RésuméC'est le premier d'une série d'articles comportant quatre textes composés entre le XIeet le XIIIesiècle, qui traitent de la proposition 5 du livre III desSphériquesde Ménélaüs. Le premier article comprend des commentaires historiques et mathématiques de l'œuvre d'Ibn ʿIrāq en géométrie sphérique et une édition critique des deux textes qu'il a consacrés à la rectification de la proposition III.5, ainsi que la traduction de ces deux textes. Le second article propose une édition critique des textes de Naṣīr al-Dīn al-Ṭūsī et (...) d'Ibn Abī Jarrāda sur le même sujet, cette fois aussi avec une traduction et des commentaires historiques et mathématiques. (shrink)

Originally, the expression “Teichmüller theory” referred to the theory that Oswald Teichmüller developed on deformations and on moduli spaces of marked Riemann surfaces. This theory is not an isolated field in mathematics. At different stages of its development, it received strong impetuses from analysis, geometry, and algebraic topology, and it had a major impact on other fields, including low-dimensional topology, algebraic topology, hyperbolic geometry, geometric group theory, representations of discrete groups in Lie groups, symplectic geometry, topological quantum field theory, theoretical (...) physics, and there are certainly others. Of course, the impacts on these various fields are not equally important, but in some cases (namely, low-dimensional topology, algebraic geometry, and physics) the impact was crucial. At the same time, Teichmüller theory established important connections between the fields mentioned. This, in part, is a consequence of the diversity and the richness of the structure that Teichmüller space itself carries. From a more subjective point of view, the result of pondering on these connections and applications demonstrates the unity of mathematics. The aim of this paper is to survey the origin of Teichmüller theory and the development of its early major ideas. (shrink)

Nicolas-Auguste Tissot published a series of papers on cartography in which he introduced a tool which became known later on, among geographers, under the name of the Tissot indicatrix. This tool was broadly used during the twentieth century in the theory and in the practical aspects of the drawing of geographical maps. The Tissot indicatrix is a graphical representation of a field of ellipses on a map that describes its distortion. Tissot studied extensively, from a mathematical viewpoint, the distortion of (...) mappings from the sphere onto the Euclidean plane that are used in drawing geographical maps, and more generally he developed a theory for the distortion of mappings between general surfaces. His ideas are at the heart of the work on quasiconformal mappings that was developed several decades after him by Grötzsch, Lavrentieff, Ahlfors and Teichmüller. Grötzsch mentions the work of Tissot, and he uses the terminology related to his name. Teichmüller mentions the name of Tissot in a historical section in one of his fundamental papers where he claims that quasiconformal mappings were used by geographers, but without giving any hint about the nature of Tissot’s work. The name of Tissot is missing from all the historical surveys on quasiconformal mappings. In the present paper, we report on this work of Tissot. We shall mention some related works on cartography, on the differential geometry of surfaces, and on the theory of quasiconformal mappings. This will place Tissot’s work in its proper context. (shrink)

RésuméDans les Sphériques, Ménélaüs ne démontre pas l'importante proposition III.5, mais propose seulement une esquisse de démonstration. Une fois le livre des Sphériques traduit en arabe, les mathématiciens, à partir de la fin du IXe siècle, ont voulu en donner une démonstration complète. Le développement de la géométrie sphérique a permis à Ibn ʿIrāq de parvenir au but. Un premier article a été consacré à sa contribution. Deux mathématiciens du XIIIe siècle – Naṣīr al-Dīn al-Ṭūsī et Ibn Abī Jarrāda – (...) ont repris la démonstration de cette même proposition, à partir de Ménélaüs et des acquis d'Ibn ʿIrāq. C'est le sujet du présent article. (shrink)