I just learned (via Facebook, no less) of a fascinating paper with the above title by Andreas Thom and coauthors.
Jost Burgi (1552-1632) was a Swiss mathematician, astronomer and clockmaker. He worked with Johannes Kepler from 1604 and is thought to have arrived at the notion of logarithms independent of Napier. He was also reputed to have constructed a table of sines by a brand new method, but until now the details of his Kunstweg (“artful method”) for computing sines were thought to have been lost. The beautiful book of van Brummelen, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry, (Princeton University Press, 2009) documents how computing trigonometric ratios was a central theoretical and practical preoccupation of ancient mathematics. We know of Burgi’s Kunstweg via a statement of his colleague and friend Nicolaus Ursus: “the calculation (of a table of sines)… can be done by a special way, by dividing a right angle into as many parts as one wants; and this is arithmetically. This has been found by Justus Burgi from Switzerland, the skilful technician of His Serene Highness, the Prince of Hesse.” But the details are not clear and apparently nobody, starting with Kepler himself, was ever able to reconstruct Burgi’s method.
Until now. Let me quote the paper:
a manuscript has survived in which urgi himself explains his method. M. Folkerts found this hitherto entirely unknown autograph of B¨urgi, with shelfmark IV Qu. 38a, in the Biblioteka Uniwersytecka in Wroclaw (Poland). From the preface of the book we know that Burgi presented it to the emperor Rudolf II (1552-1612, emperor since 1576) on 22 July 1592 in Prague and that some days later the emperor donated him the amount of 3000 Taler. Later the manuscript found its way into the library of the Augustinian monastery in Sagan (Lower Silesia; today: Zagan, Poland), the same town where Johannes Kepler lived from 1628 to 1630. In 1810, when the monastery was secularized, the manuscript was brought into the Universitatsbibiothek in Breslau (then Germany) where it fell into oblivion.
So what is the “skilful method”? Burgi’s manuscript gives instructions for carrying it out, which are equivalent to the following procedure: start with an arbitrary vector with positive entries, and iteratively apply the matrix
(this is a screen shot from the paper). By Perron-Frobenius theory, this iteration allows one to produce the unique eigenvector with a positive real eigenvalue. But some clever manipulation of trigonometric identities shows that this unique eigenvector is precisely the column vector with entries \( \sin (j\pi/2n) \) for \(j=1,\ldots,2n\). In other words, this process computes a table of sines.
Burgi does not explain why his method works (the argument above comes from Andreas Thom). How on earth did he think of it? Anyhow, I thoroughly enjoyed reading this paper with its combination of history, mathematics, and a whiff of mystery. What could be better?