348 THE PHILOSOPHY OF BIOLOGY 



We try to avoid the terms " infinitely small," " infi- 

 nitely near/' "infinitely small quantities," and so on, 

 by the device of standards of approximation. It may 

 appear to the non-mathematical reader that all this is 

 rather to be regarded as " quibbling," but the success 

 of the methods of mathematical physics should convince 

 him that such is not the case. He should also reflect 

 that clear and definite ideas on the fundamental 

 concepts of the science are just as necessary in specu- 

 lative biology as they are in mathematics. 



(Another example.) 



Let us consider the case of a stone falling from a 

 state of rest. Observations will show that when the 

 stone has fallen for one second it has traversed a space 

 of 1 6 feet ; at the end of two seconds it has fallen 

 through 64 feet ; and at the end of three seconds the 

 space traversed is 144 feet. From these and similar 

 data we can deduce the velocity of motion of the stone 

 as it passes any point in its path. 



The velocity is the space traversed in a certain time 



-. If we take any easily observable space (say five 



feet) on either side of the point chosen, and then deter- 

 mine the times when the stone was at the extremities 

 of this interval, and divide the interval of space by the 

 interval of time, we shall obtain the average velocity 

 of motion of the stone over this fraction of the whole 

 path chosen. But the velocity did not vary in a 

 constant manner during this interval (as we see by 

 considering the spaces traversed during the first three 

 seconds of the fall). Therefore our average velocity 

 does not accurately represent the velocity of the stone 

 as it passes the point at the middle of the path chosen. 

 We therefore reduce the length of the path more and 

 more so as to make the average velocity approximate 



