Jost B\"urgi's Method for Calculating Sines

TL;DR

The paper uncovers Jost B"urgi's 16th-century novel method for calculating sines, which is elementary, converges rapidly, and differs from traditional ancient procedures, with a modern proof of its correctness.

Contribution

It introduces and explains B"urgi's unique algorithm for sine calculation, providing the first detailed analysis and proof of its validity.

Findings

B"urgi's method uses only addition and halving.

The algorithm converges quickly to accurate sine values.

A modern proof confirms the method's correctness.

Abstract

For a long time it has been known that in the 16th century the Swiss mathematician Jost B\"urgi found a new method for calculating sines, but no information about the details has been available. Recently a manuscript written by B\"urgi himself has come to light in which he explains his algorithm. It is totally different from the traditional procedure which was used until the 17th century. In the first part of the article the standard method is explained which was rooted in Greek antiquity with Ptolemy's computation of chords and which was used in the Arabic-Islamic tradition and in the Western European Middle Ages for calculating chords as well as sines. The main part of the article deals with B\"urgi's way. By only using additions and halving, his procedure is elementary and it converges quickly. B\"urgi does not explain why his method is correct, but in the last part of the article a modern proof for the correctness of his algorithm is provided.