C 279 ] 
limple equations, the nature of the feries, and the 
operations above ufed, may be eafily obtained ; and 
they may be hereby enabled to make a fcientific ufe 
of logarithms in arithmetical, and of the reftification 
of the circle, in geometrical calculations ; which at 
prefent cannot be done, till a much greater progrefs 
is made in mathematical knowlege, without great 
labour j as well as to examine and corredt the printed 
tables of both forts, if dubious, or found to be erro- 
neous. 
• f 
The operation neceffary to find the coefficients of a 
feries, which will exprefs the logarithm of a given 
number. 
If the given number be reprefented by 1 -j-ff, then 
the following feries may be affumed to reprefent its 
logarithm : 
n-\-x?i* ~\-yn 3 + + unS > & c - 
And 2n 2 xn 7, + 2 yrfi -f* 2 z?2+ 2 tm s , &c. will 
reprefent the logarithm of the fquare of that number ; 
viz. of 1 -f- 2n + nn ‘ 
But, becaufe in-±nn is the excefs of 1 -f in-\- nn, 
above unity, therefore its logarithm will be alfo ex- 
preifed by - . 
2 n-\-nn + xx 2 n-\-nn z -\-y x 2/2-\-nn 3 in-y-nift, & c. 
Now 2 n-\-nn — 4 nn -f- 4 rfi 
in-\-nn = 8 -f I 2« 4 + 6n *, &c. 
2 n-\-nrf — i6n* 32ns, &c. 
2 n-\-nn s *= 3 2nS > & c * 
Therefore, 
