THE PROBLEM OF INFINITY 177 



sider, the three men B, B', B" in one row, and the three 

 men C, C, C" in the other row, are respectively opposite 

 to A, A', and A". At the very next moment, each row 

 has moved on, and now B and C" are opposite A'. Thus 

 B and C" are opposite each other. When, then, did B 

 pass C ? It must have been somewhere between the 

 two moments which we supposed consecutive, and there- 

 fore the two moments cannot really have been consecutive. 

 It follows that there must be other moments between any 

 two given moments, and therefore that there must be an 

 infinite number of moments in any given interval of time. 

 The above difficulty, that B must have passed C at 

 some time between two consecutive moments, is a genuine 

 one, but is not precisely the difficulty raised by Zeno. 

 What Zeno professes to prove is that " half of a given 

 time is equal to double that time." The most intelligible 

 explanation of the argument known to me is that of 

 Gaye. 1 Since, however, his explanation is not easy to 

 set forth shortly, I will re-state what seems to me to be 

 the logical essence of Zeno's contention. If we suppose 

 that time consists of a series of consecutive instants, and 

 that motion consists in passing through a series of con- 

 secutive points, then the fastest possible motion is one 

 which, at each instant, is at a point consecutive to that 

 at which it was at the previous instant. Any slower 

 motion must be one which has intervals of rest inter- 

 spersed, and any faster motion must wholly omit some 

 points. All this is evident from the fact that we cannot 

 have more than one event for each instant. But now, in 

 the case of our A's and B's and Cs, B is opposite a fresh 

 A every instant, and therefore the number of A's passed 

 gives the number of instants since the beginning of the 

 motion. But during the motion B has passed twice as 



1 Loc. cit., p. 105. 



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