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After reading this question and the answers to it, I am interested o know who were the first mathematicians who started classifying singular points of curves: i.e. different kind of nodes, of cusps etc., and when it reached the stage of the classification we use today? For example, a classification already appears in "The elements of the differential calculus" from 1839 of Young, but it seems to draw its conclusions from an existing investigation. If so, where did it appear earlier?

Thank you, David

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It's Julius Plücker who is at the origin of studies involving invariants relating the number of different singular points of a curve. He used duality as a mean of finding and proving his results, especialy the recently found "reciprocal polars transformation". See the (not fully satisfying) article: https://en.wikipedia.org/wiki/Pl%C3%BCcker_formula?oldid=360847971 (its french counterpart is slightly richer). It was around 1831 that he made his first publication on the subject. See also the interesting document https://tel.archives-ouvertes.fr/tel-01158895/document

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  • $\begingroup$ Thank you! But I have seen in the meantime that the book "Introduction à l'analyse des lignes courbes algébriques" of Cramer (1750) already classified (or started the classification) singulr points. I am aware of Plücker formula, but this already belongs to more "advanced" stage, I would say. Wasn't also a classification by Euler or Newton? $\endgroup$ – David Jul 27 '16 at 17:58

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