d d² d³ log(e)(x + d) = log(e)x + --- - ½ -- + (1/3) -- - &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 [pi] 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 - .1^(r)n, 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 10^5 and by 5 the number becomes 1.087678, and resolving this number into factors of the form 1 - .1^(r)n we find that 543839 = 10^5 × 5(1-.1²8)(1-.1^(4)6)(1-.1^(5)6)(1-.1^(6)3)(1-.1^(7)3) × (1-.1^(8)5)(1-.1^(9)7)(1-.1^(10)9)(1-.1^(11)3)(1-.1^(12)2), where 1-1²8 denotes 1-.08, 1-.1^(4)6 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 + .1^(n)x.