PHYSIOLOGICAL PARADOXES. 



operators, one of whom can operate twice as 

 fast as the other, so that his spot can advance 

 two yards while the other advances one ; and 

 let them have a race across the field, the fast 

 one beginning at the side and the slow one with 

 a start of less than one quarter of the distance 

 to be traversed. Since the latter moves only 

 half as fast as the other, it cannot move three- 

 quarters of the distance while the fast one 

 goes right across ; and it is therefore certain 

 to be beaten, though the fallacious argument 

 against Achilles might be adopted as follows : 



Suppose the slow one has a start of thirty- 

 two yards ; then, while the fast one covers this 

 distance, the slow one gets another start of 

 sixteen yards. While the fast one covers this 

 sixteen yards, the slow one goes eight yards 

 further in advance, and so on continually, 

 always getting a further start equal to half 

 the distance which separated them at the com- 

 mencement of each stage. 



No. When the distance between them is 

 reduced to one yard, the slow one cannot after- 

 wards get half a yard in advance, because there 

 are no half-yard boards. What happens is 

 that while the slow one goes another yard, the 

 fast one goes two and gets level. For the next 

 yard of the slow one the fast one goes two, and 

 is well ahead. 



Thus we see that, by accepting a limit to 

 the divisibility of distance or space, we are 



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