February 26, 1904.] 



SCIENCE. 



327 



who go over the subject but once and under 

 conditions not greatly superior to those of 

 your own college days should not see clearly 

 and should not use what they so little un- 

 derstand ! Because, as matters now stand, 

 the man who does not repeat his course in 

 calculus many times will fail to appreciate 

 it and use it, shall we say that it should be 

 cut out of the engineering courses and its 

 place taken by more algebra, more trigo- 

 nometry and more descriptive geometry, 

 or shall we retain it and reform its pre- 

 sentation ? The true mathematical teacher 

 will always vote for the latter proposition 

 whatever may be the attitude of the pro- 

 fessional man on the faculty or the pres- 

 sure from the outside of the practicing 

 engineer. How, then, may the higher 

 analysis in our technical schools be made 

 effective as a true means of discipline and 

 as a tool with which to equip the engineer 

 in his life of investigation? 



It is to be understood that the answer to 

 this question here is not claimed to be 

 the word nor the last word on so important 

 a topic. It is a word to be taken for what 

 it is worth. 



1. The most effective teaching of the 

 higher analysis will be possible only when 

 the reforms in mathematical instruction re- 

 ferred to earlier in this paper have per- 

 meated the principal secondary schools. 



2. The teacher should be saturated with 

 his subject. Not only should he be strong 

 and apt on the formal side, but more im- 

 portant still, its inner meaning should be 

 clear to him and its close relation to the 

 phenomena of the objective and subjective 

 life. Some contend that the only man to 

 whom the mathematics of a technical col- 

 lege can be entrusted is an engineer. Does 

 that make any difference ? Rather are not 

 these the essential questions ? Does the man 

 know his subject? In his teaching can he 

 assemble from engineering and other rec- 

 ords the material that will vitalize his 



work 1. Is he in sympathy with engineering 

 essentials and ideals? 



3. Throughout the college course the 

 teaching should be mainly concrete. The 

 problem, say from the physical sciences in- 

 cluding engineering, should first be pre- 

 sented concretely. It should then be 

 stated in mathematical symbols. The oper- 

 ations performed upon the symbols should 

 be accompanied by drawings or models, 

 the final result reduced to numerical form 

 and then intei-preted in language. Upon 

 every problem the student must bring to 

 bear the whole range of his acquired powers 

 and be taught to select the shortest method 

 within his ability. 



In other words, all typical problems 

 should receive a threefold consideration: 

 (a) Its statement in words, and the state- 

 ment in words of its solution when effected ; 

 (6) its graphical statement and solution in- 

 volving geometry and mechanical drawing 

 with squared paper; (c) its analytic state- 

 ment and solution, ending with a numerical 

 result. 



4. The purely formal should be pre- 

 sented as a necessity arising from the so- 

 called practical and in order that a body 

 of knowledge and technical ability may be 

 accumulated which will give the student 

 easy control over the practical in whatever 

 one of its various forms experience shows 

 that it may arise. 



5. The problems chosen should be pro- 

 gressive in character and their mastery 

 should amount to a complete laboratory 

 course in all that part of the higher anal- 

 ysis in which it is desirable that the engi- 

 neering student should be well versed. 



6. The course should be lecture and sem- 

 inarium and individual, more after the 

 manner of the German Technisehe Hoeh- 

 schule. The text-book should become a 

 book of reference. The instructor should 

 know clearly and be able to state accurately 

 the limitations of his methods ; but abstriise 



