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Mathematics of Computation

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Tables of Trigonometric Functions in Non-Sexagesimal Arguments

Author: Raymond Clare Archibald
Journal: Math. Comp. 1 (1943), 33-44
Corrigendum: Math. Comp. 2 (1947), 320.
Corrigendum: Math. Comp. 1 (1943), 100.
Corrigendum: Math. Comp. 1 (1943), 68.
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  • [1] S. Günther, Studien zur Geschichte der mathematischen und physikalischen Geographie. IV. Analyse einiger kosmographischer Codices der Münchener Hof- und Staatsbibliothek, Halle, a/S., 1878, p. 249; codex p. 174r-176r: ``Et notandum quod in praesenti tabula quilibet gradus et hora diyiditur in 100 minuta, et quodlibet minutum in 100 secunda et sie de aliis.'' See M. Cantor, Vorlesungen über Geschichte der Mathematik, v. 2, 2nd ed., Leipzig, 1900, p. 185. Since the destruction of the library in March 1943, the codex may well be no longer in existence.
  • [2] Trigonometria Britannica she Doctrina Triangvlorvm Libri Dvo, Gouda, 1633. vi p.; Trigon. Brit, lib. I, p. 1-60; Trig. Brit., lib. II, p. 61-110; table, 264 p. It was only the second book, on spherical trigonometry (50 p.) which was written by Gellibrand; all the rest was by Briggs; see Gellibrand's prefatory greetings to students of mathematics. Here, as usual, Delambre$ ^{9}$ is right. Unfortunately I first accepted as correct the statement of Glaisher (Report, p. 65) ``The trigonometry is by Gellibrand.'' Hence the statements in MTAC, p. 10 (1. 18), 13 (1. 8), 26 (1. 1), should be modified accordingly, and the name Briggs be substituted for Gellibrand in the footnote on p. 13. Gellibrand tells us that Briggs' ``Canon of Sines'' was his first work, prepared ``thirty years more or less'' before, and that finally, tired out by the importunities of his friends to have it published, he decided to prepare it for the press, but had completed only the first book of the introduction at the time of his death. This statement suggests that Briggs may have computed the tables of natural sines, tangents and secants about 1600, and added the much less accurate logarithms of sines and tangents (see Delambre$ ^{9}$ and (10). Andoyer) perhaps about 1620, a few years after logarithms were first conceived. The interested reader will not overlook the tribute to Briggs in Delambre's ``Rapport."4
  • [3] R. Mehmke, ``Bericht über die Winkelteilung im Namen der Tafelcommission der Deutschen Mathematiker-Vereinigung,'' D. M. V., Jahresb., v. 8, 1900, p. 139-158; see especially p. 145-146. Here are many valuable details supplementing the present article.
  • [4] Fuller information concerning matters here discussed may be found in the following sources: Riehe de Prony, (a) ``Notice sur les grandes tables logarithmiques et trigonometriques,'' and (b) ``Eclaircissemens sur un point de l'histoire des tables trigonom6triques,'' Memoires de VInstitut National des Sciences et Arts. Sciences Mathimatiques et Physiques, v. 5, Paris, 1803, p. 49-55, and 67-93. J. B. J. Delambre, ``Rapport sur les grandes tables trigonometriques decimales du Cadastre,'' idem, p. 56-66. P. A. F. Lefort, (a) ``Description des grandes tables logarithmiques et trigonomdtriques, calculees au Bureau du Cadastre sous la direction de Prony et exposition des methodes et precedes mis en usage pour leurs construction,'' Paris, Observatoire, Annales, v. 4, 1858, Supplement, p. [123]--[150]; (b) ``Note sur les deux exemplaires manuscrits des grandes tables logarithmiques et trigonometriques calculees au Bureau du Cadastre,'' Institut de France, Acad. d. Sc., Comptes Rendus, v.46, 1858, p. 994-999. See also Nouv. Ann. d. Math., v. 14, 1855, p. (14)-(17); and DeMorgan's article on Prony in Penny Cyclopadia, v. 19, 1841.
  • [5] J. W. L. Glaisher, ``On logarithmic tables,'' R. Astr. So., Mo. Notices, v. 33, 1873, p. 455 footnote.
  • [6] See ``List of logarithmic, trigonometrical, and astronomical calculations, in manuscript, by Edward Sang,'' p. 44-47 of E. M. Horsburgh, Modern Instruments and Methods of Calculation, London and Edinburgh, 1914. See also R. So. Edinb., Proc., v. 9, 1878, E. Sang, ``On the construction of the canon of sines, for the decimal division of the quadrant,'' p. 343-349; ``On the precautions to be taken in recording and using the records of original computations,'' p. 349-352. V. 12, 1884, ``On the construction of the canon of logarithmic sines,'' p. 601-619; v. 16, 1890, ``Notice of fundamental tables in trigonometry and astronomy, arranged according to the decimal division of the quadrant,'' p. 249-256. See also Proc., v. 28, 1908, p. 183-196.
  • [7] For grades, minutes and seconds we have used the notation g, ,". Fourteen other varieties of notations are exhibited by Mehmke$ ^{3}$ (p. 153). None of these are listed in Cajori's History of Mathematical Notations (1928-29).
  • [8] Institut de France, Acad. d. Sei., Comptes Rendus, v. 70, 1870, p. 1233-1236, 1390; and v. 71, 1870, p. 362-368. The astronomer Antoine Joseph Yvon Villarceau (1813-1883) to whom reference is here made, was the discoverer of the third series of circles on the torus ("Villarceau circles"; see ``Thèoréeme sur la tore,'' Nouv. Ann. Math., v. 7, 1848, p. 345 ff. for an analytic proof, and F. G. M., Exercises de Giomflrie Descriptive, fourth ed., Tours and Paris, 1909, p. 573 ff. for a geometric proof.)
  • [9] J. B. J. Delambre, Histoire de VAstronomie Moderne, v. 2, Paris, 1821, p. 76-85, 393-120.
  • [10] Institut de France, Acad. d. Sei., Comptes Rendus, v. 126, 1898, p. 192-194. We are told that the decimal hour proposal was made in 1895 and that already decimal watches and decimal chronographs were available.

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Article copyright: © Copyright 1943 American Mathematical Society