OCTOBEE 26, 1917] 



SCIENCE 



397 



related quantity proceeds in geometric pro- 

 gression, is ordinarily not brought out, nor 

 is the fact that this same situation leads to 

 a logarithmic law. 



The omission of these and similar vital 

 connections, both of mathematics to the ex- 

 terior world and of one topic in mathe- 

 matics to another, is directly responsible for 

 the failure of algebra to reach the hearts 

 of our students, and for the failure of the 

 students to gain real insight into the sig- 

 nificance of the subjects they so dully learn. 



I shall not dwell long on any one topic, 

 for I desire to emphasize the existence of 

 significance for life and society in the en- 

 tire range of mathematical courses, and I 

 desire to call your attention to the failure — 

 shall I not say our failure? — to bring to 

 light that significance. 



Let me turn to analj'tic geometry for 

 another instance of our traditional blind- 

 ness, if it be that — our sin, if it is not blind- 

 ness. Here, as before, applications abound. 

 Most of the results of scientific experiment 

 to-day are known and are recorded not by 

 algebraic formulas of traditional form, 

 but solely by curves traced in our tradi- 

 tional stj'le, showing graphically the func- 

 tional relations between two or more inter- 

 dependent variables. Laws of physics, of 

 chemistry, of every quantitative science, 

 expressed by such means abound. The ef- 

 fort of science may well be said to be to de- 

 duce from such graphical functions the 

 corresponding laws in algebraic or formal- 

 istie form. 



Yet to most students of analytic geom- 

 etry, precisely the reverse view seems to be 

 our aim. The significance of analytic geom- 

 etry as a piece of scientific machinery is 

 totally lost, and the subject sinks to the 

 level of dubious value in the minds of our 

 students and of half-informed educators. 

 In the present emergencj', popular convic- 

 tion of the real significance of analytic 

 geometry for society is being attained, and 



may be fostered, through the occurrence of 

 just such graphical laws in the dynamics 

 of airplanes, in artillery performance (bal- 

 listics) and in wireless telegraphy. Here 

 as in general in science, most of our infor- 

 mation on functions is now in graphical 

 form, and the desire to express the func- 

 tion in equation form illustrates the funda- 

 mental demand of science, and the funda- 

 mental significance of analytic geometry. 



That the calculus is regarded as dry and 

 uninteresting by many students, and that 

 its value is occasionally doubted, is the 

 strongest proof possible that its significance 

 is not grasped. Here the connection with 

 realities is so easy and so abundant that it 

 is actually a skillful feat to conceal the fact. 

 Yet it is done. I know personally of courses 

 in the calculus (and so may you) in which 

 the pressure to obtain and to enforce mem- 

 ory of formal algebraic rules has resulted 

 in absolute neglect of the idea that a deriva- 

 tive represents a rate of change ! I know 

 students whose whole conception of inte- 

 gration is the formalistie solution of inte- 

 grals of set expression by devices whose 

 complexity you well know. That an in- 

 tegral is indeed the limit of a summation, 

 and that results of science may be reached 

 through such summation is often nearly 

 ignored and not at all appreciated. That 

 the ideas of the calculus should fall so low 

 as to consist mainly in formal differentia- 

 tions and integrations of set expressions 

 must indeed astound any one to whom the 

 wonderful significance of the subject is at 

 all known. Moreover, it must convince any 

 liberally minded educator who takes our 

 own courses as a true representation of 

 mathematical values that even the calculus 

 is of no importance for real life or for so- 

 cietj'. 



I might proceed to other courses — differ- 

 ential equations as given by Forsyth, the 

 theory of equations as by Burnside and 

 Panton or as by even the most recent 



