622 



SCIENCE 



[N. S. Vol. XXXI. No. U 



so mueli more than their predecessors that the 

 work appears almost new. To what extent this 

 is true as regards the volume before us may be 

 inferred from the number of pages which the 

 French and German editions respectively de- 

 vote to the various subjects considered in the 

 published parts of this volume. The subjects 

 and numbers of pages in the two editions are 

 as follows, the first number applying to the 

 French edition: Fundamental principles of 

 arithmetic 62, 27; Combinatory analysis and 

 determinants 70, 19; Irrational numbers and 

 convergence of infinite processes with real 

 numbers, 196, 100; Ordinary and higher com- 

 plex numbers 140, 37 ; Infinite algorithms with 

 complex numbers 20, 8; Theory of sets 42, 

 24; Finite discrete groups 85, 19. Hence the 

 total number of pages devoted to arithmetic in 

 the published parts of the French edition is 

 615, while the German edition devotes only 234 

 pages in all to this fundamental subject. The 

 article on finite groups is the only one which 

 is avowedly left unfinished in the parts of the 

 French edition already issued, but additions to 

 other articles are also to be made before vol- 

 ume I. is completed. 



The list of subjects enumerated in the pre- 

 ceding paragraph constitutes arithmetic, the 

 mother queen of mathematics, according to 

 the best mathematical encyclopedia; and this 

 list should be of interest to every educated 

 person as indicative of what are regarded to 

 be the most fundamental mathematical sub- 

 jects by such an eminent tribunal. As the 

 term arithmetic is now generally employed by 

 mathematicians to include the most basal sub- 

 jects of pure mathematics, and is not restricted 

 to things which deal directly with numbers, it 

 is of great interest to compare the classifica- 

 tions by eminent authorities and to observe 

 that such new subjects as the theory of sets 

 and the theory of discrete groups of finite 

 order are accorded a place among these basal 

 sciences. It is also of interest to observe that 

 the latter of these subjects is accorded rela- 

 tively the largest increase of space in this first 

 volume of the French edition as compared 

 with the German. This is partly due to the 

 fact that the literature of this theory has 



grown very rapidly during the last decade, and 

 partly to the fact that this subject was given 

 a disproportionately small amount of space in 

 the German edition. 



It is to be hoped that the present work will 

 have a large circulation in this country, as it 

 will doubtless be a standard for many years. 

 Even those who have only a slight knowledge 

 of the French language will be able to use it 

 to great advantage, as the mathematical nota- 

 tion is practically cosmopolitan. The histor- 

 ical notes and references are especially com- 

 plete and many of those relating to elementary 

 arithmetic are of interest to teachers of this 

 subject in the secondary schools. It is scarcely 

 necessary to call the attention of the professors 

 of mathematics in our colleges and universi- 

 ties to this work, since most of them have 

 learned to appreciate the German edition and 

 can not fail to appreciate still more an edition 

 offering so many important improvements. 

 While the specialist does not always know 

 everything relating to his subject, it will prob- 

 ably be considered as almost unpardonable if 

 any scholar displays ignorance of what this 

 encyclopedia contains along the line of his 

 chief interest. 



Fortunately the volumes are sold separately 

 so that those who may not wish to subscribe 

 for the entire work can procure those volumes 

 in which they may be chiefly interested. The 

 remaining part of volume I. as well as the 

 remaining parts of the other three volumes of 

 Tome I. are, to a large extent, in press and 

 will probably be published within a few years. 

 The second and third of these volumes are 

 devoted respectively to algebra and to the 

 theory of numbers, while the fourth is devoted 

 to the calculus of probability, theory of errors 

 and diverse applications. In addition to the 

 remainder of the article on finite groups, the 

 volume under review is to contain the follow- 

 ing: Complements on all the articles in the 

 volume, bibliographical lists of the principal 

 works treated in these articles, lists of the 

 principal technical terms in the four lan- 

 guages, English, French, German and Italian, 

 and the usual subject and author indexes to- 

 gether with a preface and an introduction. 



