Page:Encyclopædia Britannica, Ninth Edition, v. 14.djvu/799

Rh 1 tot 10,000, which contained logarithms of numbers up to 10,000 to 10 places, taken from Briggs’s Arithmetica of 1624, and Gunter’s log sines and tangents to 7 places for every minute. Vlacq rendered assistance in the publication of this work, and the privilege is made out to him.

The preceding paragraphs contain a brief account of the main facts relating to the invention of logarithms. In describing the contents of the works referred to the language and notation of the present day have been adopted, so that for example a table to radius 10,000,000 is described as a table to 7 places, and so on. Also, although logarithms have been spoken of as to the base e, &c., it is to be noticed that neither Napier nor Briggs. nor any of their successors till long afterwards, had any idea of connecting logarithms with exponents.

The invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, with the exception of the Principia of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier’s Descriptio. The calculation of tables of the natural trigonometrical functions may be said to have formed the work of the last half of the 16th century, and the great canon of natural sines for every 10 seconds to 15 places which had been calculated by Rheticus was published by Pitiscus only in 1613, the year before that in which the Descriptio appeared. In the construction of the natural trigonometrical tables England had taken no part, and it is remarkable that the discovery of the principles and the formation of the tables that were to revolutionize or supersede all the methods of calculation then in use should have been so rapidly effected and developed in a country in which so little attention had been previously devoted to such questions.

The only possible rival to Napier in the invention of logarithms is Justus Byrgius, who about the same time constructed a rude kind of logarithmic or rather anti-logarithmic table; but there is every reason to believe that Napier s system was conceived and perfected before that of Byrgius; and in date of publication Napier has the advantage by six years. The title of the work of Byrgius is Arithmetische und geometrische Progress-Tabulen; in his table he has log 1 = 0 and log 10 = 230270022. The only contemporary reference to Byrgius is contained in the sentence of Kepler, &ldquo;Apices logistici Justo Byrgio multis annis ante editionem Neperianam viam prjiverunt ad hos ipsissimos logarithmos,&rdquo; which occurs in the &ldquo;Prcepta&rdquo; prefixed to the Tabul Rudolphin (1627); the apices are the signs °, ′, ″, used to denote the orders of sexagesimal fractions. The system of Byrgius is greatly inferior to that of Napier, and it is to the latter alone that the world is indebted for the knowledge of logarithms. The claims of Byrgius are discussed in Kästner’s Geschichte der Mathematik, vol. ii. p. 375, and vol. iii. p. 14; Montucla’s Histoire des Mathematiques, vol. ii, p. 10; Delambre’s Histoire de l’Astronomie moderne, vol. i. p. 560; De Morgan’s article on &ldquo;Tables&rdquo; in the English Cyclopdia; and Mr Mark Napier’s Memoirs of John Napier of Merchiston (1834).

An account of the facts connected with the early history of logarithms is given by Hutton in his History of Logarithms, prefixed to all the early editions of his logarithmic tables, and also printed in vol. i. pp. 306–340 of his Tracts (1812); but unfortunately Hutton has interpreted all Briggs’s statements with regard to the invention of decimal logarithms in a manner clearly contrary to their true meaning, and unfair to Napier. This has naturally produced retaliation, and Mr Mark Napier has not only successfully refuted Hutton, but has fallen into the opposite extreme of attempting to reduce Briggs to the level of a mere computer. It seems strange that the relations of Napier and Briggs with regard to the invention of decimal logarithms should have formed matter for controversy. The statements of both agree in all particulars, and the warmest friendship subsisted between them. Napier at his death left his manuscripts to Briggs, and all the writings of the latter show the greatest reverence for him. The words that occur on the title page of the Logarithmicall arithmetike, of 1631 are &ldquo;These numbers were first invented by the most excellent Iohn Neper, Baron of Merchiston; and the same were transformed, and the foundation and use of them illustrated with his approbation by Henry Briggs.&rdquo; No doubt the invention of decimal logarithms occurred both to Napier and to Briggs independently; but the latter not only first announced the advantage of the change, but actually undertook and completed tables of the new logarithms. For more detailed information relating to Napier, Briggs, and Vlacq, and the invention of logarithms, the reader is referred to the life of Briggs in Ward’s Lives of the Professors of Gresham College, London, 1740; Thomas Smith’s Vit quorundam eruditissimorum et illustrium virorum (Vita Henrici Briggii), London, 1707; Mr Mark Napier’s Memoirs of John Napier already referred to, and the same author’s Naperi libri qui supersunt (1839); Hutton’s History; De Morgan’s article already referred to; Delambre’s Histoire de l’Astronomie Moderne; the report on mathematical tables in the Report of the British Association for 1873; and the Philosophical Magazine for October and December 1872 and May 1873. It may be remarked that the date usually assigned to Briggs’s first visit to Napier is 1616 and not 1615 as stated above, the reason being that Napier was generally supposed to have died in 1618; but it was shown by Mr Mark Napier that the true date is 1617.

For a description of existing logarithmic tables, and the purposes for which they were constructed, the reader is referred to the article. In what follows only the most important events in the history of logarithms, subsequent to the facts connected with their invention and the original calculations, will be noticed.

Nathaniel Roe’s Tabul logarithmic (1633) was the first complete seven-figure table that was published. It contains seven-figure logarithms of numbers from 1 to 100,000, with characteristics unseparated from the mantiss, and was formed from Vlacq’s table (1628) by leaving out the last three figures. All the figures of the number are given at the heads of the columns, except the last two, which run down the extreme columns, 1 to 50 on the left hand side, and 50 to 100 on the right hand side. The first four figures of the logarithms are printed at the tops of the columns. There is thus an advance half way towards the arrangement now universal in seven-figure tables. The final step was made by John Newton in his Trigononometria Britannica (1658), a work which is also noticeable as being the only extensive eight-figure table that has ever been published; it contains logarithms of sines, &c., as well as logarithms of numbers.

In 1705 appeared the original edition of Sherwin’s tables, the first of the series of ordinary seven-figure tables of logarithms of numbers and trigonometrical functions such as are in general use now. The work went through several editions during the last century, and was at length superseded in 1785 by Hutton’s tables, which have continued in successive editions to maintain their position up to the present time.

In 1717 Abraham Sharp published in his Geometry Improv’d the Briggian logarithms of numbers from 1 to 100, and of primes from 100 to 1100, to 61 places; these were copied into the later editions of Sherwin and other works.

In 1742 a seven-figure table was published in quarto form by Gardiner, which is celebrated on account of its accuracy and of the elegance of the printing. A French edition, which closely resembles the original, was published at Avignon in 1770.

In 1783 appeared at Paris the first edition of Callet’s tables, which correspond to those of Hutton in England. These tables, which form perhaps the most complete and practically useful collection of logarithms for the general computer that has been published, have passed through many editions, and are still in use.

In 1794 Vega published his Thesaurus logarithmorum completus, a folio volume containing a reprint of the logarithms of numbers