April 6, 1900.] 



^.CIENCE. 



529 



three-quarters of a century and especially 

 the last two or three decades have witnessed 

 a marvelous change in the mathematical 

 activity of Europe. Mathematical period- 

 icals have sprung up on all sides. A number 

 of mathematical societies have been organ- 

 ized and many of the leading mathemati- 

 cians have confined their investigations to 

 comparatively small fields of mathematics. 



The rapid increase of the mathematical 

 literature ci'eated an imperative need of 

 bibliographical reviews. This need was met 

 in part by the establishment at Berlin, in 

 1869, of a year-book devoted exclusively 

 to reviews of mathematical articles, Jahr- 

 buck uher die Fortschritte der Mathematik. 

 The 28th volume of this work reached our 

 library a short time ago. It contains over 

 900 pages. With a view towards further 

 increasing the facilities to keep in touch 

 with this growing literature, the Amster- 

 dam Mathematical Society commenced the 

 publication of a semirannual review. Revue 

 Semestrielle, in 1893. In the last number, 

 236 periodicals are quoted, each of which 

 contains, at times, mathematical articles 

 that are of sufiicient merit to be noted. 

 Each of the four countries. Prance, Ger- 

 many, Italy, and America publishes over 

 thirty such periodicals. 



One of the characteristic features of our 

 times is the prominence of the spirit of co- 

 operation. The mathematical periodicals 

 and the mathematical societies are evi- 

 dences of this spirit. In quite recent years 

 international mathematical congresses have 

 given further expression of the wide-spread 

 desire to co-operate with even the most re- 

 mote workers in the same fields. The first 

 of these congresses was held in Zurich in 

 1897, and the second is to be held during 

 the coming summer in connection with 

 the Paris Exposition. The same spirit led 

 in 1894 to the starting of a periodical, 

 L'intermediare des mathematiciens, which is 

 devoted exclusively to the publishing of 



queries and answers in regard to different 

 mathematical subjects. 



This desire for extensive co-operation is 

 tending towards unifying mathematics and 

 towards laying especial stress on those sub- 

 jects which have the widest application in 

 the different mathematical disciplines. This 

 explains why the theory of functions of a 

 complex variable and the theory of groups 

 are occupying such prominent places in re- 

 cent mathematical thought.* 



Before entering upon a description of 

 some of the fields included in these sub- 

 jects and the interesting problems which 

 they present, it maj"^ be well to state ex- 

 plicitly that our remarks on mathematics 

 will have very little reference to its appli- 

 cation to other sciences. To the pure 

 mathematician a result that has extensive 

 application in mathematics is just as im- 

 portant and useful as one which applies to 

 the other sciences. Mathematics is a sci- 

 ence which deserves to be developed for its 

 own sake. The thought that some of its 

 results may find application in other sci- 

 ences is, however, a continual inspiration, 

 and those who investigate such applications 

 sometimes add materially to the develop- 

 ment of mathematics. 



The curve representing a function of a 

 single variable was the principal object of 

 investigation during the eighteenth and a 

 great part of the nineteenth century. f The 

 investigations of Abel and Cauchy on power 

 series during the early part of the nine- 

 teenth century furnished the foundation for 

 the modern theory of analytic functions — 

 a theory which has been adorned by the 

 labors of some of the most brilliant mathe- 

 maticians of the preceding generation and 

 which is claiming the attention of some of 

 the foremost thinkers of the present time. 

 Quite recently this theory has been made 



* Ct. Klein, Chicago 3fathemaiical Papers, 1893, p. 

 134. 



t Lie, Leipziger BericMe, vol. 47, p. 261. 



