March 26, 1915] 



SCIENCE 



451 



capacity as men and women, they desire to 

 learn something of this science viewed as 

 a human enterprise, as a body of human 

 achievements ; and they are willing to pay 

 the price; they are not seeking entertain- 

 ment, they are prepared to work — to listen, 

 to read and to think. 



And now we must ask : What measure of 

 mathematical training is to be required of 

 them as a preparation? In view of what 

 has just been said it is evident that such 

 training is not to be the whole of their 

 equipment nor even the principal part of 

 it, but it is an indispensable part. And 

 the question is: How much mathematical 

 knowledge and mathematical discipline is 

 to be demanded ? I have no desire to min- 

 imize my present task. I, therefore, pro- 

 pose that only so much mathematical prep- 

 aration shall be demanded as can be gained 

 in a year of collegiate study. Most of them 

 will, of course, have had more; but I pro- 

 pose as a hypothesis that the amount 

 named be regarded as an adequate mini- 

 mum. But it does not include the differ- 

 ential and integTal calculus. And is it not 

 preposterous to talk of offering graduate 

 instruction in mathematics to students who 

 have not had a first course in the calculus ? 

 I am far from thinking so. A little reflec- 

 tion will suffice to show that in the case of 

 such students as I have described it is very 

 far from preposterous. In my opinion the 

 absurdity would rather lie in demanding 

 the calculus of them. No one is so foolish 

 as to contend that a first course in the 

 calculus is a sufficient preparation for 

 undertaking the pursuit of graduate mathe- 

 matical study. But to suppose it necessary 

 is just as foolish as to suppose it sufficient. 

 There was a time when it was necessary, 

 and the belief that it is necessary now owes 

 its persistence and currency to the inertia 

 then acquired. Formerly it was necessary, 

 because formerly all advanced courses, at 



least all initial courses of the kind, were 

 either prolongations of the calculus, like 

 differential equations, for example, or else 

 courses in which the calculus played an 

 essential instrumental role as in rational 

 mechanics, or the usual introductions to 

 function theory or to higher geometry or 

 algebra. But, as every mathematician 

 knows, that time has passed. It is true that 

 courses for which a preliminary training in 

 the calculus is essential still constitute and 

 will continue to constitute the major part 

 of the graduate offer of any department of 

 mathematics. And quite apart from that 

 consideration, it seems wise, in the case of 

 intending graduate students who pvirpose 

 to specialize in mathematics, to enforce the 

 usual calculus requirement as affording 

 some slight protection against immaturity 

 and the lack of seriousness. But every 

 mathematician knows that it is now prac- 

 ticable to provide a large and diversified 

 body of genuinely graduate mathematical 

 instruction for which the calculus is strictly 

 not prerequisite. 



Fortunately it is just the material that 

 is thus available which is in itself best 

 suited for the avocational instruction we 

 are contemplating. As the calculus is not 

 to be presupposed it goes without saying 

 that this subject must find a place in the 

 scheme. For evidently an advanced mathe- 

 matical course devised and conducted in the 

 interest of general intelligence can not be 

 silent respecting "the most powerful 

 weapon of thought yet devised by the wit 

 of man. ' ' Technique is not sought and can 

 not be given. The subject is not to be pre- 

 sented as to undergraduates. For the most 

 part these gain facility with but little com- 

 prehension. It is to be presented to mature 

 and capable students who seek, not facility, 

 but understanding. Their desire is to ac- 

 quire a general conception of the nature of 

 the calculus and of its place in science and 



