4 Ellerij W. Davis 



Accordingly, we can say of these non-fractional distances on 

 the line, that they divide all fractional distances into two sets, 

 such that, 



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

 tance in the other. 



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



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

 But perhaps some one thinks that, if we were to put points 



at distances given by all integral roots of fractional distances, 

 we should get all distances — should have a continuum. At least 

 he may think that, if we were to use all sums, diflPerences, pro- 

 ducts, and ratios of these roots, we should have a continuum. 

 Let him be undeceived. A more general expression than his 

 would be, "root of an equation with rational co-efficients." It 

 has in recent years been shown that there are numbers, of 

 which our familiar friend 77 is a sample, that can be roots of no 

 such equation. Such, too, is the basa of natural logarithms, 

 and all numbers that have rational natural logarithms. These 

 are transcendents, and between any two other numbers, howso- 

 ever close together, there is an infinity of these — ,an eight-fold 

 infinity, says Herr Klein, but, as we shall show, an eight-fold 

 or a million-fold infinity is merely an infinity. 



Is there no law of distribution that gives us all distances on 

 our line? Can there not at any rate be some combination of 

 laws that will do it? The latter is only an apparent generali- 

 zation. A combination of laws is, after all, only a law [M. La 

 Place would symbolize all the relations of the universe by a 

 single equation] ; and what is the essence of law ? Why, simply 

 definition. It separates what comes under it from what does 

 not. A law of positions on our line must, then, in order to be 

 a law, fail to give some. It cannot at once single out and yet 

 give all. To get all points, you must be allowed to take them 

 where you please, at random, by chance. Thus, absolutely 

 chance distribution must enter into any complete and perfect 



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