ACHILLES AND THE TORTOISE. 



to have any objective existence for us, too small 

 not only to be measured, but even to be ex- 

 pressed, mere figments of the imagination, 

 then the imagination can follow them into their 

 differences, and while recognising that they 

 are so small that neither they nor their differ- 

 ences can have any importance in practical 

 calculations, we can nevertheless conceive of 

 their being divided into parts, and then we 

 must necessarily conceive of the whole as being 

 greater than any of its parts. 



If, therefore, we admit as mathematical 

 conventions that infinitely small things have 

 no size and are equal to one another, these are 

 only brief and convenient ways of saying that 

 such things are too small for their sizes or 

 their differences of size to be worth taking into 

 account in any practical calculations. But 

 to found an argument upon them as exact 

 expressions of strict truth is inadmissible. 



Moreover, if it were true that infinitely 

 small things are all equal, this would tell against 

 the mathematician's explanation even more 

 decidedly than for it. For at each stage of 

 the race, the infinitely small distance that the 

 tortoise went ahead would be equal to the ten 

 times greater distance covered simultaneously 

 by his antagonist, and so he would still be the 

 same distance in advance. Not only could 

 Achilles never pass him, he could not even 

 keep gaining on him as before. 



271 



