130 THE REALITIES OF MODERN SCIENCE 



size the latter separation by saying that we measure the 

 distance traversed between two points in time. The 

 ratio of the distance to the time interval is the average 

 velocity. As the time interval is reduced this ratio 

 becomes more and more nearly representative of the 

 instantaneous rate. Its value, in other words, ap- 

 proaches the value of the instantaneous rate as a limit, 

 while the time interval is caused to approach zero as 

 a limit. 



As the time interval between observations is reduced 

 the distance traversed is also reduced. The fact that 

 both the numerator and the denominator of the ratio 

 are thus made " infinitesimals " does not mean that the 

 ratio may not have a perfectly definite and finite value. 

 Of this idea, that a ratio of two magnitudes may be 

 finite even though the magnitudes concerned are physi- 

 cally very small, we have previously had an illustration 

 in our comparison of the distance ratios in the solar 

 system to the distance ratios of the electronic systems 

 of molecules. Physically speaking, with reference to 

 the diameter of the earth the diameter of an electron is 

 infinitesimal. To a mathematician, however, an in- 

 finitesimal is an abstraction, representing a quantity 

 indefinitely small, which approaches zero as its limiting 

 value. 



The fundamental ideas involved in the concept of a 

 rate are conveniently illustrated also in the case of the 

 slope or grade of a road. Such a slope is usually 

 expressed as a ratio of the vertical distance to the 

 corresponding horizontal distance, e.g. as a two per 

 cent grade. It is obviously the rate at which one 

 ascends with respect to his horizontal displacement. 



