January 14, 1892] 



NATURE 



245 



now be obtained at a sufficiently moderate cost if a 

 demand for them should arise. These hydrocarbons, 

 whilst indifferent to the action of atmospheric oxygen, 

 possess greater solvent power than any other, and on 

 this account they are also well adapted for the prepara- 

 tion of varnishes. But for this latter purpose a still more 

 suitable vehicle will be found in the amylic acetate, which 

 dissolves even the hardest copals almost entirely after 

 having been previously finely powdered and kept for 

 some time in a hot closet. In this way excellent var- 

 nishes .Tiay be produced. 



In reviewing a work dealing with so vast a number of 

 subjects, it is obviously impracticable to do more than to 

 refer to some i^vi of them ; but we think sufficient has 

 been said in the foregoing notice to show the valuable 

 character of the work before us, and although it will be 

 freely admitted that there are still a good many points 

 which demand further inquiry and elucidation, those who 

 are interested in the subject may be congratulated on 

 now possessing in an accessible form all that chemistry 

 has for the present to say on paints and painting. 



Hugo Muller. 



POINC ARE'S THERMODYNAMICS. 

 Thermodynamtque. Legons professdes . . . par H. 

 Poincard, Membre de I'lnstitut, Pp. xix., 432. (Paris : 

 Georges Carrd, 1 892.) 



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 com- 

 menced his course of Crystallography at Cambridge (after 

 seizing the chalk and drawing a diagram on the black 

 board) with the words : — "Gentlemen, let Ox,Oy, Oz be 

 the coordinate axes." 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 formulae ! 



But one learns something in forty years, and accord- 

 ingly 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 physi- 

 cal text-books few except the absolutely indispensable 

 formuUe. Take, for instance, that profound yet homely 

 and unpretentious work, Clerk-Maxwell's Theory of Heat. 

 Even his great work. Electricity, though it seems to bristle 

 with formulas, contains but few which are altogether un- 

 necessary. Compare it, in this respect, with the best of 

 NO. I 159. VOL. 45] 



more recent works on the same advanced portions of the 

 subject. 



In M. Poincarc'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 mathe- 

 matics : — and there are many other places where, probably 

 thinking such a confession would be too palpably super- 

 fluous, he does not feel constrained to make it. This 

 feature of his work, at least, is sure to render it accept- 

 able to one limited but imposing body, the Examiners 

 for the Mathematical Tripos {Part II.). 



M. Poincard 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 

 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. Poincard'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. 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 

 Poincard. The cause must therefore be traced to the 

 unnecessary display of dexterity with which, after obtain- 

 ing the equation 



Q,/Q, = I - A/(T„ T,), 



where the order of the suffixes is evidently of paramount 

 importance, M. Poincard proceeded to say " Nous 

 pouvons done dcrire 



But his unfortunate printer, not prepared for such a tour 

 de force, very naturally repeated the Qj/Qi of the first 



