August 7, 1908] 



SCIENCE 



169 



understood, the process may seem merely 

 verbal legerdemain. Properly used, this 

 phase of mathematical instruction is of 

 great advantage to the student of average 

 mathematical ability. It opens up the 

 view; it clears away uncertainties; it fixes 

 principles and concepts ; it gives life to the 

 subject. The problems used should be 

 within the field of the students' experience 

 and comprehension and may well bear some 

 relation to his future work, both in the 

 engineering class-room and beyond. And 

 the second part of this heading is not less 

 important. Mathematics is a tool for the 

 engineering student, and he must acquire 

 facility in its use. This does not mean 

 that the instructor should attempt to make 

 him a finished calculator or an expert 

 workman— time is too short— but mathe- 

 matical principles and processes must be 

 more to the student than a vague some- 

 thing which he recognizes when his atten- 

 tion is directed thereto. Instead, he must 

 have a mastery of at least the fundamen- 

 tals and he must be able to use such prin- 

 ciples and processes in his later studies 

 without having to divert his attention and 

 energy too much from the engineering fea- 

 tures involved. To acquire this facility 

 requires drill and repetition, and this drill 

 must constitute a part of the mathematical 

 training of the engineering student. The 

 multiplication table had to be learned, and 

 many other important things have to be 

 acquired in the same way. 



But it seems that this important side of 

 instruction may be abused. The student 

 who thinks that to accept facts and work 

 problems is sufScient and the instructor 

 who thinks that illustrations and practise 

 work alone constitute mathematical train- 

 ing or that mere laboratory methods suffice 

 are greatly mistaken. The mere substitu- 

 tion in formulas is only rule-of-thumb 

 work, so much decried in engineering; and 

 the mechanic who knows how to use tools. 



and no more, is not an engineer. There 

 must be a direct connection with the theory 

 and the philosophy of the subject to make 

 the practise side serve its proper purpose. 

 In teaching mathematics years ago, expres- 

 sions of approval came to me because I 

 was so "practical," but the underlying 

 purpose of the practical part was not 

 always understood, though this lack of 

 understanding did not afi'ect the results of 

 the method. Inside the "sugar coating" 

 there should always be a principle to fix, 

 a concept to illumine, a process to ex- 

 emplify, a derivation to expound. There 

 seems to be a tendency among some to over- 

 do this side of the work to the detriment 

 of the first side. While the practise fea- 

 ture is a valuable auxiliary in mathemat- 

 ical instruction, it should never be the lead- 

 ing motive. Student and instructor alike 

 should recognize this. 



3. Philosophy of the Subject. — The basis 

 on which the science rests, the underlying 

 meaning of the mathematical processes 

 used, a philosophical study of the method 

 of treatment and of the concepts used, 

 their connection with related things. This 

 is difficult to discuss in a general way, and 

 of course this phase is intimately connected 

 with the first and second. To my mind 

 this phase should not be neglected. It 

 must be apportioned according to the abil- 

 ity of the student. An understanding of 

 the philosophy of the subject will widen 

 his field of view and lessen the chances of 

 error. The better grasp of the meaning 

 will be advantageous. Its presentation 

 involves difficulties, and text-books gener- 

 ally disregard it. It must not be over- 

 emphasized, as is illustrated by the treat- 

 ment in a recent text-book in applied math- 

 ematics, where it is used largely to the ex- 

 clusion of analysis and demonstration. 



Effective methods in mathematical sub- 

 jects involve, then, the skillful selection in 

 proper proportion from these three phases, 



