MODERN MATHEMATICAL RESEARCH MILLER. 193 



master in his field, we may mention the numerous prize subjects 

 which are announced from year to year by foreign academies. 



Among the learned societies wliich announce such subjects the 

 Paris Academy of Sciences is probably most widely known, but there 

 are many others of note. The subjects announced annually by these 

 societies cover a wide range of mathematical interests, but they are 

 frequently beyond the reach of the young investigator.^ It is very 

 easy to obtain these subjects, since they generally appear in the 

 ''notes" of many mathematical journals. In our country the Bul- 

 letin of the American Mathematical Society is rendering very us'eful 

 service along this and many other lines. Wliile some of these sub- 

 jects are very general, there are others which indicate clearly the 

 particular difficulties which must be overcome before further progress 

 in certain directions seems possible and hence these subjects deserve 

 careful study, especially on the part of the younger investigators. 



As long as one is completely guided, in selecting subjects for re- 

 search, by the standing problems or by the subjects announced by 

 learned bodies and those proposed individually by prominent inves- 

 tigators, one is on safe ground. Kcal progi-css along any of these 

 hnes is welcomed by our best journals, as such progress can easily be 

 measured, and it fits into a general trend of thought which is easily 

 accessible in view of the many developed avenues of approach. 

 Not\\dthstanding these advantages, the real investigator should reach 

 the time when he can select his own problems without advice or 

 authority; when he feels free to look at the whole situation from a 

 liigher point of view and to assume the responsibility of an inde- 

 pendent choice, irrespective of the fact that an independent choice 

 may entail distrust and misgivings on the part of many who would 

 have supported him nobly if he had remained on their plane. 



In looking at the whole situation from this higher point of view 

 many new and perplexing questions confront us. Wliy should the 

 developments of the past have followed certain routes? What is 

 the probability that the development of the territory lying between 

 two such routes will exhil)it new points of contact and greater unity 

 in the whole development ? What should be some guiding principles 

 in selecting one rather than another subject of investigation? What 

 explanation can we give for the fact that some regions bear evidences 

 of great activity in the past but are now practically deserted, wliile 

 others maintained or increased their relative popularity through all 

 times ? 



One of the most important tests that can be applied to a particular 

 mathematical theory is whether it serves as a unifying and clarifying 

 principle of wide applications. Whether these applications relate to 



' For solutions of such problems in pure mathematics by Americans, see Bulletin of the American Mathe- 

 matical Society, vol. 7 (1901), p.. 190; vol. 16 (1910), p. 267. 



