350 THE PHILOSOPHY OF BIOLOGY 



We therefore think of the stone as moving in the 

 immediate vicinity of the point in the sense already 

 discussed. We say that the " immediate vicinity " is 

 an interval such that any point in it, p lt approximates 

 to the arbitrary point p which we are considering within 

 any standard of approximation : that is, no point in 

 the interval is further away from p than a certain 

 number expressing the standard of approximation, and 

 this can be any number, however small. We say the 

 same thing about the interval of time. That is to say, 

 we make the intervals as small as we like : they can be 

 smaller than any interval which will cause an error in 

 our deduced velocity, no matter how small this error 

 may be. 



The limit of the velocity of a stone falling past a 

 point in its path is, therefore, that velocity towards 

 which the mean velocities approximate within any 

 standard of approximation, when we regard the 

 interval of space as being the immediate vicinity of 

 the point, and the interval of time as being the time in 

 the immediate vicinity of the moment when the stone 



{S 



passes the point. The limit of the velocity is not ^ 



ol 



ds 

 but -j-, dt and ds being, not finite intervals of time and 



space, but " differentials." We determine this limit 

 by the methods of the differential calculus. 



FREQUENCY DISTRIBUTIONS AND PROBABILITY 



Let the reader keep a note of the number of 

 trumps held by himself and partner in a large 

 number of games of whist (the cards being cut for 

 trump). In 200 hands he may get such results as 

 the following : 



