172 



NATURE 



[April 6, 191 1 



the elegance which we expect from him ; but even he 

 can scarcely find room for any facts which are not 

 familiar to every well-informed person, or any con- 

 clusions which are not the commonplaces of every 

 journalist. 



While nothinjij but praise can be given to the authors 

 f(jr the performance of their tasks, the decision of the 

 editors that all the modern historian requires to know 

 of the science of the last century can be contained in 

 two short chapters seems at first to challenge criticism. 

 But, so far as pure science is concerned, we believe 

 that they are rij^ht. Pure science is the most esoteric 

 of all studies ; the power of appreciating the value of 

 the ideas contained in the most fundamental scientific 

 theories appears to be totally uncorrelated with any 

 other form of mental ability. It is just because the 

 scientific instinct is such a rare and peculiar gift that 

 it is so intensely valuable. Even if it were possible 

 for the mass of mankind to know truly the meaning 

 of science, it is very doubtful whether it would have 

 any effect upon those actions which history studies. 

 Science cannot define a worthy aim for action ; at 

 most it can show what aims can be attained, not what 

 ought to be attained. It vvas, of course, thought very 

 widely forty years ago that what was "natural" was 

 good, but the fallacy is quite exploded to-day. Indeed, 

 it is probable that Spencerian sociology, based on the 

 confusion of the biological and ethical meanings of 

 the word " fit," would have never received any serious 

 attention, had not the doctrine that what has survived 

 ought to have survived been so comfortable to those in 

 authority. All that the doctrine of '^volution can teach 

 us in the matter of aims is that man is master of his 

 destiny, that it is neither sufllcient nor necessary to 

 wait for the dispensations of a mysterious providence, 

 but neither the science of the last or of the next 

 century will decide the eternally disputed question of 

 what that destiny ought to be. 



However, though pure science cannot give us ends, 

 applied science can give us means, and in respect of 

 applied science "The Cambridge Modern History" 

 appears to us defective. There is no connected 

 account whatsoever of the great inventions or the 

 progress of engineering during "The Latest Age." 

 Mr. Whetham, concerned with pure science, naturally 

 only mentions discoveries which have been the by- 

 products of pure research. We have bare references 

 to photography, dynamos, telegraphy, and two pages 

 concerning medicine, surgery, and hygiene. But 

 many of the inventions which have had the greatest 

 •economic or historical effects have no immediate con- 

 nection with pure science. Can the historian of to- 

 morrow analyse the events of to-day if he has never 

 heard of such things as telephones, explosion engines, 

 •modern armaments, water-power, or electric lighting? 

 The most trivial invention in appearance may revolu- 

 tionise the world. Mr. Whetham rightly says that 

 "the locomotive engine and the electric telegraph 

 effected the great industrial and social revolution of 

 the middle of the nineteenth century " ; we suggest 

 that it would not be ridiculous to claim for an inven- 

 tion so uninteresting technically and scientifically as 

 the bicycle a comparable influence upon the end of it. 

 Besides its economic effect in increasing men's radius 

 NO. 2162, VOL. 86] 



of action, its social effect in furthering the ind^tend- 

 ence of women must surely make it worthy oi the 

 attention of a student of the modern state; and yet 

 the editors of "The Cambridge .Modern History" 

 have no official knowledii*- of it. 



THE THEORY <'/ FUNCTIONS. 

 Introduction d la thioric dcs functions d'une variable. 

 By J. Tannery. Deuxii-me ddition ; tome 2. 

 Integrales ddfinies, Ddveloppements en .S<Srie, Lan- 

 gage g^om^trique, Fonctions de Variables imagin- 

 aires. Avec une Note de M. Hadamard. Pp. iv + 

 480. (Paris: A. Hermann et fils, igifv^ I'-i'f 15 

 francs. 



THE most interesting chapter in this volume is that 

 which is entitled " Langage g^m^trique." 

 especially from a pedagogic point of view. The main 

 object of the treatise is to deduce everything from 

 purely arithmetical assumptions; but as a practical 

 teacher, Prof. Tannery was well aware of the value of 

 diagrams as an aid to the imagination, or, as he puts 

 it, for purposes of orientation. Consequently he has 

 given a series of quasi-geometrical definitions, by 

 means of which the ordinary formulae and methods of 

 analytical geometry are valid, and may be used prac- 

 tically for constructing diagrams to define boundaries 

 of aggregates, &c. In the ordinary sense, of course, 

 we thus get a locus corresponding to an equation 

 ^(A:,y) = o; but in order to emphasise the fact that 

 only arithmetical conditions are really imposed, the 

 author replaces the term " locus " (lieu) by " bond " 

 (lien), and practically confines this to the case where 

 we may put x = ^(0. .v = V' (0. *. V' being definite 

 functions for a certain range of the real continuous 

 variable t. The principal results of the chapter are 

 the proof of the existence of simple contours in a 

 plane, which separate it into two distinct continua 

 (this is given after Mr. Ames), and the further conclu- 

 sion that a domain which is (m+i) times connex can 

 be reduced to two simply connex domains by drawing 

 (m+i) simple curves. 



Another notion that occurs in this chapter is that 

 of the order of a point .\ with regard to a closed con- 

 tour C. If a point P traverses C once in the positive 

 direction, the variation of the amplitude of the vector 

 AP is of the form 2krr, where k is scMiie integer or 

 zero ; and k is called the order of A with respect to C. 

 This very important idea was generalised by 

 Kronecker, and the present volume concludes with an 

 important note by M. Hadamard (pp. 437-77) o" some 

 applications of Kronecker 's index. The main point of 

 this theor>^ is that the property of a Jacobian, that it 

 changes sign when two rows are interchanged, is 

 brought into connection with topology- (or analysis 

 situs) in a verj' general sense. Every advance in the 

 analytical treatment of this subject is noteworthy; 

 because it is here that the contrast between geo- 

 metrical intuition and analytical proof is so often a 

 glaring one. For example, take Minding's surface, 

 which is obtained by taking a strip of paper, giving 

 it a half-twist, and then pasting the free ends to- 

 gether. It is easy to see that this surface has only 



