Singular Points of Algebraic Curves: Rediscoveries of Newton's Parallelogram Method in the 2nd Half of the 19th Century

Résumé En

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 ofequations, 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.