came into very general use, Bagay’s Nouvelles tables astronomiques (1829), which also contains log sines and tangents to every second, being preferred; this latter work, which for many years was difficult to procure, has been reprinted with the original title-page and date unchanged. The only other logarithmic canon to every second that has been published forms the second volume of Shortrede’s Logarithmic Tables (1849). In 1784 the French government decided 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 à composer des tables qui ne laissassent rien à désirer quant à l’exactitude, mais à en faire le monument de calcul le plus vaste et le plus imposant qui eût jamais été exécuté ou même 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, one 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 | ” |
| Logarithms of the ratios of arcs to sines from 0q.00000 | ||
| to 0q.05000, and log sines throughout the quadrant | 4 | ” |
| Logarithms of the ratios of arcs to tangents from | ||
| 0q.00000 to 0q.05000, and log tangents throughout | ||
| the quadrant | 4 | ” |
The trigonometrical results are given for every hundred-thousandth of the quadrant (10″ centesimal 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 these tables, with an explanation of the methods of calculation, formulae employed, &c., was published by Lefort in vol. iv. of the Annales de l’observatoire de Paris. The printing of the table of natural sines was once begun, and 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 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.
As the Tables du Cadastre remained unpublished, other tables appeared in which the quadrant was divided centesimally, the most important of these being Hobert and Ideler’s Nouvelles tables trigonométriques (1799), and Borda and Delambre’s Tables trigonométriques décimales (1800–1801), both of which are seven-figure tables. The latter work, which was much used, being difficult to procure, and greater accuracy being required, the French government in 1891 published an eight-figure centesimal table, for every ten seconds, derived from the Tables du Cadastre.
Decimal or Briggian Antilogarithms.—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. Antilogarithmic tables are few in number, the only other extensive tables of the same kind that have been published occurring 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.
Hyperbolic or Napierian logarithms (i.e. to base e).—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 logarithms omitted in Schulze’s work, and which Wolfram had been prevented from computing by a serious illness, were published subsequently, and the table as given by Vega is complete. The largest hyperbolic table as regards range was published by Zacharias Dase at Vienna in 1850 under the title Tafel der natürlichen Logarithmen der Zahlen.
Hyperbolic antilogarithms are simple exponentials, i.e. the hyperbolic antilogarithm of x is ex. Such tables can scarcely be said to come under the head of logarithmic tables. See Tables, Mathematical: Exponential Functions.
Logistic or Proportional Logarithms.—The old name for what are now called ratios or fractions are logistic numbers, so that a table of log (a/x) where x is the argument and a a constant is called a table of logistic or proportional logarithms; and since log (a/x) = log a − log x it is clear that the tabular results differ from those given in an ordinary table of logarithms only by the subtraction of a constant and a change of sign. The first table of this kind appeared in Kepler’s work of 1624 which has been already referred to. The object of a table of log (a/x) is to facilitate the working out of proportions in which the third term is a constant quantity a. In most collections of tables of logarithms, and especially those intended for use in connexion with navigation, there occurs a small table of logistic logarithms in which a = 3600″ (= 1° or 1h), the table giving log 3600 − log x, and x being expressed in minutes and seconds. It is also common to find tables in which a = 10800″ (= 3° or 3h), and x is expressed in degrees (or hours), minutes and seconds. Such tables are generally given to 4 or 5 places. The usual practice in books seems to be to call logarithms logistic when a is 3600″, and proportional when a has any other value.
Addition and Subtraction, or Gaussian Logarithms.—Gaussian logarithms are intended to facilitate the finding of the logarithms of the sum and difference of two numbers whose logarithms are known, the numbers themselves being unknown; and on this account they are frequently called addition and subtraction logarithms. The object of the table is in fact to give log (a ± b) by only one entry when log a and log b are given. The utility of such logarithms was first pointed out by Leonelli in a book entitled Supplément logarithmique, printed at Bordeaux in the year XI. (1802/3); he calculated a table to 14 places, but only a specimen of it which appeared in the Supplément was printed. The first table that was actually published is due to Gauss, and was printed in Zach’s Monatliche Correspondenz, xxvi. 498 (1812). Corresponding to the argument log x it gives the values of log (1 + x−1) and log (1 + x).
Dual Logarithms.—This term was used by Oliver Byrne in a series of works published between 1860 and 1870. Dual numbers and logarithms depend upon the expression of a number as a product of 1.1, 1.01, 1.001 ... or of .9, .99, .999....
In the preceding résumé only those publications have been mentioned which are of historic importance or interest.[1] For fuller details with respect to some of these works, for an account of tables published in the latter part of the 19th century, and for those which would now be used in actual calculation, reference should be made to the article Tables, Mathematical.
Calculation of Logarithms.—The name logarithm is derived from the words λόγων ἀριθμός, the number of the ratios, and the way of regarding a logarithm which justifies the name may be explained as follows. Suppose that the ratio of 10, or any other particular number, to 1 is compounded of a very great number of equal ratios, as, for example, 1,000,000, then it can be shown that the ratio of 2 to 1 is very nearly equal to a ratio compounded of 301,030 of these small ratios, or ratiunculae, that the ratio of 3 to 1 is very nearly equal to a ratio compounded of 477,121 of them, and so on. The small ratio, or ratiuncula, is in fact that of the millionth root of 10 to unity, and if we denote it by the ratio of a to 1, then the ratio of 2 to 1 will be nearly the same as that of a301,030 to 1, and so on; or, in other words, if a denotes the millionth root of 10, then 2 will be nearly equal to a301,030, 3 will be nearly equal to a477,121, and so on.
Napier’s original work, the Descriptio Canonis of 1614, contained, not logarithms of numbers, but logarithms of sines, and the relations between the sines and the logarithms were explained by the motions of points in lines, in a manner not unlike that afterwards employed by Newton in the method of fluxions. An account of the processes by which Napier constructed his table was given in the Constructio Canonis of 1619. These methods apply, however, specially to Napier’s own kind of logarithms, and are different from those actually used by Briggs in the construction of the tables in the Arithmetica Logarithmica, although some of the latter are the same in principle as the processes described in an appendix to the Constructio.
The processes used by Briggs are explained by him in the preface to the Arithmetica Logarithmica (1624). His method of finding the logarithms of the small primes, which consists in taking a great number of continued geometric means between unity and the given primes, may be described as follows. He first formed the table of numbers and their logarithms:—
| Numbers. | Logarithms, |
| 10 | 1 |
| 3.162277... | 0.5 |
| 1.778279... | 0.25 |
| 1.333521... | 0.125 |
| 1.154781... | 0.0625 |
each quantity in the left-hand column being the square root of the one above it, and each quantity in the right-hand column being the half
- ↑ In vol. xv. (1875) of the Verhandelingen of the Amsterdam Academy of Sciences, Bierens de Haan has given a list of 553 tables of logarithms. A previous paper of the same kind, containing notices of some of the tables, was published by him in the Verslagen en Mededeelingen of the same academy (Afd. Natuurkunde) deel. iv. (1862), p. 15.
