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# Séminaire – Histoire et Philosophie des Mathématiques

## December 18, 2023 @ 9h00 - 17h30

**Algebraization of curves**

Organization Paul-Emmanuel Timotei

**Simon Gentil**(SPHERE/UPC)

*Brief overview of the use of algebra for a theory of curves between 1650 and 1750.*

Résumé

In this communication, we propose to look at the use of algebra in geometry during the early modern period, particularly with the aim of establishing a theory of curves. We will take Descartes and the publication of his « Geometry, » in French in 1637, then in Latin in 1649, and in 1659-1661 as a starting point. We will demonstrate that Descartes’ algebraic manipulations radically transform the geometric landscape of the time while following a certain tradition from the ancients. We will focus on how Descartes legitimizes and organizes the entire set of curves, rendered infinite, through his various classifications. We will also briefly discuss the popularization of the idea of a « curve in general ». In the continuation of the presentation, we will comment Leibniz’s work on the « Conic Section » to highlight some issues in Descartes’ algebraic approach, as well as Newton’s work on the classification of third-order lines to pose some epistemological questions related to the use of algebra in the context of a general discourse on curves. In particular, we will address issues of unity, link between curve, equation and coordinates, handling of specific cases, consideration of infinite elements, etc. Finally, we will look at Euler’s work, in particular his method of identifying a curve with an equation, and we will comment the distance between Descartes’ descriptive algebra and Euler’s representative algebra. It will become apparent that algebra does not play the same role in the works of the second half of the 17th century and those of the following century. Understanding this change in status is crucial for comprehending how algebra and geometry intersect, especially in the case of studies on curves.**Claire Schwartz**(Institut de Recherche Philosophique, Université Paris Nanterre)

*The correspondance between curves and equations in Reyneau’s Analyse démontrée*

Résumé :

L’Analyse démontrée, written in 1708 by C. Reyneau, a close collaborator of N. Malebranche, is one of the first textbooks including both Cartesian algebra and infinitesimal calculus. Famous geometers like A. Clairaut and J.L-R d’Alembert read it and used it to learn and practice the differential and the integral calculus.

If it is one of the first-generation textbooks about the Leibnizian calculus, it can also be considered as a second-generation treatise on Cartesian algebra that it expanded upon : two of its main features consist of a generalization of the concept of equation that is not restricted anymore to polynomial equations, and of a systematic use of Cartesian coordinates. Reyneau can rely on these two elements to develop a program that the Cartesian Geometry of 1637 started but did not fully accomplish : a systematic study of curves by their equations.

We will therefore examine the goals set by this program, its accomplishments, and the relationship between geometry and algebra it presupposes.**Thierry Joffredo**(Laboratoire d’Histoire des Sciences et de Philosophie Archives Henri-Poincaré)

*Singular points of algebraic curves : rediscoveries of Newton’s parallelogram method in the second half of the 19th century.*

Résumé :

After 1850, in England, Germany or France, some of the mathematiciens who are interested in algebraic curves and their singular points rediscover the Newton’s parallelogram method, which seems then largely neglected, even forgotten, since the past century. « How completely it has dropped out of sight will appear from the uses which can be made of it, and which, it seems to me, must have been most obvious to any writer on curves, or on the theory of

equations, who had really obtained possession of it. », said Augustus de Morgan, obviously surprised, in a lecture read in front of the members of the Cambridge Philosophical Society in 1855 and later published in the Philosophical Transactions under the title „On the Singular Points of Curves, and on Newton’s Method of Coordinated Exponents“. In this talk, we will shortly expose some of the works of these 19th century geometers on algebraic curves putting into action the Newton’s parallelogram. We will therefore show that these new uses are mostly based on new readings of Gabriel Cramer’s Introduction à l’analyse des lignes courbes algébriques, printed in Geneva in 1750, in which is made extensive use of this method to study infinite branches and singular points of curves, thus illustrating the continuities that exist between the 18th and 19th centuries in geometry.

*Les présentations seront suivies d’une table ronde avec les intervenant·es et l’audience du séminaire, animée par Karine Chemla, David Rabouin et Paul-Emmanuel Timotei.*

**Informations pratiques** : le séminaire aura lieu en Salle 628, Bâtiment Olympe de Gouges (Place Paul Ricœur, 75013 Paris) de 9h30 à 17h00. Si vous souhaitez assister y assister, vous devrez demander un badge d’accès au 6ème étage à l’accueil du bâtiment. Si vous souhaitez assister à la séance en ligne, un lien Zoom peut vous être fourni sur demande – merci d’écrire un email à **bonvoisin.clement@gmail.com**, avec comme sujet « HPM18-12-2023 ». Le lien vous sera transmis la veille du séminaire.