0)1 the Contimiity of Chance 3 



be Don -decimal distances. No distance gotten by any divi- 

 sor will be the same as a distance gotten by any other divisor. 

 To each distance corresponds a distinct point, and between any 

 two points due to a divisor, howsoever close together they may 

 be, will lie an infinite number due to that same divisor, and 

 also an infinite number due to each and every other divisor. 

 Each of this infinite number of sets of points lies on the line 

 distinct from every other set. 



Do all these points taken together form a continuum ? Not 

 if we can find a single point at another distance. Such a 

 point there is, distant one -sixth from the left hand end of our 

 line. And it is only a sample of an infinity of points that are 

 not yet marked. 



Let us, then, use all composite numbers for divisors as we 

 have used all prime numbers. We thus get all fractional 

 distances. 



And now have we a continuum? Or can still a distance be 

 found at which no point of division lies? Euclid showed that 

 there could. For example, the distance equal to the side of a 

 square, of which our inch-long line is the diagonal, would be 

 such. 



To see this, suppose there were a fraction a' h that gave the 

 distance; moreover, let it be in its lowest terms. Then tAvice 

 the square of the fraction would be unity; i. e., we should have 



Then a^ is an even number, and so a is even while h is odd. 

 Suppose o=2c, which gives 



a'=4iC" and hence 2c'=6. 



Thus the odd number 6 is also even ; or else, as stated above, 

 the distance equal to the side of the square is not given by a 

 fraction. ^ 



This is but a sample. We can, in fact, prove that between 

 any two points at fractional distances, howsoever close together, 

 can be put an infinite number at non-fractional distances. 



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