  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.
