Abstract
An effort to reorganize and systematize planar and spherical trigonometry began in the 15th century with the work of Regiomontanus, extended throughout the 16th century with work by Otho, Rheticus, Pitiscus, and Fincke, and continued into the 17th century by Napier, Torporley, Viète, and others. During the 18th and 19th century, publications by Euler, Taylor, Fourier, and Gauss extended the role of trigonometric functions into new areas including power series, and complex functions of complex variables. An analysis of De Morgan’s criticism of Napier’s and Torporley’s efforts in this area sheds light on the challenges to an historian of mathematics of one era attempting to understand the thought process of mathematicians living in earlier times. In particular, we focus on two areas: the historian’s knowledge of future mathematical developments and modes of expression unknown to those living in the earlier period, and secondly an incomplete, inaccurate, or absent knowledge on the part of the historian of definitions, references, or conventions well known to those of the earlier era. These definitions, references, and conventions were often used without comment or explanation, and occasionally used without mention, since the writer could assume them to be common knowledge to the readers of his time, and that their use would be understood by his readers, even if that use was implicit.
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Notes
- 1.
Article on Circular Parts (Napier’s), page 195.
- 2.
Note the self references in these two articles.
- 3.
Adrien Romain, of whom we have spoken in the article on Viète …published in 1609 a work under the title Triangulorum sphaericorum brevissimus. Frightened by the terrible prolixity of Rheticus and Otho, he reduced all of Spherical Trigonometry into six problems, of which all other problems were special cases. He avoids having anything to do the perpendiculars which divide any triangle in to two right angled triangles. He prefers the methods pioneered by Viète, but …In spite of these magnificent promises, the six problems, presented in a number of classes that mount up to no fewer than seventeen, form a whole which is every bit as frightening and nearly as difficult to understand as the rubbish of Rheticus.
- 4.
And following this most obscure and singular work, we come at last to another work of the same type, with which it shares nothing except its eccentricity. It is that which Torporley published under the title Diclides Coelometricae …London, 1602.
- 5.
Only homogeneous magnitudes are to be compared with one another where comparison means, on the one hand, adding and subtracting magnitudes to form algebraic expressions and, on the other, equating magnitudes or expressions with one another (Klein 1976).
- 6.
Delambre’s six-volume history of astronomy included: Histoire de l’astronomie ancienne (1817), Histoire de l’astronomie du moyen age (1819), Histoire de l’astonomie moderne. 2 volumes (1821), and Histoire de l’astronomie au dix-huitième siècle (1827). The history of astronomy in the 18th century was withheld by Delambre to be published posthumously.
- 7.
Quod si neuter angulorum a & g rectus offeratur quemadmodum ex huius trahitur, verum in triplici varietate habebuntur.
- 8.
Neutro igitur angulorum a & g recto existente, tamersi figuratione triplici utamur, syllogismus tamen erit unicus.
- 9.
Sun and moon.
References
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Silverberg, J.S. (2017). Napier, Torporley, Menelaus, and Ptolemy: Delambre and De Morgan’s Observations on Seventeenth-Century Restructuring of Spherical Trigonometry. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. CSHPM 2016. Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne d’Histoire et de Philosophie des Mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-64551-3_10
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