefinitely. The virtue of mathematical reasoning is due to this repetition. By means of the law of homogeneity, geometrical facts are apprehended. To be complete, to the law of homogeneity must be added a multitude of other laws into the details of which I do not propose to enter, but which mathematicians sum up by saying that these displacements form a group. The non-Euclidean world. If geometrical space were a framework imposed on each of our representations considered individually it would be impossible to represent to ourselves an image without this framework and we should be quite unable to change our geometry. But this is not the case. Geometry is only the summary of the laws by which these images succeed each other. There is nothing therefore to prevent us from imagining a series of representations similar in every way to our own ordinary representations but succeeding one another according to laws which differ from those to which we are accustomed. We may thus conceive that being whose education has taken place in a medium in which those laws would be so different might have a very different geometry from ours. Suppose for example a world enclosed in a large sphere and subject to the following laws. The temperature is not uniform. It is greatest at the center and gradually decreases as we move towards the circumference of the sphere where it is absolute zero. The law of this temperature is as follows. If cap r be the radius of the sphere and little r the distance of the point considered from the center the absolute temperature will be proportional to cap r squared minus little r squared. Further I shall suppose that in this world all bodies have the same coefficient of dilation so that the linear dilation of any body is proportional to its absolute temperature. Finally I shall assume that a body transported from one point to another of different temperature is instantaneously in thermal equilibrium with its new environment. There is nothing in these hypotheses either contradictory or unimaginable. The moving object will become smaller and smaller as it approaches the circumference of the sphere. Let us observe in the first place that although from the point of view of our ordinary geometry this world is finite to its inhabitants it will appear infinite. As they approach the surface of the sphere they become colder and at the same time smaller and smaller. The steps they take are therefore also smaller and smaller so that they can never reach the boundary of the sphere. If to us geometry is only the study of the laws according to which invariable solids move to these imaginary beings it will be the study of the laws of motion of solids deformed by the differences of temperature alluded to. No doubt in our world natural solids also experience variations of form and volume due to differences of temperature but in laying the foundations of geometry we neglect these variations for besides being but small they are irregular and consequently appear to us to be accidental. In our hypothetical world this will no longer be the case. The variations will obey very simple and regular laws. On the other hand the different solid parts of which the bodies of these inhabitants are composed will undergo the same variations of form and volume. Let me make another hypothesis. Suppose that light passes through media of different refractive indices such that the index of refraction is inversely proportional to cap r squared minus little r squared. Under these conditions it is clear that the rays of light will no longer be rectilinear but circular. To justify what has been said we have to prove that certain changes in the position of external objects may be corrected by correlative movements of the beings which inhabit this imaginary world and in such a way as to restore the primitive aggregate of the impressions experienced by these sentient beings. Suppose for example that an object is displaced and deformed not like an invariable solid but like a solid subjected to unequal dilations in exact conformity with the law of temperature assumed above. To use an abbreviation we shall call such a movement a non-uclidean displacement. If a sentient being be in the neighborhood of such a displacement of the object his impressions will be modified but by moving in a suitable manner he may reconstruct them. For this purpose all that is required is that the aggregate of the sentient being and the object considered as forming a single body shall experience one of those special displacements which I have just called non-uclidean. This is possible if we suppose that the limbs of these beings dilate according to the same laws as the other bodies of the world they inhabit. Although from the point of view of our ordinary geometry there is a deformation of the bodies in this displacement and although their different parts are no longer in the same relative position nevertheless we shall see that the impressions of the sentient being remain the same as before. In fact though the mutual distances of the different parts have varied yet the parts which at first were in contact are still in contact it follows that tactile impressions will be unchanged. On the other hand from the hypothesis as to refraction and the curvature of the rays of light visual impressions will also be unchanged. These imaginary beings will therefore be led to classify the phenomena they observe and distinguish among them the changes of position which may be corrected by a voluntary correlative movement just as we do. If they construct the geometry it will not be like ours which is the study of the movements of our invariable solids. It will be the study of the changes of position which they will have thus distinguished and will be quote non-uclidean displacements and this will be non-uclidean geometry. So we see that beings like ourselves educated in such a world will not have the same geometry as ours. The world of four dimensions just as we have pictured to ourselves a non-uclidean world so we may picture a world of four dimensions. The sense of light even with one eye together with the muscular sensations relative to the movements of the eyeball will suffice to enable us to conceive of space of three dimensions. The images of external objects are painted on the retina which is a plane of two dimensions. These are perspectives. But as eye and objects are movable we see in succession different perspectives of the same body taken from different points of view. We find at the same time that the transition from one perspective to another is often accompanied by muscular sensations. If the transition from the perspective A to the perspective B and that of the perspective A prime to the perspective B prime are accompanied by the same muscular sensations we connect them as we do other operations of the same nature. Then when we study the laws according to which these operations are combined we see that they form a group which has the same structure as that of the movements of invariable solids. Now we have seen that it is from the properties of this group that we derive the idea of geometrical space and that of three dimensions. We thus understand how these perspectives gave rise to the conception of three dimensions although each perspective is of only two dimensions because they succeed each other according to certain laws. Well in the same way that we draw the perspective of a three-dimensional figure on a plane. So we can draw that of a four-dimensional figure on a canvas of three or two dimensions. To a geometry this is but child's play. We can even draw several perspectives of the same figure from several different points of view. We can easily represent to ourselves these perspectives since they are of only three dimensions. Imagine that the different perspectives of one and the same object to a current succession and that the transition from one to the other is accompanied by muscular sensations. It is understood that we shall consider two of these transitions as two operations of the same nature when they are associated with the same muscular sensations. There is nothing then to prevent us from imagining that these operations are combined according to any law we choose. For instance by forming a group with the same structure as that of the movements of an invariable four-dimensional solid. In this there is nothing that we cannot represent to ourselves and moreover these sensations are those which a being would experience who has a retina of two dimensions and who may be displaced in space of four dimensions. In this sense we may say that we can represent to ourselves the fourth dimension. Conclusions. It is seen that experiment plays a considerable role in the genesis of geometry but it would be a mistake to conclude from that that geometry is even in part an experimental science. If it were experimental it would only be approximate and provisory and what a rough approximation it would be. Geometry would be only the study of movements of solid bodies but in reality it is not concerned with natural solids. Its objects is certain ideal solids absolutely invariable which are but a greatly simplified and very remote image of them. The concept of these ideal bodies is entirely mental. An experiment is but the opportunity which enables us to reach the idea. The object of geometry is a study of a particular group but the general concept of group pre-exists in our minds at least potentially. It is imposed on us not as a form of our sensitiveness but as a form of our understanding. Only from among all possible groups we must choose one that will be the standard so to speak to which we shall refer natural phenomena. Experiment guides us in this choice which it does not impose on us. It tells us not what is the truest but what is the most convenient geometry. It will be noticed that my description of these fantastic worlds has required no language other than that of ordinary geometry. Then were we transported to those worlds there would be no need to change that language. Beings educated there would no doubt find it more convenient to create a geometry different from ours and better adapted to their impressions but as for us in the presence of the same impressions it is certain that we should not find it more convenient to make a change. End of chapter four. Chapter five of science and hypothesis. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer please visit LibriVox.org. Recording by Alex Trigobi. Science and hypothesis by Henri Poncaret. Chapter five. One. I have on several occasions in the preceding pages tried to show how the principles of geometry are not experimental facts and that in particular Euclid's postulate cannot be proved by experiment. However convincing the reasons already given may appear to me I feel I must dwell upon them because there is a profoundly false conception deeply rooted in many minds. Two. Think of a material circle. Measure its radius and circumference and see if the ratio of the two lengths is equal to pi. What have we done? We have made an experiment on the properties of the matter with which this roundness has been realized and of which the measure we used is made. Three. Geometry and astronomy. The same question may also be asked in another way. If Lovachevsky's geometry is true the parallax of a very distant star will be finite. If Riemann's is true it will be negative. These are the results which seem within the reach of experiment and it is hoped that astronomical observations may enable us to decide between the two geometries. But what we call a straight line in astronomy is simply the path of a ray of light. If therefore we were to discover negative parallaxes or to prove that all parallaxes are higher than a certain limit we should have a choice between two conclusions. We could give up Euclidean geometry or modify the laws of optics and suppose that light is not rigorously propagated in a straight line. It is needless to add that everyone would look upon this solution as the more advantageous. Euclidean geometry therefore has nothing to fear from fresh experiments. Four. Can we maintain that certain phenomena which are possible in Euclidean space would be impossible in non-Euclidean space so that experiments in establishing these phenomena would directly contradict the non-Euclidean hypothesis. I think that such question cannot be seriously asked. To me it is exactly equivalent to the following the absurdity of which is obvious. There are lengths which can be expressed in meters and centimeters but cannot be measured in toise feet and inches so that experiment by ascertaining the existence of these lengths would directly contradict this hypothesis that there are toise divided in six feet. Let us look at the question a little more closely. I assume that the straight line in Euclidean space possesses any two properties which I shall call A and B. That in non-Euclidean space it still possesses the property A but no longer possesses the property B. And finally I assume that in both Euclidean and non-Euclidean space the straight line is the only line that possesses property A. If this were so experiment would be able to decide between the hypotheses of Euclidean and Lobachevsky. It would be found that some concrete object upon which we can experiment for example a pencil of rays of light possesses the property A. We should conclude that it is rectilinear and we should then endeavor to find out if it does or does not possess the property B. But it is not so. There exists no property which can, like this property A, be an absolute criterion enabling us to recognize a straight line and to distinguish it from every other line. Shall we say for instance this property will be the following. The straight line is a line such that a figure of which this line is apart can move without the mutual distances of its points varying and in such a way that all points in the straight line remain fixed. Now this is a property which in either Euclidean or non-Euclidean space belongs to the straight line and belongs to it alone. How can we ascertain by experiments if it belongs to any particular concrete object? Distances must be measured and how shall we know that any concrete magnitude which I have measured with my material instrument really represents the abstract distance. We have only removed the difficulty a little farther. In reality the property that I have just denunciated is not a property of the straight line alone. It is a property of the straight line and of distance. For it to serve as an absolute criterion we must be able to show not only that it does not also belong to any other line than the straight line and the distance but also that it does not belong to any other line than the straight line and to any other magnitude in distance. Now that is not true and if we are not convinced by these considerations I challenge anyone to give me a concrete experiment which can be interpreted in the Euclidean system and which cannot be interpreted in the system of Lobochevsky. As I am well aware that this challenge will never be accepted and may conclude that no experiment will ever be in contradiction with Euclid's postulate but on the other hand no experiment will ever be in contradiction with Lobochevsky's postulate. Five. But it is not sufficient that Euclidean or non-Euclidean geometry can ever be directly contradicted by experiment nor could it happen that it can only agree with experiment by a violation of the principle of sufficient reason and of that of the relativity of space. Let me explain myself. Consider any material system whatever. We have to consider on the one hand the state of the various bodies of the system for example their temperature their electric potential etc and on the other hand their position in space and among the data which enable us to define this position we distinguish the mutual distances of these bodies that define their relative positions and the conditions which define the absolute position of the system and its absolute orientation in space. The law of the phenomena which will be produced in this system will depend on the state of these bodies and on their mutual distances but because of the relativity and the inertia of space they will not depend on the absolute position and orientation of the system. In other words the state of the bodies and their mutual distances at any moment will solely depend on the state of the same bodies and on their mutual distances at the initial moment but will in no way depend on the absolute initial position of the system and of its absolute initial orientation. This is what we shall call for the sake of abbreviation the law of relativity. So far I have spoken as a Euclidean Geometry but I have said that an experiment whatever it may be requires an interpretation on the Euclidean hypothesis. It equally requires one on the non Euclidean hypothesis. Well we have made a series of experiments. We have interpreted them on the Euclidean hypothesis and we have recognized that these experiments thus interpreted do not violate this law of relativity. We now interpret them on the non Euclidean hypothesis. This is always possible only the non Euclidean distances of our different bodies in this new interpretation will not generally be the same as the Euclidean distances in the primitive interpretation. Will our experiment interpreted in this new manner be still in agreement with our law of relativity and if this agreement had not taken place would we not still have the right to say that experiment has proved the falsity of non Euclidean geometry? It is easy to see that this is an idle fear. In fact to apply the law of relativity in all its rigor it must be applied to the entire universe for if we were to consider only a part of the universe and if the absolute position of this part were to vary the distances of the other bodies of the universe would equally vary. Their influence on the part of the universe considered might therefore increase or diminish and this might modify the laws of the phenomena that take place in it. But if our system is the entire universe experiment is powerless to give us any opinion on its position and its absolute orientation in space. All that our instruments however perfect they may be can let us know will be the state of the different parts of the universe and their mutual distances. Hence our law of relativity may be enunciated as follows. The readings that we can make with our instruments at any given moment will depend only on the readings that we were able to make on the same instruments at the initial moment. Now such an enunciation is independent of all interpretation by experiments. If the law is true in the Euclidean interpretation it will also be true in the non-Euclidean interpretation. Allow me to make a short digression on this point. I've spoken above of the data which defined the position of the different bodies of the system I might also have spoken of those which define their velocities. I should then have to distinguish the velocity with which the mutual distances of the different bodies are changing and on the other hand the velocities of translation and rotation of the system that is to say the velocities with which its absolute position and orientation are changing. For the mind to be fully satisfied the law of relativity would have to be enunciated as follows. The state of the bodies and the mutual distances at any given moment as well as the velocities with which those distances are changing at that moment will depend only on the state of those bodies on the mutual distances at the initial moment and on the velocities with which those distances were changing at the initial moment. But they will not depend on the absolute initial position of the system nor on its absolute orientation nor on the velocities with which that absolute position and orientation were changing at the initial moments. Unfortunately the law thus enunciated does not agree with experiments at least as they are ordinarily interpreted. Supposing man were translated to a planet, the sky of which was constantly covered with a thick curtain of clouds so that he could never see the other stars. On that planet he would live as if it were isolated in space but he would notice that it revolves either by measuring its ellipticity which is ordinarily done by means of astronomical observations but which could be done by purely geodesic means or by repeating the experiment of Foucault's pendulum. The absolute rotation of this planet might be clearly shown in this way. Now here is a fact which shocks the philosopher but which the physicist is compelled to accept. We know that from this fact Newton concluded the existence of absolute space. I myself cannot accept this way of looking at it. I shall explain why in part three but for the moment it is not my intention to discuss this difficulty. I must therefore resign myself in the enunciation of the law of relativity to including velocities of every kind among the data which define the state of the bodies. However that may be the difficulty is the same for both Euclid's geometry and for Lobachevsky's. I need not therefore trouble about it further and I have only mentioned it incidentally. To sum up whichever way we look at it it is impossible to discover in geometric empiricism a rational meaning. Six experiments only teach us the relations of bodies to one another. They do not and cannot give us the relations of bodies in space nor the mutual relations of the different parts of space. Yes you reply a single experiment is not enough because it only gives us one equation with several unknowns but when I have made enough experiments I shall have enough equations to calculate all my unknowns. If I know the height of the main mast that is not sufficient to enable me to calculate the age of the captain. When you have measured every fragment of wood in a ship you will have many equations but you will be no nearer knowing the captain's age. All your measurements bearing on your fragments of wood can tell you only what concerns those fragments and similarly your experiments however numerous they may be referring only to the relations of bodies with one another will tell you nothing about the mutual relations of the different parts of space. Seven will you say that if the experiments have referenced to the bodies they at least have referenced the geometrical properties of the bodies? First what do you understand by the geometrical properties of bodies? I assume that it is a question of the relations of the bodies to space. These properties therefore are not reached by experiments which only have reference to the relations of bodies to one another and that is enough to show that it is not of those properties that there can be a question. Let us therefore begin by making ourselves clear as to the sense of the phrase geometrical properties of bodies. When I say that a body is composed of several parts I presume that I am thus enunciating a geometrical property and that will be true even if I agree to give the improper name of points to the very small parts I am considering. When I say that this or that part of a certain body is in contact with this or that part of another body I am enunciating a proposition which concerns the mutual relations of the two bodies and not the relations with space. I assume that you will agree with me that these are not geometrical properties. I am sure that at least you will grant that these properties are independent of all knowledge of metrical geometry. Admitting this I suppose that we have a solid body formed of eight thin rods. O A, O B, O C, O D, O E, O F, O G, O H connected at one of their extremities O. And let us take a second solid body for example a piece of wood on which I mark three little spots of ink which I shall call alpha, beta, and gamma. I now suppose that we find that we can bring into contact beta with A, G, O. And by that I mean alpha with A and at the same time beta with G and gamma with O. Then we can successfully bring into contact alpha, beta, gamma with B, G, O, C, G, O, D, G, O, E, G, O, F, G, O, then with A, H, O, B, H, O, C, H, O, V, H, O, E, H, O, F, H, O. And then alpha, gamma, successfully with A, B, B, C, C, D, D, E, E, F, F, A. Now these are observations that can be made without having any idea beforehand as to the form or the metrical properties of space. They have no reference whatever to the quote geometrical properties of bodies. These observations will not be possible if the bodies on which we experiment move in a group having the same structure as the Lobachevsky group. I mean according to the same laws as solid bodies in the Lobachevsky's geometry. They therefore suffice to prove that these bodies move according to the Euclidean group, or at least that they do not move according to the Lobachevsky group. That they may be compatible with the Euclidean group is easily seen, for we might make them so if the body alpha beta gamma were an invariable solid of our ordinary geometry in the shape of a right-angled triangle, and if the points A, B, C, D, E, F, G, H were the vertices of a polyhedron formed of two regular hexagonal pyramids of our ordinary geometry having A, B, C, D, E, F as their common base and having the one G and the other H as their vertices. Suppose now, instead of the previous observations, we note that we can as before apply alpha beta gamma successively to A, G, O, B, G, O, C, G, O, D, G, O, E, G, O, F, G, O, A, H, O, B, H, O, C, H, O, D, H, O, E, H, O, F, H, O, and then we can apply alpha beta and no longer alpha gamma successively to A, B, B, C, C, D, D, E, F, and F, A. These are observations that could be made if non-Euclidean geometry were true. If the body's alpha beta gamma and O, A, B, C, D, E, F, G, H were invariable solids. If the former were a right-angled triangle and the latter a double regular hexagonal pyramid of suitable dimensions. These new verifications are therefore impossible if the bodies move according to the Euclidean group, but they become possible if we suppose the bodies to move according to the Lobocheskyan group. They would therefore suffice to show, if we carry them out, that the bodies in question do not move according to the Euclidean group. And so, without making any hypothesis on the form and the nature of space, on the relations of the bodies in space, and without attributing to bodies any geometrical property, I have made observations which have enabled me to show, in one case, that the bodies experimented upon a move according to a group, the structure of which is Euclidean, and in the other case that they move in a group, the structure of which is Lobocheskyan. It cannot be said that all the first observation would constitute an experiment proving that space is Euclidean, and the second an experiment proving that space is non-Euclidean. In fact, it might be imagined, note I used the word imagined, that there are bodies moving in such a manner as to render possible the second series of observations, and the proof is that the first mechanic who came our way could construct it, if he would only take the trouble. But you must not conclude, however, that space is non-Euclidean. In the same way, just as ordinary solid bodies will continue to exist when the mechanic I constructed the strange bodies I have just mentioned, he would have to conclude that space is both Euclidean and non-Euclidean. Suppose, for instance, that we have a large sphere of radius r, that its temperature decreases from the center to the surface of the sphere, according to the law of which I spoke when I was describing the non-Euclidean world. We might have bodies whose dilation is negligible, and which would behave as ordinary and variable solids. And, on the other hand, we might have very dilatable bodies, which would behave as non-Euclidean solids. We might have two double pyramids, O, A, B, C, D, E, F, G, H, and two triangles, alpha, beta, gamma, and alpha, beta, gamma, and alpha prime, beta prime, gamma prime. The first double pyramid would be rectilinear, and the second curvilinear. The triangle alpha, beta, gamma would consist of undilatable matter, and the other, a very dilatable matter. It would then be possible to make the first observations with the double pyramid, O, A, H, and the triangle alpha, beta, gamma, and the second with the double pyramid, O prime, A prime, H prime, and the triangle, alpha prime, beta prime, gamma prime. And then experiment would seem to prove first that the Euclidean geometry is true, and then that it is false. Experiments therefore have a bearing, not on space, but on bodies. Supplement. Eight. To round the matter off, I ought to speak of a very delicate question, which will require considerable development, but I shall confine myself to summing up what I have written in the revue de metaphysique et de morale and in the monist. When we say that space has three dimensions, what do we mean? We have seen the importance of these internal changes, which are revealed to us by muscular sensations. They may serve to characterize the different attitudes of our body. Let us take arbitrarily as our origin, one of these attitudes, A. When we pass from this initial attitude to another attitude, B, we experience a series of muscular sensations, and this series, S, of muscular sensations, will define B. Observe, however, that we shall often look upon two series, S and S prime, as defining the same attitude, B. Since the initial and final attitudes, A and B, remaining the same, the intermediary attitudes of the corresponding sensations may differ. How then can we recognize the equivalence of these two series? Because they may serve to compensate for the same external change, or more generally, because, when it is a question of compensation for an external change, one of the series may be replaced by the other. Among these series, we have distinguished those which can alone compensate for an external change, and which we have called displacements. As we cannot distinguish two displacements which are very close together, the aggregate of these displacements presents the characteristics of a physical continuum. Experience teaches us that they are the characteristics of a physical continuum of six dimensions, but we do not know, as of yet, how many dimensions space itself possesses, so we must first of all answer another question. What is a point in space? Everyone thinks he knows, but that is an illusion. What we see when we try to represent to ourselves a point in space is a black spot on white paper, the spot of chalk on a blackboard, always an object. The question should therefore be understood as follows. What do I mean when I say the object B is at the point which a moment before was occupied by the object A? Again, what criterion will enable me to recognize it? I mean that although I have not moved, my muscular sense tells me this, my finger, which just now touched the object A, is now touching the object B. I might have used other criteria, for instance, another finger or the sense of sight, but the first criterion is sufficient. I know that if it answers in the affirmative, all other criteria will give the same answer. I know it from experiment. I cannot know it a priori. For the same reason I say that touch cannot be exercised at a distance. That is another way of enunciating the same experimental fact. If I say on the contrary that sight is exercised at a distance, it means that the criterion furnished by sight may give an affirmative answer, while the others reply in the negative. To sum up, for each attitude of my body, my finger determines a point, and it is that and that only which defines a point in space. To each attitude corresponds in this way a point, but it often happens that the same point corresponds to several different attitudes. In this case we say that our finger has not moved, but the rest of our body has. We distinguish therefore among changes of attitude those in which the finger does not move. How are we led to this? It is because we often remark that in these changes the object which is in touch with the finger remains in contact with it. Let us arrange then in the same class all the attitudes which are deduced one from the other by one of the changes that we have thus distinguished. To all these attitudes of the same class will correspond the same point in space. Then to each class will correspond a point, and to each point a class. Yet it may be said that what we get from this experiment is not the point, but the class of changes, or better still the corresponding class of muscular sensations. Thus when we say that space is three dimensions, we merely mean that an aggregate of these classes appears to us with the characteristics of a physical continuum of three dimensions. Then if, instead of defining the points in space with the aid of the first finger, I use for example another finger, would the results be the same? That is by no means our priori evident. But as we have seen, experiment has shown us that all our criteria are in agreement, and this enables us to answer in the affirmative. If we recur to what we have called displacements, the aggregate of which forms as we have seen a group, we shall be brought to distinguish those in which a finger does not move, and by what is preceded, those are the displacements which characterize a point in space, and their aggregate will form a subgroup of our group. To each subgroup of this kind then will correspond a point in space. We might be tempted to conclude that experiment has taught us the number of dimensions of space. But in reality our experiments have referred not to space, but to our body and its relations with neighboring objects. What is more, our experiments are exceedingly crude. In our mind the latent idea of a certain number of groups pre-existed. These are the groups with which Lie's theory is concerned. Which shall we choose to form a kind of standard by which to compare natural phenomena? And when this group is chosen, which of the subgroups shall we take to characterize a point in space? Experiment has guided us by showing us what choice adapts itself best to the properties of our body, but there its role ends. For more information or to volunteer please visit LibriVox.org. Recording by Conor Riley. Science and Hypothesis by Henri Poincaré Chapter 6 Classical Mechanics The English teach mechanics as an experimental science. On the continent it is taught always more or less as a deductive and a priori science. The English are right no doubt. How is it that the other method has been persisted in for so long? How is it that continental scientists who have tried to escape from the practice of their predecessors have in most cases been unsuccessful? On the other hand, if the principles of mechanics are of only experimental origin, are they not merely approximate and provisory? May we not be someday compelled by new experiments to modify or even to abandon them? These are the questions which naturally arise, and the difficulty of solution is largely due to the fact that treatises in mechanics do not clearly distinguish between what is experiment, what is mathematical reasoning, what is convention, and what is hypothesis. This is not all. 1. There is no absolute space, and we only conceive a relative motion, and yet in most cases mechanical facts are enunciated as if there is an absolute space to which they can be referred. 2. There is no absolute time. When we say that two periods are equal, the statement has no meaning and can only acquire a meaning by a convention. 3. Not only have we no direct intuition of the equality of two periods, but we have not even direct intuition of the simultaneity of two events occurring in two different places. I have explained this in an article entitled Measures de temps. 4. Finally, is not our Euclidean geometry in itself only a kind of convention of language? Mechanical facts might be enunciated with reference to a non-Euclidean space which would be less convenient but quite as legitimate as our ordinary space. The enunciation would become more complicated, but it would still be possible. Thus, absolute space, absolute time, and even geometry are not conditions which are imposed on mechanics. All these things no more existed before mechanics than the French language can be logically said to have existed before the truths which are expressed in French. We might endeavor to enunciate the fundamental law of mechanics in a language independent of all these conventions, and no doubt we should in this way get a clearer idea of those laws in themselves. This is what M. Andrade has tried to do, to some extent at any rate, in his leçon de mechanics physique. Of course, the enunciation of these laws would become much more complicated because all these conventions have been adopted for the very purpose of abbreviating and simplifying the enunciation. As far as we are concerned, I shall ignore all these difficulties, not because I disregard them, far from it, but because they have received sufficient attention in the first two parts of the book. Provisionally then, we shall admit absolute time and Euclidean geometry. The Principle of Inertia A body under the action of no force can only move uniformly in a straight line. Is this a truth imposed on the mind a priori? If this be so, how is it that the Greeks ignored it? How could they have believed that motion ceases with the cause of motion? Or again, that every body, if there is nothing to prevent it, will move in a circle the noblest of all forms of motion. If it be said that the velocity of a body cannot change, if there is no reason for it to change, may we not just as legitimately maintain that the position of a body cannot change, or that the curvature of its path cannot change without the agency of an external cause? Is then the Principle of Inertia, which is not an a priori truth, an experimental fact? Have there ever been experiments on bodies acted on by no forces, and if so, how did we know that no forces were acting? The usual instance is that of a ball rolling for a very long time on a marble table. But why do we say it is under the action of no force? Is it because it is too remote from all other bodies to experience any sensible action? It is not further from the earth than if it were thrown freely into the air, and we all know that in that case it would be subject to the attraction of the earth. Teachers of mechanics usually pass rapidly over the example of the ball, but they add that the principle of inertia is verified indirectly by its consequences. This is very badly expressed. They evidently mean that various consequences may be verified by a more general principle, of which the principle of inertia is only a particular case. I shall propose for this general principle the following enunciation. The acceleration of a body depends only on its position and that of neighboring bodies, and on their velocities. Mathematicians would say that the movements of all the material molecules of the universe depend on differential equations of the second order. To make it clear that this is really a generalization of the law of inertia, we may again have recourse to our imagination. The law of inertia, as I have said above, is not imposed on us a priori. Other laws would be just as compatible with the principle of sufficient reason. If a body is not acted upon by a force, instead of supposing that its velocity is unchanged, we may suppose that its position or its acceleration is unchanged. Let us for a moment suppose that one of these two laws is a law of nature, and substitute it for the law of inertia, what will be the natural generalization? A moment's reflection will show us. In the first case, we may suppose that the velocity of a body depends only on its position and that of neighboring bodies. In the second case, that the variation of the acceleration of a body depends only on the position of the body and of neighboring bodies on their velocities and accelerations. Or, in mathematical terms, the differential equations of the motion would be of the first order in the first case and of the third order in the second. Let us now modify our supposition a little. Suppose a world analogous to our solar system, but one in which, by a singular chance, the orbits of all the planets have neither eccentricity nor inclination, and further, I suppose that the masses of the planets are too small for their mutual perturbations to be sensible. Astronomers living in one of these planets would not hesitate to conclude that the orbit of a star can only be circular and parallel to a certain plane. The position of a star at a given moment would then be sufficient to determine its velocity and path. The law of inertia, which they would adopt, would be the former of the two hypothetical laws I have mentioned. Now imagine this system to be someday crossed by a body of vast mass and immense velocity coming from distant constellations. All the orbits would be profoundly disturbed. Our astronomers would not be greatly astonished. They would guess that this new star is in itself quite capable of doing all the mischief, but they would say, as soon as it has passed by, order will again be established. No doubt the distances of the planets from the sun will not be the same as before the Cataclysm, but the orbits will become circular again as soon as the disturbing causes disappeared. It would be only when the perturbing body is remote, and when the orbits, instead of being circular, are found to be elliptical, that the astronomers would find out their mistake and discover the necessity of reconstructing their mechanics. I have dwelt on these hypotheses, for it seems to me that we can clearly understand our generalized law of inertia only by opposing it to a contrary hypothesis. Has this generalized law of inertia been verified by experiment, and can it be so verified? When Newton wrote the Principia, he certainly regarded this truth as experimentally acquired and demonstrated. It was so in his eyes, not only from the anthropomorphic conception to which I shall later refer, but also because of the work of Galileo. It was so proved by the laws of Kepler. According to those laws, in fact, the path of a planet is entirely determined by its initial position and initial velocity. This, indeed, is what our generalized law of inertia requires. For this principle to be only true in appearance, lest we should fear that someday it must be replaced by one of the analogous principles which I opposed to it just now, we must have been led astray by some amazing chance such as that which has led into error our imaginary astronomers. Such a hypothesis is so unlikely that it need not delay us. No one will believe that there can be such chances. No doubt the probability that two eccentricities are both exactly zero is not smaller than the probability that one is point one and the other point two. The probability of a simple event is not smaller than that of a complex one. If, however, the former does occur, we shall not attribute its occurrence to chance. We shall not be inclined to believe that nature has done it deliberately to deceive us. The hypothesis of an error of this kind being discarded, we may admit that so far as astronomy is concerned, our law has been verified by experiment. But astronomy is not the whole of physics. May we not fear that someday a new experiment will falsify the law in some domain of physics? An experimental law is always subject to revision. We may always expect to see it replaced by some other and more exact law. But no one seriously thinks that the law of which we speak will ever be abandoned or amended. Why? Precisely, because it will never be submitted to a decisive test. In the first place, for this test to be complete, all the bodies of the universe must return with their initial velocities to their initial positions after a certain time. We ought then to find that they would resume their original paths. But this test is impossible. It can be only partially applied, and even when it is applied, there will still be some bodies which will not return to their original positions. Thus there will be a ready explanation of any breaking down of the law. Yet this is not all. In astronomy we see the bodies whose motion we are studying, and in most cases we grant that they are not subject to the action of other invisible bodies. Under these conditions, our law must certainly be either verified or not. But it is not so in physics. If physical phenomena are due to motion, it is to the motion of molecules which we cannot see. If then the acceleration of bodies we cannot see depends on something else than the positions or velocities of other visible bodies or of invisible molecules, the existence of which we have been led previously to admit. There is nothing to prevent us from supposing that this something else is the position or velocity of other molecules of which we have not so far suspected the existence. The law will be safeguarded. Let me express the same thought in another form in mathematical language. Suppose we are observing n molecules, and find that their 3n coordinates satisfy a system of 3n differential equations of the fourth order, and not of the second as required by the law of inertia. We know that by introducing 3n variable auxiliaries, a system of 3n equations of the fourth order may be reduced to a system of 6n equations of the second order. If then we suppose that the 3n auxiliary variables represent the coordinates of n invisible molecules, the result is again comfortable to the law of inertia. To sum up, this law, verified experimentally in some particular cases, may be extended fearlessly to the most general cases, for we know that in these general cases it can either be confirmed nor contradicted by experiment. The law of acceleration. The acceleration of a body is equal to the force which acts on it divided by its mass. Can this law be verified by experiment? If so, we have to measure the 3 magnitudes mentioned in the enunciation. Acceleration, force, and mass. I admit that acceleration may be measured because I pass over the difficulty arising from the measurement of time. But how are we to measure force and mass? We do not even know what they are. What is mass? Newton replies, the product of the volume and the density. It were better to say, answer Thompson and Tate, that density is the quotient of the mass by the volume. What is force? It is, replies Lagrange, that which moves or tends to move a body. It is, according to Kirchhoff, the product of the mass and the acceleration. Then why not say that the mass is the quotient of the force by the acceleration? These difficulties are insurmountable. When we say that force is the cause of motion, we are talking metaphysics, and this definition, if we had to be content with it, would be absolutely fruitless, and would lead to absolutely nothing. For a definition to be of any use, it must tell us how to measure force, and that is quite sufficient, for it is by no means necessary to tell what force is in itself, nor whether it is the cause or the effect of motion. We must therefore first define what is meant by the equality of two forces. When are two forces equal? We are told that it is when they give the same acceleration to the same mass, or when acting in opposite directions they are in equilibrium. This definition is a sham. A force applied to a body cannot be uncoupled and applied to another body, as an engine is uncoupled from one train and coupled to another. It is therefore impossible to say what acceleration such a force applies to such a body would give to another body if it were applied to it. It is impossible to tell how two forces which are not acting in exactly opposite directions would behave if they were acting in opposite directions. It is this definition which we try to materialize as it were, when we measure a force with a dynamometer or with a balance. Two forces, f and f prime, which I suppose for simplicity to be acting vertically upwards, are respectively applied to two bodies, c and c prime. I attach a body weighing p, first to c, and then to c prime. If there is equilibrium in both cases, I conclude that the two forces, f and f prime, are equal, for they are both equal to the weight of the body p. But am I certain that the body p has kept its weight when I transferred it from the first body to the second? Far from it. I am certain of the contrary. I know that the magnitude of the weight varies from one point to another, and that it is greater, for instance, at the pole than at the equator. No doubt the difference is very small, and we neglect it in practice, but a definition must have mathematical rigor. This rigor does not exist. What I say of weight would apply equally to the force of the spring of a dynamometer, which would vary according to temperature and many other circumstances. Nor is this all. We cannot say that the weight of the body p is applied to the body c, and keeps in equilibrium the force f. What is applied to the body c is the action of the body p on the body c. On the other hand, the body p is acted on by its weight, and by the reaction r of the body c on p, the forces f and a are equal, because they are in equilibrium. The forces a and r are equal by virtue of the principle of action and reaction, and finally the force r and the weight p are equal because they are in equilibrium. From these three equalities, we deduce the equality of the weight p and the force f. Thus we are compelled to bring into our definition of the equality of two forces, the principle of the equality of action and reaction, hence this principle can no longer be regarded as an experimental law, but only as a definition. To recognize the equality of two forces, we are then in possession of two rules, the equality of two forces in equilibrium, and the equality of action and reaction. But as we have seen, these are not sufficient, and we are compelled to have recourse to a third rule, and to admit that certain forces, the weight of a body for instance, are constant in magnitude and direction. But this third rule is an experimental law. It is only approximately true, it is a bad definition. We are then reduced to Kirchhoff's definition. Force is the product of the mass and the acceleration. This law of Newton in turn ceases to be regarded as an experimental law, it is now only a definition. But as a definition it is insufficient, for we do not know what mass is. It enables us no doubt to calculate the ratio of two forces applied at different times to the same body, but it tells us nothing about the ratio of two forces applied to two different bodies. To fill up the gap, we must have recourse to Newton's third law, the equality of action and reaction, still regarded not as an experimental law, but as a definition. Two bodies, A and B, act on each other. The acceleration of A, multiplied by the mass of A, is equal to the action of B on A. In the same way the acceleration of B, multiplied by the mass of B, is equal to the reaction of A on B. As by definition the action and the reaction are equal, the masses of A and B are respectively in the inverse ratio of their masses. Thus is the ratio of the two masses defined, and it is for experiment to verify that the ratio is constant. This would do very well if the two bodies were alone and could be abstracted from the action of the rest of the world, but this is by no means the case. The acceleration of A is not solely due to the action of B, but to that of a multitude of other bodies, C, D, etc. To apply the preceding rule, we must decompose the acceleration of A into many components, and find out which of these components is due to the action of B. The decomposition would still be possible if we suppose that the action of C on A is simply added to that of B on A, and that the presence of the body C does not in any way modify the action of B on A, or that the presence of B does not modify the action of C on A. That is, if we admit that any two bodies attract each other, that their mutual action is along their join, and is only dependent on their distance apart. If in a word, we admit the hypothesis of central forces. We know that to determine the masses of the heavenly bodies, we adopt quite a different principle. The law of gravitation teaches us that the attraction of two bodies is proportional to their masses. If R is their distance apart, M and M prime their masses, k a constant, then their attraction will be the quantity kM M prime divided by R squared. What we are measuring is therefore not mass, the ratio of the force to the acceleration, but the attracting mass, not the inertia of the body, but its attracting power. It is an indirect process, the use of which is not indispensable theoretically. We might have said that the attraction is inversely proportional to the square of the distance, without being proportional to the product of the masses, that it is equal to f divided by R squared, and not to kM M prime. If it were so, we should nevertheless, by observing the relative motion of the celestial bodies, be able to calculate the masses of these bodies. But have we any right to admit the hypothesis of central forces? Is this hypothesis rigorously accurate? Is it certain that it will never be falsified by experiment, who will venture to make such an assertion? And if we must abandon this hypothesis, the building which has been so laboriously erected must fall to the ground. We have no longer any right to speak of the component of the acceleration of A, which is due to the action of B. We have no means of distinguishing it from that which is due to the action of C, or of any other body. The rule becomes inapplicable in the measurement of masses. What, then, is left of the principle of the equality of action and reaction? If we reject the hypothesis of central forces, this principle must go to. The geometrical resultant of all the forces applied to the different bodies of a system abstracted from all external action will be zero. In other words, the motion of the center of gravity of this system will be uniform and in a straight line. Here would seem to be a means of defining mass. The position of the center of gravity evidently depends on the values given to the masses. We must select these values so that the motion of the center of gravity is uniform and rectilinear. This will always be possible if Newton's Third Law holds good, and it will be in general possible only in one way. But no system exists, which is abstracted from all external action. Every part of the universe is subject, more or less, to the action of the other parts. The law of motion of the center of gravity is only rigorously true when applied to the whole universe. But then, to obtain the values of the masses, we must find the motion of the center of gravity of the universe. The absurdity of this conclusion is obvious. The motion of the center of gravity of the universe will be forever to us unknown. Nothing, therefore, is left, and our efforts are fruitless. There is no escape from the following definition, which is only a confession of failure. Masses are coefficients, which it is found convenient to introduce into calculations. We could reconstruct our mechanics by giving to our masses different values. The new mechanics would be in contradiction, neither with experiment nor with the general principles of dynamics, the principle of inertia, proportionality of masses and accelerations, equality of action and reaction, uniform motion of the center of gravity in a straight line, and areas. But the equations of this mechanics would not be so simple. Let us clearly understand this. It would be only the first terms which would be simple, i.e. those we already know through experiment. Perhaps the small masses could be slightly altered without the complete equations gaining or losing in simplicity. Hertz has inquired if the principles and mechanics are rigorously true. In the opinion of many physicists, it seems inconceivable that experiment will ever alter the impregnable principles and mechanics, and yet what is due to experiment may always be rectified by experiment. From what we have just seen, these spheres would appear to be groundless. The principles of dynamics appeared to us first as experimental truths, but we have been compelled to use them as definitions. It is by definition that force is equal to the product of the mass and the acceleration. This is a principle which is henceforth beyond the reach of any future experiment. Thus it is by definition that action and reaction are equal and opposite. But then it will be said these unverifiable principles are absolutely devoid of any significance. They cannot be disproved by experiment, but we can learn from them nothing of any use to us. What then is the use of studying dynamics? This somewhat rapid condemnation would be rather unfair. There is not in nature any system perfectly isolated, perfectly abstracted from all external action, but there are systems which are nearly isolated. If we observe such a system, we can study not only the relative motion of its different parts with respect to each other, but the motion of its center of gravity with respect to the other parts of the universe. We then find that the motion of its center of gravity is nearly uniform and rectilinear in the conformity with Newton's third law. This is an experimental fact, which cannot be invalidated by a more accurate experiment. What in fact would a more accurate experiment teach us? It would teach us that the law is only approximately true, and we know that already. Thus is explained how experiment may serve as a basis for the principles of mechanics, and yet will never invalidate them. Anthropomorphic Mechanics It will be said that Kirchhoff has only followed the general tendency of mathematicians toward nominalism. From this his skill as a physicist has not saved him. He wanted a definition of force, and he took the first that came handy, but we do not require a definition of force. The idea of force is primitive, irreducible, indefinable, we all know what it is. Of it we have direct intuition. This direct intuition arises from the idea of effort which is familiar to us from childhood, but in the first place, even if this direct intuition made known to us the real nature of force in itself, it would prove to be an insufficient basis for mechanics. It would more over be quite useless. The important thing is not to know what force is, but how to measure it. Everything which does not teach us how to measure it is as useless to the mechanism as, for instance, the subjective idea of heat and cold to the student of heat. This subjective idea cannot be translated into numbers, and is therefore useless. A scientist whose skin is an absolutely bad conductor of heat, and who therefore has never felt the sensation of heat or cold, would read a thermometer in just the same way as anyone else, and would have enough material to construct the whole theory of heat. Now this immediate notion of effort is of no use to us in the measurement of force. It is clear, for example, that I shall experience more fatigue in lifting a weight of 200 pounds than a man who is accustomed to lifting heavy burdens. But there is more than this. This notion of effort does not teach us the nature of force. It is definitively reduced to a recollection of muscular sensations, and no one will maintain that the sun experiences a muscular sensation when it attracts the earth. All that we can expect to find from it is a symbol, less precise and less convenient than the arrows to denote direction used by geometers, and quite as remote from reality. Anthropomorphism plays a considerable historic role in the genesis of mechanics. Perhaps it may get furnish us with the symbol which some minds may find convenient, but it can be the foundation of nothing of a really scientific or philosophical character. The Thread School. Monsieur Andrade, in his Laissant de Mechanique Physique, has modernized anthropomorphic mechanics. To the school of mechanics with which Kirchhoff is identified, he opposes a school which is quaintly called the Thread School. This school tries to reduce everything to the consideration of certain material systems of negligible mass, regarded in a state of tension and capable of transmitting considerable effort to distant bodies, systems of which the ideal type is the fine string, wire, or thread. A thread which transmits any force is slightly lengthened in the direction of that force. The direction of the thread tells us the direction of the force, and the magnitude of the force is measured by the lengthening of the thread. We may imagine such an experiment as the following. A body A is attached to a thread. At the other extremity of the thread acts a force which is made to vary until the length of the thread is increased by alpha, and the acceleration of body A is recorded. A is then detached, and a body B is attached to the same thread, and the same or another force is made to act until the increment of length is again alpha, and the acceleration of B is noted. The experiment is then renewed with both A and B until the increment of length is beta. The four accelerations observed should be proportional. Here we have an experimental verification of the law of acceleration enunciated above. Again we may consider a body under the action of several threads in equal tension, and by experiment we determine the direction of those threads when the body is in equilibrium. This is an experimental verification of the law of composition of forces. But as a matter of fact, what have we done? We have defined the force acting on a string by the deformation of the thread, which is reasonable enough. We have then assumed that if a body is attached to this thread, the effort which is transmitted to it by the thread is equal to the action exercised by the body on the thread. In fact, we have used the principle of action and reaction by considering it, not as an experimental truth, but as the very definition of force. This definition is quite as conventional as that of Kirchhoff, but it is much less general. All the forces are not transmitted by the thread, and to compare them they would all have to be transmitted by identical threads. If we even admitted that the earth is attached to the sun by an invisible thread, at any rate it will be agreed that we have no means of measuring the increment of the thread. 9 times out of 10, in consequence, our definition will be in default. No sense of any kind can be attached to it, and we must fall back on that of Kirchhoff. But why then go on in this roundabout way? You admit a certain definition of force which has a meaning only in certain particular cases. In those cases you verify by experiment that it leads to the law of acceleration. On the strength of these experiments, you then take the law of acceleration as a definition of force in all the other cases. Would it not be simpler to consider the law of acceleration as a definition in all cases, and to regard the experiments in question not as verifications of that law, but as verifications of the principle of action and reaction, or as providing the deformations of an elastic body depend only on the forces acting on that body. Without taking into account the fact that the conditions in which your definition could be accepted can only be very imperfectly fulfilled, that a thread is never without mass, that it is never isolated from all of their forces, then the reaction of bodies attached to its extremities. The ideas expounded by Monsieur Andrade are nonetheless very interesting. If they do not satisfy our logical requirements, they give us a better view of the historical genesis of the fundamental ideas of mechanics. The reflections they suggest show us how the human mind passed from a naive anthropomorphism to the present conception of science. We see that we end with an experiment which is very particular, and as a matter of fact, very crude, and we start with a perfectly general law, perfectly precise, the truth of which we regard as absolute. We have, so to speak, freely conferred this certainty on it by looking upon it as a convention. Are the laws of acceleration and of the composition of forces only arbitrary conventions? Conventions, yes, arbitrary, no. They would be so, if we lost sight of the experiments which led the founders of the science to adopt to them, and which, imperfect as they were, were sufficient to justify their adoption. It is well, from time to time, to let our attention dwell on the experimental origin of these conventions. Chapter 7 Relative and Absolute Motion The Principle of Relative Motion Sometimes endeavors have been made to connect the law of acceleration with a more general principle. The movement of any system whatever ought to obey the same laws, whether it is referred to fixed axes or to the movable axes which were implied in uniform motion in a straight line. This is the principle of relative motion. It is imposed upon us for two reasons. The commonest experiment confirms it. The consideration of the contrary hypothesis is singularly repugnant to the mind. Let us admit it, then, and consider a body under the action of a force. The relative motion of this body with respect to an observer, moving with a uniform velocity equal to the initial velocity of the body, should be identical with what would be its absolute motion if it started from rest. We conclude that its acceleration must not depend upon its absolute velocity, and from that we attempt to deduce the complete law of acceleration. For a long time there have been traces of this proof and the regulations for the degree of Bachelor of Science. It is clear that the attempt has failed. The obstacle which prevented us from proving the law of acceleration is that we have no definition of force. This obstacle subsists in its entirety since the principle invoked has not furnished us with a missing definition. The principle of relative motion is nonetheless very interesting and deserves to be considered for its own sake. Let us try to enunciate it in an accurate manner. We have said above that the accelerations of the different bodies which form part of an isolated system only depend on their velocities and their relative positions, and not on their velocities and their absolute positions, provided that the movable axes to which the relative motion is referred move uniformly in a straight line. Or, if it is preferred, their accelerations depend only on the differences of their velocities and the differences of their coordinates, and not on the absolute values of these velocities and coordinates. If this principle is true for relative accelerations, or rather for differences of acceleration, by combining it with the law of reaction we shall deduce that it is true for absolute accelerations. It remains to be seen how we can prove that differences of acceleration depend only on differences of velocities and coordinates, or, to speak in mathematical language, that these differences of coordinates satisfy differential equations of the second order. Can this proof be deduced from experiment, or from a priori conditions? Remembering what we have said before, the reader will give his own answer. Thus enunciated, in fact, the principle of relative motion curiously resembles what I called above the generalized principle of inertia. It is not quite the same thing, since it is a question of differences of coordinates and not of the coordinates themselves. The new principle teaches us something more than the old, but the same discussion applies to it and would lead to the same conclusions. We need not recur to it. Newton's Argument Here we find a very important and even slightly disturbing question. I have said that the principle of relative motion was not for us simply a result of experiment, that a priori every contrary hypothesis must be repugnant to the mind. But then, why is the principle only true if the motion of the movable axes is uniform and in a straight line? It seems that it should be imposed upon us with the same force if the motion is accelerated, or at any rate if it reduces to a uniform rotation. In these two cases, in fact, the principle is not true. I need not dwell on the case in which the motion of the axes is in a straight line and not uniform. The paradox does not bear a moment's examination. If I am in a railway carriage and if the train, striking against any obstacle whatever, is suddenly stopped, I shall be projected on to the opposite side, although I have not been directly acted upon by any force. There is nothing mysterious in that. And if I have not been subject to the action of any external force, the train has experienced an external impact. There could be nothing paradoxical in the relative motion of two bodies being disturbed when the motion of one or the other is modified by an external cause. Nor need I dwell on the case of relative motion, referring to axes which rotate uniformly. If the sky were forever covered with clouds, and if we had no means of observing the stars, we might nevertheless conclude that the earth turns round. We should be warned of this fact by the flattening at the poles, or by the experiment of Fokalt's pendulum. And yet, would there in this case be any meaning in saying that the earth turns round? If there is no absolute space, can a thing turn without turning with respect to something? And, on the other hand, how can we admit Newton's conclusion and believe in absolute space? But it is not sufficient to state that all possible solutions are equally unpleasant to us. We must analyze in each case the reason of our dislike in order to make our choice with the knowledge of the cause. The long discussion which follows must therefore be excused. Let us resume our imaginary story. Thick clouds hide the stars from men who cannot observe them, and even are ignorant of their existence. How will those men know that the earth turns round? No doubt, for a longer period than did our ancestors, they will regard the soil on which they stand as fixed and immovable. They will wait much longer time than we did for the coming of a Copernicus. But this Copernicus will come at last. How will he come? In the first place, the mechanical school of this world would not run their heads against an absolute contradiction. In the theory of relative motion we observe, besides real forces, two imaginary forces, which we call ordinary centrifugal force, and compounded centrifugal force. Our imaginary scientists can thus explain everything by looking upon these two forces as real, and they would not see in this a contradiction of the generalized principle of inertia, for these forces would depend, the one on the relative positions of the different parts of the system, such as real attractions, and the other on their relative velocities, as in the case of real frictions. Many difficulties, however, would before long awaken their attention. If they succeeded in realizing an isolated system, the center of gravity of this system would not have an approximately rectilinear path. They could invoke, to explain this fact, the centrifugal forces which they would regard as real, and which no doubt they would attribute to the mutual actions of the bodies. Only they would not see these forces vanish at great distances. That is to say, in proportion as the isolation is better realized. Far from it. Centrifugal force increases indefinitely with distance. Already this difficulty would seem to them sufficiently serious, but would not detain them for long. They would soon imagine some very subtle medium, analogous to our ether, in which all bodies would be bathed, and which would exercise on them a repulsive action. But that is not all. Space is symmetrical. Yet the laws of motion would present no symmetry. They should be able to distinguish between right and left. They would see, for instance, that cyclones always turn in the same direction, while for reasons of symmetry they should turn indifferently in any direction. If our scientists were able by dint of much hard work to make their universe perfectly symmetrical, this symmetry would not subsist, although there is no apparent reason why it should be disturbed in one direction more than in another. They would extract this from the situation, no doubt. They would invent something, which would not be more extraordinary than the glass spheres of Ptolemy, and would thus go on accumulating complications until the long expected Copernicus would sweep them all away with a single blow, saying it is much more simple to admit that the earth turns round. Just as our Copernicus said to us, it is more convenient to suppose that the earth turns round because the laws of astronomy are thus expressed in a more simple language. So he would say to them, it is more convenient to suppose that the earth turns round because the laws of mechanics are thus expressed in much more simple language. That does not prevent absolute space, that is to say, the point to which we must refer the earth to know if it really does turn round from having no objective existence, and hence this affirmation the earth turns round has no meaning since it cannot be verified by experiment. Since such an experiment not only cannot be realized or even dreamed of by the most daring Jules Verne but cannot even be conceived of without contradiction, or in other words these two propositions the earth turns round and it is more convenient to suppose that the earth turns round have one in the same meaning. There is nothing more in one than in the other. Perhaps they will not be content with this and may find it surprising that among all the hypotheses or rather all the conventions that can be made on this subject there is one which is more convenient than the rest. But if we have admitted it without difficulty when it is a question of the laws of astronomy why should we object when it is a question of the laws of mechanics? We have seen that the coordinates of bodies are determined by differential equations of the second order and that so are the differences of these coordinates. This is what we have called the generalized principle of inertia and the principle of relative motion. If the distances of these bodies were determined in the same way by equations of the second order it seems that the mind should be entirely satisfied. How far does the mind receive this satisfaction and why is it not content with it? To explain this we have better take a simple example. I assume a system analogous to our solar system but in which fixed stars foreign to this system cannot be perceived so that astronomers can only observe the mutual distances of planets and the sun and not the absolute longitudes of the planets. If we deduce directly from Newton's law the differential equations which define the variation of these distances these equations will not be of the second order. I mean that if outside Newton's law we knew the initial values of these distances and of their derivatives with respect to time that would not be sufficient to determine the values of these same distances at an ulterior moment. A datum would be still lacking and this datum might be, for example, what astronomers call the area constant. But here we may look at it from two different points of view. We may consider two kinds of constants. In the eyes of the physicist the world reduces to a series of phenomena depending on the one hand solely on initial phenomena and on the other hand on the laws connecting consequence and antecedent. If observation then teaches us that a certain quantity is a constant we shall have a choice of two ways of looking at it. So let us admit that there is a law which requires that this quantity shall not vary, but that by chance it has been found to have had in the beginning of time this value rather than that, a value that it has kept ever since. This quantity might then be called an accidental constant. Or again let us admit on the contrary that there is a law of nature which imposes on this quantity this value and not that. We shall then have what may be called an essential constant. For example, in virtue of the laws of Newton the duration of the revolution of the earth must be constant. But if it is 366 in something, said a real days, and not 300 or 400, it is because of some initial chance or other. It is an accidental constant. If, on the other hand, the exponent of the distance which figures in the expression of the attractive force is equal to minus 2 and not to minus 3, it is not by chance, but because it is required by Newton's law. It is an essential constant. I do not know if this manner of giving to chance its share is legitimate in itself and if there is not some artificiality about this distinction. But it is certain at least that in proportion as nature has secrets she will be strictly arbitrary and always uncertain in their application. As far as the area constant is concerned we are accustomed to look upon it as accidental. Is it certain that our imaginary astronomers would do the same? If they were able to compare two different solar systems they would get the idea that this constant may assume several different values. But I supposed at the outset, as I was entitled to do, that their system would appear isolated and that they would see no star which was foreign to their system. Under these conditions they could only detect a single constant which would have an absolutely invariable unique value. They would be led no doubt to look upon it as an essential constant. One word in passing to forestall an objection. The inhabitants of this imaginary world could neither observe nor define the area constant as we do, because absolute longitudes escape their notice. But that would not prevent them from being rapidly led to remark a certain constant which would be naturally introduced into their equations and which would be nothing but what we call the area constant. But then what would happen? If the area constant is regarded as essential as dependent upon a law of nature, then in order to calculate the distances of the planets at any given moment it would be sufficient to know the initial values of these distances and those of their first derivatives. From this new point of view distances will be determined by differential equations of the second order. Would this completely satisfy the minds of these astronomers? I think not. In the first place they would very soon see that in differentiating their equations so as to raise them to a higher order these equations would become much more simple and they would be especially struck by the difficulty which arises from symmetry. They would have to admit different laws according as the aggregate of the planets presented the figure of a certain polyhedron or rather of a regular polyhedron and these consequences can only be escaped by regarding the area constant as accidental. I have taken this particular example because I have imagined astronomers who would not be in the least concerned with terrestrial mechanics and whose vision would be bounded by the solar system. But our conclusions apply in all cases. Our universe is more extended than theirs since we have fixed stars. But it too is very limited so we might reason on the whole of our universe just as these astronomers do on their solar system. We thus see that we should be definitively led to conclude that the equations which define distances are of an order higher than the second. Why should this alarm us? Why do we find it perfectly natural that the sequence of phenomena depends on initial values of the first derivatives of these distances while we hesitate to admit that they may depend on the initial values of the second derivatives? It can only be because of mental habits created in us by the constant study of the generalized principle of inertia and its consequences. The values of the distances at any given moment depend upon their initial values, on that of their first derivatives, and something else. What is that something else? If we do not want it to be merely one of the second derivatives, we have only the choice of hypotheses. Suppose as is usually done that this something else is the absolute orientation of the universe in space, or the rapidity with which this orientation varies. This may be, it certainly is, the most convenient solution for the Geometer. But it is not the most satisfactory for the philosopher, because this orientation does not exist. We may assume that this something else is the position or the velocity of some invisible body, and this is what is done by certain persons, who have even called the body Alpha, although we are destined to never know anything about this body except its name. This is an artifice, entirely analogous to that of which I spoke at the end of the paragraph, containing my reflections on the principle of inertia. But as a matter of fact, the difficulty is artificial. Provided that the future indications of our instruments can only depend on the indications which they have given us, or that they might have formally given us, such is all we want, and with these conditions, we may rest