396 



SCIENCE 



[N. S. Vol. XLVI. No. 1191 



forum for the discussion of the problems 

 of collegiate instruction in mathematics. 



As retiring president of the association, 

 I know of no more fitting topic than that I 

 have chosen. It vitally concerns us; it is 

 bound up with the functions of this asso- 

 ciation; and the times in which we live 

 seem to point forcibly toward its considera- 

 tion. I shall attempt to outline to you my 

 own views on the true significance of 

 mathematics, and to sketch what I for one 

 would be glad to see this association pro- 

 mote. 



In speaking of the significance of mathe- 

 matics, I understand that we mean not at 

 all the baser material advantage to the in- 

 dividual student, not at aU a narrow utili- 

 tarianism, but rather a comprehensive 

 grasp of the usefulness of mathematics to 

 society as a whole, to science, to engineer- 

 ing, to the nation. Any narrower view 

 would be unworthy of us ; any narrower de- 

 mand by educators means a degraded view 

 of the purposes of education in a democ- 

 racy. 



Especially under the stress of war, pub- 

 lic attention may be secured for the real 

 claim of mathematics as a public necessity, 

 not only to be employed by a few special- 

 ists, but also to influence and to determine 

 the conduct and the efficiency of thou- 

 sands. 



Thus a knowledge of trigonometry and 

 of the trigonometric theorems of geometry 

 is a prime requisite for the successful and 

 efficient conduct of our armies, not only 

 by a few engineers who are to make maps 

 and to train artillery, but also for all offi- 

 cers to whom the lives of men are en- 

 trusted. Any one of these officers, cut off 

 with his force, without a superior engineer 

 at hand, may lose his position and the lives 

 of his men if he is ignorant of the signifi- 

 cance of these propositions. Ignorance at 

 such a crisis would be next to treason; it 

 would be incompetence. 



Do we, in trigonometry, so bring out the 

 significance of the fundamental ideas on 

 right triangles that the officer who faces 

 such a test will sense the possibility of find- 

 ing a range, or estimating a distance, with- 

 out help and without instruments or tables ? 

 Frankly I do not believe that we have been 

 doing this, even in such a practical subject 

 as trigonometry. We have been too often 

 content, and too often solely seeking, even 

 here, the knowledge of intricate formalisms, 

 of formulas and rules and theorems, of 

 operations done mechanically. Too often 

 we have omitted, even here, to give insight 

 into the rather obvious significance of 

 these rules and formulas. 



On the whole, however, trigonometry is 

 the one subject in which some small meas- 

 ure of insight has usually been secured. 



If I now turn to other topics of our cur- 

 riculum, may I not name scores of equally 

 vital topic, usually studied by our students, 

 in which insight is rarely gained? Let me 

 mention some such instances: 



In algebra, as taught in colleges, among 

 the topics always considered are fractional 

 exponents, logarithms and arithmetic and 

 'geometric progression. To many, frac- 

 tional exponents remain a pure formalism, 

 learned by rote and unappreciated, con- 

 nected neither with the other topics just 

 mentioned nor -with any realities of life. 

 That fractional exponents occur in expres- 

 sions for air-resistance (as in airplanes), in 

 water resistance (as in measuring stream- 

 flow), in electricity (as in induction), 

 would surprise most students who pass our 

 courses. That these exponents are determi- 

 nable and are determined by logarithms 

 would surprise students and some teachers, 

 even if the essential equivalence of expo- 

 nents and logarithms is adequately empha- 

 sized. The idea of a compound interest 

 law, namely, that one quantity may pro- 

 ceed in arithmetic progression as another 



