Deckmbbe 22, 1911] 



SCIENCE 



859 



Tery great (cf. subcommittee 1, section V.). 

 This is peculiarly the case in those institu- 

 tions which have only recently begun a 

 policy of expansion in their graduate work, 

 where the first sign of such expansion often 

 appears in an astounding increase in the 

 number and range of courses offered, for 

 only a small part of which there are stu- 

 dents. Indeed, if students should present 

 themselves, the capacity of the teaching 

 force would be completely overtaxed. 

 This is a state of affairs which no self- 

 respecting institution should allow to con- 

 tinue, and there are signs that it is usually 

 of only a temporary nature, since with a 

 real strengthening of the mathematical de- 

 partment of such an institution this infla- 

 tion tends to disappear. We hasten to 

 add that the stronger institutions, and 

 many smaller institutions with a due sense 

 of proportion, offer admirable selections of 

 courses commensurate with their capacity 

 and the needs of their students, courses 

 which at each institution usually vary con- 

 siderably from year to year. Even in the 

 weaker institutions where a call for ad- 

 vanced instruction is hardly apparent, it 

 may often be wise to encourage instructors 

 to offer a course of a not wholly elementary 

 character, as it will frequently be found to 

 act as a tonic and, by keeping them in touch 

 with the scientific side of their subject, 

 enable them to make their elementary work 

 more vital. 



It was mentioned in section II. that no 

 sharp distinction between graduate and 

 undergraduate work in mathematics can be 

 made. Indeed it is hard to exclude en- 

 tirely from graduate work anything above 

 the first course in the calculus, now com- 

 monly taken in the second undergraduate 

 year. The actual state of affairs is best 

 expressed by regarding the group of courses 

 just following this point, such as a second 



course in the calculus, the elements of de- 

 terminants and of the theory of equations, 

 projective geometry, a first course on dif- 

 ferential equations, etc., as belonging both 

 to graduate and to undergraduate instruc- 

 tion. From this latter point of view, how- 

 ever, these courses usually appeal only to 

 the student of distinct mathematical ability 

 and seriousness of purpose, whose presence 

 in the course along with graduates does not 

 very greatly affect the character of the 

 course. 



As the external signs of success for the 

 graduate student we have the master's and 

 the doctor's degrees. The first of these is 

 commonly given for one year's graduate 

 work done largely in one subject, such as 

 mathematics or physics, and tested either 

 by course examinations in which a higher 

 standard is demanded than is accepted for 

 undergraduates, or by a single examination 

 covering the whole year 's work. A thesis is 

 also often required for the master's degree; 

 but the work done on this thesis is not com- 

 monly of the nature of research work, and 

 the degree is taken by considerable num- 

 bers of students most of whom never pro- 

 ceed further. This degree is given, and 

 properly given, by a large number of insti- 

 tutions, many of which have only a very 

 moderate strength in their graduate mathe- 

 matical work. Under these conditions sug- 

 gestions for a minimum standard for the 

 degree are not out of place, and such sug- 

 gestions will be found in the report of sub- 

 committee 1, section VII. 



The doctor's degree originally came to 

 us from Germany, but has long been nat- 

 uralized and is in all American institutions 

 of good standing distinctly a research de- 

 gree. In several of our stronger universi- 

 ties it has a standard at least as high as the 

 best German standard. The requirements 

 for the doctor's degree in universities which 



