132 THE REALITIES OF MODERN SCIENCE 



quotient of V 2 Vi and H 2 Hi. This is evidently 

 the slope of the dotted line extending from 1 to 2. As 

 the point 2 is chosen closer to 1 (that is, as the incre- 

 ment, H 2 HI, approaches zero), the dotted line be- 

 comes more nearly tangent to the curve at the point 

 1. The actual rate of change of V with respect to H, 

 therefore, at any given point is the average slope of a 

 tangent to the curve at this point. We may find the 

 rate graphically, therefore, as accurately as we can 

 construct the tangent. 1 In general, we may define 

 a rate as the limit approached by the ratio between 

 corresponding increments, in the dependent and in- 

 dependent variables re- 

 spectively, as the in- 

 crement of the latter 



approaches zero. 



In this particular case 

 of the slope of a road 

 the variables are both 

 t distances, which are 



o ti t z t a t 4 measured and plotted 



FlG - n - with reference to a com- 



mon reference point. A similar plot may be made for 

 a case involving unlike variables, e.g. that of the moving 

 automobile. For this we must assume reference points 

 both in time and in space. Corresponding to any time, 

 say t, subsequent to our assumed zero of time, the auto- 



1 Analytical methods of absolute accuracy are possible if the 

 form of the function, which one variable is of the other, may be 

 expressed in symbols. The methods are those of the calculus. 

 For example, in a function like s = at 2 /2 the rate of change of s 

 with respect to t (usually symbolized as ds/dt) may be shown to 

 be at. 



