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Jost Bürgi and the discovery of the logarithms

by J. Waldvogel

(Report number 2012-43)

Abstract
In the year 1620 the printing office of the University of Prague published a 58-page table containing the values \(a_n = (1.0001)^n\) for \(0 \le n \le 23027\), rounded to 9 decimal digits. This table had been devised and computed about 20 years earlier by the Swiss-born astronomer and watchmaker Jost Bürgi in order to facilitate the multi-digit multiplications and divisions he needed for his astronomical computations. The "Progress Tabulen", as Bürgi 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 \(a_n\), 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 Bürgi's Progress Tabulen and their application to numerical computations. We will also give a sketch of Bürgi'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 Bürgi's table. First of all, there are no systematic errors, e.g.~the crux of the table, \(1.0001{23027.0022}^ = 10\), 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.

Keywords: none

@Techreport{W12_496,
  author = {J. Waldvogel},
  title = {Jost B\"urgi and the discovery of the logarithms},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-43},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-43.pdf },
  year = {2012}
}

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