21 9 
of Logarithmic Tables. 
beginning at the top of the Table, with L. 1 999998, L. 199999b, 
&c., and setting down the remainder for the logarithms of 
the successive numbers below 100,000, viz. L. 99999, 
L. 99998, &c. 
When we have got down to 50,000, if we were to proceed 
in the same way, we should have to operate on the logarithms 
thus obtained, between 100,000 and 50,000 : If, therefore, 
we fear any accumulation of errors, we may (because 
3*49999 = 1 49997) subtract the logarithm of 3 from 
L . 149997, and from the logarithm of every third number 
going downward, and set the remainders down successively 
for the logarithms of numbers below 50,000. And thus we 
may proceed till we get somewhat below 34,000 ; then the 
logarithm of 4 will carry us down to 25,000 ; and the loga- 
rithm of 5 to 20,000, which completes the work, those below 
20,000 having been already found. 
In the great French Tables, however, it has been thought 
proper to calculate the logarithms of numbers under 10,000 
with more decimal places than the rest. These must neces- 
sarily be found independently of the others-; as they form in 
reality a separate Table. 
In the next proposition, is contained a general method of 
finding converging series for the calculation of logarithms. 
The propositions which follow this are only corollaries from 
it, and give forms for interpolation ; so that every thing relat- 
ing to the construction of logarithms is effected by one sim- 
ple and uniform process. 
