( 6zx) ) 
'a^ich the lead : but in the pre(ent cafe it happens to be the 
mod, becaufe inftead of the Difference between o, 8 and 
I, we ought flridly to have taken the difference between 
the reciprocal i , and i, which would have given the In- 
dex 10? and that would be too big, becaufe the Product 
by that means would have been bigger than i, as 1^014 
IS. Whereas this Approximation requires that the Numbers 
in the'firfl Column be alternately greater and lefs than 
I, as may be feen in the Table. 
When I have in this manner continued the Calcula- 
tion? till I have got the Numbers fmall enough, I fup- 
pofe the laft Logarithm to be equal to nothing. Which 
, gives me an Equation, from which having got away the 
Letters by means of the foregoing Equations, I have 
the relation of the Logarithms propofed. fn this man- 
ner if 1 fuppofe G =0, lhave 2136/ 2 — 643 / io = a 
Which gives the Logarithm of 1 true in feven Figures, and 
too big in the Eighth ; which happens becaufe the 
Number correfponding with G is bigger than Unite. 
There is. another Expedient Which renders this Cal- 
. dilation ftill fhorter. It is founded upon this Confidera- 
tion, that when x is very fmall i -|-xl'’is very nearly 
I Hence if i and i — are the two laft 
Numbers already got in the firft Column of the Table, 
and their Powers i -j-^rl^and i — are fuch as will 
make the Produtft i x i ^1” very near to Unite, 
w and » may be found thus : i -f- i -\-7nx, and 
I — z\" =.1 — ff Zj and confcquently* 1 xl”* x i ~ z\’* 
i -^-f»ix — itz, — X, orfnegleding 2; at) 1 -f- 
mx'—n/c- Make this equal to i, and we have 
zixi'.li — z: / I W hence a:/i — z-\- zl i -j-AT 
= 0. To give an Example of the Application of this, 
let 1,024 o» 99035'2. be the laft Numbers in the 
Table, their Logarithms being C and D. Then we have 
1, 0x4 
