( 6ip ) 
proximation ; which will be bed explain’d by an Ex- 
ample. Let it therefore be propcfed to find the Re- 
lation of the Logarithms of 2 and of lOa la order 
tothis, ItaketwoFradions^— and — , viz, and if 
100 10 lor 10;', 
whofe Numerators are Powers of and their Denomi- 
nators Powers of 10; one of them being bigger, and 
the other lefs than i . Having fee thefe down in Deci- 
mal Fraedions in the firft Column of the Table annexe, 
againfl them in the fecond Column I fet A and B for 
their Logarithms, expreffing by an Equation the manner 
how they are Compounded of the Logarithms of z and 
10, for which I write / 2 and 1 10. Then Multiplying 
the two Numbers in the firft Column together, I have 
a third Number 1,024, againft which I write C for its 
Logarithm, exprefling like wife by an Equation in what 
manner C is formed of the foregoing Logarithms A and B. 
And in the fame manner the Calculation is continued ; 
only obferving this Com^endium^ that before I Multiply 
the two lafl Numbers already got in the Table, I confl- 
der what Power of one of them muft be ufed to bring , 
the Produdi the nearefl: to Unite that can be. This is 
found, after we have gone a little way in the Table> only 
by Dividing the Differences of the Numbers from Unite 
one by the other, and caking the Quotient with the near- 
eft, for the Index of the Power wanted. Thus the two 
laft Numbers in the Table being o, 8 and i, 014, their 
Diflerences from Unit are 0, 2co and o, 024; therefore 
gives 9 for the Index; wherefore Multiplying the 
ninth Power of 1,024 by o,8, I have the next Number 
99035^03 -9, whofe Logarithm is D = 9C-]-B<, 
In feeking the Index in this manner by Divifion of the 
Diflerences, the Quotient ought generally to be taken 
with 
