227 
of Logarithmic Tables. 
l -T = 2M {Si + f (sTi)'+f (frl)’+ &c - } though, for the 
purpose of the two following propositions, they will be better 
when expanded into series of monomials; see Prop. VI. 
Prop. IV. 
8. To construct a Table of Logarithms by means of interpolation 
from the converging expressions L . «, L . a!, L . a! 1 , &c. 
When treating of the equations marked (c), we noticed a 
law to which the terms are subject ; this law affords an easy 
method of eliminating the second, third, &c. terms, and, by 
this means, we find, successively, 
L . x =L(jr— i )-f-L . « 
L(jt-{- i ) =L . x — |~L • a— j— L . cJ (f ) 
L(x-{-2) =L(x-{-l )-[-L . a-f-aL . a!f- L . a!' 
L(.r~l _ 3) :::= L( --}"2 ) -j-L . iz-f-gL ~ {-3L • / -f-L • 01,11 
L(x+b)=L(x+m— 1 )+L . *+;iL . «'+ ”-^L . *"+ 
If any one doubts whether this form is general, for every 
value-of n, let it be only a single value; and supposing «, a', 
a", &c. to become «, ct\ a", &c., by the substitution of x-f-i 
ill 
for x, we have 
L(x+«+i)=L(x+«)+L. x+nL. a'+’fe^L- «"+ 
a 1 1,2 1 
Now, if we consider the manner in which the last fractional 
factors, in the values of x, at the beginning of the last pro- 
position, were formed from one another; and the change 
which they afterwards underwent in forming equations (6), 
we shall easily perceive that 
