324 NEW SERIES for the 



continually approach when m and n are conceived to increafe 

 indefinitely, and to which each at laft comes nearer than by 

 any aflignable difference, juft as a circle is the limit to all the 

 polygons which can be infcribed in it, or defcribed about it. 



56. The two expreflions n (x" — 1), m (b m — 1), which en- 

 ter into thefe limits to the logarithm of x, and which are evi- 

 dently functions of the fame kind, have each a finite magni- 

 tude even when m or n is confidered as greater than any aflign- 

 able number ; for fince when v is greater than unity, and p any 

 whole pofitive number, we have 

 1 1 

 p (y P — 1) < v — 1, p (v p — 1) > 1 (Art. 53.) 



Therefore, fuppofing x and b both greater than 1, (which may 



always be done in the theory of logarithms), the expreffion 



1 



n {x — 1) is neceflarily contained between the limits x — 1 



1 



and 1 — - ; and in like manner, m (b"' — 1) is between b — 1 



x 



and 1 — -. 



57. As the expreffion m {b — 1) depends entirely upon the 



value of b, the number whofe logarithm is afiumed iz 1, (and 

 which is fometimes called the basis of the fyftem), the li- 

 mit to which it approaches when m increafes indefinitely will 

 be a conftant quantity in a given fyftem ; but the limit to 



which 



