from Vlacq’s Arithmetica logarithmica of 1628, and Trigonometria artificialis of 1633. The logarithms of numbers are arranged as in an ordinary seven-figure table. In addition to the logarithms reprinted from the Trigonometria, there are given logarithms for every second of the first two degrees, which were the result of an original calculation. Vega devoted great attention to the detection and correction of the errors in Vlacq’s work of 1628. He also published in 1797, in 2 vols. 8vo, a collection of logarithmic and trigonometrical tables which has passed through many editions, a very useful one volume stereotype edition having been published in 1840 by Hülsse. The tables in this work may be regarded as to some extent supplementary to those in Callet.
If we consider only the logarithms of numbers, the main line of descent from the original calculation of Briggs and Vlacq is Roe, John Newton, Sherwin, Gardiner; there are then two branches, viz. , Hutton founded on Sherwin and Callet on Gardiner, and the editions of Vega form a separate offshoot from the original tables. Among the most useful and accessible of modern ordinary seven-figure tables of logarithms of numbers and trigonometrical functions may be mentioned those of Bremiker, Schrön, and Bruhns. For logarithms of numbers only perhaps Babbage’s table is the most convenient.
In 1871 Mr Sang published a seven-figure table of logarithms of numbers extending from 20,000 to 200,000; and the logarithms of the numbers between 100,000 and 200,000 were calculated de novo by Mr Sang as if logarithms had never been computed before. In tables extending from 10,000 to 100,000 the differences near the beginning of the table are large, and they are so numerous that the proportional parts must either be very crowded, or some of them have to be omitted; and to diminish this inconvenience many tables extend to 108,000. By beginning the table at 20,000 instead of at 10,000, the differences are halved in magnitude, while the number of them in a page is quartered. In this table multiples of the differences, instead of proportional parts, are given.
As regards the logarithms of trigonometrical functions, the next great advance on the Trigonometria artificialis took place more than a century and a half afterwards, when Michael Taylor published in 1792 his seven-decimal table of log sines and tangents to every second of the quadrant; it was calculated by interpolation from the Trigonometria to 10 places and then contracted to 7. On account of the great size of this table, and for other reasons, it never came into very general use, Bagay’s Nouvelles tables astronomiques (1829), which also contains log sines and tangents to every second, being preferred, but this work is now difficult to procure. The only other logarithmic canon to every second that has been published forms the second volume of Shortrede’s Logarithmic Tables (1849). It contains also proportional parts, and is the most complete and accessible table of logarithms for every second. Shortrede’s tables originally appeared in 1844 in one volume, during the author’s absence in India; but, not being satisfied with them in some respects, he made various alterations, and published a second edition in two volumes in 1849. There have been subsequent editions of the volume containing the trigonometrical canon. The work is an important one, and the pages are clear, although the number of figures on each is very great.
On the proposition of Carnot, Prieur, and Brunet, the French Government decided in 1784 that new tables of sines, tangents, &c., and their logarithms, should be calculated in relation to the centesimal division of the quadrant. Prony was charged with the direction of the work, and was expressly required “Non seulement & composer des tables qui ne laissassent rien à desirer quant & l’exactitude, mais à en faire le monument de calcul le plus vaste et le plus imposant qui eût jamais été exécuté on meme conçu.” Those engaged upon the work were divided into three sections: the first consisted of five or six mathematicians, including Legendre who were engaged in the purely analytical work, or the calculation of the fundamental numbers; the second section consisted of seven or eight calculators possessing some mathematical knowledge; and the third comprised seventy or eighty ordinary computers. The work, which was performed wholly in duplicate, and independently by two divisions of computers, occupied two years. As a consequence of the double calculation, there are two manuscripts in existence, one of which is deposited at the Observatory, and the other in the library of the Institute, at Paris. Each of the two manuscripts consists essentially of seventeen large folio volumes the contents being as follows:
Logarithms of numbers up to 200,000 |
8 | vols. |
Natural sines |
1 | vols.„ |
Logarithms of the ratios of arcs to sines from 07.00000 to 07.05000, and log sines throughout the quadrant |
4 | vols.„ |
Logarithms of the ratios of arcs to tangents from 07.00000 to 07.05000, and log tangents throughout the quadrant |
4 | vols.„ |
The trigonometrical results are given for every hundred-thousandth of the quadrant (10″ centetimal or 3″.24 sexagesimal). The tables were all calculated to 14 places, with the intention that only 12 should be published, but the twelfth figure is not to be relied upon. The tables have never been published, and are generally known as the Tables du Cadastre, or, in England, as the great French manuscript tables.
A very full account of the Tables du Cadastre, with an explanation of the methods of calculation, formulæ employed, &c., has been published by M. Lefort in vol. iv. of the Annales de l’Observatoire de Paris. The printing of the table of natural sines was once begun, and M. Lefort states that he has seen six copies, all incomplete, although including the last page. Babbage compared his table with the Tables du Cadastre, and M. Lefort has given in his paper just referred to most important lists of errors in Vlacq’s and Briggs’s logarithms of numbers which were obtained by comparing the manuscript tables with those contained in the Arithmetica logarithmica of 1624 and of 1628. These are almost the only uses that have been made of the French tables, the calculation of which involved so great an expenditure of time and money.
It may be mentioned here that the late Mr John Thomson of Greenock made an independent calculation of the logarithms of numbers up to 120,000 to 12 places, and that the manuscript of the table was presented in 1874 to the Royal Astronomical Society by his sister. The table has been used to verify the errata which M. Lefort found in Vlacq and Briggs by means of the Tables du Cadastre. An account of Mr Thomson’s table, and of this and other comparisons between it and the printed tables, is to be found in the Monthly Notices of the Society, vol. xxxiv. pp. 447-75 (1874).
Although the Tables du Cadastre have never been published, other tables have appeared in which the quadrant is divided centesimally, the most important of these being Hobert and Ideler’s Nouvelles tables trigonometriques (1799), and Borda and Delambre’s Tables trigonometriques decimates (1800-1). The former work contains natural and log sines, cosines, tangents, and cotangents to 7 places, up to 3° (centesimal) at intervals of 10″ (centesimal), and thence to 50° at intervals of 1′. The latter gives log sines, cosines, tangents, and cosines for centesimal arguments, viz., from 0′ to 10′ at intervals of 10″, and from to 50° at intervals of 1′, to 11 places, and also, in another table, log sines, cosines, tangents, cotangents, secants, and cosecants from 0° to 3° at intervals of 10′, and thence to 50° at intervals of 1′ to 7 places. After the work was printed it was read by Delambre with the Tables du Cadastre, and a number of last-figure errors which are given in the preface were thus detected. Callet’s tables already referred to contain in a convenient form logarithms of trigonometrical functions for centesimal arguments.
Two tables of logarithms of numbers which have been recently published may be noticed, as they involve points of novelty. The first of these is Pineto’s Tables de logarithmcs (St Petersburg, 1871). The tables are intended to give in a small space (56 pages) all the results that can be obtained from a complete ten-figure table by means of the following principle:—only the logarithms of the numbers from 1,000,000 to 1,011,000 are given directly, all other numbers being brought within the range of this table by multiplication by a factor, the logarithm of which factor is to be subtracted from the logarithm in the table. A list of the most convenient factors and their logarithms is given in a separate table. The principle of multiplying by a factor which is subsequently cancelled by subtracting its logarithm is one that is frequently employed in the calculation of logarithms, but the peculiarity of the present work is that it forms part of the process of using the table. The other tables, which occupy only ten pages, were published in a tract entitled Tables de logarithmes à 12 décimales jusqu’ à 434 milliards by MM. Namur and Mansion at Brussels in 1877. The fact that the differences of the logarithms of numbers near to 434294 (these being the first figures of the modulus of the Briggian logarithms) commenced with the figures 100…, so that the interpolations in this part of the table are very easily and accurately performed, is ingeniously made use of. A table is given of logarithms of numbers near to 434294, and other numbers are brought within the range of the table by multiplication by one or two factors which are indicated.
In the ordinary tables of logarithms the natural numbers are all integers, while the logarithms tabulated are incommensurable. In an antilogarithmic table, the logarithms are exact quantities such as .00001, .00002, &c., and the numbers are incommensurable. The earliest and largest table of this kind that has been constructed is Dodson’s Antilogarithmic canon (1742), which gives the numbers to 11 places, corresponding to the logarithms from .00001 to .99999 at intervals of .00001. The only other extensive tables of the same kind that have been published occur in Shortrede’s Logarithmic tables already referred to, and in Filipowski’s Table of antilogarithms (1849). Both are similar to Dodson’s tables, from which they were derived, but they only give numbers to 7 places.
The most elaborate table of hyperbolic logarithms that exists is due to Wolfram, a Dutch lieutenant of artillery. His table gives the logarithms of all numbers up to 2200, and of primes (and also of a great many composite numbers) from 2200 to 10,009, to 48 decimal places. The table appeared in Schulze’s Neue und erweiterte Sammlung logarithmischer Tafeln (1778), and was. reprinted in Vega’s Thesaurus (1794), already referred to. Six
