PHYSIOLOGICAL PARADOXES. 



twentieth part of a yard one inch and four- 

 fifths. At the end of the fourth stage, his 

 lead is reduced to a very little less than one- 

 fifth of an inch. 



This goes on indefinitely. At the end of 

 each stage the tortoise's lead, however much 

 reduced, still exists ; and during the next 

 stage, while Achilles traverses the space which 

 constitutes this lead, the tortoise is again 

 getting one tenth of that distance in advance. 

 As he is able to do this at the end of every 

 stage, however great the number of the stages 

 may be, it follows that he can always keep in 

 front by some little distance, however infini- 

 tesimally small ; in other words, Achilles can 

 never quite catch him up, much less pass 

 him by. 



It was the unexpected appearance of this 

 theoretical difficulty that so struck the cham- 

 pion of a heap. He was presumably stronger 

 in the heels than in the head (though, indeed, 

 one of his heels was the weakest part of his 

 frame), so that the necessity of facing a mental 

 difficulty of this magnitude overwhelmed him 

 and left him so helpless that the tortoise had 

 time to reach the goal. Otherwise, the most 

 that the latter competitor could have effected 

 would have been to make a draw ; since an 

 infinite number of stages, each one-tenth of that 

 preceding it, could never have added up to 

 the half of the course, which lay between his 



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