2 Ellery W. Davis 



hand end. Not only is it true that all decimal distances are 

 given, but it is also true that there is no distance whatever but 

 can be indefinitely approximated to by these decimal distances. 

 Take any distance you please, name any limit of accuracy you 

 please, and within that limit of your distance we can find a 

 decimal distance. 



Do, then, all decimal distances form a continuum? No, says 

 the mathematician, not if there be even a single distance that 

 fails of complete and absolute representation by a decimal. No 

 degree of approximation will suffice. Such a distance is the 

 distance one-third. Every decimal distance, though there be 

 millions on millions of figures used in its expression, is either 

 lai'ger or smaller than the distance one-third. 



The mathematician amplifies this statement. "The distance 

 divides," says he, "the collectivity all-decimal-distances into 

 two sets, such that, 



"1. Every distance Id the one set is larger than every dis- 

 tance in the other. 



" 2. There is no smallest distance in the sets of larger ones. 



" 3. There is no largest distance in the set of smaller ones.'' 



Every distance, however defined, which does this is a non- 

 decimal distance, lying between the two sets it divides, and is 

 the only distance lying between those sets. For, since it is 

 hemmed in as closely as you please between them, there will 

 always be between it and any other non-decimal distance some 

 decimal distances. If some, there are an infinite number, be- 

 cause between any two decimals, howsoever close together they 

 may be, there is an infinite number of other decimals. 



Of such non-decimal distances there is no end. We can get 

 them by dividing our thirds into thirds, these thirds of thirds 

 into thirds, and so on forever. Or we can, in like manner, 

 repeatedly divide into fifths or sevenths. 



Suppose we thus use as divisors all prime numbers. All 

 distances so gotten, save only when two and five are used, will 



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