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Erwan Penchèvre

Publications and source records attributed to Erwan Penchèvre.

6 recordsLinked to original sources
arXiv2022

Quelques problèmes issus des Arithmetica Philosophica de Peter Roth

Here is a French translation and commentary of 17 problems in Peter Roth's Arithmetica Philosophica (1608). These problems are dealing with algebraic equations of degree 5 or 6; moreover, among these problems, 14 are also dealing with stereometry and polygonal numbers. These 14 problems are the only ones, in the Arithmetica Philosophica, to combine these three distinct topics in a single setting (algebra, stereometry, polygonal numbers).

math.HO↗
arXiv2021

Ibn al-Šātir, L'achèvement de l'enquête et la correction des fondements / Kitāb nihāya al-sūl fī tashīh al-'usūl. Édition, traduction et commentaire mathématique par Erwan Penchèvre

This book is a critical edition of a treatise of astronomy by the Syrian scholar Ibn al-Šātir (1304-1375). The Arabic text has been established on the basis of several manuscript copies, and it has been translated into French; a bilingual index is also provided. This treatise is one of the latest and one of the most elaborate medieval work on theoretical astronomy; it is famous today for having provided the first viable lunar model in terms of the distance Moon-Earth. Although demonstrations have been consigned to another manuscript which is now lost, we reconstruct the physical models in our mathematical commentary, and we attempt at a better historical understanding of the nascent concepts of space transformations, solid reference frames, trajectory, etc. Predictions of the models are to be compared to observations and to other contemporary models, as well as theoretical premises; in turn, when records of observations are missing, those have to be recalculated by modern means. Part of our commentary is devoted to the latitudinal movement of the five planets: Ibn al-Šātir's work on this problem belongs to a tradition initiated by Ibn al-Haytham three centuries before. Ibn al-Šātir addresses a clever criticism to the work of his predecessor Nasīr al-Dīn al-Tūsī on this problem, thus revealing a better understanding of affine rotations in space. Many other criticisms by Ibn al-Šātir, especially to the astronomers working at Marāġa during the 13th century, also contribute to a better knowledge of their own work.

math.HO↗
arXiv2016

Vénus selon Ibn al-Shatir

We attempt to grasp the mathematics behind the planetary theories of the Syrian astronomer Ibn al-Shatir (1304-1375) in his treatise Nihayat al-Sul. Following the Maragha school of astronomers, by composing circular movements with constant angular velocity, Ibn al-Shatir attains two goals. He eliminates the need of excentrics and equant points in astronomy; but he also describes longitudes and latitudes with a unique method, with no more orbs than what is strictly necessary to the longitudes. A better understanding of rotation as a spatial transformation enables this ultimate economy of thought. In our commentary, we take Venus as an example offering an interesting problem about the latitudes. It is the opportunity to give the edition of the chapter of the Nihayat al-Sul dedicated to the latitudes of Mercury and Venus.

math.HO↗
arXiv2016

Etienne Bézout on Elimination Theory

Bézout's name is attached to his famous theorem. Bézout's Theorem states that the degree of the eliminand of a system a $n$ algebraic equations in $n$ unknowns, when each of the equations is generic of its degree, is the product of the degrees of the equations. The eliminand is, in the terms of XIXth century algebra, an equation of smallest degree resulting from the elimination of $(n-1)$ unknowns. Bézout demonstrates his theorem in 1779 in a treatise entitled "Théorie générale des équations algébriques". In this text, he does not only demonstrate the theorem for $n>2$ for generic equations, but he also builds a classification of equations that allows a better bound on the degree of the eliminand when the equations are not generic. This part of his work is difficult: it appears incomplete and has been seldom studied. In this article, we shall give a brief history of his theorem, and give a complete justification of the difficult part of Bézout's treatise.

math.HO↗