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SCIENCE 



[N. S. Vol. XXVIII. No. 709 



the best books we may place in the hands 

 of a student. The simple fact is that we 

 do not follow a logical order of develop- 

 ment in acquiring knowledge. We pro- 

 ceed rather by the method of "trial and 

 error," and we often find out the facts 

 with regard to an item of learning long 

 before we become aware of the principle 

 involved. 



Hence I think the reason why few of our 

 engineers know much about the formalities 

 of mathematics and mathematical physics 

 after they get through college is plain 

 enough. They are driven over so many 

 subjects during the four years of their 

 college life that they have little or no time 

 for reflection. This latter must come later 

 in life when the mind has developed a suf- 

 ficient degree of maturity to appreciate 

 the more recondite principles which lie at 

 the foundation of all the higher learning. 

 This fact is well illustrated also by the 

 case of our friends, the humanists, who 

 have, as you know, for a long time pro- 

 posed the study of geometry for "mental 

 discipline." As a matter of fact, those 

 who have acquired anything like a grasp 

 of geometrical principles know that very 

 few students of Euclidean geometry ac- 

 quire anything like an adequa.te appre- 

 ciation of the ideas involved, and it 

 is only in the rarest instances that these 

 students pursue the subject after leaving 

 college. 



I have not much sympathy with the 

 engineers who would like to have their 

 own kind of mathematics, and I am not 

 disposed to commend very highly the 

 works on calculus and other branches of 

 pure mathematics designed especially for 

 engineers. On the other hand, our modern 

 mathematicians have generally failed to 

 imderstand the needs of the engineer. 

 Our more recent type of mathematician 

 has devoted himself too largely to the re- 

 fined questions of convergence and diverg- 



ence of series and of existence theorems to 

 properly equip him for the numerous and 

 important applications which the ideal 

 engineer should be able to make of his 

 mathematical knowledge. The modern 

 mathematician seems prone to make the 

 engineer with some degree of mathematical 

 talent afraid of himself. I have met some 

 students whose early training had filled 

 them with caution to such a degree that 

 they would not use infinite series for fear 

 that a divergent one might be encountered. 

 It is known, however, as a matter of fact, 

 that most series essential in the applica- 

 tions of mathematics to mathematical 

 physics are safe in this regard, and one 

 of the best ways for the elementary student 

 to learn of the degree of convergence is 

 to apply numerical computation to these 

 series. 



This leads me to say a few words con- 

 cerning numerical computations, in which 

 very few engineers and still fewer mathe- 

 maticians show any degree of proficiency. 

 It seems to me this is one of the most 

 lamentable defects of our elementary teach- 

 ing in mathematics, though here as else- 

 where the intrinsic difficulties are much 

 greater than we commonly suppose. This 

 fact is in evidence at almost every meeting 

 of our scientific societies, for it oftenest 

 happens that the author of a paper involv- 

 ing numerical calculation will talk of the 

 decimals involved instead of the significant 

 figures. Thus, he will say, "this result is 

 correct to five places of decimals," when 

 he should say, "this result is correct to a 

 specified number of significant figures," 

 the latter form of expression being requi- 

 site to indicate the degree of precision at- 

 tained. There is a grave defect in our 

 elementary teaching in these matters; but 

 it arises from the fact that almost none of 

 our teachers of elementary mathematics 

 are qualified to understand the refinements 

 and the difficulties of precision in compu- 



