In this chapter the contributions of the synthetists Steiner and Chasles are discussed, with their main works, respectively: Systematiche Entwicklung, and Aperçu historique sur l’origine et le dévelopepment des methodes employées en géométrie, which appeared in 1832 and 1837. In his work Steiner introduced the concepts of primitive elements, fundamental forms, and projective forms; defined the notion of double-ratio (“Doppelverhältnis”) of four elements in a projective form, and used it to develop a projective theory of conics. In particular, he showed how conics can be generated by means of two projective pencils of lines, thus generalizing Newton’s organic construction. Adopting Gergonne’s style, Steiner stated dual theorems face-to-face on double columns. All this is analysed and discussed at some length in the chapter. Later contributions by him are also taken into consideration here, and it is commented upon the reception of all his work; then some conclusions are drawn. Then Chasles’s Aperçu historique is taken into consideration. It consists of three parts: the first is the memoir he presented for the prix announced by the Belgian Academy in 1829, in which he developed the theory of reciprocal polars by means of the cross-ratio (called by him “rapport anharmonique”); the second is an extended historical account on the geometrical methods (that he wanted to add after having awarded the prix); eventually the third part: Notes, a series of mathematical explanations. The presentation of this work is preceded here by the discussion of the articles that Chasles wrote between 1828 and 1830, on the footsteps of Poncelet. Then, the Notes devoted to the anharmonic-ratio and its properties, and to the anharmonic properties of points and tangential conics (very important to understand Chalses’s line of thought), are here discussed in depth, and also the memoir on duality and homography that he presented for the prix. In the last section of the chapter, Chasles’s Traité de géométrie supérieure and Traité des sections coniques, published in 1852 and 1865 respectively, are commented at length. There he perfected, extended and applied his theory of homographic figures. At the end, it is in particular remarked that the transformations that Steiner called “projective relations” and Chasles “homographies”, both including central projection and Poncelet’s homology, had been a prelude to the general notion of “projective transformation” which became fundamental from 1870 onwards.

The Synthetic Route: The Contributions of Steiner and Chasles

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2025年

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Andrea Del Centina,
Alessandro Gimigliano

摘要

In this chapter the contributions of the synthetists Steiner and Chasles are discussed, with their main works, respectively: Systematiche Entwicklung, and Aperçu historique sur l’origine et le dévelopepment des methodes employées en géométrie, which appeared in 1832 and 1837. In his work Steiner introduced the concepts of primitive elements, fundamental forms, and projective forms; defined the notion of double-ratio (“Doppelverhältnis”) of four elements in a projective form, and used it to develop a projective theory of conics. In particular, he showed how conics can be generated by means of two projective pencils of lines, thus generalizing Newton’s organic construction. Adopting Gergonne’s style, Steiner stated dual theorems face-to-face on double columns. All this is analysed and discussed at some length in the chapter. Later contributions by him are also taken into consideration here, and it is commented upon the reception of all his work; then some conclusions are drawn. Then Chasles’s Aperçu historique is taken into consideration. It consists of three parts: the first is the memoir he presented for the prix announced by the Belgian Academy in 1829, in which he developed the theory of reciprocal polars by means of the cross-ratio (called by him “rapport anharmonique”); the second is an extended historical account on the geometrical methods (that he wanted to add after having awarded the prix); eventually the third part: Notes, a series of mathematical explanations. The presentation of this work is preceded here by the discussion of the articles that Chasles wrote between 1828 and 1830, on the footsteps of Poncelet. Then, the Notes devoted to the anharmonic-ratio and its properties, and to the anharmonic properties of points and tangential conics (very important to understand Chalses’s line of thought), are here discussed in depth, and also the memoir on duality and homography that he presented for the prix. In the last section of the chapter, Chasles’s Traité de géométrie supérieure and Traité des sections coniques, published in 1852 and 1865 respectively, are commented at length. There he perfected, extended and applied his theory of homographic figures. At the end, it is in particular remarked that the transformations that Steiner called “projective relations” and Chasles “homographies”, both including central projection and Poncelet’s homology, had been a prelude to the general notion of “projective transformation” which became fundamental from 1870 onwards.