Jost Bürgi's method for calculating sines

Abstract

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].

How Bürgi computed the sines of all integer angles simultaneously in 1586

“…As mentioned above, the convergence of the original Bürgi algorithm was already proven in [9,30]. The other assertions of Theorem 1 are new.…”

“…After consulting the hand-written catalog of the manuscripts' collection, he ordered microfilms of several works (personal communication). Folkerts looked as late as 2013 to one of those microfilms and recognized that the autograph Fundamentum Astronomiae [2], written in German by Jost Bürgi, contained the author's lost algorithm for computing sine tables (Figures 2 and 3) [8,9] Bürgi's example of his skillful method, the Artificium, explains the calculation for the multiples of 10 • , the ninth parts of the right angle. If we leave away zeros and column shifts in Figure 3, write sexagesimal numbers in base ten (for example 1 ′′ 3 ′′′ = 1 × 60 + 3 = 63 at the top of the next to rightmost column), and rotate the table by a quarter-turn, we obtain HOW BÜRGI COMPUTED THE SINES OF ALL INTEGER ANGLES SIMULTANEOUSLY IN 1586 3 the following first three lines from the three rightmost columns Sinus 1 to Sinus 2:…”

“…Only sums of integers and halving are involved before the final divisions and all sine values are determined simultaneously. The Artificium and the whole autograph were decrypted and edited by Dieter Launert [12,9]. The first proof of convergence was given by Andreas Thom using the Perron-Frobenius theorem [9].…”

“…The Artificium and the whole autograph were decrypted and edited by Dieter Launert [12,9]. The first proof of convergence was given by Andreas Thom using the Perron-Frobenius theorem [9]. Another proof with matrices by Jörg Waldvogel is more elementary and determines the rate of convergence [30].…”

Reconstruction of the genesis of a Renaissance algorithm to calculate sine tables

“…In [1] Andreas Thom has proven that Bürgis claim is true: The step from the vector (a (k) j ) 1≤j≤N to the vector (a (k+1) j ) 1≤j≤N is given by multiplication with a matrix with only positive entries. Therefore, by the Perron-Frobenius theorem of 1907/12, it has a dominant eigenvalue, to which corresponds the eigenvector (sin ∆, sin 2∆, .…”