Luigi Bianchi’s early investigations on pseudospherical surfaces

The paper provides a historical analysis of Luigi Bianchi’s early contributions to the transformation theory of pseudospherical surfaces. The origin of a geometrical construction known as complementary transformation is carefully studied by emphasizing the connection of Bianchi’s achievement with the “classical” methods of Gauss and Weingarten. Furthermore, the reception process of Bianchi’s theory is discussed in the context of Lie’s integration theory of PDE’s and of Bäcklund transformations.