C ) 
Hence the terms of any ratio being a and q becomes 
^ — ^or the difference divided by the leiTer term^ when 'tis 
an increafing ratio ; or — — wnen 'tis decrealing or as^ to a. 
Whence the Logarithm of the fame ratio may be doubly ex« 
prefl", for putting x for the difference of the terms a and 
it will be either 
I . X , x"" ^ x'' 
m 
X XX X X X X 
But if the ratio of <j to ^ be fuppofed divided into two parts, 
^i2s. into the ratio of a to the Arithmetical Mean between 
the terms, and the rati<} of the faid Arithmetical Mean to the 
other term by then will the Sum of the Logarithms of thofe 
two rationes be the Logarithm of the ratio of ato b; and fiib- 
ftitiiting 1^ inftead of | ^ the faid Arithmetical Mean, 
the Logarithms of thole rationes will be by the foregoing 
Rule, 
I X . XX x^ 
L" ^ XX ^ jc^ jc^ 
the Sum i , 2.x zx* • 2 . 2 
whereof ^'«T * +3-^' * Ac.w.ll 
be the Logarithm of the ratio of a to b^ whoft difference is 
X and Sum And this Series converges twice as fwifc as the 
former, and therefore is more proper for the Pradice of ma- 
king of Logarithms : Which it performs with that expedition, 
that where x the difference is but the hundredth part of the 
2 X 
SvLttiy the firft ftep — fuffices to feven places of the Loga- 
rithm, and the fecond ftep to twelve ; But if Briggs^'s firft 
Twenty Chiliads of Logarithms be fiippofed made, as h€ has 
very carefully computed them,to fourteen places, the firft ftep 
alone is capable to give the Logarithm of any intermediate 
Number true to all the places of thole Tables. 
L After 
