986 



SCIENCE 



[N. S. Vol. XXXIII. No. 861 



ner from many other universities and they 

 would be slightly changed at Berlin if all the 

 courses in pure mathematics had been tabu- 

 lated. They would also have been afEected by 

 the consideration of the courses on applica- 

 tions of mathematics. As is well known, a 

 large number of courses on applications of 

 elliptic functions are given at Berlin. These 

 are in addition to the courses on the theory of 

 these functions as listed above, and during the 

 six years under consideration the number of 

 lecture hours devoted to these applications in 

 Berlin University were, respectively, 6, 0, 10, 

 0, 8, 4 — making a total of 28. Hence a total 

 of 58 lecture hours for a semester were de- 

 voted to elliptic functions and their applica- 

 tions during these six years — an average of 

 nearly five hours for each semester. 



Among the other German universities which 

 maintain very strong mathematical depart- 

 ments Gottingen should perhaps be especially 

 mentioned in view of the facts that so many 

 Americans have studied there and the influ- 

 ence of Klein and Hilbert has been so great 

 in shaping our courses in higher mathematics. 

 It may also be desirable to bring the mathe- 

 matical courses of the universities of Berlin 

 and Gottingen together in view of the fact 

 that they exhibit a great difference as to em- 

 phasis on the various subjects of pure mathe- 

 matics. For instance, only two courses on 

 determinants and their applications were 

 given at Gottingen during the last six years 

 while at Berlin this course has been given 

 very frequently, as may be seen from the table 

 given above. 



At Gottingen courses on difFerential equa- 

 tions have been given very much more fre- 

 quently than at Berlin, while courses on 

 elliptic functions are much more common at 

 the latter institution than at Gottingen. 

 Judging from the number and the extent of 

 the courses, there is a very wide difference 

 between Berlin and Gottingen as regards the 

 emphasis on the subjects which are usually 

 classed under the general headings, algebra 

 and analysis; or arithmetic and algebra, and 

 analysis. At Berlin the former receive very 

 much more attention than at Gottingen and 



the predominating influence of the latter in- 

 stitution is evident in the advanced mathe- 

 matical courses of many American universi- 

 ties. A marked difference between Gottingen 

 and Berlin may also be observed with respect 

 to the tendency to give courses under a large 

 variety of names. At Gottingen we find a 

 larger number of courses under such titles as, 

 encyclopedia of geometry, encyclopedia of ele- 

 mentary mathematics and elementary mathe- 

 matics from a higher standpoint, than at 

 Berlin. 



In the following table we give again only 

 those courses to which at least twelve lecture 

 hours for one semester have been devoted 

 during the last six years, excluding courses 

 which appeared to have been devoted mainly 

 to applications, and arranging the others in 

 the order of the number of lecture hours. 



GOTTINGEN UNIVEESITT 



Subjects 



Differential equations 



Theory of functions 



Descriptive geometry 



Algebra 



Theory of numbers 



Calculus of variations 



Curves and surfaces 



Principles of mathematics 



12, 12, 12, 8, 11, 4 

 7, 0, 8, 6, 10, 8 



0, 8, 0, 8 

 4, 4, 4, 4 

 2, 4, 3, 4 



1, 4, 0, 2 



2, 0, 0, 



4, 4, 



4, 0, 



4, 0, 



4, 4, 



7, 4, 



0, 0, 4, 2, 2, 4 



A comparative study of the courses offered 

 in other German universities reveals wide dif- 

 ferences as regards emphasis on the various 

 subjects and such a study tends to explain the 

 migration of students from one institution to 

 another. Unfortunately, there is very little 

 migration in American universities, and hence 

 our students are frequently acquainted only 

 with the courses offered by one institution. 

 This makes it the more desirable that our 

 large universities should aim to offer a wide 

 range of subjects, covering the most impor- 

 tant parts of the various developed fields of 

 mathematics. It is evident, however, that it 

 would be much better if our students could 

 be induced to divide their time of graduate 

 study among different universities and to seek 

 instruction under the foremost men along the 

 lines of their chief interests. 



