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
thus
and therefore
| + | 1 | − 12 | 1 | + 13 | 1 | − &c. |
| 2,193,349 | (2,193,349)2 | (2,193,349)3 |
The first term of the series in the second line is
dividing this by 2 × 2,193,349 we obtain
and the third term is
so that the series =
whence, taking out the logarithms from Wolfram’s table,
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
| loge (x + d) = loge x + | d | − 12 | d2 | + 13 | 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:—
| 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 = log10 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 ± .1r 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 ± .1rn 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 + .1rn 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 + .1rn.
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.)
LOGAU, FRIEDRICH, Freiherr von (1604–1655), German
epigrammatist, was born at Brockut, near Nimptsch, in Silesia,
in June 1604. He was educated at the gymnasium of Brieg and
subsequently studied law. He then entered the service of the
duke of Brieg. In 1644 he was made “ducal councillor.” He
died at Liegnitz on the 24th of July 1655. Logau’s epigrams,
which appeared in two collections under the pseudonym “Salomon
von Golaw” (an anagram of his real name) in 1638 (Erstes
Hundert Teutscher Reimensprüche) and 1654 (Deutscher Sinngedichte
drei Tausend), show a marvellous range and variety of
expression. He had suffered bitterly under the adverse conditions
of the time; but his satire is not merely the outcome of
personal feeling. In the turbulent age of the Thirty Years’ War
he was one of the few men who preserved intact his intellectual
integrity and judged his contemporaries fairly. He satirized
with unsparing hand the court life, the useless bloodshed of the
war, the lack of national pride in the German people, and their
slavish imitation of the French in customs, dress and speech.
He belonged to the Fruchtbringende Gesellschaft under the name
Der Verkleinernde, and regarded himself as a follower of Martin
Opitz; but he did not allow such ties to influence his independence
or originality.
Logau’s Sinngedichte were edited in 1759 by G. E. Lessing and K. W. Ramler, who first drew attention to their merits; a second
