Order of Numerical Values of Vulgar Fractions. 177 

 of the three neighbouring fractions in the first instance with 

 denominator 1 are, 2x(l-j gg ), 2, 2x(l+jjL) ; in the 



second instance, 3 x (I-'jqq); 3 > 3 x h+—\. i n the third 



instance, 4 x (l"^)) 4, 4 x (l + ^J, &c. And with deno- 

 minator 2, the values in the first instance are - x (\ — ^ - 



2 (, 200; 



2 X ( 1 + 20o) ; in the S6C0nd instance ; \ x (}-^) 



) ay 



_\ 5 



x 



200/ 2 J 



U + 2QQh&c. And so for the other denominators. And, 



between the values thus set down, there is no other value of a 

 vulgar fraction whose numerator and denominator do not ex- 

 ceed 100. 



Now the remarkable circumstance attending these large 

 logarithmic differences is, that they all occur in the midst of 

 small differences. Thus we have 



45 

 91 

 46 

 93 

 47 

 95 

 48 

 97 

 49 



99 

 50 

 2 

 1 



99 

 49 



97 



48 



95 

 47 

 93 

 46 

 91 

 45 



•29618 

 •29628 

 •29638 

 •29648 

 •29657 

 •29667 

 •30103 

 •30544 

 •30553 

 ♦30562 

 •30572 

 •30583 



•00010 

 •00010 

 •00010 

 •00009 

 •00010 

 •00436 

 •00441 

 •00009 

 •00009 

 •00010 

 •00011 



and so for the others. Thus it appears that, though in general 



