July 23, 1915] 



SCIENCE 



111 



lish the thesis that research is ruled partly 

 by fashion; the maximum of synthetic 

 work about 1887, and its decline through 



18,0 



Fig. 9. Analysis, Pages; Titles, Dotted Curve. 

 Both show a maximum about 1886, a minimum 

 about 1895. 



the foUoMdng twenty years, is proof suffi- 

 cient. 



One graph shows the growth of what is 

 called analysis, that great body of knowl- 

 edge which takes its rise equally from 

 calculus, from the algebra of imaginaries, 

 from the intuitions and the critically refined 

 developments of geometry, and from ab- 

 stract logic : the common servant and chief 

 ruler of the other branches of mathematics. 

 The page-chart solid, the record of titles 

 broken, both show a general increase in the 

 amount of work, possibly a trebling in forty 

 years. A first maximum appears before 

 1890, probably the culmination of waves 



Pig. 10. Theory of Functions; Titles. 

 Line: Analysis (Reduced). 



Dotted 



set in motion by Weierstrass and Fuchs in 

 Berlin, by Biemann in Gottingen, by Eer- 

 mite in Paris, Mittag-Lefjler in Stockholm, 

 Dini and Brioschi in Italy. The great 

 energy of these giant intellects was directed 

 mainly to founding on a basis critically un- 

 assailable the theory of functions, before 

 this time somewhat loose and uncertain, 



and to developing its particulars from gen- 

 eral grounds, rather than by piecemeal as 

 was the necessity earlier. So in the next 

 graph we see how considerable a part of 

 the growth of analysis prior to 1887 was 

 due to activity in this — its central part, the 

 theory of functions. But the fashion 

 changes, and a new sweep of the curve up- 

 ward does not occur until after 1900, when 

 a new impulse comes from the theory of 

 integral equations (not yet recognized as a 

 separate field by the Jahrbuch editors) and 

 from the influence of another master mind, 

 Hilbert of Gottingen. In our own country 

 too this branch of science is forwarded, 

 notably at Harvard, Chicago and Yale. 



One more example is perhaps the most 

 striking of all. Algebra is shown first, 



Fig. 11. Algebra, Including Series and Groups. 

 Solid Line, Pages; Dotted Line, Titles. 



pages on a solid curve, titles on a dotted one, 

 quite nearly in accord. Each decade has 

 its chief new subject for expansion, and 

 growth is not far from steady. Equations, 

 now up to the seventh and eighth degrees : 

 determinants, invariants of linear groups of 

 operations, and substitution groups all have 

 had their years of plenty under this curve, 

 and give it a strong, well-grown look. The 

 part of this work that was shared most 

 largely by our own countrymen is the 

 theory of algebraic forms, quantics or in- 

 variants and covariants, as it is variously 

 called. Sylvester, first professor of mathe- 

 matics at Johns Hopkins University, had 

 done fundamental work in this field in his 

 youth, in the early 50 's. At Baltimore the 



