THE THEORY OF CONTINUITY 137 



time /, it must be possible to find two instants t u / 2 , one 

 earlier than / and one later, such that throughout the 

 whole time from t x to t t (both included), the particle lies 

 between P x and P 2 . And we say that this must still 

 hold however small we make the portion P x P 2 . When 

 this is the case, we say that the motion is continuous at 

 the time / ; and when the motion is continuous at all 

 times, we say that the motion as a whole is continuous. 

 It is obvious that if the particle were to jump suddenly 

 from P to some other point Q, our definition would fail for 

 all intervals P x P 2 which were too small to include Q. 

 Thus our definition affords an analysis of the continuity 

 of motion, while admitting points and instants and 

 denying infinitesimal distances in space or periods in 

 time. 



Philosophers, mostly in ignorance of the mathe- 

 matician's analysis, have adopted other and more heroic 

 methods of dealing with the prirnd facie difficulties of 

 continuous motion. A typical and recent example of 

 philosophic theories of motion is afforded by Bergson, 

 whose views on this subject I have examined elsewhere. 1 



Apart from definite arguments, there are certain 

 feelings, rather than reasons, which stand in the way of 

 an acceptance of the mathematical account of motion. 

 To begin with, if a body is moving at all fast, we see its 

 motion just as we see its colour. A slow motion, like that 

 of the hour-hand of a watch, is only known in the way 

 which mathematics would lead us to expect, namely by 

 observing a change of position after a lapse of time ; but, 

 when we observe the motion of the second-hand, we do 

 not merely see first one position and then another we 

 see something as directly sensible as colour. What is 

 this something that we see, and that we call visible mo- 



1 Monist) July 19 12, pp. 337-341. 



