326 



SCIENCE. 



[N. S. Vol. XIX. No. 478. 



vestigation that the genius of man has ever 

 contrived. 



The student of mathematics who has re- 

 flected deeply upon the meaning and in- 

 terpretation of its symbolic language knows 

 that man, in his struggle for the mastery 

 and direction of nature's laws and pro- 

 cesses, has no more subtle and no more 

 powerful ally than he finds in the calculus. 

 The other subjects leading to it are con- 

 ventional and highly artificial, but with this 

 one we return to simplicity and operate 

 with perfect ease and freedom in the realms 

 of time, space and force. 



As we find nature operating by growth, 

 and force by insensible gradations, so over 

 against that the calculus is the science of 

 continuous number. Why then does the 

 mathematician find so much in this, his 

 favorite subject, while the practical engi- 

 neer—even the one of great ability, pro- 

 ficiency and success — is inclined to think 

 that time spent upon it is wasted or at 

 least not employed to the best advantage? 

 Why this great divergency in conviction? 



No one will doubt the ability of our best 

 mathematical instructors and teachers, nor 

 their perfect familiarity with the matter 

 they are teaching. But are analytics and 

 the calculus— especially the latter— pre- 

 sented to the average student in the best 

 way? Does not the formal smother the 

 thought element and leave nothing but 

 routine machine work upon symbols? As 

 the student learns laboriously how to find 

 the first derivative of a wide range of 

 rider problems has he a faint conception 

 even of Avhat it is all about? Sir William 

 Thomson, you know, said he did not under- 

 stand an eqiiation until he could make a 

 model of it. Is the average student able 

 to make a model of his operations with the 

 ditferential calculus? And when he takes 

 x\p the integral calculus and begins his at- 

 tack upon a mass of algebraic and trans- 

 cendental functions, using at times devices 



of great complexity and extreme refine- 

 ment, does he usually walk by sight or by 

 faith ? Does he not often go forward long 

 and painful journeys in utter darkness as 

 to the meaning of it all, trusting, hoping, 

 praying that by and by his teacher and his 

 text-book will land him on solid ground 

 and in the clear light to revel and operate 

 in a new world of thought and action? 

 How many men of good natural endow- 

 ments, who are sorely needed in the higher 

 ranks of the world's workers, become ter- 

 rified in this period of distressing gloom; 

 how many have lost individual initiative 

 and independence and are content thence- 

 forward to walk, not upright, vigorous, ag- 

 gressive, daring, in the clear light of right 

 reason, but by faith, humble and submis- 

 sive? 



Wliy do practical men almost unani- 

 mously place calculus among the dispen- 

 sable elements of a technical curriculum? 



The answer, of course, is very simple; 

 they have never found any use for it, prob- 

 ably because they have never learned how 

 to use it. Yet they dare not pronounce 

 against it altogether. They know that 

 Rankine and Maxwell were master mathe- 

 maticians, and that through this mastery 

 of the most powerful of tools they were 

 able to do for terrestrial what Newton and 

 Laplace did for celestial mechanics. In 

 college the engineer has not learned to u.se 

 the modern tool called the higher analysis ; 

 it remains to him as foreign currency. Out 

 of college he has not time to learn its use. 

 Are you a teacher of mathematics and did 

 you pursue the subject under the direction 

 of a master; yet how many classes did 

 you yourself guide through the calculus 

 before its hidden meaning, its x-ange, its 

 versatility, its power, were in any adequate 

 measure revealed to you? How simple 

 and how majestic it has now become ! But 

 if you were so slow in reaching the true 

 light, is it to be wondered at that students 



