In 1904, the city of St. Louis hosted the world’s fair with many grand events including the third modern Olympics and the International Congress of Arts and Sciences. Under the Normative Science Division, the section on Applied Mathematics contained two speakers who were invited to discuss the connection between natural philosophy and applied mathematics: Ludwig Boltzmann and Henri Poincaré.
Poincaré’s talk is particularly insightful, as he gives a brief history of nineteenth century physics, followed by a discussion of current problems in theoretical physics and then by a discussion of the future of physics, which we only reproduce in part.
Poincaré beautifully explains a condensed history of early nineteenth century theoretical physics, beginning with a discussion of the Laplacian view of mathematical physics. He talks about the astronomical view of matter that all microscopic forces must follow gravitational-style force laws. While this method showed early promise, Poincaré makes it clear that its reliance on central force laws severely limited the scope and creativity of the Laplacians.
Next, Poincaré reviews the fundamental principles of physics, including the principle of relativity, the conservation of energy and the conservation of mass. He discusses how much more powerful this reasoning was, as compared to the central force methods of the Laplacians.
But there are now, as of 1904, contradictions in physics. All the principles cannot be true, and so there is a crisis in mathematical physics. Poincaré articulates this crisis beautifully, and he cites some recent ideas to resolve this crisis. All of the current investigations to which he refers foreshadow the work of Einstein. The section on the conservation of mass (Lavoisier’s Principle), for example, discusses different ideas of mass, and different types of mass, all of which we now know was barking up the wrong tree. Einstein, after all, is just about to show that E = m c2.
As you read Poincaré’s talk, appreciate how well he understood the theoretical problems that were dogging classical physics at the turn of the century. Also keep in mind that of Einstein’s famous four 1905 papers, three were accepted relatively quickly. These papers directly address Poincaré’s present crisis. They solve the problem of Brownian motion, restore the principle of relativity as fundamental, and combine the conservation of mass with that of energy.
Moreover, notice the complete absence of any discussion of the problem of blackbody radiation or the photoelectric effect. These problems were the topic of Einstein’s first 1905 paper, the one on the photoelectric effect. It is understandable that Einstein’s contemporaries regarded this paper a reckless attempt to solve a non-problem with a sweeping and unsupported hypothesis. Later, of course, he would be awarded the Nobel Prize for his work on the photoelectric effect, precisely because he not only found a solution to a known problem, but also discovered the problem in the first place.
The Present and the Future of Mathematical Physics
Professor Henri Poincaré
(Talk in association with the 1904 world fair.)
What is the present state of mathematical physics? What are its problems? What is its future? Will the object and methods of this science appear in ten years as they appear to us now? Or are we to witness a far-reaching transformation? These are the questions we are forced to face today at the outset of our inquiry.
This is easy to ask, but difficult to answer. If we feel tempted to hazard a prediction, we should stop to think of the nonsense the most eminent scholars of a hundred years ago would have spoken in answer to the question of what this science would be in the nineteenth century. They would have thought themselves bold in their predictions, and after the event how timid we should have found them! Do not expect of me any kind of prophesy.
But if, like all prudent physicians, I refuse to give a prognosis, still I cannot deny myself a little diagnosis. Well, then, yes, there are symptoms of a serious crisis, which would seem to indicate that we may expect, presently, a transformation. However, there is no cause for great anxiety. We are assured that the patient will not die, and indeed we may hope that this crisis will be salutary, since the history of the past would seem to insure that. In fact, this crisis is not the first, and in order to understand it, we must recall those that have gone before. Allow me a brief historical sketch.
A Brief History of Mathematical Physics
Mathematical physics, as we are well aware, is an offspring of celestial mechanics, which gave it birth at the end of the eighteenth century—at the moment when it had, itself, attained its complete development. The child, especially during its first years, showed a striking resemblance to its mother. The astronomical universe consists of masses, undoubtedly of great magnitude, but separated by such immense distances that they appear to us as material points; these points attract each other in the inverse ratio of the squares of their distances, and this attraction is the only force which affects their motion.
The Physics of Central Forces:
If we could measure all the details of the infinitesimal stars that are the atoms that make up all of matter, the spectacle thus disclosed would hardly differ from the one that the astronomer contemplates. There too we should see material points separated by intervals that are enormous in comparison with their dimensions, and describing orbits according to regular laws. Like the stars they attract each other—or they repel following similar laws. This attraction or repulsion, which is along the line joining them, depends only on the distance. The law according to which this force varies with the distance is perhaps not the law of Newton, but it is analogous thereto. Instead of the exponent 2, we probably have another exponent. From this diversity in the exponents proceeds all the diversity of the physical phenomena, the variety in qualities and sensations, all the world of color and sound which surrounds us; in a word, all of nature.
Such is the primitive, first half of the nineteenth century, conception in its utmost purity. Nothing remains but to inquire in the different cases, what value must be given to this exponent in order to account for all the facts. On this model, for example, Pierre Laplace constructed his beautiful theory of capillarity; he simply regards the latter as a special case of attraction, or, as he says, of universal gravitation, and no one is surprised to find it in the middle of one of the five volumes of his celestial mechanics.
More recently, Charles Briot believes he has laid bare the last secret of optics, when he has proved that the atoms of the aether attract each other in the inverse sixth power of the distance; and does not Maxwell, Maxwell himself, say somewhere that the atoms of a gas repel each other in the inverse ratio of the fifth power of the distance? We have the exponent 6 or 5, instead of the exponent 2; but it is always an exponent.
Among the theories of this period there is a single one that forms an exception, namely that of Joseph Fourier; here there are indeed atoms acting at a distance; they send each other heat, but they do not attract each other, they do not stir. From this point of view, Fourier’s theory must have appeared imperfect and provisional to the eyes of his contemporaries, and even to himself.
This conception was not without greatness. It was alluring, and many of us have not given it up. These Laplacian holdouts know that the ultimate elements of things will not be attained except by disentangling with patience the complex skein furnished us by our senses; that progress should be made step by step without neglecting any intermediate portions; that our fathers were unwise in not wishing to stop at all the stations; but they still believe that when we once arrive at these ultimate elements, we shall meet again the majestic simplicity of celestial mechanics.
Nor has this conception been useless; it has rendered us a priceless service inasmuch as it has contributed to making more precise the fundamental concept of the physical law. Let me explain.
What did the ancients understand by a law? It was to them an internal harmony, static as it were, and unchangeable; or else a model which nature tried to imitate. To us a law is no longer that at all; it is a constant relation between a phenomenon of today and that of tomorrow; in a word, it is a differential equation.
Here we have the ideal form of the physical law; and, indeed, it is Newton’s law that first gave it this form. If, later on, this form has become inured in physics, it has become so precisely by copying as far as possible this law of Newton. The similarity between the physics of atoms, and of astronomical bodies, is a result of our having used celestial mechanics as a model. Nevertheless, there came a day when the conception of central forces appeared no longer to suffice, and this is the first of the crises to which I referred a moment ago.
What was done? Abandoned was the thought of exploring the details of the universe, of isolating the parts of this vast mechanism, of analyzing one by one the forces that set them going. Rather we took certain general principles that have precisely the object of relieving us of this minute study. How is this possible?
The Physics of the Principles
Suppose we have before us any kind of machine; the part of the mechanism where the power is applied and the ultimate resultant motion alone are visible, while the transmissions, the intermediate gearing whereby the motion is communicated from one part to another, are hidden in the interior and escape our notice. We know not whether the transmissions are made by cog-wheels or by belts, by connecting-rods or other contrivances. Shall we say that it is impossible for us to learn anything about this machine unless we are allowed to take it apart?
You well know that such is not the case, and that the principle of the conservation of energy suffices to furnish us the most interesting feature. We can easily show that the last wheel turns ten times more slowly than the first, since these two wheels are visible; and we can conclude therefrom that a couple applied to the first will be in equilibrium with a couple ten times as great applied to the second. To obtain this result, it is unnecessary to look into the mechanism of this equilibrium, or to know how the forces balance in the interior of the machine.
In the case of the universe, the principle of the conservation of energy can render us the same service. This universe also is a machine, much more complicated than any in use in the industries, of which nearly all the parts are deeply hidden; but by observing the motion of those which we can see, we can by the aid of this principle draw conclusions which will remain valid no matter what the details of the invisible mechanism which actuates them.
- The principle of the conservation of energy, or Mayer’s principle, is certainly the most important, but it is not the only one; there are others from which we can derive the same advantage. These are:
- Carnot’s principle, or the principle of the dissipation of energy, or the second law of thermodynamics.
- Newton’s principle, or the principle of the equality of action and reaction.
- The principle of relativity, according to which the laws of physical phenomena must be the same for a stationary observer as for one carried along in a uniform motion of translation, so that we have no means, and can have none, of determining whether or not we are being carried along in such a motion.
- Lavoisier’s principle, or the principle of the conservation of mass.
- The principle of least action.
The application of these five or six general principles to the various physical phenomena suffices to teach us what we may reasonably hope to know about them. The most remarkable example of this new mathematical physics is without doubt Maxwell’s electromagnetic theory of light. What is the aether? How are its molecules distributed? Do they attract or repel each other? Of these things we know nothing. But we do know that this medium transmits both optical and electrical disturbances; we know that this transmission must take place in conformity with the general principles of mechanics and that suffices to establish the equations of the electromagnetic field.
These principles are the boldly generalized results of experiment; but they appear to derive from their very generality a high degree of certainty. In fact, the greater the generality, the more frequent are the opportunities for verifying them. Such verifications, as they multiply, take the most varied and most unexpected forms, and leave in the end no room for doubt.
Such is the second phase of the history of mathematical physics, and we have not yet left it. Shall we say that the first has been useless, that for fifty years [1800-1850] science was on a wrong path and that there is nothing to do but to forget all that accumulation of effort which a vicious conception from the very beginning doomed to failure? By no means! Do you think the second period could have existed without the first?
The hypothesis of central forces contained all the principles. It involved them as necessary consequences. It involved the principle of the conservation of energy, as well as that of mass, and the equality of action and reaction, and the law of least action, which appeared to be sure, not as experimental facts, but as theorems.
It is the mathematical physics of our fathers that has gradually made us familiar with these various principles, and which has taught us to recognize them in the different garbs in which they are disguised. They have been compared with the results of experiment, where it has been found necessary to change their expression in order to make them conform to the facts; thus they have been extended and strengthened. In this way they came to be regarded as experimental truths. The conception of central forces then became a useless support, or rather an encumbrance, inasmuch as it imposed upon the principles its own hypothetical character.
The bonds then are not broken, because they were elastic; but they have been extended. Our fathers who established them have not labored in vain; and in the science of today we recognize the general features of the outline they traced.
The Present Crisis of Mathematical Physics
The New Crisis
Are we now about to enter upon a third period? Are we on the eve of a second crisis? Are these principles on which we have reared everything about to fall in their turn? This has recently become a vital question.
Hearing me speak thus, you are thinking without doubt of radium, that great revolutionary of the present day; and indeed I shall return to it presently. But there is something else. It is not merely the conservation of energy that is concerned; all the other principles are in equal danger, as we shall see by successively passing them in review.
Let us begin with Carnot’s principle. It is the only one that does not present itself as an immediate consequence of the hypothesis of central forces. Quite to the contrary, indeed, it appears, if not actually to contradict this hypothesis, at least not to be reconcilable with it without some effort. If physical phenomena were due exclusively to the motion of atoms the mutual attractions of which depend only on the distance, it would seem that all these phenomena should be reversible; if all the initial velocities were reversed, these atoms, if still subject to the same forces, should traverse their trajectories in the opposite direction, just as the earth would describe backward this same elliptical orbit that it now describes forward, if the initial conditions of its motion had been reversed. Thus, if a physical phenomenon is possible, the inverse phenomenon should be equally possible, and one should be able to retrace the course of time. Now, it is not so in nature, and this it is precisely that the principle of Carnot teaches us; heat may pass from a hot body to a cold; it is impossible to compel it to take the opposite route and to reestablish differences of temperature which have disappeared. Motion can be entirely destroyed and transformed into heat by friction; the converse transformation can only occur partially.
Efforts have been made to reconcile this apparent contradiction. If the world tends toward uniformity, it is not because its ultimate parts, though diversified at the start, tend to become less and less different; it is because moving at random they become mixed. To an eye which could distinguish all the elements, the variety would remain always as great; every grain of this powder retains its originality and does not fashion itself after its neighbors; but as the mixture becomes more and more perfect, our rough senses perceive only uniformity. That is why, for example, temperatures tend to equalize themselves, without its being possible to go back.
A drop of wine, let us say, falls into a glass of water; whatever the internal motion of the liquid, we shall soon see it assume a uniformly roseate hue, and from then on no possible shaking of the vessel would seem to be capable of again separating the wine and the water. Here, then, we have what may be the type of the irreversible phenomenon of physics: to hide a grain of barley in a great mass of wheat would be easy; to find it again and to remove it is practically impossible. All this has been explained by Maxwell and Boltzmann, but the man who has put it most clearly was Gibbs in a book that is too little read because it is a little difficult to read, in his Elements of Statistical Mechanics.
To those who take this point of view, Carnot’s principle is an imperfect principle, a sort of concession to the frailty of our senses. It is because our eyes are too coarse that we do not distinguish the elements of the mixture. It is because our hands are too coarse that we cannot compel them to separate. The imaginary demon of Maxwell, who can pick out the molecules one by one, would be quite able to constrain the world to move backwards. That it should return of its own accord is not impossible; it is only infinitely improbable; the chances are that we should wait a long time for that combination of circumstances that would permit a retrogression; but, sooner or later, they will occur, after years, the number of which would require millions of figures. These reservations, however, all remained theoretical; they caused little uneasiness and Carnot’s principle preserved all of its practical value.
But now here is where the scene changes. The biologist, armed with his microscope, has for a long time noticed in his preparations certain irregular motions of small particles in suspension; this is known as Brownian motion. Brown believed at first that it was a phenomenon of life, but he soon saw that inanimate bodies hopped about with no less ardor than others; he then turned the matter over to the physicists. Unfortunately, the physicists did not become interested in the question for a long time. Light is concentrated, so they argued, in order to illuminate the microscopic preparation; light involves heat, and this causes differences in temperature and these produce internal currents in the liquid, which bring about the motions referred to.
Louis Georges Gouy had the idea of looking a little more closely, and thought he saw that this explanation was untenable; that the motion becomes more active as the particles become smaller, but that they are uninfluenced by the manner of lighting. If, then, these motions do not cease, or, rather, if they come into existence incessantly, without borrowing from any external source of energy, what must we think? We must surely not abandon on this account the conservation of energy; but we see before our eyes motion transformed into heat by friction and conversely heat changing into motion, and all without any sort of loss, since the motion continues forever. It is the contradiction of Carnot’s principle. If such is the case, we no longer need the infinitely keen eye of Maxwell’s demon in order to see the world move backward; our microscope suffices…
The Principle of Relativity
Let us consider the principle of relativity; this principle is not only confirmed by our daily experience, not only is it the necessary consequence of the hypothesis of central forces, but it appeals to our common sense with irresistible force. And yet it also is being fiercely attacked. Let us think of two electrified bodies; although they seem to be at rest, they are, both of them, carried along with the motion of the earth; Rowland has shown us that an electric charge in motion is equivalent to a current; these two charged bodies, then, are equivalent to two parallel currents in the same direction; these two currents should attract each other. By measuring this attraction we should be measuring the velocity of the earth; not its velocity relative to the sun and stars, but its absolute velocity.
I know what will be said; it is not its absolute velocity; it is its velocity relative to the aether. But, how unsatisfactory that is! Is it not clear that with this interpretation, nothing could be inferred from the principle? It could no longer teach us anything, simply because it would no longer fear any contradiction. Whenever we have succeeded in measuring anything, we would always be free to say that it is not the absolute velocity, and if it is not the velocity relative to the aether, it might always be the velocity relative to some new unknown fluid with which we might fill all space.
And then experiment, too, has taken upon itself to refute this interpretation of the principle of relativity; all the attempts to measure the velocity of the earth relative to the aether have led to negative results. Herein experimental physics has been more faithful to the principle than mathematical physics; the theorists would have dispensed with it readily in order to harmonize the other general points of view; but experimentation has insisted on confirming it. Methods were diversified; finally Michelson carried precision to its utmost limits; nothing came of it. It is precisely to overcome this stubbornness that today mathematicians are forced to employ all their ingenuity.
Their task was not easy, and if Lorentz has succeeded, it is only by an accumulation of hypotheses. The most ingenious idea is that of local time.
Let us imagine two observers, located at signal stations A and B, who wish to regulate their watches by means of optical signals. They exchange signals, but as they know that the transmission of light is not instantaneous, they are careful to cross them. When station B sees the signal from station A, its timepiece should not mark the same hour as that of station A at the moment the signal was sent, but this hour increased by a constant representing the time of transmission. Let us suppose, for example, that station A sends a signal at the moment when its timepiece marks the hour zero, and that station B receives it when its timepiece marks the hour t. The watches will be set, if the time t is the time of transmission, and in order to verify it, station B in turn sends a signal at the instant when its timepiece is at zero; station A must then see it when its timepiece is at t. Then the watches are regulated.
And, indeed, they mark the same hour at the same physical instant, but only if the two stations are stationary. Otherwise, the time of transmission will not be the same in the two directions, since the station A, for example, goes to meet the disturbance emanating from B, whereas station B flees before the disturbance emanating from A.
Watches regulated in this way, therefore, will not mark the true time; they will mark what might be called the local time, so that one will gain on the other. It matters little, since we have no means of perceiving it. All the phenomena which take place at A, for example, will be behind time, but all just the same amount, and the observer will not notice it since his watch is also behind time; thus, in accordance with the principle of relativity he will have no means of ascertaining whether he is at rest or in absolute motion.
Unfortunately this is not sufficient; additional hypotheses are necessary. We must admit that the moving bodies undergo a uniform contraction in the direction of the motion. One of the diameters of the earth, for example, is shortened by 1/200000000 as a result of our planet’s motion, whereas the other diameter preserves its normal length. Thus we find the last minute differences accounted for.
Then there is still the hypothesis concerning forces. Forces, whatever their origin, weight as well as elasticity, will be reduced in a certain ratio in a world endowed with a uniform translatory motion; or rather that would happen for the components at right angles to the direction of translation; the parallel components will not change.
Let us then return to our example of the two electrified bodies; they repel each other; but at the same time, if everything is carried along in a uniform transition, they are equivalent to two parallel currents in the same direction, which attract each other. This electrodynamic attraction is, then, subtracted from the electrostatic repulsion, and the resultant repulsion is weaker than if the two bodies had been at rest. But since we must, in order to measure this repulsion, balance it by another force, and since all these other forces are reduced in the same ratio, we observe nothing. Everything, then, appears to be in order.
But have all doubts been dissipated? What would happen if we could communicate by signals other than those of light? If, after having regulated our watches by the optimal method, we wished to verify the result by means of these new signals, we should observe discrepancies due to the common translatory motion of the two stations.
And are such signals inconceivable, if we take the view of Laplace, that universal gravitation is transmitted with a velocity a million times as great as that of light? Thus the principle of relativity has in recent times been valiantly defended; but the very vigor of the defense shows how serious was the attack.
And now let us speak of the principle of Newton, concerning the equality of action and reaction. This principle is intimately connected with the preceding and it would seem that the fall of one would involve the fall of the other. Nor must we be surprised to find here again the same difficulties.
The electrical phenomena, it is thought, are due to displacements of small charged particles called electrons that are immersed in the medium we call the aether. The motions of these electrons produce disturbances in the surrounding aether; these disturbances are propagated in all directions with the aether velocity of light, and other electrons initially at rest are displaced when the disturbance reaches the portions of the aether in which they lie.
The electrons, then, act one upon the other, but this action is not direct; it takes place by mediation of the aether. Under these conditions, is it possible to have equality between action and reaction, at least for an observer who takes account only of the motion of matter, that is of the electrons, and who ignores that of the aether that he is unable to see? Evidently not: even if the compensation were exact, it could not be instantaneous. The disturbance is propagated with a finite velocity; it reaches the second electron, therefore, only after the first has long been reduced to rest. This second electron will, then, after an interval, be subjected to the action of the first, but will certainly not at that moment react upon it, since there is no longer anything in the neighborhood of this first electron that stirs.
The analysis of the facts will allow us to become more definite. Let us imagine, for example, a Hertzian oscillator such as those used in wireless telegraphy. It sends energy in all directions; but we may attach to it a parabolic mirror, as was done by Hertz with his smallest oscillators, so as to send all the energy produced in a single direction. What then will happen according to the theory? Why, the apparatus will recoil as though it were a cannon and the projected energy a ball, and that contradicts the principle of Newton, since our present projectile has no mass; it is not matter, it is energy. It is the same, moreover, in the case of a lighthouse having a reflector, since light is merely a disturbance in the electromagnetic field. This lighthouse would recoil; as though the light it sends forth were a projectile. What is the force that must produce this recoil? It is what is known as the Maxwell-Bartholdi pressure; it is very small, and to put it in evidence caused much trouble, even with the most sensitive radiometers; but it is sufficient for our purpose that it exists.
If all the energy issuing from our oscillator strikes a receiver, the latter will act as though it had received a physical shock, which in a sense will represent the compensation of the oscillator’s recoil. The reaction will be equal to the action, but they will not be simultaneous; the receiver will advance, but not at the instant when the oscillator recoils. If the energy is propagated indefinitely without meeting a receiver, the compensation will never take place.
Shall we say that the space which separates the oscillator from the receiver, and which the disturbance must traverse in passing from one to the other, is not empty? Rather it is filled not only with aether, but with air, or even in inter-planetary space with some subtle, yet ponderable fluid. Under this hypothesis, this matter receives the shock, as does the receiver, at the moment the energy reaches it, and recoils, when the disturbance leaves it.
That would save Newton’s principle, but it is not true. If the energy during its propagation remained always attached to some material substratum, this matter would carry the light along with it and Fizeau has shown, at least for the air, that there is nothing of the kind. Michelson and Morley have since confirmed this. We might also suppose that the motions of matter proper were exactly compensated by those of the aether, but that would lead us to the same considerations as those made a moment ago. The principle, if thus interpreted, could explain anything, since whatever the visible motions we could imagine hypothetical motions to compensate them. But if it can explain anything, it will allow us to foretell nothing. It therefore becomes useless.
And then the suppositions that must be made concerning the motions of the aether are not very satisfactory. If the electric charges were doubled, it would be natural to suppose that the velocities of the atoms of the aether also became twice as great, and for the compensation it would be necessary that the mean velocity of the aether become four times as great.
This is why I have for a long time thought that these consequences of the theory, which contradict Newton’s principle, would some day be abandoned; and yet the recent experiments on the motion of the electrons emitted by radium seem rather to confirm them.
I now come to Lavoisier’s principle concerning the conservation of mass. This is certainly a principle that cannot be tampered with without shaking the science of mechanics. And still there are persons who think that it seems true to us only because in mechanics we consider only moderate velocities, and that it would cease to be so for bodies having velocities comparable with that of light. Now, such velocities are at present believed to have been realized; the cathode rays and those of radium would seem to be formed of very minute particles or electrons that move with velocities that are no doubt less than that of light, but which appear to be about one tenth or one third of it.
These rays can be deflected either by an electric or by a magnetic field, and by comparing these deflections it is possible to measure both the velocity of the electrons and their mass (or rather the ratio of their mass to their charge). But it was found that as soon as these velocities approached that of light a correction was necessary.
Since these particles are electrified, they cannot be displaced without disturbing the aether; to put them in motion, it is necessary to overcome a double inertia, that of the particle itself and that of the aether. The total or apparent mass that is measured is then composed of two parts: the real or mechanical mass of the particle and the electrodynamic mass representing the inertia of the aether.
Now, the calculations of Max Abraham and the experiments of Walter Kaufmann have shown that this mechanical mass property is nothing, and that the mass of the electrons, at least of the negative electrons, is purely of electrodynamic origin. This is what compels us to change our definition of mass; we can no longer distinguish between the mechanical mass and the electrodynamic mass, because then the first would have to vanish. There is no other mass than the electrodynamic inertia; but in this case, the mass can no longer be constant; it increases with the velocity; and indeed it depends on the direction, and a body having a considerable velocity will not oppose the same inertia to forces tending to turn it off its path that it opposes to those tending to accelerate or retard its motion.
There is indeed another resource: the ultimate elements of bodies are electrons, some with a negative charge, others with a positive charge. It is understood that the negative electrons have no mass; but the positive electrons, from what little is known of them, would seem to be much larger. They perhaps have besides their electrodynamic mass a true mechanical mass. The real mass of a body would then be the sum of the mechanical masses of its positive electrons, the negative electrons would not count; the mass defined in this way might still be constant.
Alas, this resource is also denied. Let us recall what we said concerning the principle of relativity and the efforts made to save it. And it is not simply a principle that is to be saved; the indubitable results of Michelson’s experiments are involved. Lorentz, to account for these results, was obliged to suppose that all forces, whatever their origin, are reduced in the same ratio in a medium having a uniform translatory motion. But that is not sufficient; it is not enough that this should take place for the real forces, it must also be the same in the case of the forces of inertia. It is necessary, therefore—so he says—that the masses of all particles be influenced by a translation in the same degree as the electromagnetic masses of the electrons.
Hence, the mechanical masses must vary according to the same laws as the electrodynamic; they can then not be constant.
Do I need to remark that the fall of Lavoisier’s principle carries with it that of Newton’s? The latter implies that the center of gravity of an isolated system moves in a straight line; but if there no longer exists a constant mass, there no longer exists a center of gravity; indeed the phrase would be meaningless. This is why I said above that the experiments on cathode rays seemed to justify the doubts of Lorentz concerning Newton’s principle.
From all these results, if they were to be confirmed, would issue a wholly new mechanics which would be characterized above all by this fact, that there could be no velocity greater than that of light, any more than a temperature below that of absolute zero. For an observer, participating himself in a motion of translation of which he has no suspicion, no apparent velocity could surpass that of light, and this would be a contradiction, unless one recalls the fact that this observer does not use the same sort of timepiece as that used by a stationary observer, but rather a watch giving the “local time.”
Here we are then face to face with a question, of which I shall confine myself to the mere statement. If there is no longer any mass what becomes of Newton’s law? Mass has two aspects: it is at the same time a coefficient of inertia and an attracting mass entering as a factor into Newton’s law of attraction. If the coefficient of inertia is not constant, can the attracting mass be constant? This is the question.
Mayer’s Principle: The principle of the conservation of energy at least still remained and appeared more finely established. Shall I recall to your minds how it too was thrown into discredit? That event made more noise than the preceding; the journals are full of it. Ever since the first work of Becquerel, and above all after the Curies had discovered radium, it was seen that every radioactive substance was an inexhaustible source of radiation. Its activity seemed to continue without change through months and years. That is already a strain on the principles; these radiations in fact were energy, and from the same piece of radium came forth this energy and it came forth indefinitely. But these quantities of energy were too minute to be measured; at least that was the belief, and the matter caused little uneasiness.
The scene changed when Curie thought of placing the radium in a calorimeter. It was then seen that the quantity of heat continuously generated was very considerable.
The explanations advanced were numerous; but in a case of this kind it is not possible to say that an abundance of good does no harm: as long as one explanation has not displaced the others we cannot be sure that any one of them is good. For some time, however, one of these explanations seems to be gaining the upper hand and we may reasonably hope that we hold the key to the mystery.
Sir W. Ramsey has attempted to show that radium is transformed, that it contains an enormous amount of energy, but not an inexhaustible amount. The transformation of radium must then produce a million times as much heat as any known transformation; the radium would be exhausted in 1250 years; that is not long, but you see that we are at least sure of being bound to the present state of affairs for some hundreds of years. While we wait our doubts subsist.
The Future of Mathematical Physics
In the midst of such ruin, what remains standing? The principle of least action up to now is intact, and Larmor appears to think that it will long survive the others. It is in fact more vague and even more general.
In the presence of this general collapse of principles, what attitude should mathematical physics take? First of all, before becoming too excited, it is well to ask whether all this is really true. All this disparagement of principles is encountered only in the case of the infinitely small; the microscope is needed to see Brownian motion, the electrons are rather tiny, radium is very rare and never more than a few milligrams are together; and then we can ask whether by the side of the minute thing that was observed, there was not another minute thing which was not noticed and which counterbalanced the first.
The question is surely debatable, and apparently only experiment can solve it. We should merely have to turn the matter over to the experimenters and, while waiting for them definitely to settle the controversy, not to trouble ourselves with these disquieting problems, and to keep quietly at our work, as though the principles were still unchallenged. We certainly have enough to do without leaving the domain where they can be applied with all certainty; we have enough to keep us busy during this period of doubt.
And yet is it really true that we can do nothing to relieve science of these doubts? It must indeed be said that it has not been experimental physics alone that has brought them into existence; mathematical physics has contributed its share. It was the experimenters who saw radium emit energy; but the theorists were the ones who brought to light all the difficulties inherent in the propagation of light through a moving medium. Had it not been for the mathematical physicists, the problems with light propagation probably would not have been noticed.
The physical principles, then, have done their very best to embarrass us. It is only fitting that they should help us to extricate ourselves. They must be subject to a searching criticism of all the new conceptions that I have outlined today, or they must be abandoned after a loyal effort to save them.
 Gibbs, Josiah Willard, Elementary Principles in Statistical Mechanics, ©1902 Yale University, published in 1914, Yale University Press, New Haven
 Because bodies would oppose an increasing inertia to the causes that would tend to accelerate their motion; and when approaching the velocity of light, this inertia would become infinite.