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].
The last page of Ursus's De astronomicis hypothesibus is headed: "Epigram on the new and true Ursine hypotheses." There follows an abusive couplet:As for the jealous little Dane, than whose nose nothing in the world is more revolting, may his guts burst with envy. Invidia invidulo rumpantur ut ilia Dano: Sordidius naso cuius in orbe nihil. 1After this comes an adulatory poem, introduced by a formal salutation: "To the most illustrious man, Master Nicolaus Reimarus Ursus of Dithmarschen, peerless mathematician, greetings." Reimarus, as your divine mathesis recommends you far and wide to the most eminent and distinguished men, so I, reverencer and devotee offamous men, and of you among them, ardently admire you. Because you have today condescended to visit me and we happened to shake hands, I thank you from my thankful heart with as many thanks as the silent night-sky has stars -stars which no one knows better than you, not Atlas, nor anyone who has followed him in this art.? Neither Ptolemy, nor even Copernicus himself, bold teacher of so much skill, was more learned. Indeed, whatever they taught with opposite reasons, it was granted to you first to reconcile. You are the first to show the middle way between the two, and you prove your assertions in a quite brilliant manner. So we all tum admiring eyes on you; and everyone wishes to be your eulogist. In return for such merits, Urania herself now wreathes laurel garlands for your head. Farewell! Flourish! Refreshing yourself among the sacred stars, nay one among the chorus of the gods.'
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