108 Irving Fisher — Mathematical investigations 



" mechanics " proper a few years later the same boy studies " fallings 

 bodies " he finds it helpful to use the formula v :=:gt which contrasts- 

 with the preceding formula only in that space (s) is replaced by 

 space per unit of time (v) and velocity (u) by velocity acquired per 

 unit of time (g). The increased complexity of the magnitudes 

 makes a formula relatively desirable. Yet for some minds the latter 

 formula is of no use. Experience in teaching this very subject has 

 convinced me that there are a few who understand it better without 

 the aid of the formula, but they are just those individuals whose 

 comprehension of the relations involved is the vaguest and the 

 weakest. 



The formulae, diagrams and models are the instruments of higher 

 study. The trained mathematician uses them to clarify and extend 

 his previous unsymbolic knowledge. When he reviews the mathe- 

 matics of his childhood, the elementary mechanics is to him 

 illumined by the conceptions and notation of the calculus and qua- 

 ternions. To think of velocity, acceleration, force, as fluxions is not 

 to abandon but to supplement the old notions and to think of 

 momentum, work, energy, as integrals is greatly to extend them. 

 Yet he is well aware or ought to be that to load all this on the 

 beginner is to impede his progress and produce disgust. So also the 

 beginner in economics might be mystified, while the advanced 

 student is enlightened by the mathematical method. 



§4. 



The utility of a mathematical treatment varies then according to 

 the characteristics of the u.ser, according to the degree of his mathe- 

 matical development and according to the intricacy of the subject 

 handled. There is a higher economics just as there is a higher 

 phys'cs, to both of which a mathematical treatment is appropriate. 

 It is said that mathematics has given no new theorems to economics. 

 This is true and untrue according to the elasticity^ of our terms- 

 The challenge of Cairnes might be answered by a counter challenge 

 to show the contents of Cournot, Walras, or Auspitz und Lieben in 

 any non-mathematical writer. 



If I may venture a speculation, those who frown on the mathe- 

 matical economist because he " wraps up his mysterious conclusions 

 in symbols" seem to me in some cases to point their finger at those 

 " conclusions " which when " unwrapt " of symbols they recognize as 

 old friends and lustily complain that the}" are not new; at the same 

 time they seem to ignore completely those " mysterious " conclu- 



