Spherical trigonometry before the modern era:The treatise of Nasir al-Din al-Tusi

TL;DR

This paper reviews Nasir al-Din al-Tusi's 13th-century treatise on spherical geometry, highlighting its comprehensive trigonometric formulae, historical significance, and influence on astronomy and mathematics.

Contribution

It provides a detailed analysis of al-Tusi's treatise, emphasizing its complete system of spherical trigonometry and historical context, which predates modern formulae.

Findings

Contains a complete system of spherical trigonometric formulae.

Includes proofs and historical insights on Arab mathematicians.

Highlights the treatise's influence on astronomical computations.

Abstract

This is an overview of Nasir al-Din al-Tusi's Treatise of the quadrilateral, an invaluable 13th 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 Carath{\'e}odory). 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. Nasir'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. The final version of this paper will appear in the book Spherical geometry in the eighteenth century, I: Euler, Lagrange and Lambert, edited by Renzo Caddeo and Athanase Papadopoulos, Springer, 2026.