In the decade 1810–1820, especially for the contributions of Brianchon and Poncelet, beautiful new theorems were discovered in projective geometry with astonishing ease, while the followers of Descartes’s method of coordinates seemed almost completely bogged down. So, the algebraists felt the need to renew their methods to keep up with pure geometry, and this led to reinforce the dichotomy between the analytic (or algebraic) methods and the pure (or synthetic) one. In this chapter the algebraic route which was continued by Gergonne, Möbius, and Plücker is described; each of them pushed forward new techniques or invented new tools to treat the projective properties of figures. It is seen here how Gergonne discovered the “law of duality” in the plane and in space, how he emphasized it by adopting the “two facing columns” way to state a theorem and its dual; his use of the abridged notation, invented by Lamé, which facilitating algebraic computation permitted algebraic reasoning to be brought closer to the geometric one; and how Gergonne was able to give analytical proofs of many projective theorems, as those of Pascal and Brianchon. It is commented here also his controversy with Poncelet. Then, in this chapter, it is analysed Möbius’s treatise on barycentric calculus, appeared in 1827, in which he introduced a new system of coordinates capable of taking account of point at infinity. Particular attention is given to the idea of “relation between figures”, and to the connected notions of “similarity”, “affinity” and “collinearity” that he introduced; to the concept of “Doppelschnitts-Verhältniss” or double-ratio; and the application he made of these concepts in the study of conic sections. Moreover, the reception of Möbius’s work and the influence it exerted on Plücker is commented here. In the last part of the chapter the main works of Plücker are discussed; he defined and put at work a new system of homogeneous coordinates, then perfected in System der analytischen Geometrie of 1835, and he pushed forward the use of abridged notation. Particular attention is put on the two volumes of Analytisch-geometrische Entwicklung (Analytic-geometric Developments) published in 1828 and 1831, in the second of which the use of homogeneous coordinates allowed him to develop a general theory of contacts, and the introduction of the notion of cross-ratio of four points, that he called with the same name that Möbius had given it. It is here also commented upon the above-mentioned work of 1835, in which Plücker initiated an in-depth study of singularities of higher order plane curves. Finally, the good reception that Plücker’s work had in Germany and England is described.

The Algebraic Way to Projective Geometry

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发表日期：

2025年

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Andrea Del Centina,
Alessandro Gimigliano

摘要

In the decade 1810–1820, especially for the contributions of Brianchon and Poncelet, beautiful new theorems were discovered in projective geometry with astonishing ease, while the followers of Descartes’s method of coordinates seemed almost completely bogged down. So, the algebraists felt the need to renew their methods to keep up with pure geometry, and this led to reinforce the dichotomy between the analytic (or algebraic) methods and the pure (or synthetic) one. In this chapter the algebraic route which was continued by Gergonne, Möbius, and Plücker is described; each of them pushed forward new techniques or invented new tools to treat the projective properties of figures. It is seen here how Gergonne discovered the “law of duality” in the plane and in space, how he emphasized it by adopting the “two facing columns” way to state a theorem and its dual; his use of the abridged notation, invented by Lamé, which facilitating algebraic computation permitted algebraic reasoning to be brought closer to the geometric one; and how Gergonne was able to give analytical proofs of many projective theorems, as those of Pascal and Brianchon. It is commented here also his controversy with Poncelet. Then, in this chapter, it is analysed Möbius’s treatise on barycentric calculus, appeared in 1827, in which he introduced a new system of coordinates capable of taking account of point at infinity. Particular attention is given to the idea of “relation between figures”, and to the connected notions of “similarity”, “affinity” and “collinearity” that he introduced; to the concept of “Doppelschnitts-Verhältniss” or double-ratio; and the application he made of these concepts in the study of conic sections. Moreover, the reception of Möbius’s work and the influence it exerted on Plücker is commented here. In the last part of the chapter the main works of Plücker are discussed; he defined and put at work a new system of homogeneous coordinates, then perfected in System der analytischen Geometrie of 1835, and he pushed forward the use of abridged notation. Particular attention is put on the two volumes of Analytisch-geometrische Entwicklung (Analytic-geometric Developments) published in 1828 and 1831, in the second of which the use of homogeneous coordinates allowed him to develop a general theory of contacts, and the introduction of the notion of cross-ratio of four points, that he called with the same name that Möbius had given it. It is here also commented upon the above-mentioned work of 1835, in which Plücker initiated an in-depth study of singularities of higher order plane curves. Finally, the good reception that Plücker’s work had in Germany and England is described.