 8. Energy and Thermodynamics. Energetics. The difficulties raised by the classical mechanics have led certain minds to prefer a new system which they call energetics. Energetics took its rise in consequence of the discovery of the principle of the conservation of energy. Helmholtz gave it its definite form. We begin by defining two quantities which play a fundamental part in this theory. They are kinetic energy or vis-viva and potential energy. Every change that the bodies of nature can undergo is regulated by two experimental laws. First, the sum of the kinetic and potential energies is constant. This is the principle of the conservation of energy. Second, if a system of bodies is at A at the time t0 and at B at the time ti, it always passes from the first position to the second by such a path that the mean value of the difference between the two kinds of energy in the interval of time which separates the two epics t0 and ti is a minimum. This is Hamilton's principle and is one of the forms of the principle of least action. The energetic theory has the following advantages over the classical. First, it is less incomplete. That is to say, the principles of the conservation of energy and of Hamilton teach us more than the fundamental principles of the classical theory and exclude certain motions which do not occur in nature and which would be compatible with the classical theory. Second, it frees us from the hypothesis of atoms which it was almost impossible to avoid with the classical theory. But in its turn it raises fresh difficulties. The definitions of the two kinds of energy would raise difficulties almost as great as those of force and mass in the first system. However, we can get out of these difficulties more easily at any rate in the simplest cases. Assume an isolated system formed of a certain number of material points. Assume that these points are acted upon by forces depending only on their relative position and their distances apart and independent of their velocities. In virtue of the principle of the conservation of energy, there must be a function of forces. In this simple case, the enunciation of the principle of the conservation of energy is of extreme simplicity. A certain quantity, which may be determined by experiment, must remain constant. This quantity is the sum of two terms. The first depends only on the position of the material points and is independent of their velocities. The second is proportional to the squares of these velocities. This decomposition can only take place in one way. The first of these terms, which I shall call u, will be potential energy. The second, which I shall call t, will be kinetic energy. It is true that if t plus u is constant, so is any function of t plus u, parentheses t plus u, and parentheses. But this function, parentheses t plus u, and parentheses, will not be the sum of two terms, the one independent of the velocities and the other proportional to the square of the velocities. Among the functions which remain constant, there is only one which enjoys this property. It is t plus u or a linear function of t plus u. It matters not which, since this linear function may always be reduced to t plus u by a change of unit and of origin. This then is what we call energy. The first term we shall call potential energy and the second kinetic energy. The definition of the two kinds of energy may therefore be carried through without any ambiguity. So it is with the definition of mass. Kinetic energy, or vis-viva, is expressed very simply by the aid of the masses and of the relative velocities of all the material points with reference to one of them. These relative velocities may be observed and when we have the expression of the kinetic energy as a function of these relative velocities, the coefficients of this expression will give us the masses. So in this simple case, the fundamental ideas can be defined without difficulty. But the difficulties reappear in the more complicated cases, if the forces instead of depending solely on the distances depend also on the velocities. For example, Weber supposes the mutual action of two electric molecules to depend not only on their distance, but on their velocity and on their acceleration. If material points attracted each other according to an analogous law, u would depend on the velocity and it might contain a term proportional to the square of the velocity. How can we detect among such terms those that arise from t or u? And how, therefore, can we distinguish the two parts of the energy? But there is more than this. How can we define energy itself? We have no more reason to take as our definition t plus u rather than any function of t plus u when the property which characterized t plus u has disappeared, namely that of being the sum of two terms of a particular form. But that is not all. We must take account not only of mechanical energy properly so called, but of the other forms of energy, heat, chemical energy, electrical energy, etc. The principle of the conservation of energy must be written t plus u plus q equals the constant, where t is the sensible kinetic energy, u, the potential energy of position depending only on the position of the bodies, q, the internal molecular energy under the thermal, chemical, or electrical form. This would be all right if the three terms were absolutely distinct. If t were proportional to the square of the velocities, u'd independent of these velocities and of the state of the bodies, q independent of the velocities and of the positions of the bodies and depending only on their internal state. The expression for the energy could be decomposed in one way only into three terms of this form. But this is not the case. Let us consider electrified bodies. The electrostatic energy due to their mutual action will evidently depend on their charge, that is, on their state, but it will equally depend on their position. If these bodies are in motion, they will act electrodynamically on one another. And the electrodynamic energy will depend not only on their state and their position, but on their velocities. We have, therefore, no means of making the selection of the terms which should form part of t and u and q, and of separating the three parts of the energy. If t plus u plus q is constant, the same is true of any function whatever, parentheses t plus u plus q and parentheses. If t plus u plus q were of the particular form that I have suggested above, no ambiguity would ensue. Among the functions h parentheses t plus u plus q and parentheses, which remain constant, there is only one that would be of this particular form, namely the one which I would agree to call energy. But I have said this is not rigorously the case. Among the functions that remain constant, there is not one which can rigorously be placed in this particular form. How then can we choose from among them that which should be called energy? We have no longer any guide in our choice. Of the principle of the conservation of energy, there is nothing left then but an enunciation. There is something which remains constant. In this form it in its turn is outside the bounds of experiment and reduced to a kind of tautology. It is clear that if the world is governed by laws, there will be quantities which remain constant. Like Newton's laws and for an analogous reason, the principle of the conservation of energy being based on experiment can no longer be invalidated by it. This discussion shows that in passing from the classical system to the energetic, an advance has been made, but it shows at the same time that we have not advanced far enough. Another objection seems to be still more serious. The principle of least action is applicable to reversible phenomena, but it is by no means satisfactory as far as irreversible phenomena are concerned. Helmholtz attempted to extend it to this class of phenomena, but he did not and could not succeed. So far as this is concerned, all has yet to be done. The very enunciation of the principle of least action is objectionable. To move from one point to another, a material molecule acted upon by no force, but compelled to move on a surface will take as its path the geodesic line, that is, the shortest path. This molecule seems to know the point to which we want to take it, to foresee the time that it will take to reach it by such a path, and then to know how to choose the most convenient path. The enunciation of the principle presents it to us, so to speak, as a living and free entity. It is clear that it would be better to replace it by a less objectionable enunciation, one in which, as philosophers would say, final effects do not seem to be substituted for acting causes. Thermodynamics. The role of the two fundamental principles of thermodynamics becomes daily more important in all branches of natural philosophy, abandoning the ambitious theories of 40 years ago, encumbered as they were with molecular hypotheses. We now try to rest on thermodynamics alone, the entire edifice of mathematical physics. Will the two principles of Mayer and of Clausius assure to it foundations solid enough to last for some time? We all feel it. But whence does our confidence arise? An eminent physicist said to me one day, apropos of the law of errors, everyone stoutly believes it because mathematicians imagine that it is an effect of observation, and observers imagine that it is a mathematical theorem. And this was for a long time the case with the principle of the conservation of energy. It is no longer the same now. There is no one who does not know that it is an experimental fact. But then who gives us the right of attributing to the principle itself more generality and more precision than to the experiments which have served to demonstrate it? This is asking if it is legitimate to generalize, as we do every day, empirical data, and I shall not be so foolhardy as to discuss this question after so many philosophers have vainly tried to solve it. One thing alone is certain. If this permission were refused to us, science would not exist. Or at least it would be reduced to a kind of inventory to the ascertaining of isolated facts. It would no longer be to us of any value, since it could not satisfy our need of order and harmony, and because it would be at the same time incapable of prediction. As the circumstances which have preceded any fact whatever will never again in all probability be simultaneously reproduced, we already require a first generalization to predict whether the fact will be renewed as soon as the least of these circumstances is changed. But every proposition may be generalized in an infinite number of ways. Among all possible generalizations we must choose, and we cannot but choose, the simplest. We are therefore led to adopt the same course as if a simple law were, other things being equal, more probable than a complex law. A century ago it was frankly confessed and proclaimed abroad that nature loves simplicity, but nature has proved the contrary since then on more than one occasion. We no longer confess this tendency, and we only keep of it what is indispensable, so that science may not become impossible. In formulating a general, simple and formal law, based on a comparatively small number of not altogether consistent experiments, we have only obeyed a necessity from which the human mind cannot free itself. But there is something more, and that is why I dwell on this topic. No one doubts that Mayer's principle is not called upon to survive all the particular laws from which it was deduced, in the same way that Newton's law has survived the laws of Kepler from which it was derived, and which are no longer anything but approximations if we take perturbations into account. Now why does this principle thus occupy a kind of privileged position among physical laws? There are many reasons for that. At the outset we think that we cannot reject it, or even doubt its absolute rigor, without admitting the possibility of perpetual motion. We certainly feel distrust at such a prospect, and we believe ourselves less rash in affirming it than in denying it. That perhaps is not quite accurate. The impossibility of perpetual motion only implies the conservation of energy for reversible phenomena. The imposing simplicity of Mayer's principle equally contributes to strengthen our faith. In a law immediately deduced from experiments, such as Marriott's law, this simplicity would rather appear to us a reason for distrust, but here this is no longer the case. We take elements which at the first glance are unconnected. These arrange themselves in an unexpected order, and form a harmonious whole. We cannot believe that this unexpected harmony is a mere result of chance. Our conquest appears to be valuable to us in proportion to the efforts it has caused, and we feel the more certain of having snatched its true secret from nature in proportion as nature has appeared more jealous of our attempts to discover it. But these are only small reasons. Before we raise Mayer's law to the dignity of an absolute principle, a deeper discussion is necessary. But if we embark on this discussion, we see that this absolute principle is not even easy to enunciate. In every particular case, we clearly see what energy is, and we can give it at least a provisory definition. But it is impossible to find a general definition of it. If we wish to enunciate the principle in all its generality, and apply it to the universe, we see it vanish, so to speak, and nothing is left but this. There is something which remains constant. But has this a meaning? In the determinist hypothesis the state of the universe is determined by an extremely large number n of parameters, which I shall call x1, x2, x3, and so on to xn. As soon as we know at a given moment the values of these n parameters, we also know their derivatives with respect to time, and we can therefore calculate the values of these same parameters at an anterior or ulterior moment. In other words, these n parameters specify n differential equations of the first order. These equations have it n-1 integrals, and therefore there is n-1 functions of x1, x2, x3, etc to xn, which remain constant. If we say then there is something which remains constant, we are only enunciating a totology. We would be even embarrassed to decide which among our integrals is that which should retain the name of energy. Besides, it is not in this sense that Mayer's principle is understood when it is applied to a limited system. We admit then that p of our n parameters vary independently, so that we have only n minus p relations, generally linear, between our n parameters and their derivatives. Suppose, for the sake of simplicity, that the sum of the work done by the external forces is zero, as well as that of all the quantities of heat given off from the interior. What will then be the meaning of our principle? There is a combination of these n minus p relations of which the first member is an exact differential, and then this differential vanishing in virtue of our n minus p relations, its integral is a constant, and it is this integral that we call energy. But how can it be that there are several parameters whose variations are independent? That can only take place in the case of external forces, although we have supposed, for the sake of simplicity, that the algebraic sum of all the work done by these forces has vanished. If in fact the system were completely isolated from all external action, the values of our n parameters at any given moment would suffice to determine the state of the system at any ulterior moment, whatever, provided that we still clung to the determinist hypothesis. We should, therefore, fall back on the same difficulty as before. If the future state of the system is not entirely determined by its present state, it is because it further depends on the state of bodies external to the system. But then, is it likely that there exist among the parameters x, which define the state of the system of equations independent of the state of the external bodies? And if in certain cases we think we can find them, is it not only because of our ignorance and because the influence of these bodies is too weak for our experiment to be able to detect it? If the system is not regarded as completely isolated, it is probable that the rigorously exact expression of its internal energy will depend upon the state of the external bodies. Again, I have supposed above that the sum of all the external work is zero, and if we wish to be free from this rather artificial restriction, the enunciation becomes still more difficult. To formulate Mayer's principle by giving it an absolute meaning, we must extend it to the whole universe, and then we find ourselves face to face with the very difficulty we have endeavored to avoid. To sum up and to use ordinary language, the law of the conservation of energy can have only one significance, because there is in it a property common to all possible properties, but in the determinist hypothesis there is only one possible, and then the law has no meaning. In the indeterminist hypothesis, on the other hand, it would have a meaning even if we wish to regard it in an absolute sense. It would appear as a limitation imposed on freedom. But this word warns me that I am wandering from the subject, and that I am leaving the domain of mathematics and physics. I check myself therefore, and I wish to retain only one impression of the whole of this discussion, and that is, that Mayer's law is a form subtly enough for us to be able to put into it almost anything we like. I do not mean by that that it corresponds to no objective reality, no, that it is reduced to mere totology, since in each particular case, and provided we do not wish to extend it to the absolute, it has a perfectly clear meaning. This subtlety is a reason for believing that it will last long, and as, on the other hand, it will only disappear to be blended in a higher harmony, we may work with confidence and utilize it, certain beforehand that our work will not be lost. Almost everything that I have just said applies to the principle of Clausius, what distinguishes it is that it is expressed by an inequality. It will be said perhaps that it is the same with all physical laws, since their precision is always limited by errors of observation. But they at least claim to be first approximations, and we hope to replace them little by little by more exact laws. If, on the other hand, the principle of Clausius reduces to an inequality, this is not caused by the imperfection of our means of observation, but by the very nature of the question. General conclusions on part three. The principles of mechanics are therefore presented to us under two different aspects. On the one hand, there are truths founded on experiment and verified approximately as far as almost isolated systems are concerned. On the other hand, there are postulates applicable to the whole of the universe and regarded as rigorously true. If these postulates possess a generality and a certainty which falsifies the experimental truths from which they were deduced, it is because they reduce in final analysis to a simple convention that we have a right to make, because we are certain beforehand that no experiment can contradict it. This convention, however, is not absolutely arbitrary. It is not the child of our Caprice. We admit it because certain experiments have shown us that it will be convenient, and thus is explained how experiment has built up the principles of mechanics and why moreover it cannot reverse them. Take a comparison with geometry. The fundamental propositions of geometry, for instance Euclid's postulate, are only conventions, and it is quite as unreasonable to ask if they are true or false as to ask if the metric system is true or false. Only these conventions are convenient, and there are certain experiments which prove it to us. At the first glance the analogy is complete. The role of experiment seems the same. We shall therefore be tempted to say either mechanics must be looked upon as experimental science, and then it should be the same with geometry, or on the contrary geometry is a deductive science, and then we can say the same of mechanics. Such a conclusion would be illegitimate. The experiments which have led us to adopt as more convenient the fundamental conventions of geometry refer to bodies which have nothing in common with those that are studied by geometry. They refer to the properties of solid bodies and to the propagation of light in a straight line. These are mechanical optical experiments. In no way can they be regarded as geometrical experiments, and even the probable reason why our geometry seems convenient to us is that our bodies, our hands, and our limbs enjoy the properties of solid bodies. Our fundamental experiments are preemptively physiological experiments which refer not to the space which is the object that geometry must study, but to our body, that is to say, to the instrument which we use for that study. On the other hand, the fundamental conventions of mechanics and the experiments which prove to us that they are convenient certainly refer to the same objects or to analogous objects. Conventional and general principles are the natural and direct generalizations of experimental and particular principles. Let it not be said that I am thus tracing artificial frontiers between the sciences, that I am separating by a barrier, geometry properly so cold, from the study of solid bodies. I might just as well raise a barrier between experimental mechanics and the conventional mechanics of general principles. Who does not see, in fact, that by separating these two sciences we mutilate both, and that what will remain of the conventional mechanics when it is isolated will be but very little, and can in no way be compared with that grand body of doctrine which is called geometry. We now understand why the teaching of mechanics should remain experimental. Thus only can we be made to understand the genesis of the science, and that is indispensable for a complete knowledge of the science itself. Besides, if we study mechanics it is in order to apply it, and we can only apply it if it remains objective. Now as we have seen, when principles gain in generality and certainty, they lose in objectivity. It is therefore especially with the objective side of principles that we must be early familiarized, and this can only be by passing from the particular to the general, instead of from the general to the particular. Principles are conventions and definitions in disguise. They are, however, deduced from experimental laws, and these laws have, so to speak, been erected into principles to which our mind attributes an absolute value. Some philosophers have generalized far too much. They have thought that the principles were the whole of science, and therefore that the whole of science was conventional. This paradoxical doctrine, which is called nominalism, cannot stand examination. How can a law become a principle? It expressed a relation between two real terms, A and B, but it was not rigorously true. It was only approximate. We introduce arbitrarily an intermediate term C, more or less imaginary, and C is, by definition, that which has with A, exactly the relation expressed by the law. So our law is decomposed into an absolute and rigorous principle, which expresses the relation of A to C, and an approximate experimental and revisable law, which expresses the relation of C to B. But it is clear that however far this decomposition may be carried, laws will always remain. We shall now enter into the domain of laws properly so called. End of Chapter 8 Chapter 9 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 Anna Simon. Science and Hypothesis by Henri Poincaré. Chapter 9 Hypothesis in Physics The role of experiment and generalization. Experiment is the sole source of truth. It alone can teach us something new. It alone can give us certainty. These are two points that cannot be questioned. But then, if experiment is everything, what place is left for mathematical physics? What can experimental physics do with such an auxiliary? An auxiliary, more over, which seems useless and even may be dangerous. However, mathematical physics exists. It has rendered undeniable service, and that is a fact which has to be explained. It is not sufficient merely to observe. We must use our observations, and for that purpose we must generalize. This is what has always been done. Only as the recollection of past errors has made man more and more circumspect, he has observed more and more, and generalized less and less. Every age has scoffed at its predecessor, accusing it of having generalized too boldly and too naively. Descartes used to commiserate the Ionians. Descartes in his turn makes us smile, and no doubt someday our children will laugh at us. Is there no way of getting at once to the gist of the matter, and thereby escaping the Raylary which we foresee? Cannot we be content with experiment alone? No, that is impossible. That would be a complete misunderstanding of the true character of science. The man of science must work with method. Science is built up of facts as a house is built of stones. But an accumulation of facts is no more a science than a heap of stones is a house. Most important of all, the man of science must exhibit foresight. Carlile has written somewhere something after this fashion. Nothing but facts are of importance. John Lecklin passed by here. Here is something that is admirable. Here is a reality for which I would give all the theories in the world. Carlile was a compatriot of Bacon, and, like him, he wished to proclaim his worship of the God of things as they are. But Bacon would not have said that. That is the language of the historian. The physicist would most likely have said, Don Lecklin passed by here. It is all the same to me, for he will not pass this way again. We all know that there are good and bad experiments. The latter accumulate in vain. Whether there are a hundred or a thousand, one single piece of work by a real master, by a pastor, for example, will be sufficient to sweep them into oblivion. Bacon would have thoroughly understood that, for he invented the phrase experimentum crucis. But Carlile would not have understood it. A fact is a fact. A student has read such and such a number on his thermometer. He has taken no precautions. It does not matter. He has read it, and if it is only the fact which counts, this is a reality that is as much entitled to be called a reality as the nations of King John Lecklin. What then is a good experiment? It is that which teaches us something more than an isolated fact. It is that which enables us to predict and to generalize. Without generalization, prediction is impossible. The circumstances under which one has operated will never again be reproduced simultaneously. The fact observed will never be repeated. All that can be affirmed is that under analogous circumstances and analogous fact will be produced. To predict it, we must therefore invoke the aid of analogy. That is to say, even at this stage, we must generalize. However timid we may be, there must be interpolation. Experiment only gives us a certain number of isolated points. They must be connected by a continuous line, and this is a true generalization. But more is done. The curve thus traced will pass between and near the points observed. It will not pass through the points themselves. Thus we are not restricted to generalizing our experiment, we correct it. And the physicist who would abstain from these corrections and really content himself with experiment pure and simple would be compelled to enunciate very extraordinary laws indeed. Detached facts cannot therefore satisfy us, and that is why our science must be ordered, or better still, generalized. It is often said that experiments should be made without preconceived ideas. That is impossible. Not only would it make every experiment fruitless, but even if we wish to do so, it could not be done. Every man has his own conception of the world, and this he cannot so easily lay aside. We must, for example, use language and our language is necessarily steeped in preconceived ideas. Only they are unconscious preconceived ideas, which are a thousand times the most dangerous of all. Shall we say that if we cause others to intervene, of which we are fully conscious, that we shall only aggravate the evil? I do not think so. I am inclined to think that it will serve as ample counterpoises. I was almost going to say antidotes. They will generally disagree. They will enter into conflict one with another, and ipso facto they will force us to look at things under different aspects. This is enough to free us. He is no longer a slave who can choose his master. Thus, by generalization, every fact observed enables us to predict a large number of others. Only we ought not to forget that the first alone is certain, and that all the others are merely probable. However solidly founded a prediction may appear to us, we are never absolutely sure that experiment will not prove it to be baseless if we set to work to verify it. But the probability of its accuracy is often so great that practically we may be content with it. It is far better to predict without certainty than never to have predicted at all. We should never therefore disdain to verify when the opportunity presents itself. But every experiment is long and difficult, and the laborers are few, and the number of facts which we require to predict is enormous. And besides this mass, the number of direct verifications that we can make will never be more than a negligible quantity. Of this little that we can directly attain, we must choose the best. Every experiment must enable us to make a maximum number of predictions, having the highest possible degree of probability. The problem is, so to speak, to increase the output of the scientific machine. I may be permitted to compare science to a library which must go on increasing indefinitely. The librarian has limited funds for his purchases, and he must therefore strain every nerve not to waste them. Experimental physics has to make the purchases, and experimental physics alone can enrich the library. As for mathematical physics, her duty is to draw up the catalogue. If the catalogue is well done, the library is none the richer for it, but the reader will be enabled to utilize its riches, and also by showing the librarian the gaps in his collection, it will help him to make a judicious use of his funds, which is all the more important in as much as those funds are entirely inadequate. That is the role of mathematical physics. It must direct generalization, so as to increase what I called just now the output of science. By what means it does this, and how it may do it without danger, is what we have now to examine. The unity of nature. Let us first of all observe that every generalization supposes in a certain measure a belief in the unity and simplicity of nature. As far as the unity is concerned, there can be no difficulty. If the different parts of the universe were not as the organs of the same body, they would not react one upon the other. They would mutually ignore each other, and we in particular should only know one part. We need not therefore ask if nature is one, but how she is one. As for the second point, that is not so clear. It is not certain that nature is simple. Can we without danger act as if she were? There was a time when the simplicity of Marriott's law was an argument in favor of its accuracy. When Fresnel himself, after having said in a conversation with Laplace that nature cares not for analytical difficulties, was compelled to explain his words so as not to give offense to current opinion. Nowadays, ideas have changed considerably, but those who do not believe that natural laws must be simple are still often obliged to act as if they did believe it. They cannot entirely dispense with this necessity without making all generalization and therefore all science impossible. It is clear that any fact can be generalized in an infinite number of ways, and it is a question of choice. The choice can only be guided by considerations of simplicity. Let us take the most ordinary case that of interpolation. We draw a continuous line as regularly as possible between the points given by observation. Why do we avoid angular points and inflections that are too sharp? Why do we not make our curve describe the most capricious zigzags? It is because we know beforehand or think we know that the law we have to express cannot be so complicated as all that. The mass of Jupiter may be deduced either from the movements of its satellites or from the perturbations of the major planets or from those of the minor planets. If we take the mean of the determinations obtained by these three methods, we find three numbers very close together but not quite identical. This result might be interpreted by supposing that the gravitation constant is not the same in the three cases. The observations would be certainly much better represented. Why do we reject this interpretation? Not because it is absurd but because it is uselessly complicated. We shall only accept it when we are forced to and it is not imposed upon us yet. To sum up, in most cases every law is held to be simple until the contrary is proved. This custom is imposed upon physicists by the reasons that I have indicated but how can it be justified in the presence of discoveries which daily show as fresh details, richer and more complex? How can we even reconcile it with a unity of nature? For if all things are interdependent, the relations in which so many different objects intervene can no longer be simple. If we study the history of science, we see produced two phenomena which are so to speak each the inverse of the other. Sometimes it is simplicity which is hidden under what is apparently complex. Sometimes on the contrary it is simplicity which is apparent and which conceals extremely complex realities. What is there more complicated than the disturbed motions of the planets and what more simple than Newton's law? There, as Fresnel said, nature playing with analytical difficulties only uses simple means and creates by their combination I know not what tangled skin. Here it is the hidden simplicity which must be disentangled. Examples of the contrary abound. In the kinetic theory of gases molecules of tremendous velocity are discussed whose paths deformed by incessant impacts have the most capricious shapes and plow their way through space in every direction. The result observable is Marriott's simple law. Each individual fact was complicated. The law of great numbers has reestablished simplicity in the mean. Here the simplicity is only apparent and the coarseness of our senses alone prevents us from seeing the complexity. Many phenomena obey a law of proportionality. But why? Because in these phenomena there is something which is very small. The simple law observed is only the translation of the general analytical rule by which the infinitely small increment of a function is proportional to the increment of the variable. As in reality our increments are not infinitely small but only very small. The law of proportionality is only approximate and simplicity is only apparent. What I've just said applies to the law of the superposition of small movements which is so fruitful in its applications and which is the foundation of optics. And Newton's law itself? Its simplicity so long undetected is perhaps only apparent. Who knows if it be not due to some complicated mechanism to the impact of some subtle matter animated by irregular movements and if it has not become simple merely through the play of averages and large numbers. In any case it is difficult not to suppose that the true law contains complementary terms which may become sensible at small distances. If in astronomy they are negligible and if the law thus regains its simplicity it is solely on account of the enormous distances of the celestial bodies. No doubt if our means of investigation became more and more penetrating we should discover the simple beneath the complex and then the complex from the simple and then again the simple beneath the complex and so on without ever being able to predict what the last term will be. We must stop somewhere and for science to be possible we must stop where we have found simplicity. That is the only ground on which we can erect the edifice of our generalizations. But this simplicity being only apparent will the ground be solid enough? That is what we have now to discover. For this purpose let us see what part is played in our generalizations by the belief in simplicity. We have verified a simple law in a considerable number of particular cases. We refuse to admit that this coincidence so often repeated is a result of mere chance and we conclude that the law must be true in the general case. Kepler remarks that the positions of a planet observed by Tycho are all on the same ellipse. Not for one moment does he think that by a singular freak of chance Tycho had never looked at the heavens except at the very moment when the path of the planet happened to cut that ellipse. What does it matter then if the simplicity be real or if it hide a complex truth? Whether it be due to the influence of great numbers which reduces individual differences to a level or to the greatness or the smallness of certain quantities which allow of certain terms to be neglected in no case is it due to chance. This simplicity real or apparent has always a cause. We shall therefore always be able to reason in the same fashion and if a simple law has been observed in several particular cases we may legitimately suppose that it still will be true in analogous cases. To refuse to admit this would be to attribute an inadmissible role to chance. However there is a difference. If the simplicity were real and profound it would bear the test of the increasing precision of our methods of measurement. If then we believe nature to be profoundly simple we must conclude that it is an approximate and not a rigorous simplicity. This is what was formally done but it is what we have no longer the right to do. The simplicity of Kepler's laws for instance is only apparent but that does not prevent them from being applied to almost all systems analogous to the solar system though that prevents them from being rigorously exact. Role of Hypothesis. Every generalization is a hypothesis. Hypothesis therefore plays a necessary role which no one has ever contested. Only it should always be as soon as possible submitted to verification. It goes without saying that if it cannot stand this test it must be abandoned without any hesitation. This is indeed what is generally done but sometimes with a certain impatience. Ah well this impatience is not justified. The physicist who has just given up one of his hypotheses should on the contrary rejoice for he found an unexpected opportunity of discovery. His hypothesis, I imagine, had not been lightly adopted. It took into account all the known factors which seem capable of intervention in the phenomenon. If it is not verified it is because there is something unexpected and extraordinary about it because we are on the point of finding something unknown and new. Has the hypothesis thus rejected been sterile? Far from it. It may be even said that it has rendered more service than a true hypothesis. Not only has it been the occasion of a decisive experiment but if this experiment had been made by chance without the hypothesis no conclusion could have been drawn. Nothing extraordinary would have been seen and only one fact that more would have been catalogued without deducing from it the remotest consequence. Now under what conditions is the use of hypothesis without danger? The proposal to submit all to experiment is not sufficient. Some hypotheses are dangerous first and foremost those which are tacit and unconscious and since we make them without knowing them we cannot get rid of them. Here again there is a service that mathematical physics may render us. By the precision which is its characteristic we are compelled to formulate all the hypotheses that we would unhesitatingly make without its aid. Let us also notice that it is important not to multiply hypotheses indefinitely. If we construct a theory based upon multiple hypotheses and if experiment condemns it, which of the premises must be changed? It is impossible to tell. Conversely if the experiment succeeds must be supposed that it has verified all these hypotheses once. Can several unknowns be determined from a single equation? We must also take care to distinguish between the different kinds of hypotheses. First of all there are those which are quite natural and necessary. It is difficult not to suppose that the influence of very distant bodies is quite negligible, that small movements obey a linear law and that effect is a continuous function of its cause. I will say as much for the conditions imposed by symmetry. All these hypotheses affirm, so to speak, the common basis of all the theories of mathematical physics. They are the last that should be abandoned. There is a second category of hypotheses which I shall qualify as indifferent. In most questions the analyst assumes at the beginning of his calculations either that measure is continuous or the reverse that it is formed of atoms. In either case his results would have been the same. On the atomic supposition he has a little more difficulty in obtaining them, that is all. If then experiment confirms his conclusions, will he suppose that he has proved, for example, the real existence of atoms? In optical theories two vectors are introduced, one of which we consider as a velocity and the other as a vortex. This again is an indifferent hypothesis since we should have arrived at the same conclusions by assuming the former to be a vortex and the letter to be a velocity. The success of the experiment cannot prove therefore that the first vector is really a velocity, it only proves one thing, namely that it is a vector, and that is the only hypothesis that has really been introduced into the premises. To give it the concrete appearance that the fallibility of our mind's demands, it was necessary to consider it either as a velocity or as a vortex. In the same way it was necessary to represent it by an x or y, but the result will not prove that we were right or wrong in regarding it as a velocity nor will it prove we are right or wrong in calling it x and not y. These indifferent hypotheses are never dangerous provided their characters are not misunderstood. They may be useful either as artifices for calculation or to assist our understanding by concrete images to fix the ideas as we say. They need not therefore be rejected. The hypotheses of the third category are real generalizations. They must be confirmed or invalidated by experiment. Whether verified or condemned they will always be fruitful, but for the reasons I've given they will only be so if they are not too numerous. Origin of mathematical physics. Let us go further and study more closely the conditions which have assisted the development of mathematical physics. We recognize at the outset that the efforts of man of science have always tended to resolve the complex phenomenon given directly by experiment into a very large number of elementary phenomena and that in three different ways. First with respect to time. Instead of embracing in its entirety the progressive development of a phenomenon we simply try to connect each moment with the one immediately proceeding. We admit that the present state of the world only depends on the immediate past without being directly influenced so to speak by the recollection of a more distant past. Thanks to this postulate instead of studying directly the whole succession of phenomena we may confine ourselves to writing down its differential equation. For the laws of Kepler we substitute the law of Newton. Next we try to decompose the phenomena in space. What experiment gives us is a confused aggregate of facts spread over a scene of considerable extent. We must try to deduce the elementary phenomenon which will still be localized in a very small region of space. A few examples perhaps will make my meaning clear. If we wish to study in all its complexity the distribution of temperature in a cooling solid we could never do so. This is simply because if we only reflect that a point in the solid can directly impart some of its heat to a neighboring point it will immediately impart that heat only to the nearest points and it is but gradually that the flow of heat will reach other portions of the solid. The elementary phenomenon is the interchange of heat between two contiguous points. It is strictly localized and relatively simple if, as is natural, we admit that it is not influenced by the temperature of the molecules whose distance part is small. I bend a rod. It takes a very complicated form, the direct investigation of which would be impossible. But I can attack the problem however if I notice that its flexure is only the resultant of the deformations of the very small elements of the rod and that the deformation of each of these elements only depends on the forces which are directly applied to it and not on the least on those which may be acting on the other elements. In all these examples which may be increased without difficulty it is admitted that there is no action at a distance or at great distances. That is a hypothesis. It is not always true as the law of gravitation proves. It must therefore be verified. If it is confirmed even approximately it is valuable for it helps us to use mathematical physics at any rate by successive approximations. If it does not stand the test we must seek something else that is analogous for there are other means of arriving at the elementary phenomenon. If several bodies act simultaneously it may happen that their actions are independent and may be added one to the other either as vectors or as scalar quantities. The elementary phenomenon is then the action of an isolated body. Or suppose again it is a question of small movements or more generally of small variations which obey the well-known law of mutual or relative independence. The movement observed will then be decomposed into simple movements. For example sound into its harmonics and white light into its monochromatic components. When we have discovered in which direction to seek for the elementary phenomena by what means may we reach it. First it will often happen that in order to predict it or rather in order to predict what is useful to us it will not be necessary to know its mechanism. The law of great numbers will suffice. Take for example the propagation of heat. Each molecule radiates towards its neighbor. We need not inquire according to what law and if we make any supposition in this respect it will be an indifferent hypothesis and therefore useless and unverifiable. In fact by the action of averages and thanks the symmetry of the medium all differences are leveled and whatever the hypothesis may be the result is always the same. The same feature is presented in the theory of elasticity and in that of capillarity. The neighboring molecules attract and repel each other we need not inquire by what law. It is enough for us that this attraction is sensible at small distances only and that the molecules are very numerous that the medium is symmetrical and we have only to let the law of great numbers come into play. Here again the simplicity of the elementary phenomenon is hidden beneath the complexity of the observable result and phenomenon. But in its turn this simplicity was only apparent and disguised a very complex mechanism. Evidently the best means of reaching the elementary phenomenon would be experiment. It would be necessary by experimental artifices to dissociate the complex system which nature offers for our investigations and carefully to study the elements as dissociated as possible. For example natural white light would be decomposed into monochromatic lights by the aid of the prism and into polarized lights by the aid of the polarizer. Unfortunately that is neither always possible nor always sufficient and sometimes the mind must run ahead of experiment. I shall only give one example which has always struck me rather forcibly. If I decompose white light I shall be able to isolate a portion of the spectrum but however small it may be it will always be a certain width. In the same way the natural lights which are called monochromatic give us a very fine array but a Y which is not however infinitely fine. It might be supposed that in the experimental study of the properties of these natural lights by operating with finer and finer rays and passing on at last the limit so to speak we should eventually obtain the properties of rigorously monochromatic light. That would not be accurate. I assume that two rays emanate from the same source that they are first polarized in planes at right angles that they are then brought back again to the same plane of polarization and that we try to obtain interference. If the light were rigorously monochromatic there would be interference but with our nearly monochromatic lights there will be no interference and that however narrow the ray may be. For it to be otherwise the ray would have to be several million times finer than the finest known rays. Here then we should be led astray by proceeding to the limit. The mind has to run ahead of the experiment and if it has done so with success it is because it has allowed itself to be guided by the instinct of simplicity. The knowledge of the elementary fact enables us to state the problem in the form of an equation. It only remains to reduce from it the observable and verifiable complex fact. That is what we call integration and it is the province of the mathematician. It might be asked why in physical science journalization so readily takes the mathematical form? The reason is now easy to see. It is not only because we have to express numerical laws. It is because the observable phenomenon is due to the superposition of a large number of elementary phenomena which are all similar to each other and in this way differential equations are quite naturally introduced. It is not enough that each elementary phenomenon should obey simple laws. All those that we have to combine must obey the same law. Then only is the intervention of mathematics of any use. Mathematics teaches us in fact to combine like with like. Its object is to divine the result of a combination without having to reconstruct that combination element by element. If we have to repeat the same operation several times mathematics enables us to avoid this repetition by telling the result beforehand by a kind of induction. This I have explained before in the chapter on mathematical reasoning. But for that purpose all these operations must be similar. In the contrary case we must evidently make up our minds to working them out in full one after the other and mathematics will be useless. It is therefore thanks to the approximate homogeneity of the matter studied by physicists that mathematical physics came into existence. In the natural sciences the following conditions are no longer to be found. Homogeneity relative independence of remote parts simplicity of the elementary fact and that is why the student of natural science is compelled to have recourse to other modes of generalization. End of chapter 9 Chapter 10 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 Science and Hypothesis by Henri Poincaré Chapter 10 The Theories of Modern Physics Significance of Physical Theories The ephemeral nature of scientific theories takes by surprise the man of the world. Their brief period of prosperity ended he sees them abandoned one after another. He sees ruins piled upon ruins. He predicts that the theories in fashion today will in a short time succumb in their turn and he concludes that they are absolutely in vain. This is what he calls the bankruptcy of science. His skepticism is superficial. He does not take into account the object of scientific theories and the part they play or he would understand that the ruins may still be good for something. No theory seemed established on firmer ground than Fresnel's, which attributed light to the movements of the ether. Then if Maxwell's theory is today preferred, does that mean that Fresnel's work was in vain? No. For Fresnel's object was not to know whether there really is an ether, if it is or is not formed of atoms. If these atoms really move in this way or that, his object was to predict optical phenomena. This Fresnel's theory enables us to do today, as well as it did before Maxwell's time. The differential equations are always true. They may be always integrated by the same methods and the results of this integration still preserve their value. It cannot be said that this is reducing physical theories to simple practical recipes. These equations express relations and if the equations remain true, it is because the relations preserve their reality. They teach us now, as they did then, that there is such and such a relation between this thing and that, only the something which we then called motion we now call electric current. But these are merely names of the images we substituted for the real objects which nature will hide forever from our eyes. The true relations between these real objects are the only reality we can attain, and the sole condition is that the same relations still exist between these objects as between the images we are forced to put in their place. If the relations are known to us, what does it matter if we think it convenient to replace one image by another? That a given periodic phenomenon, an electric oscillation for instance, is really due to the vibration of a given atom which behaving like a pendulum is really displaced in this manner or that. All this is neither certain nor essential. But that there is between the electric oscillation, the movement of the pendulum, and all periodic phenomena, an intimate relationship which corresponds to a profound reality, that this relationship, this similarity, or rather this parallelism, is continued in the details, that it is a consequence of more general principles such as that of the conservation of energy and that of least action, this we may affirm, this is the truth, which will ever remain the same in whatever garb we may see fit to close it. Many theories of dispersion have been proposed. The first were imperfect and contained but little truth. Then came that of Helmholtz, and this in its turn was modified in different ways. Its author himself conceived another theory, founded on Maxwell's principles. But the remarkable thing is that all the scientists who followed Helmholtz obtained the same equations, although their starting points were to all appearance widely separated. I venture to say that these theories are all simultaneously true, not merely because they express a true relation, that between absorption and abnormal dispersion. In the premises of these theories, the part that is true is the part common to all. It is the affirmation of this or that relation between certain things, which some call by one name and some by another. The kinetic theory of gases has given rise to many objections, to which it would be difficult to find an answer where it claimed that the theory is absolutely true. But all these objections do not alter the fact that it has been useful, particularly in revealing to us one true relation which would otherwise have remained profoundly hidden, the relation between gaseous and osmotic pressures. In this sense, then, it may be said to be true. When a physicist finds a contradiction between two theories which are equally dear to him, he sometimes says, Let us not be troubled, but let us hold fast to the two ends of the chain lest we lose the intermediate links. This argument of the embarrassed theologian would be ridiculous if we were to attribute to physical theories the interpretation given them by the man of the world. In case of contradiction, one of them, at least, should be considered false. But this is no longer the case if we only seek in them what should be sought. It is quite possible that they both express true relations, and that the contradictions only exist in the images we have formed to ourselves of reality. To those who feel that we are going too far in our limitations of the domain accessible to the scientists, I reply, these questions which we forbid you to investigate, in which you so regret, are not only insoluble, they are illusory and devoid of meaning. Such a philosopher claims that all physics can be explained by the mutual impact of atoms. If he simply means that the same relations obtain between physical phenomena as between the mutual impact of a large number of billiard balls, well and good, this is verifiable and perhaps is true. But he means something more, and we think we understand him because we think we know what an impact is. Why? Simply because we have often watched a game of billiards. Are we to understand that God experiences the same sensations in the contemplation of his work that we do in watching a game of billiards? If it is not our intention to give his assertion this fantastic meaning, if we do not wish to give it the more restrictive meaning I have already mentioned, which is the sound meaning, then it has no meaning at all. Hypotheses of this kind have therefore only a metaphorical sense. The scientist should no more banish them than a poet banishes metaphor, but he ought to know what they are worth. They may be useful to give satisfaction to the mind, and they will do no harm as long as they are only in different hypotheses. These considerations explain to us why certain theories that were thought to be abandoned and definitively condemned by experiment are suddenly revived from their ashes and begin a new life. It is because they expressed true relations and had not ceased to do so when for some reason or other we felt it necessary to enunciate the same relations in another language. Their life had been latent as it were. Barely fifteen years ago was there anything more ridiculous, more quaintly old-fashioned, than the fluids of Coulomb, and yet here they are reappearing under the name of electrons. In what do these permanently electrified molecules differ from the electric molecules of Coulomb? It is true that in the electrons the electricity is supported by a little, a very little matter, in other words they have mass. Yet Coulomb did not deny mass to his fluids, or if he did it was with reluctance. It would be rash to affirm that the belief in electrons will not also undergo an eclipse, but it was nonetheless curious to note this unexpected renaissance. But the most striking example is Carnot's principle. Carnot established it, starting from false hypotheses. When it was found that heat was indestructible and may be converted into work, his ideas were completely abandoned. Later Clausius returned to them, and to him is due their definitive triumph. In its primitive form Carnot's theory expressed in addition to true relations, other inexact relations, the debris of old ideas. But the presence of the latter did not alter the reality of the others. Clausius had only to separate them, just as one lops off dead branches. The result was the second fundamental law of thermodynamics. The relations were always the same, although they did not hold, at least to all appearance, between the same objects. This was sufficient for the principle to retain its value. Nor have the reasonings of Carnot perished on this account. They were applied to an imperfect conception of matter, but their form, that is, the essential part of them, remained correct. What I have just said throws some light at the same time on the role of general principles, such as those of the principle of least action or of the conservation of energy. These principles are a very great value. They were obtained in the search for what there was in common in the annunciation of numerous physical laws. They thus represent the quintessence of innumerable observations. However, from their very generality results a consequence to which I have called attention in Chapter 8, namely that they are no longer capable of verification. As we cannot give a general definition of energy, the principle of the conservation of energy simply signifies that there is a something which remains constant. Whatever fresh notions of the world may be given us by future experiments, we are certain beforehand that there is something which remains constant and which may be called energy. Does this mean that the principle has no meaning and vanishes into a tautology? Not at all. It means that the different things to which we give the name of energy are connected by a true relationship. It affirms between them a real relation. But then if this principle has a meaning, it may be false. It may be that we have no right to extend indefinitely its applications. And yet it is certain beforehand to be verified in the strict sense of the word. How then shall we know when it has been extended as far as is legitimate? Simply when it ceases to be useful to us, that is, when we can no longer use it to predict correctly new phenomena. We shall be certain in such a case that the relation affirmed is no longer real, for otherwise it would be fruitful. Experiment without directly contradicting a new extension of the principle will nevertheless have condemned it. Physics and mechanism Most theorists have a constant predilection for explanations borrowed from physics, mechanics, or dynamics. Some would be satisfied if they could account for all phenomena by the motion of molecules attracting one another according to certain laws. Others are more exact. They would suppress attractions acting at a distance. Their molecules would follow rectilinear paths, from which they would only be deviated by impacts. Others again, such as Hertz, suppress the forces as well. But suppose their molecules subjected to geometrical connections analogous for instance to those of articulated systems. Thus they wish to induce dynamics to a kind of kinematics. In a word they all wish to bend nature into a certain form, and unless they can do this they cannot be satisfied. Is nature flexible enough for this? We shall examine this question in Chapter 11, Upper Pro of Maxwell's Theory. Every time that the principles of least action and energy are satisfied, we shall see that not only is there always a mechanical explanation possible, but that there is an unlimited number of such explanations. By means of a well-known theorem due to conics, it may be shown that we can explain everything in an unlimited number of ways, by connections after the manner of Hertz, or again by central forces. No doubt it may be just as easily demonstrated that everything may be explained by simple impacts. For this let us bear in mind that it is not enough to be content with the ordinary matter of which we are aware by means of our senses, and the movements of which we observe directly. We may conceive of ordinary matter as either composed of atoms whose internal movements escape us, our senses being able to estimate only the displacement of the whole, or we may imagine one of those subtle fluids which under the name of either or other names have from all time played so important a role in physical theories. Often we go further, and regard the ether as the only primitive, or even as the only true matter. The more moderate consider ordinary matter to be condensed ether, and there is nothing startling in this conception. But others only reduce its importance still further, and see in matter nothing more than the geometrical locus of singularities in the ether. Lord Kelvin, for instance, holds what we call matter to be only the locus of those points, at which the ether is animated by vortex motions. Rhymon believes it to be the locus of those points at which the ether is constantly destroyed. To Weichert or Lemor, it is the locus of the points at which the ether has undergone a kind of torsion of a very particular kind. Taking any one of these points of view, I ask by what right do we apply to the ether the mechanical properties observed in ordinary matter, which is but false matter. The ancient fluids, caloric, electricity, etc., were abandoned when it was seen that heat is not indestructible. But they were also laid aside for another reason. In materializing them, their individuality was, so to speak, emphasized. Gaps were opened between them. And these gaps had to be filled in when the sentiment of the unity of nature became stronger, and when the intimate relations which connect all the parts were perceived. In multiplying the fluids, not only did the ancient physicists create unnecessary entities, but they destroyed real ties. It is not enough for a theory not to affirm false relations. It must not conceal true relations. Does our ether actually exist? We know the origin of our belief in the ether. If light takes several years to reach us from a distant star, it is no longer on the star, nor is it on the earth. It must be somewhere, and support it, so to speak, by some material agency. The same idea may be expressed in a more mathematical and more abstract form. What we know are the changes undergone by the material molecules. We see for instance that the photographic plate experiences the consequences of a phenomenon of which the incandescent mass of a star was the scene several years before. Now, in ordinary mechanics, the state of the system under consideration depends only on its state at the moment immediately proceeding. The system therefore satisfies certain differential equations. On the other hand, if we did not believe in the ether, the state of the material universe would depend not only on the state immediately proceeding, but also on much older states. The system would satisfy equations of finite differences. The ether was invented to escape this breaking down of the laws of general mechanics. Still, this would not only compel us to fill the interplanetary space with ether, but not to make it penetrate into the midst of material media. The So's experiment goes further. By the interference of rays which have passed through the air or water in motion, it seems to show us two different media penetrating each other and yet being displaced with respect to each other. The ether is all but in our grasp. Experiments can be conceived in which we come closer still to it. Assume that Newton's principle of the equality of action and reaction is not true if applied to matter alone and that this can be proved. The geometrical sum of all the forces applied to all the molecules would no longer be zero. If we did not wish to change the whole of the science of mechanics, we should have to introduce the ether in order that the action which matter apparently undergoes should be counterbalanced by the reaction of matter on something. Or again, suppose we discover that optical and electrical phenomena are influenced by the motion of the earth. It would follow that those phenomena might reveal to us not only the relative motion of material bodies, but also what would seem to be their absolute motion. Again it would be necessary to have an ether in order that these so-called absolute movements should not be their displacements with respect to empty space but with respect to something concrete. Will this ever be accomplished? I do not think so, and I shall explain why, and yet it is not absurd for others have entertained this view. For instance, if the theory of Lorenz of which I shall speak in more detail in Chapter 13 were true, Newton's principle would not apply to matter alone, and the difference would not be very far from being within reach of experiment. On the other hand, many experiments have been made on the influence of the motion of the earth. These results have always been negative. But if these experiments have been undertaken, it is because we have not been certain beforehand, and indeed, according to current theories, the compensation would be only approximate, and we might expect to find accurate methods giving positive results. I think that such a hope is illusory. It was nonetheless interesting to show that a success of this kind would, in a certain sense, open to us a new world. And now allow me to make a digression. I must explain why I do not believe in spite of Lorenz that more exact observations will ever make evident anything else but the relative displacements of material bodies. Experiments have been made that should have disclosed the terms of the first order. The results were nougatory. Could that have been by chance? No one has admitted this. A general explanation was sought and Lorenz found it. He showed that the terms of the first order should cancel each other, but not the terms of the second order. Then more exact experiments were made, which were also negative. Neither could this be the result of chance. An explanation was necessary and was forthcoming. They always are. Hypotheses are what we lack the least. But this is not enough. Who is there who does not think that this leaves to chance far too important a role? Would it not also be a chance that this singular concurrence should cause a certain circumstance to destroy the terms of the first order and that a totally different but very opportune circumstance should cause those of the second order to vanish? No. The same explanation must be found for the two cases and everything tends to show that this explanation would serve equally well for the terms of the higher order and that the mutual destruction of these terms will be rigorous and absolute. The present state of physics. Two opposite tendencies may be distinguished in the history of the development of physics. On the one hand, new relations are continually being discovered between objects which seem destined to remain forever unconnected. Scattered facts cease to be strangers to each other and tend to be marshaled into an imposing synthesis. The march of science is towards unity and simplicity. On the other hand, new phenomena are continually being revealed. It will be long before they can be assigned their place. Sometimes it may happen that to find them a place a corner of the edifice must be demolished. In the same way, we are continually perceiving details ever more varied in the phenomena we know where our crude senses used to be unable to detect any lack of unity. What we thought to be simple becomes complex and the march of science seems towards diversity and complication. Here then are two opposing tendencies each of which seems to triumph in turn which will win if the first wins science is possible but nothing proves this a priori and it may be that after unsuccessful efforts to bend nature to our ideal of unity in spite of herself we shall be submerged by the ever rising flood of our new riches and compelled to renounce all idea of classification to abandon our ideal and to reduce science to the mere recording of innumerable recipes. In fact, we can give this question no answer. All that we can do is to observe the science of today and compare it with that of yesterday. No doubt after this examination we shall be in a position to offer a few conjectures. Half a century ago hopes ran high indeed. The unity of force had just been revealed to us by the discovery of the conservation of energy and of its transformation. This discovery also showed that the phenomena of heat could be explained by molecular movements. Although the nature of these movements was not exactly known, no one doubted but that they would be ascertained before long. As for light, the work seemed entirely completed. So far as electricity was concerned, there was not so great in advance. Electricity had just annexed magnetism. This was a considerable and a definitive step towards unity. But how was electricity in its turn to be brought into the general unity? And how was it to be included in the general universal mechanism? No one had the slightest idea. As to the possibility of the inclusion all were agreed, they had faith. Finally, as far as the molecular properties of material bodies are concerned, the inclusion seemed easier, but the details were very hazy. In a word, hopes were vast and strong, but vague. Today, what do we see? In the first place, a step in advance, immense progress. The relations between light and electricity are now known. The three domains of light, electricity and magnetism, formally separated, are now one. And this annexation seems definitive. Nevertheless, the conquest has caused us some sacrifices. Optical phenomena became particular cases in electric phenomena. As long as the former remained isolated, it was easy to explain them by movements which were thought to be known in all their details. That was easy enough. But any explanation to be accepted must now cover the whole domain of electricity. This cannot be done without difficulty. The most satisfactory theory is that of Lorentz. It is unquestionably the theory that best explains the known facts, the one that throws into relief the greatest number of known relations, the one in which we find most traces of definitive construction. That it still possesses a serious fault I have shown above. It is in contradiction with Newton's law that action and reaction are equal and opposite. Or rather, this principle according to Lorentz cannot be applicable to matter alone. If it be true, it must take into account the action of the ether on matter and the reaction of the matter on the ether. Now in the new order it is very likely that things do not happen in this way. However, this may be it is due to Lorentz that the results of Fizzo on the optics of moving bodies, the laws of normal and abnormal dispersion and of absorption are connected with each other and with the other properties of the ether by bonds which no doubt will not be readily severed. Look at the ease with which the new Zeeman phenomena found its place and even aided the classification of Faraday's magnetic rotation which had defied all Maxwell's efforts. This facility proves that Lorentz's theory is not a mere artificial combination which must eventually find its solvent. It will probably have to be modified but not destroyed. The only object of Lorentz was to include in a single hole all the optics and electrodynamics of moving bodies. He did not claim to give a mechanical explanation. Larmore goes further keeping the essential part of Lorentz's theory he graphs upon it so to speak. McCullough's ideas on the direction of the movement of the ether. McCullough held that the velocity of the ether is the same in magnitude and direction as the magnetic force. In genius as is this attempt the fault in Lorentz's theory remains and is even aggravated. According to Lorentz we do not know what the movements of the ether are and because we do not know this we may suppose them to be movements compensating those of matter and reaffirming that action and reaction are equal and opposite. According to Larmore we know the movements of the ether and we can prove that the compensation does not take place. If Larmore has failed as in my opinion he has does it necessarily follow that a mechanical explanation is impossible? Far from it I said above that as long as a phenomenon obeys the two principles of energy in least action so long it allows an unlimited number of mechanical explanations and so with the phenomena of optics and electricity but this is not enough for a mechanical explanation to be good it must be simple to choose it from among all the explanations that are possible there must be other reasons than the necessity of making a choice. Well we have no theory as yet which will satisfy this condition and consequently be of any use. Are we then to complain? That would be to forget the end we seek which is not the mechanism the true and only aim is unity. We ought therefore to set some limits to our ambition let us not seek to formulate a mechanical explanation let us be content to show that we can always find one if we wish in this we have succeeded. The principle of the conservation of energy has always been confirmed and now it has a fellow in the principle of least action stated in the form appropriate to physics this has also been verified at least as far as concerns of reversible phenomena that obey Lagrange's equations in other words which obey the most general laws of physics. The irreversible phenomena are much more difficult to bring into line but they too are being coordinated and tend to come into unity. The light which illuminates them comes from Carnot's principle. For a long time thermodynamics was confined to the study of the dilatations of bodies and of their change of state. For some time past it has been growing bolder and has considerably extended its domain. We owe to it the theories of the Voltaic cell and of their thermoelectric phenomena. There is not a corner in physics which it has not explored and it has even attacked chemistry itself. The same laws hold good. Everywhere disguised in some form or other we find Carnot's principle. Everywhere also appears that eminently abstract concept of entropy which is as universal as the concept of energy and like it seems to conceal a reality. It seemed that radiant heat must escape but recently that too has been brought under the same laws. In this way fresh analogies are revealed which may be often pursued in detail. Electric resistance resembles the viscosity of fluids. Historesis would rather be like the friction of solids. In all cases friction appears to be the type most imitated by the most diverse irreversible phenomena and this relationship is real and profound. A strictly mechanical explanation of these phenomena has also been sought but owing to their nature it is hardly likely that it will be found. To find it it has been necessary to suppose that the irreversibility is but apparent that the elementary phenomena are reversible and obey the known laws of dynamics but the elements are extremely numerous and become blended more and more so that to our crude site all appears to feted towards uniformity that is all seems to progress in the same direction and that without hope of return. The apparent irreversibility is therefore but an effect of the law of great numbers only a being of infinitely subtle senses such as Maxwell's demon could unravel this tangled skein and turn back the course of the universe. This conception which is connected with the kinetic theory of gases has cost great effort and has not on the hull been fruitful it may become so. This is not the place to examine if it leads to contradictions and if it is in conformity with the true nature of things. Let us notice however the original ideas of Mr. Guille on the Brownian movement. According to this scientist this singular movement does not obey Carnot's principle the particles which it sets moving would be smaller than the meshes of that tightly drawn net they would thus be ready to separate them and thereby to set back the course of the universe. One can almost see Maxwell's demon at work. Footnote one Clerk Maxwell imagine some supernatural agency at work sorting molecules in a gas of uniform temperature into A those possessing kinetic energy above the average and B those possessing kinetic energy below the average and of note. To resume phenomena long known are gradually being better classified but new phenomena come to claim their place and most of them like the Zeeman effect find it at once. Then we have the cathode rays the x-rays uranium and radium rays in fact a whole world of which none has suspected the existence. How many unexpected guests to find a place for? No one can yet predict the place they will occupy but I do not believe they will destroy the general unity. I think that they will rather complete it. On the one hand indeed the new radiation seem to be connected with the phenomena of luminosity not only do they excite fluorescence but they sometimes come into existence under the same conditions as that property. Neither are they unrelated to the cause which produces the electric spark under the action of ultraviolet light. Finally and most important of all it is believed that in all these phenomena there exist ions animated it is true with velocities far greater than those of electrolytes. All this is very vague but it will all become clearer. Phosphorescence and the action of light on the spark were regions rather isolated and consequently somewhat neglected by investigators. It is to be hoped that a new path will now be made which will facilitate their communications with the rest of science. Not only do we discover new phenomena but those we think we know are revealed and unlooked for aspects. In the free ether the laws preserve their majestic simplicity but matter properly so-called seems more and more complex all we can say of it is but approximate and our formulae are constantly requiring new terms but the ranks are unbroken. The relations that we have discovered between objects we thought simple still hold good between the same objects when their complexity is recognized and that alone is the important thing. Our equations become it is true more and more complicated so as to embrace more closely the complexity of nature but nothing is changed in the relations which enable these equations to be derived from each other. In a word the form of these equations persists. Take for instance the laws of reflection. Fresnel established them by a simple and attractive theory which experiments seem to confirm. Subsequently more accurate researchers have shown that this verification was but approximate. Traces of elliptic polarization were detected everywhere but it is owing to the first approximation that the cause of these anomalies was found in the existence of a transition layer and all the essentials of Fresnel's theory have remained. We cannot help reflecting that all these relations would never have been noted if there had been doubt in the first place as to the complexity of the objects they connect. Long ago it was said if Tycho had had instruments ten times as precise we would never have had a Kepler or a Newton or astronomy. It is a misfortune for a science to be born too late when the means of observation have become too perfect. That is what is happening at this moment with respect to physical chemistry. The founders are hampered in their general grasp by third and fourth decimal places. Happily they are men of robust faith. As we get to know the principles of matter better we see that continuity reigns. From the work of Andrews and Vanderwalls we see how the transition from the liquid to the gaseous state is made and that it is not abrupt. Similarly there is no gap between liquid and solid states and in the proceedings of a recent congress we see memoirs on the rigidity of liquids side by side with papers on the flow of solids. With this tendency there is no doubt a loss of simplicity. Such and such an effect was represented by straight lines. It is now necessary to connect these lines by more or less complicated curves. On the other hand unity is gained. Separate categories quieted but did not satisfy the mind. Finally a new domain that of chemistry has been invaded by the method of physics and we see the birth of physical chemistry. It is still quite young but already it has enabled us to connect such phenomena as electrolysis, osmosis and the movement of ions. From this cursory exposition what can we conclude? Taking all things into account we have approached the realization of unity. This has not been done as quickly as was hoped fifty years ago and the path predicted has not always been followed but on the whole much ground has been gained. End of chapter 10