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SCIENCE 



[N. S. Vol. XLI. No. 1056 



for example, made the doctrine of limits, 

 analytical geometry, projective geometry, 

 and the notions of the derivative and the 

 integral available for presentation to col- 

 lege freshmen or even to high-school pupils. 

 It is not to condemn that kind of popu- 

 larization which is so natural a process that 

 it actually goes on in a thousand ways all 

 about us without our deliberate coopera- 

 tion, without our intention or our consent, 

 and has enriched the common sense and 

 common knowledge of our time with count- 

 less precious elements from among the sci- 

 entific and philosophic discoveries made by 

 other generations of men. 



Finally it remains to mention the impor- 

 tant type of specialist in whom strongly 

 predominates the predilection for research 

 as distinguished from exposition. He 

 knows, as every one knows, that through 

 what is called practical applications of sci- 

 ence many a scientific discovery is made to 

 serve innumerable human beings who do 

 not understand it and innumerable others 

 who never can. He may or may not be- 

 lieve in avocational instruction; he may or 

 may not regard intelligence as an ultimate 

 good and an end in itself; he may or may 

 not think that the arts and agencies for the 

 dissemination of knowledge, as distin- 

 guished from the discovery and practical 

 applications of truth, are important; he 

 may or may not know that the art and the 

 gifts of the great expositor are as important 

 and as rare as those of the great investi- 

 gator and less often owe their success to the 

 favor of accident or chance. He may not 

 even have seriously considered these things. 

 He does know his own predilection ; and so 

 strong is his inclination towards research 

 that for him to engage in exposition, espe- 

 pecially in popular exposition, in avoca- 

 tional instruction for laymen, would be to 

 sin against the authority of his vocation. 

 This man, if he have intellectual powers 

 fairly corresponding to the seeming author- 



ity and urgence of his inner call, belongs to 

 a class whose rights are peculiarly sacred 

 and whose freedom must be guarded in the 

 interest of all mankind. It is not contended 

 that every representative of a given sub- 

 ject is under obligation to expound it for 

 the avocational interest and enlightenment 

 of laymen. The contention is that such 

 exposition is so important a service that 

 any university department should contain 

 at least one man who is at once willing and 

 qualified to render it. 



I come now to the keeping of my promise. 

 It is to be shown that the service is prac- 

 ticable in the subject of mathematics and 

 how it is so. Let us get clearly in mind 

 the kind of persons for whom the instruc- 

 tion is to be primarily designed. They are 

 to be students of "maturity and power"; 

 they do not intend to become teachers, 

 much less producers, of mathematics ; they 

 are probably specializing in other fields ; 

 they do not aim at becoming mathemati- 

 cians; their interest in mathematics is not 

 vocational, it is avocational; it is the inter- 

 est of those whose curiosity transcends the 

 limits of any specific profession or any 

 specific form or field of activity; each of 

 them knows that, whatever his own field 

 may be, it is penetrated, overarched, com- 

 passed about by an infinitely vaster world 

 of human interests and human achieve- 

 ments; they feel its immense presence, the 

 poignant challenge of it all; as specialists 

 they will win mastery over a little part, but 

 they have heard the call to intelligence and 

 are seeking orientation in the whole; this 

 they know is a thing of mind; they are 

 aware that the essential environment of a 

 scholar's life is a spiritual environment — 

 the invisible and intangible Avorld of ideas, 

 doctrines, institutions, sciences and arts; 

 they know or they suspect that one of the 

 great components of that world is mathe- 

 matics ; and so, not as candidates for a 

 profession or a degree, but in their higher 



