[ 2 7 ° ] 
prefs any quantity at all ; becaufe after the 5th term 
the coefficients begin to increafe, and they afterwards 
increal'e at a greater rate than what can be compcn- 
fated by the increafe of the powers of z, though z 
reprefent a number ever fo large; as will be evident 
by confidering the following manner in which the 
coefficients of that feries may be formed. Take 
a — T-j ^b=d', ycz=zzba y gd~z ca-\-b\ lie — 
zdaA r 2cby 2 ea-\- 2 db\-c*y 1 5 g — 2 y'rf q- 
ze b-\-zdc, and fo on ; then take A —a, B— 2 b, Cz=z 
2 x 3 X 4Cy D— 2 x 3 X 4 X 5 X 6 d, E—z x 3 X 4 X 5 
X 6 x 7 X 8 e and fo on, and A, B, C, D, K, b , 6cc. 
will be the coefficients of the foregoing feries : from 
whence it eafily follows, that if any term in the feries 
after the 3 firft be called y, and its diftance from the 
firft term n> the next term immediately following 
will be greater than Whereforeatlength 
the fubfequent terms of this feries are greater than 
the preceding ones, and increafe in infinitum, and 
therefore the whole feries can have no ultimate value 
whatfoever. 
Much lefs can that feries have any ultimate value, 
which is deduced from it by taking z= 1, and is 
fuppofed to be equal to the logarithm of the fquare 
root of the periphery of a circle whofe radius is 
unity ; and what is faid concerning the foregoing fe- 
ries is true, and appears to be fo, much in the lame 
manner, concerning the feries for finding out the fum 
of the logarithms of the odd numbers 3. 5.7. &c....z, 
and thofe that are given for finding out the fum of 
the infinite progreihons, in which the leveral terms 
have the fame numerator whilft their denominators 
are 
