Chasles’ Foundational Programme for Geometry

After an introductory section where I summarize the most important contributions given by Chasles to projective geometry in the period 1827–1850, which support my interpretation concerning the existence of his foundational programme, this chapter presents four sections, divided into several subsections. One of my main theses is that Chasles tried to explain the metric properties within a projective context. He performed this task through a particular polarity he called “parabolic transformation”. Therefore, in the first section I expand and comment on the contributions given by Chasles to the theory of polarity and his new reading of this transformation. The second section is dedicated to the development of Chasles’ projective geometry and, specifically, to the notion of anharmonic ratio, today better known as cross ratio. The explanation of the way in which this concept enriched and specified his foundational programme is offered. The Swiss mathematician Jakob Steiner developed a programme that, to some extent, was similar to Chasles’. On the other hand, there are also remarkable differences in the approach of these two mathematicians to projective geometry and to the relation between projective and metric geometry. Therefore, in the third section, a close comparison between the two authors’ conceptions is proposed. In the Conclusive considerations I develop a comparison between my interpretation of Chasles’ works and those proposed by other scholars.