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[N. S. Vol. XLII. No. 1073 



eager requests of students led him tack 

 to that line of research, and many joined 

 him in producing extensive and valuable 

 work; much of it published in the Amer- 

 ican Journal of Mathematics, which he 

 founded. In Germany the same subject 

 engaged attention, at the same time or a 

 little earlier, through the efforts of Aron- 

 hold at Berlin — a true pioneer; of Clehsch 

 at Gottingen and Gordan at Erlangen, the 

 last two founding in 1869 a new journal, 

 the Mathematische Annalen, to facilitate 

 the publication needed for the work of 

 themselves and their enlarging circle of 

 progressives. 



The rapid rise, the climax and the decline 

 to a low normal, in this theory of forms or 

 invariants, is shown in a graph which 

 amply repays this rapid statistical study. 



ispo 



!PlG. 12. Algebraic Forms, Invariants, etc.: 

 Titles. Algebra, including Number-theory and 

 Groups, Above (Reduced). 



The solid curve shows the annual number 

 of titles, while the dotted graph above, on 

 smaller scale, shows how the general divi- 

 sion, algebra, was fluctuating. Here the 

 fashion reaches its acme before 1890, an in- 

 crease twice as rapid as that of the main 

 division; and declines most surprisingly. 

 Sylvester returned to England in 1884 ; and 

 in Germany a climactic series of discoveries 

 by Hilbert set a temporary high mark, 

 discouraging further effort for the time. 

 But a later maximum, 1905, has in it a 

 guarantee that growth is not impossible; 



this I do not pretend to explain, but the 

 fact is obvious. 



The field of differential equations has al- 

 ways held attraction for mathematicians, 

 principally because of its close contact with 

 physics and geometry. Its development 

 naturally waited for that of the theory of 

 functions. We see the researches in this 



Fig. 13. Differential Equations: Titles. The 

 dotted curve is averaged for three years. 



Fig. 14. Differential Equations Above; Below, 

 the Eesidue of Analysis, Number of Titles. 



department increasing in number slowly 

 from 1870, under the combined influence of 

 Weierstrass, Darboux and Lie; and note a 

 slight decline about 1886, followed by a 

 marked recovery and advance during the 

 publication of lectures by Forsyth, Pieard, 

 Goursat and Painleve. It is of interest to 

 see the relative variation in differential 

 equations on the one hand, and all the rest 

 of analysis on the other. 



Finally we examine in a separate diagram 

 the fluctuation in absolute quantity of work 

 on the mathematical theory of electricity 

 and magnetism, and its ratio to the whole of 

 applied mathematics. It remains less than 

 one fourth of the whole, but rises after 1873 



