194 ANNUAL EEPOET SMITHSONIAN INSTITUTION, 1912. 



pure mathematics only or to related fields seems less important. In 

 fact, the subjects of application may have to be developed. If this 

 is the case, it is so much the better provided always that the realm 

 of thought whose relations are exhibited by the theory is extensive 

 and that the relations are of such a striking character as to appeal to 

 a large number of mathematical intellects of the present or of the 

 future. Some isolated facts may be of great interest, but as long as 

 they are isolated they have little or no real mathematical interest. 

 One object of mathematics is to enable us to deal with infinite sets 

 with the same ease and confidence as if they were individuals. In 

 this way only can our finite mind treat systematically some of the 

 infinite sets of objects of mathematical thought. 



In comparatively recent years the spirit of organization has made 

 itself felt among mathematicians with rapidly increasing power, and 

 it has already led to many important results. Beginning with small 

 informal organizations in which the social clement was often most 

 prominent, there have resulted large societies, national and even 

 international, with formal organizations and with extensive publica- 

 tions. In reference to one of these early organizations, the mathe- 

 matical society of Spitalfields in London, which lasted for more than 

 a centuiy (1717-1845), it is said that each mem])er was expected to 

 come to the meetings vdth. his pipe, his mug, and his problem.^ 



The modern mathematical society is dominated by a different 

 sphit. It generally supports at least one organ for publication, and 

 scholarly publicity develops scholarly cooperation as well as scholarly 

 ambitions. This cooperation has led to movements which could not 

 have been undertaken by a few individuals. One may recall here the 

 Revue Semestrielle, published under the auspices of the Amsterdam 

 Mathematical Society; the extensive movement to examine and 

 compare methods and courses of mathematical instruction in various 

 countries, inaugurated at the fourth international congress, held at 

 Home in 1908; and, especially the great mathematical encj^clopedias, 

 whose start was largel}'^ influenced by the support of the deutschen 

 Mathematikcr-Vereinigung as expressed at the Vienna meeting in 

 1894. The French edition of the latter work, which is now in the 

 course of publication, is expected to include 34 large volumes, besides 

 those which are to be devoted to questions of the philosoph}', the 

 teaching, and the history of mathematics. 



These encyclopedias and other large works of reference are doing 

 much to expedite travel in the mathematical field. In fact, it would 

 probably not be exaggerating if we shoidd say that by these encyclo- 

 pedias alone the distances in time and eft'ort between many points 

 of the mathematical field have been cut in two. In this connection 

 it may be fitting to i-ecall with a deep sense of obligation the great 



» "Eswurdevonjedemerwartet, dass er seine Pfeife,seinen Krugiinisein Problem mitbringe." Cantor, 

 "Vorlesungen ueber Geschichte der Mathematik," vol. 4, 1908, p. 59. 



