Tycho Brahe and Mathematical Techniques

During the course of his life, Tycho was involved in several disputes over intellectual property and claims to priority. The greatest of his antagonists was the Imperial Mathematician Nicolai Reymers Baer, called Ursus. In a lengthy and bitter dispute, Tycho's main charge was that Ursus had plagiarised his system of the world. Also at stake, however, was the credit for the development of a new mathematical technique, the method of prosthaphaeresis. Tycho claimed that he had played a key role in this mathematical discovery, and that Ursus' publication of the method constituted another theft; Ursus advanced the counter-claim that it had been the work of the itinerant mathematician Paul Wittich (d. 1586) and the instrument-maker (and co-inventor of logarithms) Jost Bürgi (1552-1632), from whom he had obtained it, and that Tycho was a poor mathematician. In this matter, posterity has sided with Ursus rather than Tycho.

If Tycho had little to do with the invention of prosthaphaeresis, he certainly recognised its importance for astronomy. Named after the Greek words for addition and subtraction, the method simplified calculations in spherical trigonometry by substituting these procedures for the multiplication and division of trigonometrical functions. It was therefore of considerable use in reducing the large quantities of astronomical data generated on Hven. Following WittichÍs visit to Uraniborg in 1580, Tycho made use of the prosthaphaeretic identity which is expressed in modern notation as sin A sin B = 1/2(cos (A-B) - cos (A+B)). A second identity, cos A cos B = 1/2(cos (A-B) + cos (A+B)), appeared in a trigonometric manual from the observatory dated to 1591, and may have been imparted to Tycho by Bürgi, who seems to have discovered it and the proofs of both equations after himself meeting with Wittich. Following Ursus' publication of the formulae in 1588, the prosthaphaeretic method was further developed by other mathematicians, including the Jesuit Christoph Clavius.

As various scholars have suggested, Wittich's own contribution to the development of prosthaphaeresis may have been to do little more than convey to Tycho and Bürgi a formula extracted from the unpublished manuscripts of the Nuremberg mathematician Johannes Werner (1468-1528). But the question of why Tycho should have made any claim to a role in the invention of prosthaphaeresis, which he did as early as the end of 1580, has not been adequately answered.

Recommended Reading

J. Dreyer, "On Tycho Brahe's Manual of Trigonometry", The Observatory 39 (1916): 127-131

V. Thoren, "Prosthaphaeresis Revisited", Historia Mathematica 15 (1988): 32-39

Full Bibliography