544 



SCIENCE 



[N. S. Vol. LIII. No. 1381 



Wedderburn and Rice in the theory of equa- 

 tions, matrices and determinants; Miss 

 Hazlett and Glenn in invariants and in mod- 

 ular analysis; W. A. Manning and H. H. 

 Mitchell in the theory of groups; E. B. Wilson 

 and Shaw in vector theory and higher complex 

 algebras; Rietz and Dodd in probabilities and 

 statistics; White, Coble and Cole in combina- 

 tory analysis. 



Under the heading of analysis proper, in 

 which the notion of limit plays a role, we 

 find the theory of functions of complex vari- 

 ables taking first place. Our progress along 

 these lines is largely due to Osgood, although 

 there is also found a gratifying variety of con- 

 tributions on conformal mapping, the theory 

 of algebraic functions, and special analytic 

 functions by Lefschetz, Gronwall, Haskins, 

 and others. The theory of functions of a real 

 variable would normally come higher on the 

 list but for the fact that certain topics usually 

 here included have been separated out, such as 

 Fourier series, point-sets, etc. The theory of 

 functions of a real variable is characterized 

 by the fact that it has a larger number of 

 individual contributors than the other topics, 

 although the work of Blumberg deserves spe- 

 cial notice. The field of difEerential equations, 

 apart from Sturmian problems, has had, ex- 

 cept for three fundamental papers of Birkhoff, 

 comparatively little and scattered attention. 

 Macmillan and Lipka have, however, written 

 interesting papers on this topic. In the field 

 of Sturmian problems including boundary 

 value problems, oscillation and expansion 

 problems, we may take distinct satisfaction 

 in the valuable work of Bocher, Richardson, 

 Birkhoff, Jackson, and their pupils and fol- 

 lowers. The calculus of variations, once 

 characterized by Schwarz as the most inter- 

 esting and difficult branch of mathematics, 

 has had comparatively few devotees, but con- 

 tributions of importance have been made by 

 Bliss, certain of his pupils, by Dresden and 

 E. Y. Mi]es. 



I wish now to speak briefly of certain com- 

 paratively new branches of analysis. Professor 

 White once pointed out in an interesting sta- 



tistical review of mathematical development^ 

 a distinct tendency to follow fashions. The 

 reflection that men are apt to be stimulated by 

 each other's work may rob this fact of some 

 of its surprise, but the substantiality of the 

 fact can not be denied. Of course when a 

 new domain is opened up by fundamental dis- 

 coveries, it is to be expected that sooner or 

 later the event will be followed by a wide- 

 spread and rapid development of that domain. 

 An interesting example of a delay in such de- 

 velopment is found in the fact that Fredholm's 

 paper on integral equations, above alluded 

 to, lay for two years unnoticed until the labors 

 of Hilbert gave it its due prominence. ITew 

 domains of the sort alluded to are at present: 

 the still vital subject of integral equations, 

 the related field of functions of infinitely many 

 variables (though Hill and von Koch con- 

 siderably antedate Fredholm), the theory of 

 generalized integrals opened up by Lebesgue, 

 general analysis, due to E. H. Moore, Frechet 

 and Volterra, and one or two other fields to 

 be mentioned presently. It seems to me that 

 in view of the general attention being given 

 to these subjects, American interest in them 

 has been distinctly less than it should have 

 been. General analysis leads, with 2.9 per 

 cent, of the total number of papers. The gen- 

 eral analysis of Moore has been ably culti- 

 vated by his pupils, Hildebrand, Chittenden, 

 and others, while the calcul fonctionel has 

 had fewer devotees. But along the latter 

 lines should be mentioned the papers of 

 C. A. Fischer, Evans, and the two articles 

 of Bliss inspired by his work in bal- 

 listics. The cultivation of integral equations 

 in this country has been due to several in- 

 fiuences. Besides the general theory of Moore, 

 wc find interest in the subject stimulated by 

 Bocher and Volterra, the contributions com- 

 ing mainly from the pens of Mrs. Pell, Hur- 

 witz, and Evans, respectively. The theory 

 of functions of infinitely many variables re- 

 ceives more than one contribution each from 

 but two authors. Hart and Daniell. 



1 have given a special place to analysis situs 



2 Science, new series, Vol. 42, pp. 105-113, 

 1915. 



