Page:Encyclopædia Britannica, Ninth Edition, v. 14.djvu/803

and he has deduced the value of log, 10 and its reciprocal $\mathrm {M}$, the modulus of the Briggian system of logarithms. The value of the modulus found by Professor Adams is

$\mathrm {M} =$.	43429	44819	03251	82765	11289
	18916	60508	22943	97005	80366
	65661	14453	78316	58646	49208
	87077	47292	24949	33843	17483
	18706	10674	47663	03733	64167
	92871	58963	90656	92210	64662
‍
	81226	58521	27086	56867	03295
	93370	86965	88266	88331	16360
	77384	90514	28443	48665	76864
	65860	85135	56148	21234	87653
	43543	43573	17247	48649	05993
	55353	05

The values of the other logarithms are given in the paper referred to.

If the logarithms are Briggian all the series in the preceding formulæ must be multiplied by $\mathrm {M}$, the modulus; thus, for example,

$\log _{10}\left(1+x\right)=\mathrm {M} \left(x-{\frac {1}{2}}x^{2}+{\frac {1}{3}}x^{3}-{\frac {1}{4}}x^{4}+\And c.\right)$,

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 diviscurs 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 = 1/36(23 × 47 × 2029 + 1),

and therefore

$\log _{e}43,867=\log _{e}23+\log _{e}47+\log _{e}2029-\log _{e}50$

$+{\frac {1}{2,193,349}}-{\frac {1}{2}}{\frac {1}{\left(2,193,349\right)^{2}}}+{\frac {1}{3}}{\frac {1}{\left(2,193,349\right)^{3}}}-\And c.$

The first term of the series in the second line is

0.00000 04559 23795 07319 6286;

dividing this by 2 × 2,193,349 we obtain

0.00000 00000 00103 93325 3457,

and the third term is

0.00000 00000 00000 00003 1590,

so that the series =

0.00000 04559 23691 13997 4419;

whence, taking out the logarithms from Wolfram’s table,

log_e 43,867 = 10.68891 76079 60568 10191 3661.

The principle of the method is to multiply the given prime (supposed to consist of 4, 5, or 6 figures) by such a factor that the product may be a number within the range of the factor tables, and such that, when it is increased by 1 or 2, the prime factors may all be within the range of the logarithmic tables. The logarithm is then obtained by use of the formula

$\log _{e}\left(x+d\right)=\log _{e}x+{\frac {d}{x}}-{\frac {1}{2}}{\frac {d^{2}}{x^{2}}}+{\frac {1}{3}}{\frac {d^{3}}{x^{3}}}=\And c.$,

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 $\mathrm {M}$.

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 - 1⁴n, 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 − .1⁴n we find that

$543839=10^{5}$	$\times 5\left(1-.1^{3}8\right)\left(1-.1^{4}6\right)\left(1-.1^{5}6\right)\left(1-.1^{6}3\right)\left(1-.1^{7}3\right)$
	$\times \left(1-.1^{8}5\right)\left(1-.1^{9}7\right)\left(1-.1^{10}9\right)\left(1-.1^{11}3\right)\left(1-.1^{12}2\right)$

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 − .1^rn 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 Mr 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 discovered a method of applying a table of Hearn’s form (i.e., of factors of the form 1 + (.1^rn) to the construction of logarithms, and calculated a table of logarithm’s 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 Mr 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 Mr Gray’s tables we obtain the required logarithm by addition as follows:

522	878	745	280	337	562	704	972	= colog3
119	255	889	277	936	685	553	913	= log (1)
	3	040	050	733	157	610	239	= log (2)
		259	708	022	525	453	597	= log (3)
			338	749	695	752	424	= log (4)
					868	588	964	= log (5)
					261	445	278	= log (6)
						178	929	= log (7)
							148	= log (8)
‍
4.642	137	934	655	780	757	288	464	= log10 43,867

In Shortrede’s Tables there are tables of logarithms and factors of the form 1 ± (.01)^rn to 16 places and of the form 1 ± (.1)^rn to 25 places; and in his Tables de Logarithmes à 27 Décimales (Paris, 1867) Fédor Thoman gives tables of logarithms of factors of the form 1 ± .1^r n. In the Messenger of Mathematics, vol. iii. pp. 66-92, 1873, Mr Henry Wace gave a simple and clear account of both the logarithmic and antilogarithmic processes, with tables of both Briggian and hyperbolic logarithms of factors of the form 1 ±.1^rn to 20 places.

Although the method is usually known by the names of Weddle and Hearn, it is really, in its essential features, due to Briggs, who gave in the Arithmctica logarithmica of 1624 a table of the logarithms of 1 + .1^rn up to r = 9 to 15 places of decimals. It was first formally proposed as an independent method, with great improvements, by Robert Flower in The Radix, a new way of making Logarithms, which was published in 1771; and Leonelli, in his Supplément logarithmique (1802-3), already noticed, referred to Flower and reproduced some of his tables. A complete bibliography of this method has been given by Mr A. J. Ellis in a paper “on the potential radix as a means of calculating logarithms” printed in the Proceedings of the Royal Society, vol. xxxi., 1881, pp. 401-407, and vol. xxxii., 1881, pp. 377-379. Reference should also be made to Hoppe’s Tafeln zur dreissigstelligen logarithmischen Rechnung (Leipsic, 1876), which give in a somewhat modified form a table of the hyperbolic logarithm of 1 + .1^rn.

The preceding methods are only appropriate for the calculation of isolated logarithms. If a complete table had to be reconstructed, or calculated to more places, it would undoubtedly be most convenient to employ the method of differences. A full account of this method as applied to the calculation of the Tables du Cadastre is given by M. Lefort in vol. iv. of the Annales de l’Observatoire de Paris. (J. W. L. G.)