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
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.
This method of calculating logarithms by the resolution of numbers
into factors of the form 1 - .1^(r)n 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)^(r)n 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)^(r)n, so that n had the values 1, 2, ... 99; and
subsequently he constructed a similar table for factors of the form 1
+ (.01)^(r)n. He also devised a method of applying a table of Hearn's
form (i.e. of factors of the form 1 +.1^(r)n) to the construction of
logarithms, and calculated a table of logarithms of factors of the
form 1 + (.001)^(r)n 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)^(4)002
(2) 1.000007 (6) 1.(001)^(5)602
(3) 1.(001)²598 (7) 1.(001)^(6)412
(4) 1.(001)³780 (8) 1.(001)^(7)340
Taking the logarithms from Gray's tables we obtain the required
logarithm by addition as follows:--
522 878 745 280 337 562 704 972 = colog 3
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 = log(10)43,867
In Shortrede's _Tables_ there are tables of logarithms and factors of
the form 1 ± (.01)^(r)n to 16 places and of the form 1 ± (.1)^(r)n 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, 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^(r)n
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 _Arithmetica logarithmica_ of 1624 a table of the logarithms of 1
+ .1^(r)n 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 _Supplement logarithmique_
(1802-1803), already noticed, referred to Flower and reproduced some
of his tables. A complete bibliography of this method has been given
by 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_ (Leipzig, 1876), which
give in a somewhat modified form a table of the hyperbolic logarithm
of 1 + .1^(r)n.
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
Lefort in vol. iv. of the _Annales de l'Observatoire de Paris_.
(J. W. L. G.)