## 2-62-2. Peter Guthrie Tait: Review of Poincaré, Thermodynamique

[14.01.1892]11endnote: 1 Book review, published in Nature 45 on 14.01.1892 (Tait 1892).

Thermodynamique. Leçons professées …par H. Poincaré, Membre de l’Institut. Pp. xix, 432. (Paris : Georges Carré, 1892.)

The great expectations with which, on account of the well-won fame of its Author, we took up this book have unfortunately not been realized. The main reason is not far to seek, and requires no lengthened exposition. Its nature will be obvious from the following examples.

The late Prof. W.H. Miller, as able a mathematician as he was a trustworthy experimenter, regularly commenced his course of Crystallography at Cambridge (after seizing the chalk and drawing a diagram on the black board) with the words: “Gentlemen, let O$x$, O$y$, O$z$ be the coordinate axes.”22endnote: 2 William Hallowes Miller (1801–1880). And, some forty years ago, in a certain mathematical circle at Cambridge, men were wont to deplore the necessity of introducing words at all in a physico-mathematical text-book: the unattainable, though closely approachable, Ideal being regarded as a work devoid of aught but formulæ!

But one learns something in forty years, and accordingly the surviving members of that circle now take a very different view of the matter. They have been taught, alike by experience and by example, to regard mathematics, so far at least as physical inquiries are concerned, as a mere auxiliary to thought: of a vastly higher order of difficulty, no doubt, than ordinary numerical calculations, but still to be regarded as of essentially the same kind. This is one of the great truths which were enforced by Faraday’s splendid career.

And the consequence, in this country at least, has been that we find in the majority of the higher class of physical text-books few except the absolutely indispensable formulæ. Take, for instance, that profound yet homely and unpretentious work, Clerk-Maxwell’s Theory of Heat.33endnote: 3 Maxwell 1888. Even his great work, Electricity, though it seems to bristle with formulæ, contains but few which are altogether unnecessary.44endnote: 4 Maxwell 1891. Compare it, in this respect, with the best of more recent works on the same advanced portions of the subject.

In M. Poincaré’s work, however, all this is changed. Over and over again, in the frankest manner (see, for instance, pp. xvi, 176), he confesses that he lays himself open to the charge of introducing unnecessary mathematics: and there are many other places where, probably thinking such a confession would be too palpably superfluous, he does not feel constrained to make it. This feature of his work, at least, is sure to render it acceptable to one limited but imposing body, the Examiners for the Mathematical Tripos (Part II.).

M. Poincaré not only ranks very high indeed among pure mathematicians but has done much excellent and singularly original work in applied mathematics: all the more therefore should he be warned to bear in mind the words of Shakespeare55endnote: 5 Isabella to Angelo, in Measure for Measure, act 2, scene 2.

“Oh ! it is excellent
To have a giant’s strength; but it is tyrannous
To use it like a giant.”

From the physical point of view, however, there is much more than this to be said. For mathematical analysis, like arithmetic, should never be appealed to in a physical inquiry till unaided thought has done its utmost. Then, and not till then, is the investigator in a position rightly to embody his difficulty in the language of symbols, with a clear understanding of what is demanded from their potent assistance. The violation of this rule is very frequent in M. Poincaré’s work and is one main cause of its quite unnecessary bulk. Solutions of important problems, which are avowedly imperfect because based on untenable hypotheses (see, for instance, §§ 284–286), are not useful to a student, even as a warning: they are much more likely to create confusion, especially when a complete solution, based upon full experimental data and careful thought, can be immediately afterwards placed before him. If something is really desired, in addition to the complete solution of any problem, the proper course is to prefix to the complete treatment one or more exact solutions of simple cases. This course is almost certain to be useful to the student. The whole of M. Poincaré’s work savours of the consciousness of mathematical power: and exhibits a lavish, almost a reckless, use of it. Todhunter’s favourite phrase, when one of his pupils happened to use processes more formidable than the subject required, was “’Hm: — breaking a fly on the wheel!” He would have had frequent occasion to use it during a perusal of this volume. An excellent instance of the dangerous results of this lavish display of mathematical skill occurs at pp. 137–38, the greater part of which (as printed) consists of a mass of error of which no one, certainly, would accuse M. Poincaré. The cause must therefore be traced to the unnecessary display of dexterity with which, after obtaining the equation

 $Q_{2}/Q_{1}=1-Af(T_{1},T_{2}),$

where the order of the suffixes is evidently of paramount importance, M. Poincaré proceeded to say “Nous pouvons donc écrire

 $Q_{1}/Q_{2}=\Phi(T_{1},T_{2})\text{.''}$

But his unfortunate printer, not prepared for such a tour de force, very naturally repeated the $Q_{2}/Q_{1}$ of the first equation, with the result of wholly falsifying all that follows. On the other hand, we must fully recognize that, when more formidable analysis is really required (as, for instance, in the treatment of v. Helmholtz’s monocyclic and polycyclic systems), M. Poincaré seems to feel so thoroughly at home as to criticize with freedom.

One test of the soundness of an author, writing on Thermodynamics, is his treatment of temperature, and his introduction of absolute temperature. M. Poincaré gets over this part of his work very expeditiously. In §§ 15–17 temperature, $t$, is conventionally defined as in the Centigrade thermometer by means of the volume of a given quantity of mercury; or by any continous function of that volume which increases along with it. Next (§ 22) absolute temperature, $T$, is defined, provisionally and with a caution, as $273+t$; from the (so-called) laws of Mariotte and Gay-Lussac. Then, finally (§ 118), absolute temperature is virtually defined afresh as the reciprocal of Carnot’s function. [We say virtually, as we use the term in the sense defined by Thomson. M. Poincaré’s fonction de Carnot is a different thing.] But there seems to be no hint given as to the results of experiments made expressly to compare these two definitions. Nothing, for instance, in this connection at all events, is said about the long-continued early experimental work of Joule and Thomson, which justified them in basing the measurement of absolute temperature on Carnot’s function.

In saying this, however, we must most explicitly disclaim any intention of charging M. Poincaré with even a trace of that sometimes merely invidious, sometimes purely Chauvinistic, spirit which has done so much to embitter discussions of the history of the subject. On the contrary, we consider that he gives far too little prominence to the really extraordinary merits of his own countryman Sadi Carnot. He writes not as a partisan but rather as one to whom the history of the subject is a matter of all but complete indifference.66endnote: 6 Maxwell, in the first part of his review in Nature of the second edition of Tait’s Sketch of Thermodynamics (Tait 1877, Maxwell 1878a), approved of Tait’s account of the history of thermodynamics, not for its accuracy, but for its capacity to “rouse the placid reader, and startle his thinking powers into action.” So far, in fact, does he carry this that the name of Mayer, which frequently occurs, seems to be spelled incorrectly on by far the greater number of these occasions!77endnote: 7 Julius Robert von Mayer (1814–1878). He makes, however, one very striking historical statement (§ 95): “Clausius …lui donna le nom de Principe de Carnot, bien qu’il l’eût énoncé sans avoir connaissance des travaux de Sadi Carnot.”

Still, one naturally expects to find, in a Treatise such as this, some little allusion at least to Thermodynamic Motivity; to its waste, the Dissipation of Energy; and to the rest of those important early results of Sir W. Thomson, which have had such immense influence on the development of the subject. We look in vain for any mention of Rankine or of his Thermodynamic Function; though we have enough, and to spare, of it under its later alias of Entropy.88endnote: 8 Brush notes that several writers, including J. T. Merz, J. Swinburne, and V. V Raman, consider Rankine’s formulation of “thermodynamic function” to be equivalent to Clausius’ entropy, as Rankine claimed it to be in 1865, although the physical equivalence of the two concepts was unclear circa 1854; see Brush (1974, 31). The word dissipation does indeed occur, for we are told in the Introduction that the Principe de Carnot is “la dissipation de l’entropie.”99endnote: 9 Poincaré 1892, 1.

We find Bunsen and Mousson cited, with regard to the effect of pressure upon melting points, almost before a word is said of James Thomson; and, when that word does come, it wholly fails to exhibit the real nature or value of the great advance he made.1010endnote: 10 Robert Bunsen, Albert Mousson (1805–1890).

Andrews again, à propos of the critical point, and his splendid work on the isothermals of carbonic acid, comes in for the barest mention only after a long discussion of those very curves, and of the equations suggested for them by Van der Waals, Clausius, and Sarrau: though his work was the acknowledged origin of their attempts.1111endnote: 11 Thomas Andrews (1813–1885), J.D. van der Waals, Rudolf Clausius, Émile Sarrau.

The reason for all this is, as before hinted, that M. Poincaré has, in this work, chosen to play almost exclusively the part of the pure technical analyst; instead of that of the profound thinker, though he is perfectly competent to do that also when he pleases. And, in his assumed capacity, he quite naturally looks with indifference, if not with absolute contempt, on the work of the lowly experimenter. Yet, in strange contradiction to this, and still more in contradiction to his ascription of the Conservation of Energy to Mayer, he says of that principle: “personne n’ignore que c’est un fait expérimental.”

Even the elaborate thermo-electric experiments of Sir W. Thomson, Magnus, &c., are altogether ignored. What else can we gather from passages like the following?

(§ 287) “Sir W. Thomson admet qu’il existe une force électromotrice au contact de deux portions d’un même conducteur à des températures différentes; il assimile donc ces deux portions à deux conducteurs de nature différente, assimilation qui paraît très vraisemblable.”
(§ 291) “…si l’effet Thomson a pu être mis en évidence par l’expérience, on n’a pu jusqu’ici constater l’existence des forces électromotrices qui lui donnent naissance.”

Everyone who comes to this work of M. Poincaré fresh from the study of Clerk-Maxwell’s little treatise (or of the early papers of Thomson, to which it owed much) will feel as if transferred to a totally new world. Let him look, for instance, at Maxwell’s treatment of the Thermodynamic Relations, Intrinsic and Available Energy, &c., and then turn to pp. 148–150 of M. Poincaré’s work. There he will find at least a large portion of these most important matters embodied in what it seems we are now to call the Fonctions caractéristiques de M. Massieu!1212endnote: 12 The functions to which Tait objects are those François Jacques Dominique Massieu (1832–1896) presented in two notes to the Paris Academy of Science on the “characteristic functions of diverse fluids” (Massieu 1869b, 1869a). In the entropy representation of thermodynamics, a Massieu function (also known as a “Massieu-Gibbs”, “Gibbs”, or “characteristic” function) is defined by a Legendre transformation in which one extensive variable (or several) is replaced by the conjugate intensive variable. For Poincaré (1892, 150), a “great simplification” may be realized in certain cases, as the equations of thermodynamics may be expressed in terms of Massieu functions and their derivatives alone.

But the most unsatisfactory part of the whole work is, it seems to us, the entire ignoration of the true (i.e. the statistical) basis of the second Law of Thermodynamics. According to Clerk-Maxwell (Nature, xvii. 278)1313endnote: 13 Maxwell 1878b, published 2 July, 1878.

“The touch-stone of a treatise on Thermodynamics is what is called the second law.”

We need not quote the very clear statement which follows this, as it is probably accessible to all our readers.1414endnote: 14 Maxwell argued for a statistical interpretation of the second law, as follows: “The second law must either be founded on our actual experience in dealing with real bodies of sensible magnitude, or else deduced from the molecular theory of these bodies, on the hypothesis that the behaviour of bodies consisting of millions of molecules may be deduced from the theory of the encounters of pairs of molecules, by supposing the relative frequency of different kinds of encounters to be distributed according ot the laws of probability.
The truth of the second law is therefore a statistical, not a mathematical, truth, for it depends on the fact that the bodies we deal with consist of millions of molecules, and that we never can get hold of single molecules.” (Maxwell 1878b, 279)
It certainly has not much resemblance to what will be found on the point in M. Poincaré’s work: so little, indeed, that if we were to judge by these two writings alone it would appear that, with the exception of the portion treated in the recent investigations of v. Helmholtz, the science had been retrograding, certainly not advancing, for the last twenty years.1515endnote: 15 Poincaré did not agree with Tait’s assessment of his work; see his letter to the editor of Nature, published on 03.03.1892 (§ 2-55-1).

P. G. T.

PD. Tait 1892.

Time-stamp: " 1.09.2019 20:33"

### Notes

• 1 Book review, published in Nature 45 on 14.01.1892 (Tait 1892).
• 2 William Hallowes Miller (1801–1880).
• 3 Maxwell 1888.
• 4 Maxwell 1891.
• 5 Isabella to Angelo, in Measure for Measure, act 2, scene 2.
• 6 Maxwell, in the first part of his review in Nature of the second edition of Tait’s Sketch of Thermodynamics (Tait 1877, Maxwell 1878a), approved of Tait’s account of the history of thermodynamics, not for its accuracy, but for its capacity to “rouse the placid reader, and startle his thinking powers into action.”
• 7 Julius Robert von Mayer (1814–1878).
• 8 Brush notes that several writers, including J. T. Merz, J. Swinburne, and V. V Raman, consider Rankine’s formulation of “thermodynamic function” to be equivalent to Clausius’ entropy, as Rankine claimed it to be in 1865, although the physical equivalence of the two concepts was unclear circa 1854; see Brush (1974, 31).
• 9 Poincaré 1892, 1.
• 10 Robert Bunsen, Albert Mousson (1805–1890).
• 11 Thomas Andrews (1813–1885), J.D. van der Waals, Rudolf Clausius, Émile Sarrau.
• 12 The functions to which Tait objects are those François Jacques Dominique Massieu (1832–1896) presented in two notes to the Paris Academy of Science on the “characteristic functions of diverse fluids” (Massieu 1869b, 1869a). In the entropy representation of thermodynamics, a Massieu function (also known as a “Massieu-Gibbs”, “Gibbs”, or “characteristic” function) is defined by a Legendre transformation in which one extensive variable (or several) is replaced by the conjugate intensive variable. For Poincaré (1892, 150), a “great simplification” may be realized in certain cases, as the equations of thermodynamics may be expressed in terms of Massieu functions and their derivatives alone.
• 13 Maxwell 1878b, published 2 July, 1878.
• 14 Maxwell argued for a statistical interpretation of the second law, as follows: “The second law must either be founded on our actual experience in dealing with real bodies of sensible magnitude, or else deduced from the molecular theory of these bodies, on the hypothesis that the behaviour of bodies consisting of millions of molecules may be deduced from the theory of the encounters of pairs of molecules, by supposing the relative frequency of different kinds of encounters to be distributed according ot the laws of probability. The truth of the second law is therefore a statistical, not a mathematical, truth, for it depends on the fact that the bodies we deal with consist of millions of molecules, and that we never can get hold of single molecules.” (Maxwell 1878b, 279)
• 15 Poincaré did not agree with Tait’s assessment of his work; see his letter to the editor of Nature, published on 03.03.1892 (§ 2-55-1).

## References

• S. G. Brush (1974) The development of the kinetic theory of gases, VIII: randomness and irreversibility. Archive for History of Exact Sciences 12 (1), pp. 1–88. Cited by: endnote 8.
• F. Massieu (1869a) Addition au précédent Mémoire sur les fonctions caractéristiques. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 69 (21), pp. 1057–1061. Cited by: endnote 12.
• F. Massieu (1869b) Sur les fonctions caractéristiques des divers fluides. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 69 (16), pp. 858–862. Cited by: endnote 12.
• J. C. Maxwell (1878a) Tait’s ‘Thermodynamics’ I. Nature 17 (431), pp. 257–259. Cited by: endnote 6.
• J. C. Maxwell (1878b) Tait’s ‘Thermodynamics’ II. Nature 17 (432), pp. 278–280. Cited by: endnote 13, endnote 14.
• J. C. Maxwell (1888) Theory of Heat. Longmans, Green and Co, London. Cited by: endnote 3.
• J. C. Maxwell (1891) A Treatise on Electricity and Magnetism. Oxford University Press, Oxford. Cited by: endnote 4.
• H. Poincaré (1892) Thermodynamique. Georges Carré, Paris. Cited by: endnote 12, endnote 9.
• P. G. Tait (1877) Sketch of Thermodynamics. David Douglas, Edinburgh. Cited by: endnote 6.
• P. G. Tait (1892) Poincaré’s Thermodynamics. Nature 45, pp. 245–246.