164 



SCIENCE 



[N. S. Vol. XXVIII. No. 710 



cients, total and partial differentials, and 

 maxima and minima. 



Throughout the entire course in rational 

 mechanics the writer either demanded from 

 the students or gave them demonstrations 

 of all difficult or important formulae; and 

 the students in explaining their blackboard 

 work were repeatedly asked to state the 

 denominations, not only of the equations as 

 a whole, but also of their factors and com- 

 ponent parts. The answers to such ques- 

 tions evidenced clearly whether the student 

 had a true conception of the mathematical 

 work he was doing, or whether he had 

 merely memorized certain manipulations of 

 quantities. 



It was the writer's custom also to supple- 

 ment as much as possible all analytical 

 work by graphical demonstrations; and if 

 he were to resume the teaching of me- 

 chanics, he would adhere to this method. 



In teaching technical mechanics the 

 writer followed only to a certain extent the 

 manner of instruction just described ; for 

 by the time his students had reached the 

 technical studies, they were so well drilled 

 and weeded out that constant quizzing on 

 fundamentals was no longer necessary; 

 nevertheless the question, "what is the 

 denomination of that equation or of that 

 quantity," was one that was very likely to 

 be asked any student who gave his demon- 

 strations haltingly or who evidenced at all 

 a lack of conception of the principles in- 

 volved. 



In the writer's opinion, the manner of 

 teaching pure mathematics to engineering 

 students should differ materially from that 

 usually employed in academic courses; for 

 while in the latter case it suffices if the 

 instructors be good mathematicians, in the 

 former they should also be engineers, and 

 should have taught, or at least should have 

 studied specially, both rational and tech- 

 nical mechanics. 



Some institutions still adhere to the anti- 



quated custom of teaching pure mathe- 

 matics by lectures. This method has 

 always appeared to the writer to be per- 

 fectly absurd; for the primary benefit to 

 be obtained from the study of mathematics 

 is mental training ; and the student can get 

 this only by severe effort, and not by hav- 

 ing another man's mind do the reasoning 

 for him. Midnight oil and the damp towel 

 are for most students necessary accessories 

 to the courses in pure mathematics. 



The writer believes that the only legiti- 

 mate lectures in pure-mathematical courses 

 for engineering students are as follows : 



First: A short opening lecture to outline 

 the work that is to be covered in the course 

 and to explain how best to study the sub- 

 ject. 



Second: Frequent informal talks to in- 

 dicate the application of the mathematics 

 studied to engineering practise, to explain 

 clearly the meaning of all equations, 

 factors and terms, and to show the true 

 raison d'etre of all that is being done. 



Third: A concluding lecture in the na- 

 ture of a resume to call attention to what 

 has been accomplished during the entire 

 course and to the importance thereof. 



Fourth : Pei-sonal and forcible lectures to 

 lazy students so as to give them clearly to 

 understand that they must either study 

 harder or drop out of the class. 



All mathematical work done by engineer- 

 ing students should be so thorough and 

 complete that the subject shall be almost as 

 much at command as the English language 

 or the four simple rules of arithmetic. 

 Only such thorough knowledge will enable 

 the engineer to use mathematics readily as 

 a tool, rather than as a final resource to be 

 employed solely in extreme need. 



Analytical geometry should be taught 

 graphically as well as analytically in order 

 that the student shall comprehend it fully 

 and shall realize that the work is real and 

 tangible and that the equations represent 



