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In the first part, I will present the definition of a mathematical curve that we will use, as well as the challenges related to the conceptual history of mathematics. In the second part, I will focus more specifically on my research topic—the development of the notion of a general curve in Europe between 1650 and 1750—and the issues I address in my thesis: (1) the distinction between algebraic and non-algebraic curves introduced by Descartes in La Géométrie, (2) the methods devised by his successors to study non-algebraic curves, and (3) the use of projective principles and tools for the study of algebraic curves.