Mathematische Semesterberichte

, Volume 65, Issue 1, pp 15–34 | Cite as

How Bürgi computed the sines of all integer angles simultaneously in 1586

Mathematik in historischer und philosophischer Sicht
  • 50 Downloads

Abstract

We present an algorithm discovered by Jost Bürgi around 1586, lost until 2013, and proven in 2015. Bürgi’s method needs only sums of integers and divisions by \(2\) to compute simultaneously and with any desired accuracy the sines of the \(n\)th parts of the right angle. We explain why it works with a new proof using polygons and discrete Fourier transforms.

Keywords

Jost Bürgi Sine table Discrete Fourier transformation Finite difference Planar polygon Napoleon’s theorem Petr–Douglas–Neumann theorem 

References

  1. 1.
    Berlekamp, E.R., Gilbert, E.N., Sinden, F.W.: A polygon problem. Am. Math. Mon. 72, 233–241 (1965)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bürgi, J.: Fundamentum Astronomiæ, 1592. Digital Library of the University of Wrocław. http://www.bibliotekacyfrowa.pl/dlibra/doccontent?id=45624. Accessed: 27 October 2017
  3. 3.
    Chang, G., Davis, P.J.: A circulant formulation of the Napoleon-Douglas-Neumann theorem. Linear Algebra Appl. 54, 87–95 (1983)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Clark, K.: Jost Bürgi’s Aritmetische und geometrische Progreß Tabulen (1620). Edition and commentary. Science Networks, Historical Studies 53. Birkhäuser, Springer, New York (2015)CrossRefMATHGoogle Scholar
  5. 5.
    Darboux, G.: Sur un problème de géométrie élémentaire, Bull. Sci. Math. Astr. 2e Sér. 2, pp. 298–304 (1878). http://archive.numdam.org/ARCHIVE/BSMA/BSMA_1878_2_2_1/BSMA_1878_2_2_1_298_1/BSMA_1878_2_2_1_298_1.pdf. Accessed: 27 October 2017Google Scholar
  6. 6.
    Douglas, J.: On linear polygon transformations. Bull. Am. Math. Soc. 46, 551–560 (1940)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Fisher, J.C., Ruoff, D., Shilleto, J.: Perpendicular polygons. Am. Math. Mon. 92, 23–37 (1985)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Folkerts, M.: Eine bisher unbekannte Schrift von Jost Bürgi zur Trigonometrie. In: Gebhardt, R. (ed.), Arithmetik, Geometrie und Algebra in der frühen Neuzeit, pp. 107–114. Adam-Ries-Bund, Annaberg-Buchholz (2014)Google Scholar
  9. 9.
    Folkerts, M., Launert, D., Thom, A.: Jost Bürgi’s method for calculating sines. Hist. Math. 43(2), 133–147 (2016)CrossRefMATHGoogle Scholar
  10. 10.
    Grünbaum, B.: Lecture notes on modern elementary geometry. EPrint Collection. University of Washington, Washington (1999). http://hdl.handle.net/1773/15592 Google Scholar
  11. 11.
    Kepler, J.: Gesammelte Werke. Bayerische Akademie der Wissenschaften, Munich (1937–2017). http://kepler.badw.de/kepler-digital.html. Accessed: 27 October 2017
  12. 12.
    Launert, D.: Nova Kepleriana: Bürgis Kunstweg im Fundamentum Astronomiæ, Entschlüsselung seines Rätsels. Verlag der Bayerischen Akademie der Wissenschaften, Munich (2015)Google Scholar
  13. 13.
    List, M., Bialas, V.: Nova Kepleriana: Die Coss von Jost Bürgi in der Redaktion von Johannes Kepler. Verlag der Bayerischen Akademie der Wissenschaften, Munich (1973)MATHGoogle Scholar
  14. 14.
    Martini, H.: On the theorem of Napoleon and related topics. Math. Semesterber. 43, 47–64 (1996)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Neumann, B.H.: Some remarks on polygons. J. Lond. Math. Soc. 16, 230–245 (1941)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Neumann, B.H.: A remark on polygons. J. Lond. Math. Soc. 17, 165–166 (1942)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Nicollier, G.: Convolution filters for polygons and the Petr–Douglas–Neumann theorem. Beitr. Algebra Geom. 54, 701–708 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Nicollier, G.: Convolution filters for triangles. Forum Geom. 13, 61–85 (2013)MathSciNetMATHGoogle Scholar
  19. 19.
    Nicollier, G.: Some theorems on polygons with one-line spectral proofs. Forum Geom. 15, 267–273 (2015)MathSciNetMATHGoogle Scholar
  20. 20.
    Nicollier, G.: A characterization of affinely regular polygons. Beitr. Algebra Geom. 57, 453–458 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Nicollier, G., Stadler, A.: Limit shape of Iterated Kiepert triangles. Elem. Math. 68, 61–64 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pech, P.: The harmonic analysis of polygons and Napoleon’s theorem. J. Geom. Graph. 5, 13–22 (2001)MathSciNetMATHGoogle Scholar
  23. 23.
    Petr, K.: O jedné větě pro mnohoúhelníky rovinné (On a Theorem for Planar Polygons). Čas. Mat. Fys. 34, 166–172 (1905). http://www.digizeitschriften.de/dms/toc/?PPN=PPN31311028X_0034 Google Scholar
  24. 24.
    Roegel, D.: Some remarks on Bürgi’s interpolations. Technical report. LORIA, Nancy (2016). http://locomat.loria.fr/locomat/reconstructed.html Google Scholar
  25. 25.
    Schoenberg, I.J.: The finite Fourier series and elementary geometry. Am. Math. Mon. 57, 390–404 (1950)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Schuster, W.: Polygonfolgen und Napoleonsätze. Math. Semesterber. 41, 23–42 (1994)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Shapiro, D.B.: A periodicity problem in plane geometry. Am. Math. Mon. 91, 97–108 (1984)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Staudacher, F.: Jost Bürgi, Kepler und der Kaiser, 3rd edn. NZZ Libro, Zurich (2016)Google Scholar
  29. 29.
    Vartziotis, D., Wipper, J.: On the construction of regular polygons and generalized Napoleon vertices. Forum Geom. 9, 213–223 (2009)MathSciNetMATHGoogle Scholar
  30. 30.
    Waldvogel, J.: Jost Bürgi’s Artificium of 1586 in modern view, an ingenious algorithm for calculating tables of the sine function. Elem. Math. 71, 89–99 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.University of Applied Sciences of Western SwitzerlandSionSwitzerland

Personalised recommendations