44 UNIVERSITY OF MISSOURI BULLETIN 



but it seems most logical to do so. The notion of a function 

 is none other than the very common notion of the interdepen- 

 dence of quantities which occur in everyday life, — in other 

 words, the all-pervading question of cause and effect, or more 

 exactly, the effect produced by a given cause. In science, 

 quantites which depend upon each other occur in practically 

 every experiment. The length of a pendulum determines 

 the time of vibration, the thickness of a wire determines the 

 musical note which it will produce, other circumstances re- 

 maining the same : — in fact, every scientific question must 

 have to do with dependence, with cause and effect. 



This notion reduced to its utmost simplicity is the function 

 notion of mathematics. I may say roughly that one thing is a 

 function of another if it depends upon that other. Thus, 

 the time of vibration of a pendulum is a function of its length ; 

 the price of wheat is a function of the amount harvested; 

 and so on. In fact, all of the statistics and other quantities 

 so far considered are functions of something else, and the 

 geometrical figures for temperature, for price of wheat, etc., 

 suggested above are nothing but the illustration of the growth 

 and fall of some quantity depending upon some other: — that 

 is, they are geometrical pictures of functions. 



In mathematics proper we study a few very special func- 

 tions of the simplest type which we can devise, in order to 

 find out something about the general properties of functions, 

 about the general properties of things which depend on other 

 things. Thus, we study x-, which clearly depends on x; and 

 we call x^ a function of x. We study other powers, and other 

 longer expressions, and we go further to study different kinds 

 of variation, as when we say one thing varies as another, — 

 for example, that the force exerted on a spring varies as 

 its extension. The common impression is that variation is 

 always equivalent to proportion, which we sometimes call 

 linear variation, and it is commonly expected that everything 

 will work according to the rule of three. If one pound of 



