860 



SCIENCE 



[N. S. Vol. XXXIV. No. f 



have been given to any extent during the 

 last ten years are tolerably uniform (of. the 

 report of subcommittee 2), but in this mat- 

 ter so much depends on the unwritten 

 standards of individual professors or de- 

 partments that there still remains a great 

 difference in the ease with which the degree 

 can be obtained at different institutions. 

 It is for this reason that the suggestion 

 which is sometimes made that it would be 

 well to attempt to formulate definite stand- 

 ards for the doctor's degree, to which the 

 universities of the country should conform, 

 seems to be of slight practical value. 



In school and college work America 

 adopts in one respect a very different 

 standpoint from France and Germany, and 

 this has a certain indirect influence on 

 graduate work. We refer here to the fact 

 that in the last-named countries a pupil 

 will not be allowed to proceed from class to 

 class, and, particularly, will not be allowed 

 to pass the great educational landmarks 

 (for instance graduation from the gym- 

 nasium in Germany) without conforming 

 to a very exacting standard which a con- 

 siderable percentage of each class fails to 

 attain. In America, on the other hand, the 

 teacher who tries to impede seriously the 

 progress of any but the unusually lazy 

 or stupid soon makes himself impossible. 

 This is not the place to discuss the respec- 

 tive merits of these two points of view in 

 the secondary school or even in the college ; 

 but when we come to the graduate student 

 of mathematics it seems clear that the 

 American attitude must be modified, and, 

 as a matter of fact, in all the stronger in- 

 stitutions of the country a much greater 

 ability and earnestness of purpose is de- 

 manded for passing examinations and se- 

 curing degrees in the graduate school than 

 would be allowed to pass muster in under- 

 graduate work. Nevertheless, it is to be 



hoped that something more will be accom- 

 plished in this direction, and that, in par- 

 ticular, candidates for the doctor's degree 

 will be made to feel that success for them 

 at an institution of good standing is not a 

 mere matter of time and patience. It is 

 the more important to insist on this, since, 

 as has just been said, the whole current of 

 secondary and college education runs in 

 another channel. 



rV. TEACHERS 



We must be concerned with this subject 

 for two different reasons, first, because the 

 great majority of graduate students of 

 mathematics ultimately become teachers in 

 secondary schools, colleges or universities; 

 and, secondly, because on the quality and 

 efficiency of the teachers in the graduate 

 school itself (professors, instructors, etc.) 

 depends to such a large degree the quality 

 of the school. 



It is a favorable sign of the gradual ele- 

 vation of the profession of secondary-school 

 teacher that of late years many persons 

 wishing to adopt this profession spend a 

 year in study in a gradviate school. It is 

 true that this time is frequently not spent 

 in the study of a single subject ; but for the 

 future teacher of mathematics (or of math- 

 ematics and some other subject) to have 

 had a couple of graduate courses in mathe- 

 matics, usually in the intermediate group, 

 is a very substantial gain over the condi- 

 tions of twenty years ago. It is to this 

 class of students that the courses on the 

 teaching of mathematics, which are now 

 given at many colleges and universities, 

 mainly appeal. 



If we except this group who go into sec- 

 ondary-school teaching, and a second 

 group who study mathematics as a tool for 

 use in some other science, such as physics, 

 it may be said with almost absolute pre- 



