Author’s abstract: 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 (1830–1892), also provided a general definition of transfirmation, which is the same that Giovanni Virginio Schiaparelli (1835–1910) refers to in 1862 as conic transfirmation.
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.
Reviewer’s remarks: One of the principal aims of the author is to unearth the value/importance of Bellavitis’ research, set out in his paper of 1838. His work introduces the notion of “equipollence. [Two line segments of equal length and also orientated in the same direction are called “equipollent”. One recognizes here the notion of “vector”, as later studied, for instance by Grassmann et al.] See also the book by M. J. Crowe [A history of vector analysis. The evolution of the idea of a vectorial system. Notre Dame-London: University of Notre Dame Press (1967; Zbl 0165.00303)], in particular pp. 53–54. The author does not mention this important source in the references. He gives on pages 82 and 83 an example as worked out in Bellavitis’ 1838 treatise. As a matter of fact, it concerns an earlier theorem due to Carnot (1803); it deals with three non-pairwise parallel lines intersected by means of an ellipse, and gives connections on the line segments involved inter alea. It is not clear whether Bellavitis knew of this earlier result; the author does not spend a word on it.
The list of the references is an eye opener. The interested reader should also consult the MacTutor History of Mathematics Archive as well as Wikipedia, concerning the merits of Bellavitis. The author did good work in order to brings to light the (relatively forgotten) credits of Bellavitis’ research.