196 ANNUAL KEPORT SMITHSONIAN INSTITUTION, 1912. 



and safeguard his freedom with the zeal of his European colleague. 

 It is too commonly assumed that loyalty implies lying. 



The investigators in pure mathematics form a small army of about 

 2,000 men and a few women. ^ The question naturally arises, What 

 is this httle army trying to accomphsh? A direct answer is that they 

 are trying to find and to construct paths and roads of thought which 

 connect with or belong to a network of thought roads commonly 

 knov/n as mathematics. Some are engaged in constructing trails 

 through what appears an almost impassable region, while others are 

 mdening and smoothing roads which have been traveled for centuries. 

 There are others who are engaged in driving piles in the hope of secur- 

 ing a sohd foundation through regions where quicksand and mire 

 have combined to obstruct progress. 



A characteristic property of mathematics is that by moans of cer- 

 tain postulates its thought roads liave been proved to be safe and 

 they always lead to some prominent objective points. Hence they 

 primarily serve to economize thought. The number of objects of 

 mathematical thought is infinite, and these roads enable a finite mind 

 to secure an intellectual penetration into some parts of this infinitude 

 of objects. It should also be observed that mathematics consists of 

 a connected network of thought roads, and mathematical progress 

 means that other such connected or connecting roads are being estab- 

 lished which either lead to new objective points of interest or exhibit 

 new connections between laiown roads. 



The network of thought roads called mathematics furnishes a very 

 interesting chapter in the intellectual history of the world, and in 

 recent years an increasing number of investigators have entered the 

 field of mathematical history. The results are very encouraging. 

 In fact, there are very few other parts of mathematics where the 

 progress during the last 20 years has been as great as in this history. 

 This progress is partly reflected by special courses in this subject in 

 the leading universities of the world. While the earliest such course 

 seems to have been given only about 40 years ago, a considerable 

 number of universities are now offering regular courses in this subject, 

 and these courses have the great advantage that they establish another 

 point of helpful contact between mathematics and other fields. 



Mathematical thought roads may be distinguished by the facts 

 that by means of certain assumptions they have been 'proved to lead 

 safely to certain objective points of interest, and each of them con- 

 nects, at least in one point, with a network of other such roads which 



1 Between 5 and 10 per cent of tlie members of the American Mathematical Society are women, but the 

 per cent of women in the leading foreign mathematical societies is much smaller. Less than 2 per cent 

 of the members of the national mathematical societies of France, Germany, and Spain are women, according 

 to recent lists of members. The per cent of important mathematical contributions by women does not 

 appear to be larger, as a rule, than that of their representation in the leading societies. The list of about 

 300 collaboralors on the great new German and French matliematical encyclopedias does not seem to 

 include any woman. Possibly women do not prize sufficiently intellectual freedom to become good mathe- 

 matical investigators. Some of them exhibit excellent ability as mathematical students. 



