[ ] 
XXIII. The Principles and Illuf ration of an advantageous Me- 
thod of arranging the Differences of Logarithms , on lines 
graduated for the Purpofe of Computation . By Air. William 
Nicholfon ; communicated hy Sir Jofeph Brinks, Bart . 
P. R. S. 
Read March 29, 1787. 
1. 1 TF two geometrical feries of numbers, having the fame 
JL common ratio, be placed in order with the terms oppo- 
f te each other ; the ratio, between any term in one feries and 
its oppofite in the other, will be conftant *. 
2. And like wife the ratio of a term in one feries to any term 
in the other, will be the fame as obtains between any other two 
terms having the fame relative pofition and diftance -f. 
3. In all fuch pairs of geometrical feries, as have the fame 
common ratio, the laft mentioned property obtains, though the 
frft antecedent and confequent be taken in one pair, and the 
fecond in any other pair J. 
'* Geom. 
feries ' 
a an an an 5 an 4 
b bn brl brfi bn 4 
Then a : b :: an : bn :: an 2 : bn z } See. 
•f In the foregoing feries a : bn z :: an : bn 4 :: an : bn^ t &e. 
J Geo. feries 
an an 2 an i 3 
an 1 
Geo, feries 
bn bn z bn 3 bn* 
d dn dn ? dn 3 dn* 
bd bdn bdn z bdn 3 bdn * 
a a a a a 
i % j bdrp 
Then a : bn dn : , &c. 
a 
4- If 
