the aim of the course is to provide an account on the development of the main research lines of Western mathematics up to early 20th century. A more in-depth study of one or more of the following topics will be offered: the rise of infinitesimal calculus, the arithmetization of analysis throughout 19th century (Cauchy, Dirichlet, Riemann and Weierstrass), the emergence of non-Euclidean geometries and the history of differential geometry.