On the Teaching of Mathematical Physics

Vito Volterra delivered the plenary lecture On the Teaching of Mathematical Physics: and Some Points of Analysis on Monday 27 September at 10.30 at the 1920 International Congress of Mathematicians held in Strasbourg. We give a version of this talk below.

On the Teaching of Mathematical Physics: and Some Points of Analysis

By Vito Volterra.

In previous International Congresses of Mathematicians, which took place in France and Italy, I had the honour of giving Plenary Lectures. I then dealt with historical questions and I tried to set out the evolution which has taken place in Italy in analysis and in the whole of mathematical research during these last years. Allow me to leave the historical questions in the lecture which I have the honour to give today, and to deal with a question of teaching. At the present time, it seems, we have to look at the future more than the past.

I wish to speak of the teaching of mathematical physics and of questions which are specially connected with it from the point of view of the analytical part of this science.

The work done over the past century and a half has made it possible to unify and systematise research that forms an organic whole, which one could call Analytical Physics. It is the counterpart of Analytical Mechanics. In this the main role is played by ordinary differential equations, while in analytical physics the instrument which is used the most is constituted by partial differential equations.

I will dedicate this lecture to the presentation of the programme, which I think it would be useful to develop. I think that a course in analytical physics is essential as a course in general mathematics and a course in analytical mechanics have both become necessary. This course will serve to provide the body of knowledge essential to more modern chapters and to future developments in mathematical physics. The concepts that we can draw on are scattered everywhere, but there may be new work to do in their presentation, some guiding ideas to link together the various subjects discussed so that they are not separated from each other.

The homogeneity thus obtained leads to leaving aside certain branches of the science which occupies us; this is why, I admit right now, a course of this kind will not be a complete course in mathematical physics. We will see later what questions do not fit into the framework that we are going to determine and what advantages we find in placing them in another.

At what historical moment can we place the creation of mathematical physics, and what were the needs that gave birth to it?

We can recall in this regard that dynamics was formed in the Renaissance period, and that infinitesimal calculus is intimately linked to its creation and development. You will realise this if you think that it was the invention of gunpowder that gave rise to the problems of ballistics. The ancients did not need the dynamics, which is why they left it aside and did not believe it necessary to deepen the laws of movement.

But when the development of artillery showed the need to study the movement of heavy bodies, infinitesimal methods were used from the very beginning for these studies by Galileo. Then the laws of dynamics and the first examples of integration of ordinary differential equations were the work of Newton. Later still analytical mechanics and the calculus of variations were created by Lagrange.

We can fix in the second half of the penultimate century [the 18th C] and the beginning of the century past [the 19th C], the time when mathematical physics begins its grandiose evolution.

It would naturally be possible to trace the origins of Mathematical Physics further. If we wanted to take the trouble to delve into the history of mathematics, we could find at almost all times, even in very remote times, the trace of physics questions dealt with by mathematics, and that we could thus consider in a general sense as entering the field of mathematical physics; but these attempts have been in no way systematic, and it was not until d'Alembert, Fourier, Poisson, and Cauchy began to make common use of partial differential equations, that the really fertile period of mathematical physics has started.

I have just pronounced the word: differential equations with partial derivatives. Indeed, the development of this branch of analysis, and all that relates to it, has constituted the most powerful instrument which has served for the theoretical development of elasticity, acoustics, hydrodynamics, optics, electricity, and the propagation of heat.

It was precisely in the first years of the past century that the need to deepen these different chapters of physics by means of analysis was most felt.

The art of construction made it necessary to deepen knowledge of the resistance of materials; it was required, while ensuring solidity, to achieve the greatest possible economy of materials. Large iron constructions, and the constructions of large machines were not long in starting. The need was felt to calculate the dimensions of the different parts by calculation and to know the laws which regulate the elasticity of the bodies. Galileo's first tests on the bending of beams, Boyle's ideas on the elastic properties of gases, and Euler's studies gradually led to the general theory of elasticity.

Likewise, the ever more widespread use of steam engines made it necessary to perfect methods of determining the temperature, conductivity and calorific capacity of bodies. Geophysical questions were also essential. All of this research was to lead on one hand, under the impetus of Sadi Carnot's conceptions, to thermodynamics, on the other hand to the theory of propagation of heat. This one, which had only been preceded by timid attempts by Lambert, was the work of Fourier. It even establishes today, after so many discoveries and works, the most beautiful construction of all mathematical physics. This is the model on which the other classical theories were built. We will always have this in mind during this lecture.

But I believe that the most active push which acted on the development of mathematical physics is due to the enormous extension acquired by the theory and the practical applications of electricity. It is not necessary to recall here the role that electricity plays in all modern life. Whichever way you look, you see this energy of nature helping us at all times and in all of life's circumstances. However, it was only after Coulomb had given us the fundamental laws of electrostatics and that Volta had invented the battery in 1800, that the applications of electricity followed one another with ever increasing speed. Now most of the discoveries of electricity, having a double experimental and mathematical source, were the work of scientists, and the applications resulted most of the time from theoretical work. This is why important analytical developments followed one another until they led to the electromagnetic theory of light by which optics became a chapter of electrodynamics.

I said earlier that dynamics only began to exist the day the artillerymen posed the problem of the trajectories of projectiles. Would we have such advanced mathematical physics, if electrical engineering had not always had new questions to ask mathematicians? We can go further: dynamics became the basis of cosmological theories when Newton conceived the problem of the movement of the planets as a large ballistic problem. Likewise electricity is becoming, through the work of modern physicists, the basis of all molecular and atomic theories, and therefore the basis of the constitution of the universe.

Although fairly recent, the development of mathematical physics can be divided into three phases.

Its first age, which includes the last years of the eighteenth century and the first years of the nineteenth, is its heroic age. It is certainly in this period that the fundamental works are placed which gave the first impulse to all the subsequent researches which marked the art of discovery in this branch of natural philosophy.

It was in the following period, which took place in the middle of the past century, that the honour of having perfected the different chapters that had been created took place, and the much greater honour of having linked some of them and to have stated new general principles. It goes without saying that between the first and the second period there is no clear dividing line. The works of Maxwell, Lord Kelvin, Stokes, Helmholtz, Riemann, Kirchhoff, Lamé, De Saint-Venant, Mossotti and Betti are too closely linked to those of their predecessors Laplace, Lagrange, Ampère, Green, Fourier, Poisson, Cauchy, Gauss, to make it possible to separate them.

Similar considerations can be repeated if we move to more recent times. Any attempt at demarcation would be even more difficult, because recent work touches us more closely and thus, being closer to us, the perspective is more difficult to grasp. However, if we compare Poincaré's works on mathematical physics with those of the scientists I have just named, we see that they stand out clearly. This is why it must be said that a third period begins around the time when Poincaré began his courses and his work in mathematical physics. Indeed, by this time Maxwell's ideas, while still presenting difficulties, had gained ground and were beginning to become classic; the general principles of thermodynamics were now acquired in science. At the same time the analysis, thanks to delicate criticism, reformed demonstrations which were no longer sufficient and sought new bases for the old and the new theories. A fruitful period of preparation was closed, and a new phase, marked by the discoveries on electric waves, X-rays, radium, electron theory, relativity, began. At the dawn of the new century, Poincaré inaugurated in Paris the Congress of Physicists with a conference that has become classic, in which he showed the role of mathematical physics, its limits and all the importance that this branch of natural philosophy had in the past and the one it was going to have in the future. It was at the same time a summary of past work and a programme for future work.

This very simple overview has already given us an account of the reasons why classical mathematical physics was created and the branches it includes, and finally the analytical means it uses.

We have already said that the set of concepts that we can draw from the course, the programme of which we are tracing here, is not new; but it is important to arrange them in the most economical and systematic way possible.

And let's first see what notions will find a place in it. They are of two kinds, physical and mathematical concepts.

Let's examine which branches of physics should be exposed. We have already named them, but it will be useful to recall them: it is the propagation of heat, elasticity, acoustics, optics, electricity and magnetism, hydrodynamics, including also the general theory of potential.

On the other hand, we can enumerate the analytical notions that will be necessary: these are first the ordinary differential equations and with partial derivatives, then the integral and integro-differential equations, finally relations of a more general type.

These different physical and mathematical theories are so closely related to each other that it is impossible to separate them, without danger of losing the unity of an organism created by long effort. We first see historical links, then intimate links between the fundamental and guiding thoughts that gave birth to these different theories. They intertwine so that while seeing the need to unravel them we understand that nothing should be cut. How can we overcome these difficulties?

Let us recall what is normally done in courses and in treatises. Mathematical physics courses are generally of monograph type. Indeed, in this teaching one does not generally seek to give an overall idea on the various subjects, but one deepens a special chapter. This is an example that is presented to the minds of the students; if they can appropriate the methods that we expose, we are of the opinion that they are capable of studying by themselves the other chapters afterwards. If the pupils can follow several lessons, they see before their eyes one after the other the different parts of mathematical physics. This is how we have developed optics courses, elasticity courses, electricity courses, etc. All this is very useful, but in this way, if each chapter is very suitably deepened, we obviously lose sight of the whole and the unity of analytical physics.

There is also another method. We can first present the analytical notions by developing the theory of partial differential equations, their integration processes, the integral equations and the theories which result from their study, and those which constitute their logical continuation. We can then deal with series developments and the special functions that we use in mathematical physics. It is only after having given the tools that must be used, that we can attack the problems that arise in physics and that we can seek to solve them.

It goes without saying that this route has considerable advantages. It is indeed the most methodical route, where one goes from the known to the unknown, nothing being unforeseen. But it is not without drawbacks, we have to say that a good pupil who has full confidence in his teacher will be convinced that everything he presents to him will be useful in the rest of the course, and that he will not get lost in too subtle theoretical questions. If the student does not understand the reason for the theories he sees developing before his eyes, if he does not realise the root causes that led to the different processes being created in building the different mathematical theories, and to make classifications that seem artificial to him, or to create difficulties that he would never have proposed, he is supported by faith that at the end of the course, seeing the outcome, he will understand the role played by each particular subject.

You will already suspect that, while understanding the benefits that can be drawn from it, I am far from being in favour of this method. Here we touch on a general question of the pedagogy of science. In my opinion, the ideal would be to follow exactly the opposite path. We cannot completely assimilate any branch of science if we have not had the opportunity to see the main difficulties which have opposed its development, and if we do not know the intimate reason for which we followed one direction rather than another. Now, if it is not possible to follow the historical path, because it is too long, if it is not possible to start from the practical questions which have gradually imposed themselves and which have led to theoretical developments, if it is also impossible to follow the tests that have been carried out step by step until the moment when the classic methods have been definitively established, it is however necessary to get as close as possible to this ideal.

I return to the great work of Lagrange, the best model we know of a scientific treatise. In his analytical mechanics, he does not approach his subject by considering the different problems of mechanics, one after the other, as we did before him and he does not use different methods for the various questions. Furthermore, he does not start with the study of analytical questions and he does not first prepare the reader by setting out theories on differential equations and their integration.

But after a short and marvellous historical overview which highlights the most salient points in the evolution of the principles of mechanics and the stages through which these principles have passed, Lagrange very quickly arrives at the general equations which embrace the various questions, either static or dynamic. All the questions are linked by these equations which in themselves summarise the principles, and the questions can be grouped and analysed by these equations. Mechanics is thus reduced to their study, because by keeping to it, we consider all the problems at the same time, this is why it is not necessary to use a different analysis specific to each question, nor to repeat for each subject considerations which have already been made, nor to borrow the solution of a particular problem from the solution of another problem.

With this model in mind, one may wonder if it is possible to get close to it when dealing with the much more difficult and more complicated subject of mathematical physics.

In my opinion this is possible, if we limit ourselves to the branches of mathematical physics that I have mentioned, the partial differential equations which relate to them establishing between them a link which presents a close analogy with that of the general equations of analytical mechanics.

Thus is justified the limitation of the subject to which I alluded earlier and we see at the same time the unity of the discipline in question and its parallelism with analytical mechanics, which justifies the chosen denomination of analytical physics.

So, next to the two methods that I mentioned earlier, there appears a third one whose general lines I have just sketched. I will now enter the heart of my subject.

It is first possible to obtain very quickly the equations of the propagation of heat, the elasticity of solid and fluid bodies, their vibratory movements and the equations of the non-vortex movements of liquids and to solve the problems that follow it. I will not go into details on this point, but for example for the theory of the propagation of heat, one easily passes from laws of the uniform propagation of heat in a wall of finite thickness to an infinitely small element. In elasticity it suffices to establish the consequences of Hooke's law.

For the general equations of electrostatics and electromagnetism, the question is less easy. Many difficulties arise. Too many ideas have followed one another each quite different from the other; the design of electrical flow, for example, that of electrical displacements and so on. Most of the time these ideas are contradictory to each other. They brought words which, having a definite meaning when they relate to a certain theory, lose all meaning when they are transported into another. However these words remained close to each other, both those which are still current and those which are not. This constitutes a complication that one cannot overcome without briefly recalling the phases through which the different theories have passed. So it is only by following a rapid presentation of a historical type analogous to that of Lagrange, and which leads to general principles, that one can arrive in a clear manner at the fundamental equations of electrical phenomena.

We thus manage to write the general differential equations of the various disciplines envisaged, either the equations which we call indefinite, or those which are valid at the frontiers of time and space, and this is how we can put into equations the different problems posed by physics.

The first part of the course ends at this time. We can see that questions of a very different nature from a physical point of view lead to identical or analogous equations in the analytical domain.

One thus realises the advantage of treating them simultaneously, by establishing uniform principles, methods and general theorems which apply to all. Other principles or methods apply differently depending on the different types into which the equations can be classified. Likewise one can be persuaded of the remarkable economy obtained by the plan of the work. At the same time, analytical analogies make provision for hidden analogies of a much greater scope which go beyond the domain of analysis.

But before leaving this preliminary part I must say a few words on a question which I have passed over in silence so far. Perhaps you will be surprised that I have not yet spoken about it, because many attach great importance to it. And those who have followed me so far probably want to ask me: Will you use vectors, or will you not use them? I hope that I will not shock you too much by saying that I have a secondary interest in this subject. I think it is almost indifferent to general theories and the explanation of integration methods to use vector methods, or coordinate methods, or both simultaneously. By considering the vectors themselves instead of constantly defining them by their components, we do not acquire new analytical power. I think that if I wrote my treatise using coordinates, there would be no more difficulty translating it into vector language or vice versa than translating it into English.

The fact remains that the language of vectors offers considerable formal and conceptual advantages. First it is simpler and more concise. Sometimes very long operations are carried out by the vectors with a stroke of the pen. More importantly, you never give up on the very entity you are calculating. But it is obviously useless to continue this discussion. The use of vectors has now spread, and I will not hesitate to use them from the start. I think it is necessary to make another fundamental remark here. In the questions we have just mentioned, we can do without molecular or atomic hypotheses and assume at first sight that phenomena take place in a continuous medium. It is only a first approximation that we make, but one which accounts for a large number of phenomena.

Mathematical physics thus conceived can be called continuous physics. This observation leads us to give special importance to the shape of the domain and its borders, which is why it is at the very moment when the first part of the course ends that topological considerations would find their place. They will be applied and used later.

Having reduced everything to linear equations with three or four independent variables, we must begin by classifying these equations.

In my opinion there is only one way to obtain their logical classification; is to consider their characteristics.

We thus fall on the three fundamental types elliptical, hyperbolic and parabolic, that is to say, on the types with characteristics, real, imaginary or multiple.

The classification into types of the equations leads to the classification of the problems which one must consider, because without even tackling the question of existence theorems, one easily succeeds in establishing which elements can be used to determine the solutions, and moreover in which regions of space and for which values of time; they are thus defined. Questions about the past or the future that Paul Appell has so well discussed would find a place here. The classification of these problems, linked to the physical questions from which the equations originate, sheds new light on this.

We have all heard with great pleasure the beautiful lecture given by Sir Joseph Larmor. He rose to consider the highest philosophical conceptions. One had the impression that all of our analytical considerations materialised and came to acquire a kind of reality which made them intuitive. While he was delivering his lecture, I translated his words into another language. Do not think that I translated into Italian what he said in English, but I translated into the language of characteristics what he represented to us by such striking images, which made us grasp the way of producing phenomena.

But one must study the classification of the types of problems with the most attentive care, because it is necessary to be wary of a too perfect parallelism between these and the types of differential equations. One runs the risk of making a mistake if one classifies the problems envisaged exclusively according to the type of corresponding differential equations. Without going into details, which would take us too long, it suffices to consider examples which are very striking.

We learned these days from a deep lecture by Jacques Hadamard that problems related to hyperbolic equations can present themselves in a manner analogous to those found in studying the Laplace equation. The work on this subject is classic and I will not add anything to what he has been presented so well to us. But I only allow myself to illustrate what I said earlier with another example.

Take the problem of seiches in lakes. It depends on the Laplace equation, however it admits periodic solutions whose periods are the roots of a transcendent equation. How is it that a problem dependent on the fundamental elliptic equation gives rise to solutions analogous to those of the questions of vibrations which depend on hyperbolic equations?

It suffices to examine the conditions on the free surface of the liquid to recognize that the type of solution, far from being linked to the indefinite differential equation, follows from the conditions at the contour. The more complicated problem of tides gives rise to similar considerations.

We therefore see that the types of solutions are the consequence of complex circumstances in which indefinite differential equations do not always play the main role.

I also believe that an important distinction between the different differential equations can be obtained by considering their relationships and their dependence on the problems of the calculus of variations. We cannot stress enough the interest that should be attached to the development of this point.

In this connection, I would like to come back once more to Sir Joseph Larmor's lecture. He showed us the link between the Hamiltonian principle and actions at a distance. The deep words he spoke on this subject open up new horizons for us on the links between the calculus of variations and the equations of mathematical physics.

As soon as we tackle the question of characteristics, we are naturally led to consider four-dimensional spaces. Indeed, in the case of one-dimensional bodies, the characteristics are lines. They are surfaces for two-dimensional bodies and, when we consider three-dimensional bodies, they are three-dimensional spaces forming part of a four-dimensional space because we must consider, in addition to the three coordinates, a fourth coordinate time which plays a role analogous to the first three. The transition from one type of equation to another is also obtained immediately, by considering imaginary values of time. Here the introduction of four-dimensional vectors, as well as the extension of the basic operations of vectors to hyperspace becomes a necessity and at the same time it is easy and intuitive.

The changes of coordinates, Lorentz transformations, Einsteinian kinematics present themselves in such a natural way that some of the difficulties of the theory of relativity (at least of the special theory of relativity) are completely eliminated. We can also add that this whole chapter subsists independently of the philosophical substratum of relativity, and if by chance this theory were to be abandoned, this chapter would remain in all its integrity.

As we have established a classification of differential equations and fundamental problems, we must establish a classification of integration processes. In my opinion, they can be divided into three categories: Green's method, that of characteristics, and that of simple Fourier solutions. To avoid any misunderstanding, I will make it clear at once that the three methods I have just named should be considered as typical procedures, but that in particular questions one can use mixed methods which involve all three types at the same time.

By introducing the term "Green's method", we broaden its ordinary meaning, because not only do we mean by this the method of integrating the Laplace equation for different contour conditions, but also analogous methods of integrating the equations of the elliptical type (as for example that of Betti for the equations of elasticity) and even those equations of hyperbolic and parabolic type, where one does not use the characteristics explicitly. Also the formulas relating to the vibrations which are linked to that of Kirchhoff would fit into the domain of Green's methods. Basically, Green's method relates to the use of fundamental solutions, in all cases where the characteristics are not explicitly used.

The characteristics method is really only a modification of that of Green, but, in my opinion, it is useful to distinguish the two methods from each other. The interest in the notion of characteristics is so great, they play such an important role in the integration of equations, that a separation is necessary which is not artificial, but which corresponds to something substantial.

Finally, the Fourier method includes all those where simple solutions are used, and the general solution is obtained by means of series of these solutions.

The different methods being thus classified, we can proceed to their development. To use Green's methods and characteristics, we must first look for a reciprocity theorem. There is no difficulty in obtaining it in general and in clarifying the meaning it takes in the different branches of physics, which clarifies the whole of the theory.

I take permission, in this connection, to recall a recent result which relates to the Hall phenomenon. It is precisely a reciprocity theorem which, interpreted in this theory, reveals to us a remarkable property of currents, produced in a bismuth laminar subjected to the action of a magnetic field. Suppose that we make the current enter by a point A and leave by a point B and that we determine the potential difference at two points C and D. Let us reverse and let enter the current by C and leave by D. So that we find that for the same potential difference in A and B as before, we must invert the magnetic field. This theorem was used by Orso Mario Corbino to obtain many practical results relating to the actions of magnetic fields on currents.

Since I am talking about this problem, it is interesting to note that in this case there are mixed problems with respect to the contour conditions, which can be solved by a method that I gave several years ago. It does not extend beyond two-dimensional domains. But using a very ingenious method based on different principles, Léon Brillouin gave the general solution in the case of spaces with any number of dimensions.

But the most difficult and delicate part of integration consists in the search for fundamental solutions. Their calculation and the determination of their properties, their classification, the different types that can be obtained, compared with the different types of differential equations, and the different types of problems to which they relate, as well as their physical meanings, all this constitutes a set of concepts of very great interest and the exposure of a fairly large field of research which has given rise to great analytical difficulties.

From the simple expression of the inverse of the distance between two points, which constitutes the fundamental integral of the Laplace equation, to the fundamental integral for double refraction, given, a few years ago, by Nils Zeilon, there is a long way to go, where difficulties are encountered at every step.

I would be tempted to go into a few details on this subject, but I am running out of time. I will only note that we cannot avoid talking about the different kinds of singularities that we find in fundamental integrals, because it is only by very wise use of all their singularities that we succeed in obtaining the best possible advantage from their use.

I will cite in connection with polydrome solutions those of double refraction; and for the case of parabolic equations, the fundamental solution of the heat propagation equation.

It is sometimes necessary to know how to reject those whose polydromy makes their use impossible and to know how to replace them by others skilfully found, while in other cases, it is their polydromy which is the source of the most hidden and most fertile results. In my opinion, the study of fundamental solutions is far from being exhausted; on the contrary, although they have been used at all times, they have not been considered sufficiently as a whole and we have not yet systematised their general study enough. This is due, most probably, as I have said elsewhere, to the methods that we have followed most of the time in mathematical physics, to study the different branches separately, without considering them side by side as a whole, like a body of doctrines.

On the other hand, purely analytical research too often departs from the applications that you have to have in view. By losing contact with reality, we are no longer fed by the richest source of discovery.

Once the fundamental solutions have been found, they must be used taking into account the reciprocity theorems. We thus come across general formulas which are of great interest from a physical point of view. From an analytical point of view, they do not completely solve the problems posed, but they lead to other questions, which can be classified as another type of physics problem, and where a new tool must be brought into play: integral equations or even equations of a more complicated type. I am of the opinion that we must gradually store up the questions on our way which belong to a new field, beyond differential equations, and then devote part of the course to their study.

I will be brief on the development of the characteristics method. As it is closely related to Green's method, much of what we have already presented relates to it.

I will recall in this connection the first solution given by Émile Picard of the telegrapher's equations by using the method of characteristics. It has been the starting point for a great number of researches and it has had, and still retains, a great deal of interest from a theoretical point of view and from applications.

The method should not be separated from the characteristics of a few particular processes which constitute one of the most elegant sides of mathematical physics. One can cite as a typical type the method of images. It was discovered by Lord Kelvin to give a simple and intuitive form to the solution of the Poisson problem of electrical induction of spheres, but Sir George Gabriel Stokes soon made it pass into hydrodynamics. Then it was introduced into the theory of magnetism, and later into the theory of vibrations.

This is perhaps one of the most conclusive examples of the usefulness of considering the three types of equations simultaneously. One sees there, in an extremely striking way, the modifications which the same fundamental idea must undergo in order to comply with the analytical necessities of the different cases. The particularities and the role of characteristics appear very clearly there. It is also a very instructive exercise on the metrics of hyperbolic spaces, which is so closely related to considerations leading to relativity.

In the same order of considerations are transformations of equations into themselves. That by reciprocal vector rays is not only applicable to the case of the Laplace equation, but also to other elliptic equations, and to hyperbolic equations. For these we have transformations where time and space enter, as in the Lorentz transformation, and playing a role which has not yet been exploited.

Finally arriving at the simple solutions method, we have a vast domain to consider, because from Fourier series to new series of orthogonal functions, many of the most modern theories fit into it. It suffices to recall Poincaré's research on the equations of mathematical physics, and the fundamental results, obtained by Émile Picard, which are closely related to the theory of integral equations.

I believe that the second part of the course should be limited to what we have just quickly summarised. We have left all the details aside, which is why we have not touched on a host of very interesting issues. Thus we have not spoken of the distinctions to be made in the different problems according to the connection envisaged between the fields. However, I believe that the character of this second part of the course is quite clear from the above.

If in the first part, taking physical problems as a starting point, we have obtained the differential equations, in the second part we have classified these equations and presented general integration methods. By considering them all at the same time we were able to synthesize, which simplified the presentation and gave an integral whole to a set of doctrines whose unity must always be noted.

But at the point where we have arrived many physical problems are not yet completely resolved, and many questions remain; the analysis that has been developed is not sufficient in itself to resolve them all. It cannot be separated from an analysis which completes and integrates it and which imposes itself at first sight. And it is really a question here not of a method but a new analysis.

The existence theorems, the definitive problem solved by contour conditions and a large number of other questions of the same kind, require a calculus in which we consider all the values of certain functions in a certain domain at the same time; other problems depend on the form of the domain which one considers, or on its boundary. All these questions as well as those reserved in the second part of the course and which depend on the resolution of integral equations should be studied in the third part.

The guiding concept would therefore be the notion of functions depending on all the values of other functions. Basically, the integrals of partial differential equations depend on the one hand on independent variables, but on the other hand they depend on arbitrary functions. And, in fact, in analysis as soon as we needed to consider the integrals of partial differential equations, we encountered arbitrary functions. What role do the values of arbitrary functions play? Are they not an infinite and continuous number of independent variables, on which the integrals depend? It was in an unconscious way that they played this role, until the day when we saw that it was necessary to create a special analysis, specific to this kind of questions. Let me recall the following words with which, in 1887, I began my research on the functions which depend on other functions or the functions of lines: "In many questions of physics and mechanics and in the integration of the equations with partial derivatives, we must consider quantities which depend on all the values of one or more functions of a variable. Thus, for example, the temperature at a point on a conductive plate depends on all the values of the temperature at the contour, the infinitely small displacement of a flexible and inextensible surface depends on the projections on a certain direction of displacement contour points."

On the other hand, since the mathematical physics that we are developing is precisely that of the continuous, and since this continuum is variable, it is obvious that we cannot do without considering a continuum as a variable element. One could object that the separation of the second and the third part of our course is not philosophical, because one should not separate, from each other, the different types of functions and that it would be useful to directly attack the questions with the new analysis; but the same objection could be raised against the distinction between differential calculus and integral calculus, a distinction which, however, continues because of the well-known advantages which it brings.

While not being a treatise on functions which depend on other functions, but a chapter devoted to their applications to analytical physics, the third part should, in my opinion, begin by setting out their fundamental properties. We would then go on to the general process of passing from the finite to the infinite in this order of questions. It is reduced practically to two fundamental rules which consist first of replacing a finite number of indices by one or more continuous variables, secondly of replacing the sums made with respect to these indices by integrals.

Nothing simpler than these rules which bring us from all ordinary problems of algebra and differential and integral calculus to increasingly difficult problems and which also lead from solutions of ordinary problems to solutions of new problems, as well as theorems or properties about new principles. One can thus pass from algebraic equations to integral equations of the most general type which are distinguished by the existence or not of exceptional points, from where their classification is indicated by Émile Picard as equations of first, second or third species with fixed or variable limits.

Ordinary differential equations lead by the same process of extension to ordinary integro-differential equations, while partial differential equations can be generalised in three directions by leading either to integral differential equations with partial derivatives, or to functional equations of the type of with total differentials, or to functional differential equations proper.

The applications of linear integral equations to existence theorems and to the definitive resolution of the problems posed by Green's methods and characteristics, as well as their use in series developments and in the theory of simple solutions are so classic that it is not necessary to play it up here.

On the other hand, the studies on the functional differential equations developed by Jacques Hadamard and Paul Lévy, which relate to Green's functions, are among the modern results which have most struck and interested me.

It is also in this last part that the study of the modifications that must be made to classical theories would be found if we want to correct the solutions, taking into account heredity.

But I think I can dispense with insisting on this research here, especially in the University of Strasbourg where it is so well represented by Maurice Fréchet and Joseph Pérès.

The third part of the course could be limited to what I just spoke about. It would be the last part of the course. It is now very easy to list the branches of mathematical physics which have not been part of our study; it is also easy, after what we have said, to understand the reasons why they stayed outside.

Pure Thermodynamics develops without needing to deepen the theory of partial differential equations; therefore it does not fit into analytical physics; capillarity, which is a fairly limited branch of physics, employs particular methods. But there is an extremely vast field which embraces the most modern and interesting theories and which also remains outside the framework which we have sketched. You could call it statistical physics. Its methods, which were created by James Clerk Maxwell, are indeed analytical methods of the highest scope and of the greatest difficulty; but these are methods very different from those that we considered previously. It does not enter into the physics of the continuous, this is why, by the spirit which animates it and by the processes which it employs, it constitutes a body distinct from the one whose methods we have studied. It is enough to read the works that Émile Borel and Paul Langevin devoted to it, to be persuaded by it.

The systematic constitution of a physics of probability marks a profound transformation in natural philosophy because it brings about the order of disorder, and the laws derive only from the defect of laws. The physics of probability overturns all the methods of analytical physics and in some cases it enters much more intimately into the very essence of matter and phenomena by giving results closer to reality.

Analytical physics while not old, and while being far from being one of the dry branches of the big tree of mathematics, is however no longer young. It is precisely because of this that it is ripe for its systematisation and unification. Before the progress made recently it would not have been possible to reach this goal for lack of being able, for example, to count and specify the postulates which it implies, or to demonstrate exactly certain necessary propositions, or to state with precision to what extent it represents reality and what are its limits. The path we have adopted leads to these results, which are of considerable philosophical interest.

Even devoting all efforts to appropriate and advance the physics of probability, it is essential that physicists, mathematicians, and engineers know the fundamental principles of analytical physics. How can we bring to the attention of such a large audience such a vast set of concepts? A distinction must be made here between a course that is delivered to the public as a printed work and a course that is exhibited orally. In a course constituting a treatise, we could develop with all the details, the programme that I have just quickly outlined; in a delivered course, it would be better to confine oneself to the essential points.

It takes a very fine art to achieve these two goals. To write a treatise (as to teach a course) it is not enough to have scientific knowledge of the subject, but it also requires a certain artistic sense. Books that have survived the centuries are works of art as much as works of science.

Euclid, who systematised and unified the geometry created by his predecessors, was a scientist, but he possessed the same artistic taste that made Homer and Phidias famous.

Last Updated January 2020