282 LIMITS AND FLUXIONS 



velocity is ds / dt=^gt. At the end of the first 

 second, the velocity is ^^. If we ask ourselves, How 

 far does the body move with the velocity gì we 

 must admit that no distance can be assigned. We 

 cannot say that the body moves from a certain point 

 to the point immediately beneath ; there is no such 

 point immediately beneath. For, as soon as we try 

 to locate such a point, it occurs to us that we can 

 imagine at least one point located between the two 

 points under consideration. This intermediate point 

 serves our purposes no better, for a fourth point 

 located between it and the initial point is easily 

 detected, and so on, without end. Thus it is seen 

 that no distance, however small, can be assigned 

 through which a body falls with a given velocity. 

 We are thus compelled to reject variable velocity 

 as a physical fact. What, then, is ds j dt=gtì 

 Clearly it is merely a limit, a mathematical concept, 

 useful in mathematical analysis, but without physical 

 reality. To say that ds / dt represents the distance 

 a body would fall in unit time after the instant 

 indicated by /, is to assign it merely hypothetical 

 meaning, destitute of concreteness. While these 

 considerations in themselves may not debar the use 

 of velocity as a mathematical concept upon which to 

 build the calculus, they show that the concept is not 

 as simple as it would seem to be at first approach. 



The reader will have observed that in ali discus- 

 sion of limits during the eighteenth century the 

 question of the existence of a limit of a convergent 

 sequence was never raised ; no proof was ever given 



