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 ds a circle is the limit to all the 
polygons which can be infcribed in it, or defcribed about it. 
T Tt 
56. THE two expreflions (x — 1), m(b"—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 » 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 
Ws I 
ee =e r) Ey pee a * (ot) S 1—*, (Art. 53+) 
Therefore, fuppofing x and 4 both greater than 1, (which may 
always be done in the theory of logarithms), the expreffion 
I 
nu . . . . . 
n(x —1) is neceflarily contained between the limits x — 1 
I 
I = ofge m : 
and 1—-3 and in like manner, m (d"— 1) is between b—1 
x 
I 
and I— >. 
b 
Tt 
57. As the expreffion m cu" — 1) depends entirely upon the 
value of J, the number whofe logarithm is aflumed = 1, (and 
which is fometimes called the dasis 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 
ete 
