ACHILLES AND THE TORTOISE. 



starting-point and the goal. He could never 

 have got more than a fractionally small dis- 

 tance beyond E in the diagram. 



Now, the question propounded is what 

 is the fallacy in the foregoing argument which 

 professes to prove that Achilles, travelling 

 ten times as fast as the tortoise, could never 

 overtake it ? What is the precise point in 

 which it goes wrong ? 



Learned mathematicians, when asked this 

 question, usually give a reply which, in one 

 form or another, amounts to the following argu- 

 ment. 



Let it be granted that Achilles can run the 

 distance in x seconds (x would be about ten if 

 ancient times were something like as good as 

 modern). Then, since Achilles is ten times as 

 swift as the tortoise, the latter will require 

 TOX seconds for the course, and therefore 5# 

 seconds for half the course, which is the actual 

 distance he has to go. But 5 x is greater than 

 x, therefore he will arrive later than Achilles. 

 That is, Achilles must win. Therefore the 

 above argument to prove that he cannot pass 

 the tortoise must be wrong. 



Wonderful mathematician ! Even the way- 

 faring man, with the well-known qualification, 

 could not miss his way to that conclusion. 

 It is plain to all of us that Achilles will win. 

 We are none of us in danger of putting our 

 money on the tortoise. We know that the 



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