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Napier, Torporley, Menelaus, and Ptolemy: Delambre and De Morgan’s observations on seventeenth-century restructuring of spherical trigonometry. (English) Zbl 1390.01009

Zack, Maria (ed.) et al., Research in history and philosophy of mathematics. The CSHPM 2016 annual meeting in Calgary, Alberta, Canada, May 2016. Cham: Birkhäuser (ISBN 978-3-319-64092-1/hbk; 978-3-319-64551-3/ebook). Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne d’Histoire et de Philosophie des Mathématiques, 149-168 (2017).
Nathaniel Torporley and John Napier, in the first and second decades respectively of the seventeenth century, presented rules for dealing with the trigonometry of spherical triangles. Over two hundred years later, Augustus De Morgan discussed these results, (1) expressing a low opinion of their value and (2) raising the possibility that Napier had based his work on Torporley’s. The present paper argues against De Morgan on both counts.
For the entire collection see [Zbl 1390.01007].

MSC:

01A45 History of mathematics in the 17th century
01A55 History of mathematics in the 19th century
51-03 History of geometry
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[1] Augustus De Morgan. Circular parts (Napier’s). The Penny Cyclopaedia for the Diffusion of Useful Knowledge, VII:195, 1837.
[2] Augustus De Morgan. On the invention of the circular parts: To the editors of the philosophical magazine and journal. Philosophical Magazine and Journal, Series 3, Volume 22(146):350-353, 1843a.
[3] Augustus De Morgan. Nathaniel Torporley. The Penny Cyclopaedia for the Diffusion of Useful Knowledge, XXVI:312, 1843b.
[4] Jean Baptiste Joseph Delambre. Histoire de l’astronomie moderne. Courcier, 1821.
[5] J. Klein. Greek Mathematical Thought and the Origin of Algebra. MIT Press Classics. MIT Press, 1976. ISBN 9780262610223.
[6] John Napier. Mirifici logarithmorum canonis constructio, et eorum ad naturales ipsorum numeros habitudines; vna cum appendice, de alia eaque præstantiore logarithmorum specie condenda. quibus accessere propositiones ad triangula sphaerica faciliore calculo resoluenda: vnà cum annotationibus aliquot doctissimi d. henrici briggi in eas, et memoratam appendicem. authore et inuentore ioanne nepero,.., 1614a.
[7] John Napier. Mirifici logarithmorum canonis descriptio, ejusque usus, in utraque trigonometria, ut etiam in omni logistica mathematica, amplissimi, facillimi, et expeditissimi explicatio. ex officina Andrea Hart biblipola, Edinburgi, 1614b.
[8] John Napier. A Description of the Admirable Table of Logarithms translated in English by Edward Wright. All persued and aproued by the Authour, and published since the death of the Translator. Simon Waterson, London, 1618.
[9] Claudius Ptolemy. Almagest [Translated by G.J. Toomer]. Princeton University Press, 1998.
[10] Claudius Ptolemy, Luca Gaurico, and George. Clavdii Ptolemaei …Almagestvm Sev Magnae Constrvctionis Mathematicae Opvs Plane Divinvm Latina Donatvm Lingva. [& calcographica Luceantonii Iuta officina Aere proprio, ac Typis excussa], [In Vrbe Veneta], 1528.
[11] Johannes Regiomontanus. De triangulis omnimodis libri quinque. John Petreus, Nurnberg, 1533. URL https://books.google.com/books?id=lVxRAAAAcAAJ.
[12] Joel Silverberg. Napier’s Rules of Circular Parts. In CSHPM, University of British Columbia, Vancouver BC, Canada, volume 21, pages 160-174. Canadian Society for History and Philosophy of Mathematics, 2008. URL https://www.academia.edu/5870432.
[13] Joel Silverberg. Nathaniel Torporley and his Diclides Coelometricae (1602)-a preliminary investigation. In CSHPM, Memorial University, St.John’s, Newfoundland, Canada, volume 22, pages 143-154. Canadian Society for History and Philosophy of Mathematics., 2009. URL https://www.academia.edu/5890336. CSHPM, Memorial University, St Johns, Newfoundland, Canada.
[14] Nathale Torporlaeo. Diclides coelometricae seu Valuae astronomicae vniuersales omnia artis totius munera pseuphophoretica in sat modicis finibus duarum tabularum. Excudebat Felix Kingston, Londini, 1603.
[15] Glen Van Brummelen. the Mathematics of the Heavens and the Earth, The early history of trigonometry. Princeton University Press, 2009. · Zbl 1160.01003
[16] Glen Van Brummelen. Heavenly mathematics: The forgotten art of spherical trigonometry. Princeton University Press, 2013. · Zbl 1273.51003
[17] Glen Van Brummelen.
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