546 



SCIENCE 



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



For one third of the twelve most productive 

 mathematicians have papers scattered over a 

 half dozen different fields each. Doubtless the 

 advice given is particularly apt in cases where 

 industry is a more predominant characteristic 

 than elements of genius. But it does seem as 

 if actual productive experience in different do- 

 mains did, in some cases, add to the mathe- 

 matician's power, by suggesting ideas, anal- 

 ogies and methods. It seems to me also to 

 have a steadying effect on one's sense of val- 

 ues. He who sits in judgment on the value 

 of scientific work is in a precarious position. 

 To be sure, it must be done, by editors, if not 

 by others. I have sought for some time a satis- 

 fying criterion of values. Probably no abso- 

 lute criterion exists. The best working test 

 I have been able to find, both in my own judg- 

 ment and in that of those mathematicians 

 with whom I have discussed the question, is to 

 be found in the degree of relationship of the 

 investigation to be judged with other branches 

 of mathematical or allied sciences whose vi- 

 tality and interest are recognized. If this 

 solution is at all an acceptable one, it is at 

 once clear how experience in different fields 

 may enhance the value of the worker's product. 



It may be of interest to note, in these times 

 of agitation for cooperative research, that less 

 than 3 per cent, of the titles listed were of 

 joint papers, and of these not one bore evi- 

 dence of being inspired by the movement. 

 While cooperative investigation in mathe- 

 matics should have all the ti-ial it can get, it 

 is evident that men are not likely to take to it 

 naturally in any great degree, though there 

 have clearly been instances in which investi- 

 gators, brought together by community of in- 

 terest, have distinctly enhanced their product 

 by collaboration. 



In addition to publications in journals, 

 there are the books which have appeared dur- 

 ing the decade. The number of books on 

 higher mathematics which Americans have 

 published in this time barely exceeds three 

 score. Titles of American books in the lists 

 of current publications in higher mathematics 

 are as needles in a haystack. The value to 

 American mathematics of authoritative and 



up to date handbooks by American authors has 

 been sufiiciently emphasized to need no further 

 comment here. The books which have ap- 

 peared include a sufficient number of treatises 

 of such excellence as to leave no doubts as to 

 the capabilities of authorship in this country. 

 Three, at least, of them, have been translated 

 into French or German; two, written in these 

 languages, enjoy large sales here and abroad. 

 The problem is an economic one, and the de- 

 sirability of subsidy encouragement has been 

 pointed out. All I wish to do here, is to 

 suggest the help each individual can give by 

 buying such books whenever possible, and by 

 recommending their purchase by libraries. 



Our sketch of the decade should not termi- 

 nate without mention of the fact that a half 

 dozen Americans have been elected to foreign 

 academies, and that in three instances Ameri- 

 cans have been the recipients of prizes or 

 medals from such organizations, and in a 

 further instance, of an honorable mention. 

 The foreign recognition in these cases has been 

 amply merited. The point is, of course, that 

 by reason of national pride, of habit, of lan- 

 guage barriers, recognition must of necessity 

 lag behind merit. But some conscious effort 

 on our part may weU be directed toward the 

 attainment of deserved recognition. It seems 

 incontrovertible that if a bit of mathematics 

 is worth writing, it is worth writing to be 

 read. Otherwise the author is guilty of usurp- 

 ing pages in the journals and space on library 

 shelves to no purpose but the gratification of 

 vanity. This is not the place to enter upon a 

 discussion of style, but one or two aspects of 

 the matter have forced themselves upon my 

 attention in connection with the present study. 

 Style as a whole is, and doubtless should be, 

 individual, and its development is largely 

 merely a matter of conscious purpose. Its 

 fundamental element for the mathematician 

 is, of course, clarity. But when one looks 

 over the standard reviews of mathematical 

 literature, and notices the extent to which the 

 reviewer takes his cue from the author's open- 

 ing lines — frequently contenting himself with 

 citing the author's own estimate of his work, 

 it becomes clear that particular emphasis in 



