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.