Laurea magistrale in Matematica

Meaning and significance of the history of mathematics for mathematicians and teachers;

The major stages in the history of mathematical analysis;

The concept of proof in the Greek sense: analytical methods and synthetic methods;

Euclid’s theory of proportions and comparison with Richard Dedekind’s theory of real numbers;

The method of exhaustion in Greek mathematics (Euclid, Archimedes);

Archimedes’ “method of mechanical theorems” and current use of infinity;

The isoperimetric problem in the Greek world and the progress made by of Joseph Steiner in the 19th century;

The birth of analytic geometry (René Descartes, Pierre Fermat);

Various methods (algebraic, analytic, kinematic) for finding the tangent to a curve in the 17th century;

Method of indivisibles to calculate areas and volumes;

Infinitesimal calculus in the works of Isaac Newton (method of fluxions, method of first and last ratios, methods of series, the “fundamental theorem of integral calculus”, integration of functions, integration of differential equations);

Infinitesimal calculus in Gottfried W. Leibniz (differential calculus, integral calculus, the quadratrix curve and the “fundamental theorem of integral calculus”, integration of differential equations);

The comparison between the School of Leibniz and that of Newton;

Spread and developments of Leibnizian analysis in the 18th century;

The origin of the calculus of variations (Newton, Johann and Jacob Bernoulli, Leonhard Euler, Joseph-Louis Lagrange);

Differential equations in the 18th century;

A didactic treatise: Instituzioni analitiche ad uso della gioventù italiana (1748) by Maria Gaetana Agnesi;

Euler’s Introductio in analysin infinitorum (1748);

The evolution of the concept of function;

A textbook for teaching: Lagranges’s Principj di Analisi sublime (1759);

Lagrange’s Théorie des fonctions analytiques (1797) and the algebraization of analysis;

The beginning of the process of rigorization of analysis: Augustin-Louis Cauchy’s Cours d’analyse (1821) and Résumés des leçons données à l'Ecole royale polytechnique sur le calcul infinitésimal (1823);

The evolution of the concept of integral;

The treatises on Analysis published abroad and their influence in Italy: Silvestre François Lacroix’s Traité de calcul différentiel et de calcul intégral (1828).

The problem of foundations in Karl Weierstrass’s classes in infinitesimal calculus in Berlin;

“Modern Analysis” in Italy: the courses given by Angelo Genocchi, Felice Casorati and Ulisse Dini;

Giuseppe Peano’s research in the foundations of infinitesimal calculus; the Genocchi-Peano treatise (1884) and famous counter examples;

The definition of area of a curved surface and the measure theory  according to Peano-Jordan: the treatise Applicazioni geometriche del calcolo infinitesimale (1887);

The theory of functions of a real variable and its relationships to the set theory and functional analysis: outline of the contributions of Giulio Vivanti, Giuseppe Vitali, Salvatore Pincherle, Vito Volterra.