ACHILLES AND THE TORTOISE. 



possible smallness of distances in actual fact, 

 though not in our imagination ; and it may be 

 that when the distance between the runners has 

 been reduced to a certain degree of smallness, 

 it will be impossible for any smaller distance to 

 exist, and so the scheme on which we have made 

 the tortoise able to keep in front becomes im- 

 possible. 



A diagram will make the matter clearer. 

 Take the race at a point where Achilles has so 

 nearly overtaken the tortoise that the distance 

 between them is so excessively small as to be 



i 1 . . > i j i i t M. 



B 'A T DC 



FIG. 60.-8PACE AND LIMITS OF DIVISIBILITY. 



approaching the size which we call infinitely 

 small. In order to examine the state of things, 

 then, we shall have to view a part of the course 

 enormously magnified, so that it may appear, 

 as in the diagram, some millions of times its 

 actual length. Let A (Fig. 60) be the position 

 of Achilles, and T that of his competitor. 

 While the tortoise travelled his last stage from 

 A to T, Achilles had travelled ten such stages 

 namely, B A and nine equal stages preceding 

 it. Then, says the argument, while Achilles 

 travels the next equal stage AT, the tortoise 

 goes on one-tenth of that distance from T to D. 

 Quite right. And then again, proceeds the 

 argument, while Achilles travels from T to D, 



s 273 



