274 PETER GUTHRIE TAIT 



M. Poincare' 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 Shakespeare : 



"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 

 enquiry 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. Poincare'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. Poincare's work savours of the 

 consciousness of mathematical power : and exhibits a lavish, almost a reckless, use of it. 



One test of the soundness of an author, writing on Thermodynamics, is his 

 treatment of temperature, and his introduction of absolute temperature. M. Poincare 

 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 continuous 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 Marriotte 

 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. Poincare"'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. Poincare" 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. 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 



