172 SCIENTIFIC METHOD IN PHILOSOPHY 



any distance, however small, can be halved. From this 

 it follows, of course, that there must be an infinite 

 number of points in a line. But, Aristotle represents 

 him as arguing, you cannot touch an infinite number of 

 points one by one in a finite time. The words " one by 

 one' are important, (i) If all the points touched are 

 concerned, then, though you pass through them continu- 

 ously, you do not touch them " one by one." That is 

 to say, after touching one, there is not another which you 

 touch next : no two points are next each other, but 

 between any two there are always an infinite number of 

 others, which cannot be enumerated one by one. (2) 

 If, on the other hand, only the successive middle points 

 are concerned, obtained by always halving what remains 

 of the course, then the points are reached one by one, 

 and, though they are infinite in number, they are in fact 

 all reached in a finite time. His argument to the contrary 

 may be supposed to appeal to the view that a finite time 

 must consist of a finite number of instants, in which case 

 what he says would be perfectly true on the assumption 

 that the possibility of continued dichotomy is undeniable. 

 If, on the other hand, we suppose the argument directed 

 against the partisans of infinite divisibility, we must 

 suppose it to proceed as follows : x " The points given by 

 successive halving of the distances still to be traversed 

 are infinite in number, and are reached in succession, each 

 being reached a finite time later than its predecessor ; 

 but the sum of an infinite number of finite times must be 

 infinite, and therefore the process will never be completed." 

 It is very possible that this is historically the right inter- 

 pretation, but in this form the argument is invalid. If 

 half the course takes half a minute, and the next quarter 



1 C/ Mr C. D. Broad, " Note on Achilles and the Tortoise," Mind, N.S., 

 vol. xxii. pp. 318-9. 



