On Duhem’s Energetics or General Thermodynamics

Abstract

Pierre Duhem is an unavoidable figure if one wants to scrutinize the progress in the mixed science of mechanics and thermodynamics in the period 1880–1920. He is a prolific writer and a never tired propagandist of the global science of energetics. Here we examine his main contribution, its novelty and its inherent limitations in the light of two remarkable synthetic and/or critical works, his treatise on energetics (Traité d’énergétique ou de thermodynamique générale. Gauthier-Villars, Paris, 528+504 pages, 1911) and his series of papers (Duhem in L’évolution de la mécanique. A. Joanin, Paris, 1903) on the “Evolution of Mechanics” (of which we also provide a partial translation). These works are replaced in their socio-scientific background with its main sources (Gibbs and Helmholtz) and its possible interaction with, and influence on, contemporary scientists. A particular emphasis is put on Duhem’s style and interests that are strongly influenced by his combined epistemological, philosophical and historical vision. We concentrate on the specialized fields examined and tentatively improved by Duhem in the “Evolution of mechanics”, with a personal interest in those “nonsensical branches”—friction, false equilibria, permanent alterations, hysteresis—that Duhem tries to attach to the former Gibbs’ statics and Helmholtz’ dynamics by way of subtle generalizations. In this analysis we account for the enlightening comments of contemporaries (E. Picard, J. Hadamard, O. Manville), of his various biographs, and of Duhem’s own perusal (Duhem in Notice sur les titres et travaux de Pierre Duhem, 1913) of his oeuvre. We conclude with modern developments which provide answers to queries of Duhem that now appear as too much in advance on their time.

Appendices

Appendix A

Partial English Translation of P. Duhem, “Traité d’énergétique ou thermodynamique générale”, Gauthier-Villars, Paris, 1911, by Gérard A. Maugin [Only the Introduction and small parts of Chapter I: “Définitions préliminaires”, are translated in order to give a flavour of Duhem’s style and exposition. Original footnotes, if any, are reported to the end and numbered consecutively. Translator’s remarks are placed within square brackets in the main text. This is a verbatim translation without any ambition of literary prowess.

Traité d’énergétique ou thermodynamique générale

Treatise of energetics or general thermodynamics

By Pierre DUHEM

Introduction

• 1. Of thermodynamics or energetics

Theoretical physics represents by means of quantities (grandeurs) the properties of the bodies that it studies. Methods of measurement allow one to place in correspondence, with a more or less broad approximation, each intensity of a property with a particular determination of the quantity that represents this property. Through the methods of measurement, each physical phenomenon corresponds to a set of numbers, each physical law corresponds to one or several algebraic relations between various quantities, each set of concrete bodies to a system of quantities, to an abstract and mathematical scheme.

Theoretical physics has constantly to solve the following problem: From given physical data, extract new physical laws; either it proposes to show that the latter, already directly known, are none other than consequences of the former, or it proposes to announce laws that the experimentalist has not yet observed.

To treat this problem theoretical physics combines given laws together, that concern particularly certain physical properties and certain bodies, in agreement with rules issued from general principles that are supposed to hold true for all properties and all bodies.

For example, it wants to show that if we know the law of pressure of the saturated vapour of a liquid, and the laws of compressibility and dilatation of a liquid and its vapour, then one can fix the law according to which the heat of vaporisation varies; to this purpose, its combines the first laws along the rules of the principle of conservation of energy and the principle of Carnot, principles that are supposed to apply to all bodies and all of their properties.

It is the system of these general principles that we propose to expose here.

For a long time, physicists have assumed that all properties of bodies reduced, in the last analysis, to combinations of figures [geometrical forms] and local motions. The general principles to which all physical properties must obey, were none other than the principles that govern the local motions, i.e., the principles that compose rational mechanics. Rational mechanics was the code for the general principles of physics.

The reduction of all physical properties to combinations of figures and motion or, following commonly used denomination, the mechanical explanation of the Universe, seems today to be condemned. It is not so for a priori reasons, whether metaphysical or mathematical. It is condemned because it has been so far just a project, a dream, and not a reality. Despite immense efforts, physicists never succeeded to conceive an arrangement of figures and of local motions that, treated following the rules of rational mechanics, give a satisfactory representation of a somewhat extended set of physical laws.

Is the attempt at a reduction of all physics to rational mechanics, an always vain attempt in the past, destined to succeed one day? Only a prophet could answer this question positively or negatively. Without prejudging the meaning of this answer, it appears wiser, provisionally, to renounce their efforts, fruitless until now, towards the mechanical explanation of the Universe.

Therefore, we are going to attempt at a formulation of the corpus of general laws to which all properties must obey, without assuming a priori that these properties are all reducible to a geometrical figure and a local motion. Accordingly, the corpus of these general laws will no longer reduce to rational mechanics.

In truth, the geometrical figure and the local motion remain physical properties; they are in fact those that are the most immediately accessible. Our corpus of general laws must apply to these properties, and, being applied to the latter, it must recover the rules that govern the local motion, the rules of rational mechanics. The latter must, therefore, result from the corpus of general laws that we propose to constitute; it must be what follows when we apply these general laws to particular systems where we account only for the figure of bodies and their local motion.

The code of the general laws of physics is known nowadays under two names: the name of thermodynamics and the name of energetics.

The name of thermodynamics is intimately attached to the history of this science; its two main principles, the principle of Carnot and the principle of conservation of energy, were discovered when studying the motive power in machines exploiting fire. This name is also justified by the fact that the two notions of work and quantity of heat are constantly at play in the reasoning through which this theory develops.

The name of energetics is due to Rankine¹; the idea of energy being the first that this theory has to define, the one to which most other used notions are attached; this name seems to us as well chosen as that of thermodynamics.

Without deciding which naming is preferable to the other, we shall use both as equivalent to one another.

• 2. On the logical significance of the principles of energetics

The logical character of the principles that we are going to formulate and group together must be borne in mind².

These principles are pure postulates; we can state them as we please, on the condition that the statement of each of them is not self contradictory, and that the statements of the various principles are not in reciprocal contradiction.

The character with which we recognize that the whole set of the so formulated principles constitute a good theoretical physics is the following one: applying this set of principles to formulas that represent exact experimental laws, we can deduce new formulas which, in turn, represent other exact experimental laws.

The experimental control of the set of principles of energetics is thus the only criterion of truth of this theory.

Indeed, this control can be done only on the whole set of principles of energetics taken in its totality or, for the least, on extended parts of this set. It would be impossible to submit to the control of experiment one isolated among these principles, or even a small number of these principles. Any experiment, simple as it may be, involves in its interpretation very many and diverse principles. We will often have the opportunity to recognize this fact in the course of this exposition.

The experimental control can only concern the whole set of ultimate consequences of the theory; it estimates if this set of consequences provides, or does not provide, a satisfactory representation of these experimental data; but in so far as the theory has not produced the set of these last consequences, we cannot call for this control, as this would be premature; hence the following rule which will be of frequent usage in the sequel of our studies: In the course of it exposition, a theoretical physics is free to choose the path that it likes, in so far as it avoids any logical contradiction; in particular, it does not need to account for any experimental fact; it is only when it has reached the end of its development that its ultimate consequences can and must be compared to experimental laws.

To say that the principles of energetics are pure postulates and that no logical constraint limits our right to choose them arbitrarily, is not to say that we will formulate them by chance. On the contrary, we shall be very strictly guided in the choice of these statements, knowing well that it would suffice to alter any thing for the experimental check of the consequences to become at fault at some point.

We are assured by this guideline by our knowledge of the past of science. Principles have been formulated that were proved to be in gross contradiction with experiment; other principles were then substituted to them, which have received a partial confirmation, although an imperfect one; then they were modified, corrected, guaranteeing for each change a more exact agreement of their corollaries with facts. We are assured that the clothing of which we cut the forms will fit exactly the body it must dress because the blueprint has been tried and retouched many times.

Each of the principles that we shall state presents thus no logical proof; but it would carry a historical justification; we could, before stating it, enumerate the principles of differing forms that were tried before it, and which could not fit exactly reality, that we have been forced to reject or to correct until the whole system of energetics adapts in a satisfactory manner to the set of physical laws. The fear of an excessive length will forbid us the exposition of this historical justification.

• Notes by Duhem

  • 1. J. Macquorne Rankine, Outlines of the science of energetics (Glagow Philosophical Society Proceedings, Vol. III, no.6, 2 May 1855). – J. Macquorne Rankine, Miscellaneous scientific papers, p. 209.

  • 2. We will limit ourselves to giving here a very concise résumé of what we expanded in the following book: La théorie physique, son objet et sa structure, Paris, 1906 [English translation: The aim and structure of physical theory, Princeton University Press, New Jersey, 1954; paperback reprint, 1991]. This work can be viewed as a kind of logical introduction to the present treatise.

Appendix B

Partial English Translation of P. Duhem, “L’évolution de la mécanique – Part VII – Les branches aberrantes de la thermodynamique”, Revue genérale des sciences, pp. 416-429, Paris, 1903, by Gérard A. Maugin [Only small parts of this lengthy contribution are translated; Original footnotes, if any, are reported to the end and numbered consecutively. Translator’s remarks are placed within square brackets in the main text. This is a verbatim translation without any ambition of literary prowess.

Duhem P. (1903), L’évolution de la mécanique (published in seven parts in: Revue générale des sciences, Paris; as a book, A. Joanin, Paris) [There exists already an English translation: The evolution of mechanics, Sijthoof and Noordhoff, 1980, to which we had no access].

L’évolution de la Mécanique

The Evolution of Mechanics

VII - The nonsensical branches of thermodynamics

I.- Friction and chemical false equilibria

From the original text, p. 418:

That outside systems of which the equilibrium states can always be classified in reversible changes, there exists an infinity of other systems of which the statics is not that of Gibbs, and the dynamics not that of Helmholtz, and that, among such systems, are precisely those exhibiting friction?

Therefore, the laws according to which systems with friction evolve or remain in equilibrium, require a specific formulation. This formulation, we will not ask it to chance. The formulation imposed to statics by Gibbs and to dynamics by Helmholtz was shown to be admirably fruitful; it is natural to conserve its type as much as possible; to deduce the new formulation from the old one by means of additions and modifications as light as possible; this is the idea that guided us when we built the mechanics of systems with friction.

It would not be easy to expose the latter with entering details that the present writing should not involve. However, let us try to draw a summary sketch and, to that purpose, we restrict ourselves to the study of a system such that only one normal variable, apart from temperature, suffices for its definition. Let α represent this unique variable. If F, A, J, v are, respectively, the internal potential [energy], the external action, the inertial force, and the action of viscosity, then according to Helmholtz’ dynamics at each instant we can write the equality

$$A + J + v = \frac{\partial F}{\partial \alpha }.$$

(3)

This equality, the general law of motion [probably “evolution” would be better] of the system, implies the law of its equilibria, a law in conformity with Gibbs’ statics.

The equilibrium of systems with friction does not agree with Gibbs’ statics; Equality (3) does not apply to them; but we can try to modify it in such a way that it will be extended to such systems.

To that purpose, we continue to attach to each state of the system a quantity F that is determined without ambiguity through the knowledge of this state. To this quantity that we still call internal potential, we will continue to attach internal energy and entropy by means of previously known relations; the external action, the inertial force and the action of viscosity will remain defined just as before; but these elements will no longer be sufficient to set forth the equation governing the system. It will be necessary to know a new element, the action of friction f.

This action, always positive, will depend, just like the action of viscosity, on the absolute temperature, the variable α and the general velocity $\dot{\alpha } = d\alpha /dt$; but contrary to what happens for the generalized velocity, it will also depend also on the external action A; furthermore, it will not vanish with the generalized velocity; the latter going to zero, the action of friction will tend to a positive value g.

In order to govern the motion of the system, we will no longer have a unique equation, but two distinct equations; the first of these should be used only when the generalized velocity $\dot{\alpha } = d\alpha /dt$ is positive; it will take the following form:

$$A + J + v - f = \frac{\partial F}{\partial \alpha }.$$

(4)

The second of these equations will read:

$$A + J + v + f = \frac{\partial F}{\partial \alpha }.$$

(4b)

This one will be reserved to the case when the generalized velocity $\dot{\alpha } = d\alpha /dt$ is negative.

As to the equilibrium condition, it will no longer be represented by an equality, but by a double inequality that expresses that the absolute value of the difference A − ∂F/∂α is not larger than g:

$$- g \le A - \frac{\partial F}{\partial \alpha } \le g.$$

(5)

We rapidly go over the equation of living forces [equation of kinetic energy]; we can only repeat here practically all what was said when studying the dynamics of Helmholtz; it is only necessary to add the work of friction to the work of viscosity. The former, like the latter, is always negative. We also do not deal with the Clausius inequality which remains exact in the new dynamics. Here also, the work of friction is just being added to the work of viscosity. Other consequences of the laws just formulated, and more particularly the condition of equilibrium, will require a little more attention.

Gibbs’ statics would require the difference A − ∂F/∂α to vanish, and therefore having value between −g and +g. The equilibrium states predicted by this Statics, and that are usually called states of true equilibrium, are thus among those that are predicted by the new Statics; But the latter announces the existence of an infinity of other equilibrium states, that we designate by the name of false equilibria.

If the value of g is large, then the states of false equilibrium spread on both sides of those of true equilibrium, in a large domain. They will shrink close to the states of true equilibrium whenever the value of g is small. If this value becomes sufficiently small, then the states of false equilibrium will be so close to those of true equilibrium that experiments would no longer distinguish them; practically, the Statics of systems with friction would be undistinguishable from Gibbs’ statics.

This is only a particular application of the following remark: Gibbs’ Statics and Helmholtz’ Dynamics are limit forms of the Statics and Dynamics of systems with friction; these tend to those when the action of friction becomes infinitesimally small.

This remark is not a simple view of the mind; it acquires a particular interest in the study of chemical equilibria¹.

Note 1. We have given an exposition of the theory of chemical equilibria accounting for friction and the principal applications of this theory in the following works: Théorie thermodynamique de la viscosité, du frottement et des faux équilibres chimiques, Paris, 1896 – Traité élémentaire de mécanique chimique fondée sur la thermodynamique, Vol. II, T. I, Paris, 1897; Thermodynamque et chimie, leçons élémentaires à l’usage des chimistes; Leçons XVII, XIX and XX, Paris, 1902.

[In the rest of this Chapter Duhem expands an example from chemical physics].

II. Permanent alterations and hysteresis

[Here Duhem first gives a general idea of what permanent alterations are. He emphasizes the role of infinitesimally slow evolutions, adapting accordingly temperature and external actions]

⋯⋯

[From page 421]

The theory of systems capable of permanent alterations will thus be distinct from general mechanics for which, with Gibbs and Helmholtz, we have sketched the principles; but it will also differ from the mechanics of systems with friction; it will be a new branch of mechanics.

How is this new mechanics to be constituted?

Only the principal thought is of concern to us here; the detail of formulas is not needed; we restrict ourselves to the study of a simple case that will allow for a better transparency of the frame of ideas. As object of our analysis, let us choose a system defined by a single normal variable, apart from temperature; for example a thread [wire] under tension of which the length will be this normal variable, while the pulling weight will be the corresponding external action.

First let us give certain infinitesimal variations to the temperature and the pulling weight; the length of the thread will suffer an infinitesimally small increase. Then let us give to both temperature and pulling weight variations equal in absolute value to the preceding ones, but with opposite sign so that these two quantities recover their original value. The length of the thread is reduced, but this decrease does not have the same absolute value as the preceding lengthening, because the thread suffers a permanent deformation. Thus, in the course of an infinitely slow alteration, a linear algebraic relation determines the infinitely small variation of the length of the thread when we impose infinitely small variations to temperature and pulling weight; but this relation must not have the same form when the thread elongates or when it contracts; a certain equality must be written when the normal variable suffers a positive variation, and another one when this variation is negative.

What guide will help us to discover the form of these two equalities? It is the theory itself, which cannot be sufficient to treat permanent alterations, but which proved to be so fruitful in the study of systems with reversible modifications. We shall look for a construction of this new mechanics in such a way that it is as close as possible to that theory, that it follows from it by a very slight transformation, that it be one of its generalizations, that the Statics and Dynamics of systems admitting no permanent alterations be regarded as limit forms of the Statics and Dynamics of systems with very weak permanent alterations. In a nutshell, we shall follow the same method as that which was given by the theory of systems with friction.

When a system presenting no permanent alteration is subjected to an infinitely slow modification, i.e., a reversible evolution, the equilibrium conditions are satisfied at each instant; if the state of the system depends on a single normal variable α, then at each instant the external action A equals the derivative of the internal potential F with respect to α; this we are taught by equality (1) [i.e., A = ∂F/∂α]

For coordinated infinitely small variations of the temperature, the external action and the normal variable, there therefore exists the relation

$$dA = d\frac{\partial F}{\partial \alpha },$$

(6)

by virtue of the fact that the always equal quantities A and ∂F/∂α suffer simultaneously equal increases. According to this relation, if we change the sign of these variations without changing their absolute value, then we change the sign of the variation suffered by the normal variable without changing its own absolute value; this way we express the reversibility of the infinitely slow modification.

These peculiarities cannot be met in a system capable of permanent alterations; each of the elements of which the succession composes an infinitely slow modification cannot be governed by equality (6); we must substitute to this equality two distinct relations, one valid when the normal variable increases, and the other valid when this variable decreases.

In the first case, we substitute to equality (6) the relation:

$$dA = d\frac{\partial F}{\partial \alpha } + h\,d\alpha ,$$

(7)

In the second case, we substitute to (6) the relation

$$dA = d\frac{\partial F}{\partial \alpha } - h\,d\alpha .$$

(7b)

The quantity h, of which the introduction in these equations distinguishes systems capable of permanent alterations from those not capable of these, depends on the state of the system, and also on the external action A.

It is obvious that it suffices to give to this quantity h a very small value so that inequalities (7) and (7b) differ very little from the equality (6); the permanent alterations of the system then are very little sensitive, and its infinitely slow modifications are almost reversible; thus, systems without permanent alterations and capable of reversible modifications are precisely limit forms of systems subjected to small permanent alterations.

For systems without permanent alterations, a simple rule allows us to deduce from the internal potential the knowledge of the internal energy, and thus, the quantity of heat involved in an infinitely slow modification. Nothing forbids the extension of this rule to systems with permanent alterations. Joint to what was previously given, it will provide the essential principles on which the Statics of such systems relies². With some accessory hypotheses, all inspired by the desire to make the small branch (“rameau”) of Thermodynamics as similar as possible as its master branch, this will complement these principles.

Note 2. We have devoted six memoirs to this Statics under the general title: Les déformations permanentes et l’hystérésis (Mémoires in-4° de l’Académie de Belgique, t. LIV, 1895; t. LVL, 1898; t. LXII, 1902) and eight memoirs published under the title: Die dauernden Aenderungen und die Thermodynamik, Zeitschrift für physikalishe Chemie, Bd. XXI, XXIII, 1897; XXVIII, 1899; XXXIV, 1900; XXXVII, 1901), etc.

What are the applications of this new Statics?

A first category of permanent alterations is formed by elastic deformations. The traction, torsion and flexion cause deformations that do not disappear with the cause that produced them; these deformations, known and observed at all times since Antiquity, find in the above given principles, their theoretical explanation.

[Here Duhem mentions examples from residual magnetization, magnetic hysteresis, and analogous properties for electric polarisation in dielectrics].

……

[continued from p. 422].

Essential as it is in the study of elasticity and the theory of magnetism, hysteresis is destined to play a very important role in “chemical mechanics” ……

It is without doubt to permanent alterations of this kind that must be attached the effects of tempering, annealing, and hardening that complicate so strangely the study of metals and their industrial combinations. Quite often, these effects result from both elastic hysteresis and chemical hysteresis; only the simultaneous consideration of these two hystereses, can somewhat untangle these phenomena, in appearance inextricable, that present some bodies; such as Nickel based steels of which M. Ch.-E. Guillaume has analyzed the strange properties, or the platinum-silver alloy of which the electric resistance manifests so curious residual variations, as observed by M. H. Chevallier.

This superposition of chemical hysteresis and elastic hysteresis makes the laws of dilatation of glass singularly complex; the observation of the displacement of the zero point of thermometers had not revealed, first to Desprez and then to M. Ch.-Edmond Guillaume much more than this extreme complexity; numerous and patient measurements, guided by the thermodynamics of permanent modifications, have finally allowed M. L. Marchis to put some order in this chaos [Guillaume, Chevallier, Desprez and Marchis were doctoral students of Duhem in Bordeaux].

……

[continued from page 423]

……

In a system subjected to permanent alterations, the quantity h, that we name coefficient of hysteresis, does not vanish in general. The two equalities (7) and (7b) are thus distinct from one another; if we suppose that the system suffers, with an infinite slowness an infinitely small alteration due to certain variations in the temperature and the external action, we will not be able, by inverting those variations, to reverse the modification and to bring the system back to its initial state.

But what is not generally true may become true in certain particular cases; by associating in an appropriate manner the values of the normal variable, of temperature and of the external action, we can make the coefficient of hysteresis vanish; when these values are thus associated, we say that the system is placed in a natural state; in general, if we take the system in any state, defined by a certain value of the normal variable and a certain value of temperature, we can submit it to an external action so that this state becomes a natural one.

[Here Duhem goes more deeply in the notion of natural state and that of residual fields, and then he discusses the possible dynamics in which the standard application of d’Alembert’s principle is at fault, referring to works by Henri Bouasse (1866-1958; a French physicist in Toulouse, South-west of France) and the noted German physicist Max Wien (1866-1938)].

III. Electrodynamics and electromagnetism

[In this long section, pp. 424-427, Duhem considers the general thermodynamics of electromagnetic bodies. He ponders the notions of electromagnetic energy, electric displacement, electromagnetic induction, electrodynamic forces, properties of the system which have no inertia associated with the relevant variables, in spite of the existence of generalized velocities duly associated with such variables, electrodynamic potential, electrokinetic energy, Ohm’s effect considered as a viscosity. He pays an emphatic tribute to Helmholtz. He clearly expresses his irreconcilable appraisal of Maxwell’s vision. This, of course, appears to be outdated and certainly not very objective].

Conclusion (pages 427-429).

[Here Duhem offers a rather literary synthesis of the contents of the whole work. He goes all the way to compare the construction of his successive theories to the “sanguine sketches” of Raphael on exhibit at the Louvre [museum] where you can see the work by successive approximations of this master painter, starting from a rough sketch and then improving the details at each successive picture and finally producing a masterpiece that finally causes the admiration of all. This is the way his new Mechanics, unique though complex, emerges from his own mind].

[continued from page 428, 2^nd column]:

The old Mechanics pushed to the extreme the simplification of its fundamental hypotheses. It had condensed these hypotheses in a unique presupposition: All systems are reducible to a set of material points and solid bodies which move in agreement with Lagrange’s equations. And even more with Hertz, it went further by erasing real forces from its equations.

The new Mechanics [i.e., Duhem’s] is not imbued of such a simplification of its principles; it does not avoid to increase the complication of its fundamental hypotheses; it admits terms of varying nature and form in its equation, terms of viscosity, friction, hysteresis, electro-kinetic energy, while the old Mechanics excluded from its formulas such symbols, as in contradiction with its unique principle.

But reality is more complex, infinitely so; each new improvement in experimental methods, by scrutinizing more thoroughly the facts, discovers in them new complications. Human mind, in its weakness, although trying hard to work toward a simple representation of the external world, suffices to place the image in front of the object and to compare them in good faith in order to realize that this simplicity, so forcefully desired, is an un-captured chimera, an unrealizable utopia.

……

[continued from page 429, first column]:

This capability to mould facts and to capture their finest detail was acquired by the new Physics by giving up some of the requirements that rigidified the old Mechanics. Among these requirements, the first and most essential one was the one that intended to reduce all properties of bodies to quantities, figures and local motions; the new Physics rejects totally this requirement; it admits in its reasoning the consideration of these qualities; it endows the notion of motion with the generality that Aristotle granted to it. This is the secret of its marvellous compliance. Indeed, with this it gives up the consideration of hypothetical mechanisms that the natural philosophy of Newton disliked, the research of the masses and hidden motions of which the only object is to explain the qualities in geometrical terms. Freed from this work that Pascal proclaimed uncertain, painstaking and useless, it can, in all freedom, devotes its efforts to more fruitful endeavours.

……

The creation of this Mechanics, based on thermodynamics, is thus a reaction against atomistic and Cartesian ideas – not foreseen by those who most contributed to it – to the deepest principles of peripatetician doctrines.

……

The expansion of Mechanics is thus a true evolution; each stage of this evolution is the natural corollary of previous stages; it is pregnant of future stages. The meditation of this law must me the comfort of the theoretician. It would be presumptuous to imagine that the system toward which he contributes will escape the common fate of systems that were before, and will deserve to last longer than them; But, without any vain verbiage, he is right in thinking that his efforts will not be sterile, that across centuries the ideas that he sowed and made germinate, will continue to grow and bear fruits.

P. Duhem,

Corresponding member of the Institute [Academy of Sciences, Paris],

Professor of Theoretical physics at the Science Faculty in Bordeaux.