Historical Origins of the nine-point conic -- The Contribution of Eugenio Beltrami

TL;DR

This paper traces the historical development of the nine-point conic, highlighting Beltrami's contributions and the evolution of geometric methods from Euclidean to projective geometry, emphasizing algebraic and birational transformations.

Contribution

It provides a detailed historical analysis of the nine-point conic's development, emphasizing Beltrami's innovative use of quadratic transformations and birational geometry.

Findings

Beltrami's work marked a significant advancement in geometric methods.

The evolution from Euclidean to projective geometry is exemplified through the nine-point conic.

Correspondence between Beltrami and Cremona offers insights into the development of algebraic geometry.

Abstract

In this paper, we examine the evolution of a specific mathematical problem, i.e. the nine-point conic, a generalisation of the nine-point circle due to Steiner. We will follow this evolution from Steiner to the Neapolitan school (Trudi and Battaglini) and finally to the contribution of Beltrami that closed this journey, at least from a mathematical point of view (scholars of elementary geometry, in fact, will continue to resume the problem from the second half of the 19th to the beginning of the 20th century). We believe that such evolution may indicate the steady development of the mathematical methods from Euclidean metric to projective, and finally, with Beltrami, with the use of quadratic transformations. In this sense, the work of Beltrami appears similar to the recent (after the anticipations of Magnus and Steiner) results of Schiaparelli and Cremona. Moreover, Beltrami's methods are closely related to the study of birational transformations, which in the same period were becoming one of the main topics of algebraic geometry. Finally, our work emphasises the role played by the ninepoint conic problem in the studies of young Beltrami who, under Cremona's guidance, was then developing his mathematical skills. To this end, we make considerable use of the unedited correspondence Beltrami-Cremona, preserved in the Istituto Mazziniano, Genoa.