| 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.)
[1] Dr Thomas Smith thus describes the ardour with which Briggs studied the Descriptio: “Hunc in deliciis habuit, in sinu, in manibus, in pectore gestavit, oculisque avidissimis, et mente attentissima, iterum iterumque perlegit,...” Vitae quorundam eruditissimorum et illustrium virorum (London, 1707).
[2] William Lilly’s account of the meeting of Napier and Briggs at Merchiston is quoted in the article [Napier].
[3] It was certainly published after Napier’s death, as Briggs mentions his “librum posthumum.” This liber posthumus was the Constructio referred to later in this article.
[4] Frisch’s Kepleri opera omnia, ii. 834. Frisch thinks Bramer possibly relied on Kepler’s statement quoted in the text (“Quibus forte confisus Kepleri verbis Benj. Bramer....”). See also vol. vii. p. 298.