From various sources we know that the Swiss instrument maker and mathematician Jost Bürgi (1552-1632) found a new way of calculating any sine value. Many mathematicians and historians of mathematics have tried to reconstruct his so-called "Kunstweg" ("skillful / artful method"), but they did not succeed. Now a manuscript by Bürgi himself has been found which enables us to understand his procedure. The main purpose of this article is to explain Bürgi's method. It is totally different from the conventional way to calculate sine values which was used until the 17th century. First we will give a brief overview of the early history of trigonometry and the traditional methods for calculating sines.
Historical remarks on trigonometry and on trigonometrical tablesThe main purpose of the following remarks is to show how values of trigonometric functions, especially of chords and sines, were calculated before the time of Bürgi. For this reason it is necessary to go back to Greek antiquity, the Arabic-Islamic tradition and the Western European Middle Ages. Of course this is not the place to present an extensive history of trigonometry 1 .In Greek antiquity, trigonometric functions were used in different contexts in order to calculate triangles and quadrangles: in geodesy particularly for determining heights and distances and in astronomy for calculating spherical triangles. In geodesy the main method was to use the proportion of the catheti in plane orthogonal triangles, i.e., in modern terms, the tangent. In astronomy the central problem was to find relations between the chords and the radius of the circle, today expressed by the sine.When simple geodetic measurements had to be carried out, properties of similar triangles were used to find the fourth proportional with the help of 1 This can be found in [Van Brummelen, 2009].
This article describes how the decimal place value system was transmitted from India via the Arabs to the West up to the end of the fifteenth century. The arithmetical work of al-Khwārizmī's, ca. 825, is the oldest Arabic work on Indian arithmetic of which we have detailed knowledge. There is no known Arabic manuscript of this work; our knowledge of it is based on an early reworking of a Latin translation. Until some years ago, only one fragmentary manuscript of this twelfth-century reworking was known (Cambridge, UL, Ii.6.5). Another manuscript that transmits the complete text (New York, Hispanic Society of America, HC 397/726) has made possible a more exact study of al-Khwārizmī's work. This article gives an outline of this manuscript's contents and discusses some characteristics of its presentation.
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