log10 (1 + x) = M (x − 1⁄2x2 + 1⁄3x3 − 1⁄4x4 + &c.),
and so on.
As has been stated, Abraham Sharp’s table contains 61-decimal Briggian logarithms of primes up to 1100, so that the logarithms of all composite numbers whose greatest prime factor does not exceed this number may be found by simple addition; and Wolfram’s table gives 48-decimal hyperbolic logarithms of primes up to 10,009. By means of these tables and of a factor table we may very readily obtain the Briggian logarithm of a number to 61 or a less number of places or of its hyperbolic logarithm to 48 or a less number of places in the following manner. Suppose the hyperbolic logarithm of the prime number 43,867 required. Multiplying by 50, we have 50 × 43,867 = 2,193,350, and on looking in Burckhardt’s Table des diviseurs for a number near to this which shall have no prime factor greater than 10,009, it appears that
2,193,349 = 23 × 47 × 2029;
thus
43,867 = 1⁄50 (23 × 47 × 2029 + 1),
and therefore
loge 43,867 = loge 23 + loge 47 + loge 2029 − loge 50
| + | 1 | − 1⁄2 | 1 | + 1⁄3 | 1 | − &c. |
| 2,193,349 | (2,193,349)2 | (193,349)3 |
The first term of the series in the second line is