Giusto Bellavitis and its `Geometria di Derivazione

Giusto Bellavitis (1803-1880), now mainly regarded for his theory of equipollences, was among the first Italian geometers, in 19th century, who devoted himself to the study of modern theories on projective transformations that were spreading among the transalpine geometers. His studies on geometric transformations - to which he refers with the term of derivations, since they make it possible to derive the properties of one figure from those of another one - are generally unknown. It is remarkable that Bellavitis, in Saggio di Geometria derivata (1838), besides having suggested the idea of quadric inversion, which would be studied about twenty years later by Thomas Archer Hirst (18301892), also provided a general defini- tion of transformation, which is the same that Giovanni Virginio Schiaparelli (1835-1910) refers to in 1862 as conic transformation.In this paper, Bellavitis' work is taken as a case-study of the importance, in the history of mathematics, to conceive concepts and theories within a context of scientific isolation, before their spreading within the scientific community. This paper also tries to shed light on the reasons why Bellavitis' works remained unknown for a long time. Finally, the paper tries to assess to what extent, some decades after their discovery, Bellavitis' works have contributed, although not decisively, to rethink, from an historical and critical standpoint, the development of the theory of geometric transformations.