June 17, 1921] 



SCIENCE 



545 



because it seems to me that the work of Poin- 

 care and others, including Birkhoff in this 

 country, has emphasized the growing impor- 

 tance of qualitative mathematics in dynamical 

 problems. The showing here is good, in- 

 cluding interesting contributions from Veb- 

 len, Alexander, BirkhofF, E. L. Moore, and 

 Kline. The closely related topic of point sets, 

 whose vital connection with geometry and 

 dynamics was forcefully pointed out by Van 

 Vleck in his address as retiring president of 

 the American Mathematical Society in 1915, 

 claims but one per cent, of the total num- 

 ber of titles, the articles on this subject 

 coming from Van Vleck, E. L. Moore, Blum- 

 berg and Kline. In the theory of integration 

 we find 1.2 per cent, of the titles, Bliss and 

 Daniell being the principal contributors. 



While geometry does not seem to have had 

 its full share of attention, we are well repre- 

 sented in differential geometry, largely be- 

 cause of the labors of Eisenhart, Wylczynski,. 

 Kasner, G. M. Green, Graustein and others. 

 Further branches of geometry in which Ameri- 

 cans have labored productively are: the geom- 

 etry of algebraic varieties, in which Lefschetz 

 has done notable work ; the geometry of special 

 classes of curves and surfaces, cultivated by 

 Snyder, White, Emch, Sisam, Eanum, Eoe and 

 others ; the geometry of forms ; modular geom- 

 etry, by Dickson, Glenn and Coble; the geom- 

 etry of hyperspace, by C. L. Moore and Eies- 

 land; transformations, by Snyder, Sharpe and 

 others ; non-Euclidean geometry ; while a num- 

 ber of memoirs on different aspects of geom- 

 etry have been contributed by Coolidge. 



The work in mathematical physics has been 

 due almost solely to four men: Bateman, 

 Gronwall, Webster and Eoever. Progress in 

 celestial mechanics is to be credited almost 

 entirely to F. E. Moulton and his pupils and 

 to Birkhoff. The work of Birkhofi ia the 

 field of dynamical systems has been conspicu- 

 ous. 



In postulate theory, the papers of Hunting- 

 ton, E. L. Moore, and Scheffer have aroused 

 mucli interest. 



The above review lays claim only to being 

 a sketch, and doubtless overlooks single papers 



of real importance. It does, however, give 

 some idea of the fields being cultivated, and 

 of the more prominent figures in them. Not 

 as an afterthought, but with singular pleasure, 

 do I allude to several developments which 

 are peculiarly American, in that they were 

 largely initiated and cultivated on this side 

 of the ocean: to Moore's general analysis al- 

 ready touched upon, and to Wilczynski's pro- 

 jective differential geometry, ably initiated 

 by him and carried on by himself and pupils, 

 and to the all too short-lived Green — to 

 Kasner's geometrical mechanics, and to the 

 theory of linear difference equations in the 

 hands of Birkhoff, Carmichael and their pu- 

 pils. While one might wish a more extensive 

 cultivation of such branches as are largely 

 indigenous, it seems to me that their very ex- 

 istence furnishes some evidence of the vitality 

 of American mathematics, and a foundation 

 for predictions that its importance is on the 

 increase. Doubtless there are other evidences 

 of the same sort of thing that have escaped 

 my attention. 



A few further remarks on the statistics 

 gathered may be of interest. The 1,258 titles 

 found were the contributions of 325 persons. 

 ^Nearly half of this number contributed but 

 one paper each. I think it fair to assume that 

 two thirds of the latter had recently received 

 their doctorates and were writing their first 

 and last paper at the same time. This large 

 " mortality " indicates a great waste of intel- 

 lectual capital, and deserves careful considera- 

 tion. Some of it means the diversion of en- 

 ergy of able investigators into the instruction 

 of pupils who might be predicted to be un- 

 productive, and some of it is due to the crush- 

 ing out of scientific enthusiasm in really able 

 young mathematicians by an unsympathetic 

 or over-exacting environment. The " treat- 

 ment indicated " must be decided upon in the 

 individual cases. 



I have heard the advice given to young 

 scientists, that if they wish to show the great- 

 est productivity, the best way to accomplish 

 this is by a high degree of specialization. The 

 results of the present study bear this out, 

 though not to the extent one might anticipate. 



