| loge (x + d) = loge x + | d | − 1⁄2 | d2 | + 1⁄3 | d3 | − &c., |
| x | x2 | x3 |
in which of course the object is to render d/x as small as possible. If the logarithm required is Briggian, the value of the series is to be multiplied by M.
If the number is incommensurable or consists of more than seven figures, we can take the first seven figures of it (or multiply and divide the result by any factor, and take the first seven figures of the result) and proceed as before. An application to the hyperbolic logarithm of π is given by Burckhardt in the introduction to his Table des diviseurs for the second million.
The best general method of calculating logarithms consists, in its simplest form, in resolving the number whose logarithm is required into factors of the form 1 − .1rn, where n is one of the nine digits; and making use of subsidiary tables of logarithms of factors of this form. For example, suppose the logarithm of 543839 required to twelve places. Dividing by 105 and by 5 the number becomes 1.087678, and resolving this number into factors of the form 1 − .1rn we find that
| 543839 = 105 | × 5(1 − .128) (1 − .146) (1 − .156) (1 − .163) (1 − .173) |
| × (1 − .185) (1 − .197) (1 − .1109) (1 − .1113) (1 − .1122), |
where 1 − 128 denotes 1 − .08, 1 − .146 denotes 1 − .0006, &c., and so on. All that is required therefore in order to obtain the logarithm of any number is a table of logarithms, to the required number of places, of .n, .9n, .99n, .999n, &c., for n = 1, 2, 3, ... 9.
The resolution of a number into factors of the above form is easily performed. Taking, for example, the number 1.087678, the object is to destroy the significant figure 8 in the second place of decimals; this is effected by multiplying the number by 1-.08, that is, by subtracting from the number eight times itself advanced two places, and we thus obtain 1.00066376. To destroy the first 6 multiply by 1 − .0006 giving 1.000063361744, and multiplying successively by 1 − .00006 and 1 − .000003, we obtain 1.000000357932, and it is clear that these last six significant figures represent without any further work the remaining factors required. In the corresponding antilogarithmic process the number is expressed as a product of factors of the form 1 + .1nx.
This method of calculating logarithms by the resolution of numbers into factors of the form 1 − .1rn is generally known as Weddle’s method, having been published by him in The Mathematician for November 1845, and the corresponding method for antilogarithms by means of factors of the form 1 + (.1)rn is known by the name of Hearn, who published it in the same journal for 1847. In 1846 Peter Gray constructed a new table to 12 places, in which the factors were of the form 1 − (.01)rn, so that n had the values 1, 2, ... 99; and subsequently he constructed a similar table for factors of the form 1 + (.01)rn. He also devised a method of applying a table of Hearn’s form (i.e. of factors of the form 1 + .1rn) to the construction of logarithms, and calculated a table of logarithms of factors of the form 1 + (.001)rn to 24 places. This was published in 1876 under the title Tables for the formation of logarithms and antilogarithms to twenty-four or any less number of places, and contains the most complete and useful application of the method, with many improvements in points of detail. Taking as an example the calculation of the Briggian logarithm of the number 43,867, whose hyperbolic logarithm has been calculated above, we multiply it by 3, giving 131,601, and find by Gray’s process that the factors of 1.31601 are
| (1) 1.316 | (5) 1.(001)4002 |
| (2) 1.000007 | (6) 1.(001)5602 |
| (3) 1.(001)2598 | (7) 1.(001)6412 |
| (4) 1.(001)3780 | (8) 1.(001)7340 |
Taking the logarithms from Gray’s tables we obtain the required logarithm by addition as follows:—