Jost Bürgi and the discovery of the logarithms

Abstract

In the year 1620 the printing office of the University of Prague published a 58- page table containing the values an = (1.0001)n for 0 ! n ! 23027, rounded to 9 decimal digits. This table had been devised and computed about 20 years earlier by the Swiss-born astronomer and watchmaker Jost B¨urgi in order to facilitate the multi-digit multiplications and divisions he needed for his astronomical computations. The “Progreß Tabulen”, as B¨urgi called his tables, are considered to be one of the two independent appearances of the logarithms in the history of mathematics - the other one, due to John Napier (1550-1617), appeared in 1614. There are only a few copies of the original printing extant: one of them is now in the Astronomisch-Physikalisches Kabinett in Munich. Based on a copy of this original, the terminal digits of all table entries were extracted and compared with the exact values of an, a matter of a split second on a modern computer. In this presentation we give a brief account of the mathematical environment at the end of the 16th century as well as a detailed description of B¨urgi’s Progreß Tabulen and their application to numerical computations. We will also give a sketch of B¨urgi’s remarkable life and of his numerous achievements besides the discovery of the logarithms. Our main purpose, however, is to analyze the numerical errors in B¨urgi’s table. First of all, there are no systematic errors, e.g. the crux of the table, 1.000123027.0022 = 10, is correct with all digits given. 91.5% of the table entries are entirely correct, and 7.3% of the values show round-off errors between 0.5 and 1 unit of the least significant digit. The remaining 1.17% table errors are mainly errors of transcription and illegible digits. Statistics of the round-off errors leads to interesting conclusions concerning Bürgi’s algorithms of generating his table and on his handling of the round-off errors, as well as on the computational effort involved.