The Concept of Tangent line: historical and didactical aspects in Portugal (18th century)

The concept of a tangent line to a curve is, from the beginning, present in the history of mathematics. The study of the historical evolution of this concept is a source of knowledge and enlightenment that can lead to a better understanding of the concept. By 1772 Marquês de Pombal reorganized the University of Coimbra, establishing in Portugal the first Faculty of Mathematics in the world. The Statutes that ruled that Faculty stated, in particular, that the concept of a tangent line to a curve was to be taught in the first and second years of the Mathematics degree. The books for these years included the Portuguese translation of Euclid’s Elements and translations, made by José Monteiro da Rocha, of the work of Etienne Bezout. By then, José Anastácio da Cunha wrote a manuscript “Principios de Geometria tirados dos de Euclides” where he strongly criticizes the Euclidean definition of a tangent line, presenting an alternative definition in his treat “Principios Mathematicos”. In this article we approached the teaching of the concept of a tangent line in the Faculty of Mathematics at University of Coimbra and we focus on the alternative solution presented by the Portuguese mathematician José Anastácio da Cunha