OCTOBEB 19, 1906.] 



SCIENCE. 



495 



acquaintance with such a strong tool as 

 vector analysis. Moreover, a vector has 

 so many physical applications that the line 

 representing it becomes full of meaning to 

 the student, and geometric constructions 

 thus admit of interpretations which escape 

 the notice of the student whose geometrical 

 training has proceeded along the old lines. 



The plea for an early introduction of the 

 methods of analytic geometry has historical 

 support since the 'Tractatus de latitudi- 

 nibus formarum' of Oresme was popular 

 during the period of the early development 

 of European universities when very little 

 of Euclid's 'Elements' were taught and the 

 mental advancement of the students was 

 low. The graphic ~ representation of a 

 function upon which so much stress has 

 been laid in recent years was relatively 

 farther advanced in these early days than 

 it is at the present time. The thought of 

 direct usefulness is again assuming a more 

 dominating influence in early mathematical 

 instruction. We are laying more stress 

 upon the student's ability to use a theorem 

 intelligently than upon hU giving a fault- 

 less demonstration. In fact, a prominent 

 Harvard professor is said to have told his 

 students that the demonstration of a 

 theorem is no evidence that it is under- 

 stood, but the intelligent use of the theorem 

 constitutes such evidence. 



A fundamental aim in early education 

 is to give the student a clear comprehension 

 of the world in which he lives and to fur- 

 nish him with the necessary knowledge to 

 make a wise use of his faculties. In view 

 o' the fact that the sciences and their ap- 

 plications are continually playing a more 

 prominent role in the civilized world and 

 that international relations are becoming 

 more and more important, it becomes neces- 

 sary to readjust the machinery of educa- 

 tion from time to time. Many of the neces- 

 sary changes should be regarded as steps 

 towards a proper readjustment rather than 



as fundamental progress. In view of the 

 great conservatism in mathematics the re- 

 adjustment along this line is apt to be slow 

 and to require unusual perseverance. 



When such strong men as Klein, of Ger- 

 many, Forsyth and Sir Oliver Lodge, of 

 England, Moore and Fiske, of America, 

 take active part in reform movements they 

 are certain to be effective. The discus- 

 sions in Germany in recent years have been 

 so active that Gutzmer describes them as 

 stormy at times.^ In France the discus- 

 sions have received less attention, but there 

 has been a decided movement towards 

 graphic methods and especially towards 

 the early use of the differential and in- 

 tegral calculus. In England Professor 

 John Perry, of the Royal College of Sci- 

 ence in London, has probably done the 

 most radical and effective work, so that the 

 reform movement is sometimes known as 

 the 'Perry movement.' 



While these tendencies in elementary 

 mathematics have some influence on mathe- 

 matical instruction in the universities, yet 

 they do not affect this work nearly so much 

 as the elective system in secondary educa- 

 tion. In view of the diversity of the 

 courses pursued by students before enter- 

 ing the universities, the departments where 

 the work has to proceed step by step, as in 

 mathematics, are compelled either to make 

 provision for more elementary courses or to 

 exclude a large number of students. This 

 difficulty exists not only in our institutions 

 but also abroad, and Klein, for instance, 

 urges that the universities should meet the 

 difficulty by offering a greater variety of 

 courses. He discourages the efforts on the 

 part of friends of the real gymnasium who 

 have tried to abolish some elementary 

 courses in languages at the universities 

 with a view to securing a more general 



- Jahresbericht der Deutschen Mathematiker 

 Vereinigung, Vol. 13, p. 517. 



