NATURE 



3^3 



THURSDAY, AUGUST 3, 1905. 



RECENT FRENCH MATHEMATICAL WORKS. 

 La Philosophie naturelle int^grale et les Rudiments 



des Sciences exactes. By Dr. A. Rist. Part i. 



Pp. vi+132. (Paris: A. Hermann, 1904.) Price 



3.50 francs. 

 Etude sur le Developpemcnl des Mi}thodes geo- 



m^tfiques. By Gaston Darboux. Pp. 34. (Paris : 



Gauthier-\'illars, 1904.) Price 1.50 francs. 

 Sur le Diveloppement de I'Analyse et ses Rapports 



avec diverses Sciences. By Emile Picard. Pp. 16S. 



(Paris : Gauthier-Villars, 1905.) Price 3.50 francs. 

 Introduction a la Geomitrie gdnerale. By Georges 



Lechalas. Pp. ix + 65. (Paris: Gauthier-Villars, 



1404.) Price 1.50 francs. 

 Introduction a la Theoric des Fonctions d'une 



Variable. By Jules Tannery. Vol. i. Second 



edition. Pp. ix + 422. (Paris: A. Hermann, 1904.) 

 Correspondance d'Hermite et de Stieltjes. Edited by 



B. Baillaud and H. Bourget. Vol. i. Pp. xxi + 



477. (Paris : Gauthier-Villars, 1905.) Price ifa 



francs. 



THE part which France has played in the de- 

 velopment of modern mathematical methods, 

 especially in connection with geometry and analysis, 

 is well known to every mathematician. Of recent 

 years, however, the trend of mathematical thought 

 has considerably changed in every country, and while 

 France has produced a large school of writers on the 

 philosophy of mathematics, it is in the opinion of the 

 present reviewer doubtful whether this school can 

 forge more than a very small link in the chain of 

 mathematical development. The doubts which arose 

 in the minds of mathematicians regarding Euclid's 

 eleventh axiom led to the new science of non-Euclidean 

 geometry, but it was not so much the mere philo- 

 sophical speculations concerning the axiom itself as 

 the examination of the consequences of making 

 alternative assumptions that led to substantial pro- 

 gress being made. The discovery that we cannot be 

 sure that two and two make four except as the result 

 of experience is undoubtedly of importance, but it is 

 in the development of the consequences of a inore 

 extended hypothesis, of which this one is or is not a 

 particular case, that substantial progress must be 

 sought. 



Dr. Rist's book may be taken as affording a good 

 example of the kind of philosophical speculations 

 which arise when we try to analyse the why and 

 wherefore of the various processes and operations 

 occurring in even so elementary a subject as arith- 

 metic. It contains chapters on the prolegomena of 

 both geometry and arithmetic, but it is in connec- 

 tion with the latter subject that the discussion is 

 most extended. The mere act of counting forms the 

 subject of a number of paragraphs of which the 

 general character may be fairly understood from an 

 enunciation of the headings : — " The number con- 

 sidered as the result of an act," "What do we 

 count?" "Why do we count?" "The different 

 modes of counting." F"rom counting the author 

 NO. 1866, VOL. 72] 



proceeds to calculation, and in the following chapter 

 gives a detailed discussion of the various processes 

 and svmbols involved in the two operations of addition 

 and subtraction. One would naturally expect multi- 

 plication and division to be treated in the same way, 

 but instead. Dr. Rist sets out an alternative method 

 of approaching this study, and this first volume closes 

 with a chapter showing how numbers serve for 

 evaluations. 



The book seems to appeal more particularly to 

 elementarv teachers who only possess a rudimentary 

 training in algebra and geometry, for there is little 

 or nothing in it which assumes more than an elemen- 

 tary knowledge of these subjects. The highly trained 

 mathematician would hardly benefit by reading such 

 a book, as he would probably have already formed 

 ideas of his own on the subject, and in all likelihood 

 would consider the treatment to be unsatisfactory in 

 a good many respects. 



Of the useful purpose that can be served by 

 popular addresses containing the survey of wide 

 regions of mathematical thought we have two excel- 

 lent examples before us. .\merica, with that spirit 

 of internationalism the absence of which from our 

 islands is so greatly to be regretted, loses no chance 

 of picking the brains of the world's greatest mathe- 

 maticians, irrespective of nationality. Prof. Dar- 

 boux's pamphlet and the second part of Prof. Picard's 

 contain the substance of addresses delivered at St. 

 Louis last vear. The two addresses are to a great 

 extent complementary. Prof. Darboux treats of the 

 development of geometry during the nineteenth 

 century, and Prof. Picard gives a historical account, 

 similar in character, of the development of analysis, 

 with especial reference to its relations with geometry, 

 mechanics, and mathematical physics. Prof. Picard's 

 .St. Louis address also forms a sequel to the series of 

 three lectures delivered by him in 1899 ^^ Clark 

 University which form the first part of the same 

 book. The first of these deals with the gradual 

 extension of the meaning attached to the word " func- 

 tion " during the last century, and the numerous new 

 regions of mathematical thought opened up by this 

 development. The second deals with the theory of 

 differential equations, and the third with analytic and 

 certain other functions. In concluding, M. Picard 

 advises students not to specialise in mathematics at 

 too earl)' a stage, but to endeavour to form a general 

 survey of different branches of the science first, and 

 his lectures afford an excellent preliminary step 

 towards the formation of such a survey in the case 

 of analysis. 



An English translation of M. Darboux's addresses 

 has appeared in recent numbers of the Mathematical 

 Gazette. 



M. Lechalas 's small volume in the series of 

 " .\ctualites scientifiques " deals with Euclidean and 

 non-Euclidean geometry. The subject is introduced by 

 a chapter on Euclidean geometry of one, two, and 

 three dimensions. The geometry of Riemann's space 

 is deduced from the Euclidean geometry of four 

 dimensions. That the properties of a Riemann plane 

 ;ind a Euclidean sphere are identical so long as only 



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