3H 



NA TURE 



[August 3, 1905 



the surface itself is concerned is admitted, but whether 

 the Riemann space is identical with, or only analogous 

 to, .spherical space in a hyperspace of four dimensions 

 retndins a subject of controversy between the author 

 of ■ the book and M. Mansion. At any rate, M. 

 Lefhalas does not discuss space of positive curvature 

 independently of its connection with four-dimensional 

 Euclidean space, and accordingly the book contains 

 only one more chapter devoted to the geometry of 

 Lobatchefsky and Bolyai. In this respect the treat- 

 ment is analogous to that given in some books on 

 conies where the properties of the ellipse are proved 

 by three-dimensional methods (orthogonal projection) 

 and those of the hyperbola by plane geometry. 

 Whether this is the best plan is open to question ; 

 many mathematicians seem to prefer it, and an author 

 cannot please everybody. 



In his preface, which is printed in italics, M. 

 Tannery fairly well defines the scope and object of 

 his book. Although this is a second edition, it has 

 been entirely re-written. It is primarily intended for 

 readers who do not possess a very extended know- 

 ledge of mathematics. It covers mainly those por- 

 tions of analysis which are commonly found in 

 English te.xt-books on higher algebra, viz. properties 

 of irrational numbers, continued fractions, aggre- 

 gates, convergency and divergency of series and of 

 infinite ■ products, the binomial theorem, the ex- 

 ponential 'and logarithmic series, and expansions of 

 trigonometric functions treated without the aid of 

 imaginaries. Finally, we have a chapter oh derived 

 functions containing applications of the formula 



l{x + h)~f{x) = hf(x + eh), 



and an illustration of functions which have no 

 differential coefficient. The subject-matter may all be 

 included under the heading " functions of real vari- 

 ables treated algebraically," as M. Tannery has 

 avoided the use of geometrical methods in the present 

 volume. .A second volume is promised dealing with 

 functions of complex variables, in which geometrical 

 methods are to be freely used. 



The treatment is clear and full, and the book gives 

 the inipression of being as good an exposition of the 

 subject as could well be written on the lines laid 

 down by the author. It does not profess to give 

 historical' or 'bibliographical information, for which 

 the reader is referred to the " Mathematical Encyclo- 

 paedia," of which the French edition is now coming 

 out. 



.\n interesting -insight into the thoughts of two 

 eminent mathematicians is afforded by the first 

 volume of correspondence between Hermite and 

 Stieltjes, covering wie' period 1882-1889. The in- 

 timacy seems -to- have arisen in 1882, out of a letfer 

 addressed by Stieltjes to Hermite dealing with a 

 theorem of M.- Tisserand relating to the expansion 

 of the disturbing force when the mutual inclination 

 of two orbits is considerable. ■ The subject-matter of 

 this letter' (wfhichi's missing from the collection) was 

 published in the Comptes rendus for November 13, 

 1882. 



At this time Thonias Jean Stieltjes was attached 

 NO. 1866, VOL, 72] 



to the Observatory of Leyden, and the influence of 

 Hermite doubtless accounts in large measure for his 

 activity in mathematical research during the years 

 which followed, culminating in his migration to 

 France in 1S85, after his failure to obtain a mathe- 

 matical chair in his own country. 



A noteworthy feature of. Stieltjes's . work is his 

 partiality for simple arithmetical tests of general 

 theorems. The value of his examinations of 

 numerical details must have been enormous to a man 

 of Hermite 's calibre. It seems as if Hermite in 

 many cases furnished the ideas which Stieltjes 

 elaborated and extended. It was not with Stieltjes 

 alone that Hermite carried on an extensive corre- 

 spondence, for he was evidently fond of writing 

 letters, and even many of his contributions to journals 

 appeared in epistolary form. But among his various 

 correspondents Stieltjes played a prominent part, and 

 it was Hermite's own wish that the letters of his 

 colleague should be published after the premature 

 death of the latter in 1894. One thing is unfortu- 

 nately wanting. Hermite was to have written an 

 introduction, but he did not live to do so. In its place 

 we have a preface by M. Picard and a biographical 

 notice by M. H. Bourget, who, in conjunction with 

 M. Baillaud, were colleagues of Stieltjes in the Uni- 

 versity of Toulouse from 1886 until his death, and 

 who have jointly edited the present volume. 



It would be difficult to give a general summary of 

 the subject-matter of this correspondence, which deals 

 with continued , fractions, hypergeometric series, 

 Legendre's functions, semi-convergent series, and, 

 indeed, analysis generally. Portraits of Hermite and 

 Stieltjes complete the volume. There is a certain 

 brightness and freshness about the way one of the 

 two mathematicians writes to the other announcing 

 some new result and the second takes up the clue 

 and develops it, and one can imagine the delight that 

 [the two kindred spirits must have had in working 

 (together. 



, While the volumes before us are widely different in 

 icharacter, it may be well to warn the busy reader, as 

 has been done on previous occasions, that they all 

 possess one objectionable feature in common. While 

 jthe guillotine was originally invented in France, the 

 modern instrument of that name has not been applied 

 to its proper use on the pages of any one of the series, 

 consequently readers, unless they are prepared to set 

 lup a private guillotine, are compelled to waste hours 

 in hacking and jagging the leaves with a paper knife, 

 [producing a very untidy result. 

 I , G. H. B. 



\THE : MUTATION-. THEORY OF THE ORIGIN 



r : ■■ ; .QF SF.ECIES. 



Species q,nd ■V^arj.^tiesj.-iikeir Origin by Mutation. By 



Hugo. ,de Vries, . Edited, by D. T. MacDougal. 



Pp. .x.vji.i+-8.i).7.,, .(London.: Kegan Paul and Co., 

 .Ltd., 1905.), .. • •■ 



AT .th'e. .pr.esent- itime., w-hen naturalists are be- 

 , _ . gfinning^-to turn , again -,10 the problem of the 

 origin of species, .(this account ofj Prof, de Vries's 

 theories and experiments is sure- of- a welcome, partly 



