58o 



NATURE 



[February 29, 1912 



iOfff 



(8) Die Prinzipien der Mechanik fUr eine oder 

 mehrcre von den rdumlichcn Koordinaten und der 

 Zeit abhdngige Variabeln, II. By Leo Konigs- 

 berper. Pp. 24. (Heidelberg: Carl Winter, 191 1.) 



(9) Theoretische Mechanik. By Prof. R. Marcolongo. 

 Autorisierte deutsche Bearbeitung. By Prof. H. E. 

 Timerdingf. Erster Band, Kinematik und Statik. 

 Pp. viii + 346. (Leipzig and Berlin : B. G. Teubner, 

 191 1.) Price 10 marks. 



(10) Sur la notion de Coiirbure, et sur quclques points 

 de Geomitrie infinitdsimale non euclidienne. By 

 C. Cailler (M^moires de la Soci6t6 physique et 

 d'Histoire naturelle de Geneve xxxvii, 2.) Pp. 62. 

 (Geneve: Georg et Cie., 191 1.) Price 5 francs. 



(11) Proceedings of the London Mathematical Society. 

 Second series. Vol. ix. Pp. xvi + 48g. (London : 

 Francis Hodgson, 191 1.) 



(12) Bulletin of the Calcutta Mathematical Society. 

 Vol. i.. No. 3 (October, 1909). Pp. 70. (Calcutta : 

 Mathematical Society, Senate House, Calcutta, 

 191 1.) Price :o rupees per year. 



(13) Rcvista de la Sociedad matemdtica espaiiola, 1-5 

 (Mayo-Diciembre, 191 1). Pp. 40-76. (Madrid: 

 Dr. Jos6 Nungot, Universidad Central, 191 1.) 



ONE of the most important facts which modern 

 mathematicians now realise is that the prin- 

 ciples of the differential and integral calculus can be 

 taught to beginners by simple methods involving only 

 a knowledge of the rudiments of algebra, and later 

 on of trigonometry. There are not many teachers 

 still in the dark on these points, though a remarkable 

 exception occurred recently w^hen an anonymous 

 author wrote a book intended to show that certain 

 fools called mathematicians had made the calculus 

 unnecessarily hard, and proved his point, not in the 

 way he probably contemplated, but by establishing the 

 fundamental formulae with a wasteful luxuriance of 

 infinite series and disregard of small quantities which 

 would have formed a more fitting subject for a book 

 entitled "Calculus Made Difficult." The change has 

 been marked in England by the appearance of a flood 

 of school calculuses, reminding one of the former flood 

 of school geometries. 



(i) In these circumstances English teachers will 

 derive considerable interest from studying the elemen- 

 tary but rigorous treatment of the subject in Dr. 

 Scheffer's " Lehrbuch der Mathematik." The first 

 chapter introduces the arc or radian measure of 

 angles, and deals with functions in general. It opens 

 with a careful comparison of the uses of graphic 

 and analytical methods of solution, which should be 

 a lesson to those teachers who try to hide up the 

 shortcomings of constructive geometry by making 

 fheir pupils use hard pencils. The notion of a differ- 

 ential coefficient is introduced in the first instance by 

 consideration of linear and quadratic functions. Then 

 follow the formulae for algebraic functions deduced 

 from the sum and product rules, and the rule for 

 functions of functions, which is here enunciated as 

 the chain rule. Differentiation of a power is deduced 

 from the product rule by mathematical induction, as 

 it should be, the use of the binomial theorem being 

 avoided. The elements of the integral calculus are 

 NO. 2209, VOL. 88] 



treated next, and it is only after this that the noti< 

 of a natural logarithm is shown to follow directly 

 from the calculus, exponential functions being taken; 

 in the succeeding chapter, and trigonometric functioni 

 coming next. The chapter dealing with these contains 

 a synopsis of trigonometry. For English readers the 

 hyperbolic notation (" fin, co8," printed in Germanr 

 type) is perhaps inconvenient. Then follow succes-; 

 sive differentiation, maxima and minima, curvature: 

 and evolutes, particle dynamics, Lagrange's and Tay— 

 lor's formulae, miscellaneous methods of integration, 

 Fourier's series, and partial differentiation, .\t the 

 end is a useful collection of tables and integration 

 formulae. A large number of applications are given* 

 in the form of examples, some completely worked out, 

 others left to the reader. 



It need scarcely be pointed out that no two writers 

 would agree as to what should be included in a book' 

 of this kind, and what should be omitted. On the 

 whole, this book tends on the side of thoroughness, 

 rigorous development, and careful discussion of points; 

 of detail, notably in dealing with such matters as con- 

 tinuity. Possibly there may be few students of physics 

 in this country who do not have to skip over and take 

 for granted some of the arguments. But in such 

 matters as order of treatment and rendering the sub- 

 ject independent of an extensive previous knowledge 

 of algebra and geometry, the book pretty nearly' 

 reaches the goal towards which modern teachers have 

 been striving. 



The influence of pure mathematics on the progress 

 of mathematical physics is well shown by the next ' 

 group of books under review. The partial differential 

 equations of physical problems have received so much 

 attention at the hands of both mathematicians and 

 physicists that we had begun to think that their study 

 had reached a stage of finality in which nothinj:: 

 further of importance remained to be done. But 

 during the last ten years harmonic analysis has bf • 

 completely revolutionised by the development of i 

 theory of integral equations, which places a new aiul 

 powerful' weapon in the hands of the applied mathe- 

 matician. This modern theory traces its origin back 

 to Fredholm's paper of 1900 on Dirichlet's problem. 

 and the most important subsequent works are thi 

 due to Stekloff, Hilbert, and Schmidt; in particul... 

 Hubert's " Foundations of a General Theor}' of Linear 

 Integral Equations." 



(2) Dr. Kneser's contribution to the new subj- 

 is exactly described by the following notice of the , 

 publishers : — • 



"The present work develops the theory of integral 

 equations not from the starting point of analytical , 

 generalities but from the theory of heat-conduction, i 

 of free and forced oscillations and of the potential. 

 The author thus hopes to meet the requirements 

 those mathematicians and physicists who wish 

 apply the new analytical method to concrete qu- — 

 tions." 



Now, in endeavouring to get the whole hang of the | 

 problem, so to speak, condensed in a nutshell, or, in ' 

 other words, to find out what the investigations ar.^ 

 driving at, and to put the matter in a form in which 

 it could be explained to a pupil in ten minutes, the 



