Page:Encyclopædia Britannica, Ninth Edition, v. 14.djvu/798

774 first published table of decimal or common logarithms, is only a small octavo tract of sixteen pages, and gives the logarithms of numbers from unity to 1000 to 14 places of decimals. There is no author’s name, place, or date. The date of publication is, however, fixed as 1617 by a letter from Sir Henry Bourchier to Ussher, dated December 6, 1617, containing the passage&mdash;&ldquo;Our kind friend, Mr Briggs, hath lately published a supplement to the most excellent tables of logarithms, which I presume he has sent to you.&rdquo; Briggs’s tract of 1617 is extremely rare, and has generally been ignored or incorrectly described. Hutton erroneously states that it contains the logarithms to 8 places, and his account has been followed by most writers. There is a copy in the British Museum.

Briggs continued to labour assiduously at the calculation of logarithms, and in 1624 published his Arithmetica logarithmica, a folio work containing the logarithms of the numbers from 1 to 20,000, and from 90,000 to 100,000 (and in some copies to 101,000) to 14 places of decimals. The table occupies 300 pages, and there is an introduction of 88 pages relating to the mode of calculation of the tables, and the applications of logarithms.

There was thus left a gap between 20,000 and 90,000, which was filled up by Adrian Vlacq, who published at Gouda, in Holland, in 1628, a table containing the logarithms of the numbers from unity to 100,000 to 10 places of decimals. Having calculated 70,000 logarithms and copied only 30,000, Vlacq would have been quite entitled to have called his a new work. He designates it, however, only a second edition of Briggs’s Arithmetica logarithmica, the title running Arithmetica logarithmica sive logarithmorum chiliades centum,&hellip; Editio secunda aucta per Adrianum Vlacq, Goudanum. This table of Vlacq s was published, with an English explanation prefixed, at London in 1631 under the title Logarithmicall Arithmetike &hellip; London, printed by George Miller, 1631. There are also copies with a French title page and introduction (Gouda, 1623).

Briggs had himself been engaged in filling up the gap, and in a letter to Pell, written after the publication of Vlacq s work, and dated October 25, 1628, he says:&mdash;

&ldquo;My desire was to have those chiliades that are wanting betwixt 20 and 90 calculated and printed, and I had done them all almost by my selfe, and by some frendes whom my rules had sufficiently informed, and by agreement the busines was conveniently parted amongst us; but I am eased of that charge and care by one Adrian Vlacque, an Hollander, who hathe done all the whole hundred chiliades and printed them in Latin, Dutche, and Frenche, 1000 bookes in these 3 languages, and hathe sould them almost all. But he hathe cutt off 4 of my figures throughout; and hathe left out my dedication, and to the reader, and two chapters the 12 and 13, in the rest he hath not varied from me at all.

The original calculation of the logarithms of numbers from unity to 101,000 was thus performed by Briggs and Vlacq between 1615 and 1628. Vlacq’s table is that from which all the hundreds of tables of logarithms that have subsequently appeared have been derived. It contains of course many errors, which have gradually been discovered and corrected in the course of the two hundred and fifty years that have elapsed, but no fresh calculation has been published. The only exception is Mr Sang’s table (1871), part of which was the result of an original calculation.

The first calculation or publication of Briggian or common logarithms of trigonometrical functions was made in 1620 by Gunter, who was Briggs’s colleague as professor of astronomy in Gresham College. The title of Gunter’s book, which is very scarce, is Canon triangulorum, and it contains logarithmic sines and tangents for every minute of the quadrant to 7 places of decimals.

The next publication was due to Vlacq, who appended to his logarithms of numbers in the Arithmetica logarithmica of 1628 a table giving log sines, tangents, and secants for every minute of the quadrant to 10 places; these were obtained by calculating the logarithms of the natural sines, &c., given in the Thesaurus Mathematicus of Pitiscus (1613).

During the last years of his life Briggs devoted himself to the calculation of logarithmic sines, &c., and at the time of his death in 1631 he had all but completed a logarithmic canon to every hundredth of a degree. This work was published by Vlacq at his own expense at Gouda in 1633, under the title Trigonometria Britannica. It contains log sines (to 14 places) and tangents (to 10 places), besides natural sines, tangents, and secants, at intervals of a hundredth of a degree. In the same year Vlacq published at Gouda his Trigonometria artificialis, giving log sines and tangents to every 10 seconds of the quadrant to 10 places. This work also contains the logarithms of the numbers from unity to 20,000 taken from the Arithmetica logarithmica of 1628. Briggs appreciated clearly the advantages of a centesimal division of the quadrant, and by dividing the degree into hundredth parts instead of into minutes, made a step towards a reformation in this respect, and but for the appearance of Vlacq’s work the decimal division of the degree might have become recognized, as is now the case with the corresponding division of the second. The calculation of the logarithms not only of numbers but also of the trigonometrical functions is therefore due to Briggs and Vlacq; and the results contained in their four fundamental works&mdash;Arithmetica logarithmica (Briggs), 1624; Arithmetica logarithmica (Vlacq), 1628; Trigonometria Britannica (Briggs), 1633; Trigonometria artificialis (Vlacq), 1633&mdash;have never been superseded by any subsequent calculations.

A translation of Napier’s Descriptio was made by Edward Wright, whose name is well known in connexion with the history of navigation, and after his death published by his son at London in 1616 under the title A Description of the admirable Table of Logarithms (12mo); the edition was revised by Napier himself. Both the Descriptio (1614) and the Constructio (1619) were reprinted at Lyons in 1620 by Bartholomew Vincent, who thus was the first to publish logarithms on the Continent.

Napier calculated no logarithms of numbers, and, as already stated, the logarithms invented by him were not to base e. The first logarithms to the base e were published by John Speidell in his New Logarithmes (London, 1619), which contains hyperbolic log sines, tangents, and secants for every minute of the quadrant to 5 places of decimals.

In 1624 Benjamin Ursinus published at Cologne a canon of logarithms exactly similar to Napier’s in the Descriptio of 1614, only much enlarged. The interval of the arguments is 10&Prime;, and the results are given to 8 places; in Napier’s canon the interval is 1&prime;, and the number of places is 7. The logarithms are strictly Napierian, and the arrangement is identical with that in the canon of 1614. This is the largest Napierian canon that has ever been published.

Kepler took the greatest interest in the invention of logarithms, and in 1624 he published at Marburg a table of Napierian logarithms of sines, with certain additional columns to facilitate special calculations.

The first publication of Briggian logarithms on the Continent is due to Wingate, who published at Paris in 1625 his Arithimetique logarithmétique, containing seven-figure logarithms of numbers up to 1000, and log sines and tangents from Gunter’s Canon (1620). In the following year, 1626, Denis Henrion published at Paris a Traicté des Logarithmes, containing Briggs’s logarithms of numbers up to 20,001 to 10 places, and Gunter’s log sines and tangents to 7 places for every minute. In the same year De Decker also published at Gouda a work entitled Nieuwe Telkonst, inhoudende de Logarithms voor de Ghetallen beginnende van