Spherical Geometry in the Eighteenth Century I

Renzo Caddeo, born in Sardinia, is a retired professor of geometry and former director of the Department of Mathematics at the University of Cagliari, where he was the organizer of a Masters course dedicated to the visualization of mathematical objects. Before teaching at Cagliari, he studied in Romania and France. His research interests are in differential geometry, in particular in biharmonic maps between Riemannian manifolds and in the surfaces of certain remarkable three-dimensional spaces. He collaborated with Alfred Gray to realize the extended Italian version of the book Modern Differential Geometry of Curves and Surfaces and with Athanase Papadopoulos on the two-volume work Leonhard Euler, Écrits Sur la Musique and the book Christiaan Huygens, Écrits Sur la Musique et le Son. Athanase Papadopoulos (born 1957) is Directeur de Recherche at the French Centre National de la Recherche Scientifique. His main fields of interest are geometry and topology, the history and philosophy of mathematics, and mathematics and music. He has held visiting positions at the Institute for Advanced Study, Princeton (1984–1985 and 1993–1894), USC (1998–1999), CUNY (Ada Peluso Professor, 2014), Brown University (Distinguished visiting professor, 2017), Tsinghua University, Beijing (2018), and the State University of Saint Petersburg (as the Lamé Chair) (2019), and has had several long visits to the Max-Plank Institute for mathematics (Bonn), the Erwin Schrödinger Institute (Vienna), the Graduate Center of CUNY (New York), the Tata Institute (Bombay), Galatasaray University (Istanbul), the University of Florence (Italy), Fudan University (Shanghai), Gakushuin University (Tokyo) and Presidency University (Calcutta) and Banaras Hindu University. He is the author of more than 220 published articles and 50 monographs and edited books.

• 1 Introduction by Renzo Caddeo and Athanase Papadopoulos.- Part I: Essays.- 2 Euler’s work on spherical geometry: An overview with comments by Athanase Papadopoulos and Vladimir Turaev.- 3 An introduction to spherical geometry based on fundamental works of Euler and Lagrange by Charalampos Charitos.- 4 On Euler’s memoir A construction relative to a problem of Pappus of Alexandria by Guillaume Théret.- 5 Notes on angles and solid angles, in relation with Euler’s memoir De mensura angulorum solidorum by Stelios Negrepontis and Athanase Papadopoulos.- 6 On the life and work of Johann Heinrich Lambert by Athanase Papadopoulos.- 7 A review of Johann Heinrich Lambert’s memoir Theorie der Parallellinien by Athanase Papadopoulos and Guillaume Théret.- 8 The pentagramma mirificum from Napier to Lambert: Notes on Lambert’s memoir on spherical trigonometry by Annette A’Campo-Neuen.- 9 Spherical trigonometry before the modern era: The treatise of Naṣīr al-Dīn al-Ṭūsī by Athanase Papadopoulos.- 10 The polar triangle, from Ibn ʿIrāq to Euler and Lagrange by Athanase Papadopoulos.- Part II: Sources.- 11 Principles of spherical trigonometry deduced from the method of maxima and minima by Leonhard Euler. Translated from the French by Athanase Papadopoulos and Alena Zhukova.- 12 General spherical trigonometry, deduced from fundamental principles in a brief and clearmanner by Leonhard Euler. Translated from the Latin by Renzo Caddeo.- 13 Various speculations on the area of spherical triangles by Leonhard Euler. Translated from the Latin by Renzo Caddeo.- 14 On the measure of solid angles by Leonhard Euler. Translated from the Latin by Renzo Caddeo.- 16 A construction relative to a problem of Pappus of Alexandria by Leonhard Euler. Translated from the Latin by Renzo Caddeo.- 17 Algebraic solution of a problem in geometryby Joseph-Louis Lagrange. Translated from the French by Athanase Papadopoulos.- 18 Solution of some problems relative to spherical triangles with a complete analysis of these triangles by Joseph-Louis Lagrange. Translated from the French by Vincent Alberge and Athanase Papadopoulos.- 19 Notes and Comments on Trigonometry by Johann-Heinrich Lambert. Translated from the German by Annette A’Campo Neuen.- 15 Some questions of the geometry of the plane and of the sphere by Leonhard Euler. Translated from the Latin by Renzo Caddeo.