NATURE 



THURSDAY, Of fOBER 17, 1918. 



THE SAL] It; I: OF N1NETEENTH- 

 CENTl R\ SCIENi E. 

 Theory of Functions o\ ,1 Complex Variable. B3 

 Prof. A. I\. Forsyth. Third edition, l'p. xxiv-t 

 855. (Cambridge: At the University Press, 

 1918.) Prici ; os. net. (Firsl edition, 1893, 

 pp. xxii + 682; second edition, 1900, pp. xxiv + 

 782.) 

 "TO anyone interested in tin- progress of British 

 -*■ science the appearance oi .1 third edition of 

 this spacious volume, well-nigh two hundred pages 

 longer than the firsl edition, must be a very wel- 

 come event. All those who know the con- 

 spicuous services which the authot has rendered 

 to mathematical learning will wish to congratulate 



liim. \n<l those who have known the stimulus of 



nal contact with him, who can recognise, 

 beneath the happy diction with which the book 

 is everywhere written, the sympathetic teacher, 

 always able and willing to realise the learner's 

 point of view, hut eager to inform with a wealth 

 o| detail that is truly wonderful, will remember 

 and he grateful. For the book stands between 

 a time when, largely by the exigencies of a 

 certain examination, a proof was soundest if it 

 involved a considerable piece of algebra, and a 

 time when the youngest student can prove any- 

 thing by a judicious arrangemenl of arrow-heads. 

 And the multiplicity of its content, who shall 

 describe? Nor is it possible to say that it is loo 

 long it it he remarked that the index contains, for 

 example, neither the entry "aggregate" nor the 

 entry " enumei able. " 



It is easy, of course, for a reviewer, taking 

 sections o! the subject over which he happens to 

 have pondered more intently than wisely, maybe, 

 to explain how much better the book' could have 



been, 01 1, . exemplify what mathematical students 

 do not usually know the number of amendments 

 sin to the finished form ol .1 mathematical 

 •rem. And some indications may he given of 

 how the present reviewer would discharge this 

 traditional ungrateful duty if he were compelled 

 to it. Hut they may be brief, and limited to the 

 earlier parts of the book. \s regards nomen- 

 clature, there was an opportunity for rendering 

 the use of 1 he words analytic and monogenic more 

 uniform. The French use (Cauchy, Picard, 

 Goursat, etc.) differs from the lust German use. 

 Compare Weierstrass 's statement in regard to a 

 construct ("Werke," iv., p. 1 3 ; or hi., p. 101): 

 "und bezeichne dasselbe als ein m mogenes weil es 

 in seinem ganzen Umfange durch irg< nd eins seiner 

 Elemente vollstandig bestimmt ist." As regards 

 uniform convergence there is some change in the 

 present edition, the remarks on p. 92 (c/. the 



theorem, p. 150) differing from 'lie statement on 

 pp. 83, 84 ol the second edition, and still more 

 from those on |>. 127 of the lirsl edition, where the 

 phrase " fiir jedes dem Bereiche angehorige Werth- 

 system " is untranslated. The difference between 

 NO. -555. vol. 102] 



ase of a seie unctions and that of a 



series of functions ol a comj >lex ariable might use- 

 full) be remarked. In regard to the definition of 

 a function of a complex variable there is no sub- 

 stantial change; there is no refer nc e to the ques- 

 tion w hethei- the derivatives of the function need 

 I" assumed continuous; and a holomorphic func- 

 tion is both monogenic and contini And in 

 regard to the fundamental question ot tlie inte- 

 gration of a function of the complex variable there 

 still remains what is surely a very su 1 d in- 

 completeness. If it be held that the curve of inte- 

 gration need not be recti fi able, and the com 

 of the function need not be uniform, it should 

 surely be so slated. As examples of romp', 

 that may be put forward for later pages, We ~ ! 

 onl\ three. The statement on p. J4.S, that the 

 integral can be made to assume any value, is 

 incorrect. The proof on p. 243, for any three 

 periods of a doubly periodic function, does not 

 seem to carry Corollary II. without further ampli- 

 fication. The footnote on p. 344 might now be 

 supplemented by reference to Painleve, "Acta 

 Math.," xxvii. (1903), and Camb. Phil. Proc, xii. 

 ,), [). 235. 

 But all this class ol criticism seems impertinent 

 to such a corpus of learning. There are two other 

 reflections which are suggested by its perusal. 

 When one turns over its brilliant pages and in- 

 quires of the history, one has sorrowfully to con- 

 fess that not any substantial or path-breaking 

 development has been made by Britisn thinkers. 

 All this illuminating theory, so important for the 

 history of human thought, we owe to Frenchmen 

 or Germans, or others. The great names are 

 Cauchy, Abel, Riemann, Weierstrass, and Poin- 

 cai e. it is a very interesting question: Why is 

 this so? In the second place, nearly all this matter 

 is the work of the nineteenth century. When 

 the present war shall finally cease, how long will 

 it be before mankind will be able again to turn 

 from the inevitable necessity for the production 

 of commodities to putting together such another 

 bodv of clarifying thought? Well indeed is it that 

 such a summary as this volume constitutes has 

 been made, and very grateful should we be to 

 the author — for such work remains among the im- 

 perishable records of human endeavour, a real joy 

 for ever but will a similar, or even a more pro- 

 ductive, salvage be possible- in 2018? 



I TEXT-BOOK OF PLANT PHYSIOLOGY. 



Plant Physiology. B3 Prof. V. I. Palladin. 

 Authorised English edition. Edited by Prof. 

 B. hi. Livingston. Pp. xxv + 320. (Phila- 

 delphia : P. Blakiston's Son and Co., 1918.) 

 Price 3 dollars net. 



IT has been a matter for surprise to those who 

 were familiar, through the German edition, 

 wnli Prof. Palladin's text-book of plat 

 logy that it had not hitherto been available in 

 English. The German edition, which was based 

 on the- sixth Russian edition, appeared in 1911, 

 so that we have had to wait unduly long for the 



H 



