Spherical Trigonometry Before the Modern Era: The Treatise of Naṣı̄r al-Dı̄n al-Ṭūsı̄

Athanase Papadopoulos ()
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Athanase Papadopoulos: Université de Strasbourg et CNRS, Institut de Recherche Mathématique Avancée

Chapter Chapter 9 in Spherical Geometry in the Eighteenth Century I, 2026, pp 205-221 from Springer

Abstract: Abstract This is an overview of Naṣı̄r al-Dı̄n al-Ṭūsı̄’s Treatise of the quadrilateral, an invaluable thirteenth century document on spherical geometry which was translated into French in 1891. The title we are using here is the one given by the translator (Alexandre Carathéodory). A title which is closer to the original Arabic is Disclosing the secrets of the secant figure. The term “secant figure”, to which the title refers is the so-called “complete (spherical) quadrilateral”, that is, the figure that underlies what we call today Menelaus’ Theorem. This theorem gives a formula that was extensively used by astronomers in their computations and the establishment of their tables since the first century AD, notably by Ptolemy, in the absence of the spherical trigonometric formulae that were discovered later. Naṣı̄r al-Dı̄n al-Ṭūsı̄’s treatise contains much more than Menelaus’ theorem, since we find there a complete system of spherical trigonometric formulae, with complete proofs. The treatise includes at the same time invaluable historical information on the discovery of the trigonometric formulae by the Arab mathematicians of the Middle-Ages and the transformation of the field of spherical trigonometry that this discovery led to.

Keywords: Naṣı̄r al-Dı̄n al-Ṭūsı̄; Spherical trigonometry; Spherical geometry; Quadrilateral; Compounded ratio; Complete spherical quadrilateral; Sector figure; Menelaus theorem; Discovery of the sine rule; 01A20; 01A30; 01A35 (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-12466-1_9

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DOI: 10.1007/978-3-032-12466-1_9

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