June 30, 1911] 



SCIENCE 



985 



This great number of titles of courses im- 

 plies that many of them are devoted to com- 

 paratively narrow fields and raises the ques- 

 tion whether we are not in danger of special- 

 izing too much and of depriving our students 

 of a thorough training in the more general 

 and fundamental methods of wide contact. 

 It would probably be generally conceded that 

 we have gone further in the line of special 

 courses than the universities of Germany, 

 which were pioneers in this direction. On 

 the other hand, it may be observed that the 

 large German and French mathematical en- 

 cyclopedias which are in the course of publi- 

 cation treat pure mathematics under about 

 one hundred general headings, each of which 

 would be sufficiently extensive for a course 

 suitable for graduate students. 



This great extent of available material 

 makes it impossible for one student to cover 

 even superficially the entire field during his 

 college days and hence it tends to increase his 

 interest in a wise choice of his courses, espe- 

 cially as regards such courses as are most gen- 

 erally given and are most likely to be very 

 useful in his later mathematical development. 

 One of the most reliable sources of informa- 

 tion along this line is furnished by the num- 

 ber and the extent of the courses devoted to 

 different subjects in the various leading 

 universities. 



Various journals, including the Bulletin of 

 the American Mathematical Society, are pub- 

 lishing, from time to time, lists of announce- 

 ments of the courses offered by a large num- 

 ber of different universities of Europe and 

 America. A comparative study of these an- 

 nouncements for a period of years involves a 

 large amount of labor in view of the changes 

 from year to year and the uncertainty as to 

 the real significance of some of the titles. 

 As American mathematical activity has been 

 so largely influenced by Germany, especially 

 during recent years, and as the University of 

 Berlin occupies such a prominent place among 

 the greatest centers of learning, it may be 

 fitting to begin with some results drawn from 

 the announcements of this university during 

 the last six years. 



The following table gi .■ se courses 



in pure mathematics to whicij. equivalent 



of at least twelve lecture hours for one 

 semester have been devoted during the last 

 six years. A large number of special courses 

 have thus been excluded. For instance, one 

 of the courses offered only once during the 

 period covered by the table bears the title, 

 " Transcendental Nature of -k and e," another 

 is announced under the title " Influence of 

 Euler's Work on Modern Mathematics," while 

 still another bears the title " Problems in 

 Maxima and Minima treated by Methods of 

 Elementary Geometry." No course bearing 

 the title " Differential Geometry " was offered 

 at Berlin during the period under considera- 

 tion, and only one course entitled " Linear 

 Substitutions." A large part of the subject 

 matter of such courses was, of course, given 

 in the courses under more general titles. 

 Similar remarks apply to a large number of 

 other subjects. The number of hours de- 

 voted to seminars and colloquia was not in- 

 cluded in the following table as the subject 

 matter of these exercises was not always 

 clearly announced. 



BERLIN UNIVEKSITT 



Subjects 



Curves and surfaces 



Theory of functions 



Determinants, theory and 



applications 



Theory of numbers 



Elliptic functions 



Algebraic equations 



Differential equations 



Algebra 



Calculus of variation 



It may be observed that the total number 

 of hours devoted to the four subjects belong- 

 ing to arithmetic and algebra is 106, while the 

 total numbers of hours devoted to analysis 

 and geometry are 98 and 52, respectively. As 

 we shall see later, these results are not in 

 accord with those obtained in a similar man- 



^ The last of these numbers applies to the cur- 

 rent year, and the numbers which precede relate 

 to the preceding years in order. 



