Many collections of tables give the logarithmic trigonometrical canon to 7 places (e.g., Schron, Bruhns, etc.) for every sexagesimal minute, or for every io sexagesimal seconds, or for every centesi mal minute, or for every 10 centesimal seconds. Bauschinger and Peters (Leipzig, 1921) in Logarithmisch-trigonometrische Tafeln give the logarithmic trigonometrical functions for • every sexa gesimal second to 8 places, while Peters in another work (Leipzig, I91I) gives a similar table to 7 places. In 1911 Andoyer at Paris published a table as a result of an original calculation giving these functions to 14 decimals with differences for every 10 sexagesimal seconds; Nouvelles Tables Trigonometriques Fondamentales. In the second volume of his Zelinstellige Logarithmentafel, Peters gives an extended canon to 10 decimal places. For the centesimal division of the quadrant Hobert and Ideler, Nouvelles Tables Trigonometriques (Berlin, 1799) and Borda and Delambre, Tables Trigonometriques Decimales (Paris, 1801), give the canon to 7 places for every centesimal minute quadrant). Mendizabal Tamborrel in his tables, already mentioned, gives an 8-place canon for every gone (about 1.3 sexagesimal seconds) (A gone= 36o°). Becker and Van Orstrand, Smithsonian Mathematical Tables (Washington, 1909) give these functions to 5 places for every o.00r radian in the first quadrant.
Natural Trigonometrical Functions.—The greatest corn puter of pure trigonometrical tables was Rheticus, whose work has never really been superseded. His celebrated io-decimal canon, the Opus Palatinum was published at Neustadt in 1596 and in 1613 his I 5-decimal tables of sines by Pitiscus at Frankfort with title Thesaurus Mathematicus. This wonderful achievement was overshadowed by the invention of logarithms by Napier in 1614, for the natural trigonometrical functions gave way to the logarithmic. The Opus Palatinum contains the io-decimal trig onometrical functions for every io seconds with differences. The Thesaurus gives the sines to i5 places with differences to the third. These tables are the fundamental tables for practically all natural trigonometrical tables up to the present day. The number of tables of natural trigonometrical functions published since Rheticus is not large. Natural sines and tangents to 8-places for every sexa gesimal second have been published by Gifford, Natural Sines (Manchester, 1914) and Natural Tangents (Manchester, 1920). Briggs in his Trigonometria gives natural sines to 15 places, tangents and secants to IC places for every o.or degree. Hobert Ideler in work already quoted gives a 7-decimal canon for every centesimal minute. Becker and Van Orstrand have published an interesting table with interval of argument o.00i radian, while Burrau (Berlin, 1907) gives a 6-place table for interval o.or radian. Andoyer, Nouvelles Tables, Trigonometriques Fondamen tales contenant les valeurs naturelles (Paris, Hermann, 1915-18) published tables of the natural functions to 15 places for every lc) seconds, which he had calculated de novo during the years 1910-14; the tables occupy three large volumes. He also gives
tables for the centesimal division of the quadrant to 20 decimals.
Antilogarithms.--A table of antilogarithms gives as the tabular results the number whose logarithm is equal to the argu ment. By inverse entry and interpolation, tables of logarithms can be used as tables of antilogarithms, so that few antilogarith mic tables have been published. The methods mentioned above for the determination of logarithms to a large number of places can, in general, be applied inversely for antilogarithms. The largest and earliest usable table of antilogarithms is Dodson's Anti logarithmic Canon (London, 1742), giving II-figure numbers cor responding to the logarithms 0.00000(0.00001)0.99999 with differ ences. In 1849 Filipowski in A Table of Antilogarithms (London) and Shortrede in Logarithmic Tables (Edinburgh) give 7-figure antilogarithms for 0.00000(0.00001)0.99999. Dietrichkeit (Berlin, 1906) gives a similar table. In 1908 Borgen published a table and method for calculating logarithms to 1 i or 10 places in Logarith misch-trigonometrische Tafel (Leipzig, Engelmann, 1908) and his main table gives 1-figure antilogarithms for logarithms o.0000 (0.0001)0.9999. A similar table is given by Guillemin, Tables de Logarithmes (Paris, 1912) to 13 figures for 0.0000(0.00000.700o and to 12 figures for 0.7001(0.0001)0.9999.
Napierian Logarithms in their original form have passed com pletely out of use and are only of historic interest. (See NAPIER.) Hyperbolic Logarithms.—The first publication of a table which can be interpreted as a hyperbolic logarithm table in the modern sense is New Logarithmes (1619) by J. Speidell. The most extensive table was computed by Dase, Tafel der natiir lichen Logaritlimen der Zahlen (Vienna, 1850) ; it gives the 7-place logarithms of r (I )1000 and it:m(0.0'0,50o. Barlow in New Mathematical Tables (London, 1814) gives the 8-place logarithms of all integers to io,000. Schulze in his Neue und erweiterte Sammlung logarithmischer Tafeln (Berlin, 1778) includes a table of hyperbolic logarithms to 48 places of all integers to 2,200 and of the primes and some other numbers to io,000. This table was calculated by Wolfram, who was not able to complete the work. The incomplete table was given by Schulze, but Vega in the Thesaurus (1794) completed it. An 8-figure abridgment of it was included in Vega's collection of tables (1797) and later edi tions. Barlow used Wolfram's table in his calculations. Thiele in Tafel der Wolframschen Hyperbolischen 48-stelligen Logarith men (Dessau, 1908) extended the table to all numbers to 5,000 and the primes to io,000. Salomon, Logarithmische Tafeln (Vienna, 1827) gives io-place logarithms to ',coo and of primes to 10,333 ; Callet, 48-place logarithms to ioo and primes to 1,097; Hutton, 7-place to 1,200, also Willich (1852) ; Rees's Cyclopaedia (1819) art. "Hyperbolic Logarithms," r (I )io,000 to 8-places. Vega in Tabukze Logarithmico-trigonometricae (Leipzig, 1797) and Kohler in Logarithmisch-trigonometrisches Handbuch (Leip zig, 1848) give 8-place logarithms to 1,000 and primes to 10,000.