Jost Bürgi’s methods of calculating sines, and possible transmission from India

  • Roy Wagner
  • Samuel HunzikerEmail author


A few years ago, a manuscript by Jost Bürgi (1552–1632) was brought to scholarly attention, which included an ingenious sine calculation method. The purpose of this paper is to discuss two aspects of this manuscript. First, we wish to improve the current understanding of Bürgi’s method of sine calculation, especially with respect to the calculation of sines at a resolution of 1 min. Second, we wish to suggest a possible transfer of knowledge between India’s Kerala School of mathematical astronomy and Bürgi. The evidence for the latter seems to be stronger than the evidence for other available case studies, but still revolves mainly around analogies, and can therefore not be considered as conclusive proof of transmission. We also append a translation of the relevant chapter of Bürgi’s treatise.


Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. Bala, Arun. 2006. The dialogue of civilizations in the birth of modern science. New York: Palgrave Macmillan.CrossRefGoogle Scholar
  2. Bürgi, Jost. 1592. “Fundamentum Astronomiae.” Manuscript no. IV Q 38 a. Biblioteka Uniwersytecka in Wrocław., Accessed July 3 2019.
  3. Clark, Kathleen M. 2015. Jost Bürgi’s Arithmetische Und Geometrische Progreß Tabulen (1620). Basel: Birkhäuser.CrossRefzbMATHGoogle Scholar
  4. Davids, Karel. 2005. Craft secrecy in Europe in the early modern period: A comparative view. Early Science and Medicine 10 (3): 341–348.CrossRefGoogle Scholar
  5. Divakaran, P.P. 2018. The Mathematics of India: Concepts, Methods, Connections. Singapore: Springer.CrossRefzbMATHGoogle Scholar
  6. Folkerts, Menso, Dieter Launert, and Andreas Thom. 2016. Jost Bürgi’s method for calculating sines. Historia Mathematica 43 (2): 133–147. Scholar
  7. Gaulke, Karsten (ed.). 2007. Der Ptolemäus von Kassel. Landgraf Wilhelm IV. von Hessen-Kassel und die Astronomie. Kassel: Museumslandschaft Hessen-Kassel.Google Scholar
  8. Gingerich, Owen, and Robert S. Westman. 1988. The Wittich connection: Conflict and priority in late sixteenth-century cosmology. Transactions of the American Philosophical Society 78 (7): i–i148. Scholar
  9. Hanschke, Ulrike. 1991. Die Gartenanlagen der Landgrafen Wilhelm IV. und Moritz in Kassel im Spiegel handschriftlicher Quellen. Die Gartenkunst 3 (2): 175–188.Google Scholar
  10. Hayashi, Takao. 1997. Āryabhata’s rule and table for sine-differences. Historia Mathematica 24 (4): 396–406. Scholar
  11. Jaeger, Friedrich (ed.). 2007. Enzyklopädie der Neuzeit, vol. 5. Stuttgart: J.B. Metzler.Google Scholar
  12. Joseph, George Gheverghese. 2009a. A passage to infinity: Medieval Indian mathematics from Kerala and its impact. New Delhi: Sage Publications.zbMATHGoogle Scholar
  13. Joseph, George Gheverghese (ed.). 2009b. Kerala mathematics: History and its possible transmission to Europe. Delhi: B.R. Publishing Corporation.Google Scholar
  14. Lach, Donald F. 1994. Asia in the making of Europe, Book 1, vol. 1. Chicago: University of Chicago Press.Google Scholar
  15. Launert, Dieter. 2015. Bürgis Kunstweg im Fundamentum Astronomiae: Entschlüsselung seines Rätsels. München: Bayerische Akademie der Wissenschaften München. Accessed July 3 2019.Google Scholar
  16. List, Marta, and Volker Bialas. 1973. Die Coss von Jost Bürgi in der Redaktion von Johannes Kepler: Ein Beitrag zur frühen Algebra. München: Bayerische Akademie der Wissenschaften München. Accessed July 3 2019.zbMATHGoogle Scholar
  17. Nicollier, Grégoire. 2018. How Bürgi computed the sines of all integer angles simultaneously in 1586. Mathematische Semesterberichte 65 (1): 15–34. Scholar
  18. Plofker, Kim. 2009. Mathematics in India. Princeton: Princeton University Press.zbMATHGoogle Scholar
  19. Plofker, Kim. 2011. Why did sanskrit mathematics ignore Asakrt methods? In Scientific literature in Sanskrit, vol. 1, ed. S.R. Sarma and G. Wojtilla, 61–76. Delhi: Motilal Banarsidass.Google Scholar
  20. Raju, C.K. 2007. Cultural foundations of mathematics: The nature of mathematical proof and the transmission of the calculus from India to Europe in the 16th c. CE.. Delhi: Pearson Longman.Google Scholar
  21. Ramasubramanian, K., and M.S. Sriram. 2010. Tantrasangraha of Nilakantha Somayaji. London: Springer.zbMATHGoogle Scholar
  22. Roegel, Denis. 2015. Jost Bürgi’s skillful computation of sines. hal-01220160. Accessed July 3 2019.
  23. Roegel, Denis. 2016. A preliminary note on Bürgi’s computation of the sine of the first minute. hal-01316358. Accessed July 3 2019.
  24. Sarma, K.V., K. Ramasubramanian, M.D. Srinivas, and M.S. Sriram. 2009. Ganita-Yukti-Bhasa of Jyesthadeva. Berlin: Springer.Google Scholar
  25. Scaliger, Joseph Justus. 1583. Opus Novum de Emendatione Temporum. Basel: Roberti Stephani.Google Scholar
  26. Šíma, Zdislav. 1993. Prague sextants of Tycho Brahe. Annals of Science 50: 445–453. Scholar
  27. Staudacher, Fritz. 2018. Jost Bürgi, Kepler und der Kaiser: Uhrmacher, Instrumentenbauer, Astronom, Mathematiker, Erz-Metallurgist, 4th ed. Zürich: NZZ Libro.Google Scholar
  28. Ullrich, Peter. Forthcoming. An educated guess how Jost Bürgi may have come to his ‘Artificium’ for the calculation of values of the sine function.Google Scholar
  29. Ullrich, Peter. 2016. The mathematics behind Jost Bürgi’s method for calculating sine tables. Proceedings in Applied Mathematics and Mechanics 16 (1): 891–892. Scholar
  30. von Mackensen, Ludolf. 1979. Die erste Sternwarte Europas mit ihren Instrumenten und Uhren: 400 Jahre Jost Bürgi in Kassel. Munich: Callwey Verlag.Google Scholar
  31. Wagner, Roy. 2015. Citrabhānu’s twenty-one algebraic problems in Malayalam and Sanskrit. Historia Mathematica 42 (3): 263–279. Scholar
  32. Waldvogel, Jörg. 2016. Jost Bürgi’s Artificium of 1586 in modern view, an ingenious algorithm for calculating tables of the sine function. Elemente der Mathematik 71 (3): 89–99. Scholar
  33. Werner, Petra. 2013. Die Menagerie des Landgrafen Karl: Ein Beitrag zur Einheit von Natur und Kunst im Barcokzeitalter. Ph.D. dissertation, University of Kassel.Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Chair of History and Philosophy of Mathematical SciencesETH ZürichZurichSwitzerland

Personalised recommendations