July 31, 190S] 



SCIENCE 



135 



looked, that the fundamental ideas in- 

 volved in the mathematics and in the 

 mathematical physics essential to the pre- 

 liminary training of a prospective engi- 

 neer are far more difficult of compre- 

 hension than we are wont to suppose. As 

 a rule, I think we begin our elementary 

 mathematics somewhat too early for the 

 average mind. The result is that our 

 students acquire a mere literary knowledge 

 of the subject without grasping the basic 

 ideas essential to clear thought and espe- 

 cially essential to applications. I am go- 

 ing to give you some illustrations of this 

 fact. They will show how difficult it is 

 for the average mind to attain a proper 

 understanding of mathematico-physieal 

 concepts. The difficulties here are much 

 the same as the difficulties of grammar. 

 As you know, children learn to speak, and 

 often speak very well, long before they 

 know anything of formal grammar, and 

 this is the natural mode of development, 

 for the logic and subtleties of grammar 

 can be appreciated only by rather mature 

 minds. 



But if the concepts which belong to the 

 study of language and of grammar are 

 rather formidable, those which belong to 

 the higher mathematics and mathematical 

 physics are profoundly more difficult of 

 adequate comprehension. Let me illus- 

 trate this point by a citation from experi- 

 ence furnished by the case of a graduate 

 from one of our universities who pre- 

 sented himself to me a few years ago, while 

 I was dean of a graduate school of Co- 

 lumbia University, as a candidate for a 

 higher degree in mathematical physics. 

 This student had studied mechanics and 

 had attained a degree in engineering. In 

 order to learn something of the breadth 

 and depth of his knowledge, I asked him 

 what it is that makes the trolley car run 

 after the current is cut off. He answered, 

 "It is the force of the momentum of the 



power of the energy of the car." There is 

 no reason to suppose that he had not re- 

 ceived good mathematical and physical 

 training, and yet it is plain from the 

 answer he gave me that he knew next to 

 nothing of the meaning of the terms he 

 used. I may cite another case of a suc- 

 cessful practising engineer, who was a 

 pupil of no less authorities in mechanics 

 and engineering than Lord Kelvin and 

 Rankine. This man wrote me a letter in 

 which he sought to convince me that 

 Newton and his followers are all wrong 

 with regard to the parallelogram of im- 

 pulses. "Thus," he said in his letter, "if 

 a particle start out from a given point 

 under the simultaneous action of two im- 

 pulses, it will not move in the parallelo- 

 gram of the impulses, but it will move in a 

 tautochronous, brachistoehronic, plane cate- 

 nary curve of a resilient character." 



These illustrations show how extremely 

 difficult it is to master the fundamental 

 ideas which belong to a great science; and 

 the difficulties are so great that I am dis- 

 posed to excuse, or at any rate palliate, the 

 blunders made by our average student. 

 He is, in fact, with all his blunders, not 

 very far behind many of his teachers, for 

 it is not uncommon for them to use in 

 their lectures and text-books words not at 

 all free from ambiguity. "Witness, in fact, 

 the loose use of such words as force, power, 

 pressure, stress, and strain in some of the 

 best text-books and treatises of the nine- 

 teenth century. The word "power," for 

 example, is often used in two radically 

 different senses in the same sentence. 



These difficulties and ambiguities lead 

 me to suggest, in opposition to the precepts 

 laid down by a previous speaker, that we 

 may well consider the desirability of print- 

 ing mathematical books free from demon- 

 strations but containing plain statements 

 of facts. I have used such books myself 

 and am disposed to think they are amongst 



