Between algebra and geometry : undulations and cusps of algebraic curves in the correspondence of Gabriel Cramer

Abstract : Gabriel Cramer publishes his well-known treatise on algebraic curves Introduction à l’analyse des lignes courbes algébriques in 1750, in which he establishes a classification of the lines of the first five orders based on the number and position of their infinite branches. Two whole chapters of this book are devoted to the study of their multiple and singular points : in the first part of the 18th century, the mathematical knowledge about some of these kinds of points is still questioned, and one of Cramer’s goal is to give accurate answers on this subject. In this talk, after considering how undulations ("points de serpentement") and cusps ("points de rebroussement") are defined in Cramer’s treatise, I will chiefly examine the way that the reality of these two mathematical objects is discussed by Cramer in his correspondence with Euler and d’Alembert (based on a selection of few letters written between 1744 and 1751). I will show how their exchanges reveal tensions between geometrical and analytical approaches, how these transfers of mathematical knowledge contributed to clarify the points of view of each of them, and how d’Alembert eventually chose to present these notions in the Encyclopédie.
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Contributor : Thierry Joffredo <>
Submitted on : Wednesday, December 2, 2015 - 10:14:13 PM
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Thierry Joffredo. Between algebra and geometry : undulations and cusps of algebraic curves in the correspondence of Gabriel Cramer. 26th Novembertagung on the History of Mathematics, Nov 2015, Turin, Italy. ⟨halshs-01237223⟩

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