6 Ellery W. Davis 



line been a mile or a thousand miles long instead of one inch, 

 the same correspondence could have been effected; nor would 

 we have had to use any numbers save those between zero and 

 one. But, if we can have a one-to-one correspondence be- 

 tween the numbers zero to one, and either all the points on 

 the inch line or all those on the thousand-mile line, then also 

 we can establish a one-to-one correspondence between the 

 points of the two lines. This correspondence is easily shown 

 geometrically. On the thousand-mile line as base construct 

 a triangle and fit the inch line within the vertical angle. Im- 

 agine now a ray through the vertex to sweep over the two 

 lines. It passes over every point of each and in every posi- 

 tion joins a distinct pair of points on the two. 



We can even establish a one-to-one correspondence between 

 the points of an infinite line and our inch line. For, bend the 

 inch line into a circle and let a tangent ray roll around it. 

 As the ray rolls, it sweeps over the whole of any infinite line 

 without the circle, and in each position joins a distinct pair of 

 points on line and circle. 



More than this. We can establish a one-to-one correspond- 

 ence between the points on the inch line and all the points on 

 an infinite number of infinite lines. To do this, bend the line 

 into a spiral, taking half the line for the first turn, a quarter 

 of the line for the second turn, and an eighth for the third, and 

 so on. We get an infinite number of turns, and between the 

 points of each and those of an infinite line a one-to-one corre- 

 spondence. Let the infinite lines be successive parallels 

 forming a continuous surface, and we have a one-to-one corre- 

 spondence between the points of the inch and all the points of 

 an infinitely extended plane. 



Equally well can we have a one-to-one correspondence be- 

 tween the points on the inch line and all the points in 

 infinite three-way space. In fact, each of the infinite number 

 of turns of the spiral can itself be bent into an infinite number 



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