This chapter, the more extensive in the book, is entirely dedicated to Poncelet and his Traité des propriétés projectives des figures, published in 1822, which marked the birth of Projective Geometry. This work had a long gestation, and originated from the seven Notebooks that he composed in Saratov on the banks of the Volga, during his captivity as a prisoner of war, following the Russian campaign of the Napoleonic army to which he took part as a lieutenant of engineers in 1812. After commenting upon the first articles that Poncelet wrote still as a student at the École polytechnique, these Notebooks are here analysed and studied in detail; and it is showed how the main ideas, techniques, and results, later expounded in the Traité, developed in his mind. A particular attention is given to the “principle of continuity”, that Poncelet put at the basis of his geometrical thought, but distinguishing it from Carnot’s principle of correlation; and to the “principles of projection”, by which he formalised the projection method already used by Brianchon. To develop this programme, aiming to confer to synthetic geometry the same generality that algebra offered to the analytic one, Poncelet introduced two fundamental concepts, namely the ones of “ideal secant” and of “ideal common chord” of two or more conic sections, by means of which he was able to manage imaginary elements without using algebra. All this is discussed at length, and it occupies the first one third of the chapter. Then, the articles that are considerevole are the ones that Poncelet published in Gergonne’s Annales after his return to France, to forward his ideas and methods. As it is seen in this chapter, also the correspondence he had with Terquem, Servois, and Brianchon, that Poncelet published in 1864 at the end of his life, helps to understand how the project of the Traité matured in the years 1818–1820. As he later showed in his treatise, Poncelet refused the slightest hint of analytic geometry in favour of a purely synthetic presentation. In fact, his principal aim, already emerged in Saratov, was the study of the graphical properties of figures, which he defined as those properties that do not involve distances and angles (which he regarded as contingent properties), whereas the graphical ones remain unchanged by the operations of projection and section. This led him to contrast Gergonne, defender of the analytic approach, on the roles that synthetic and algebraic techniques had to play in geometry. To present and to comment upon Poncelet’s works in the years 1815–1820 is the theme of the central part of the chapter. The last one third of the chapter is devoted to illustrate and to comment upon the contents of the Traité. In particular, the salient topics of the treatise are discussed, as his foundational general principles, the theory of reciprocal pole and polars, the centres of similitude and homology, the ideal common chords and the theory of double contacts, and the discovery of cyclic points. Finally, his theory of polygons inscribed in a conic and circumscribed about another, which led him to state the porism today known with his name, is commented at some length.

Poncelet, the Projective Properties of Figures

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发表日期：

2025年

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Andrea Del Centina,
Alessandro Gimigliano

摘要

This chapter, the more extensive in the book, is entirely dedicated to Poncelet and his Traité des propriétés projectives des figures, published in 1822, which marked the birth of Projective Geometry. This work had a long gestation, and originated from the seven Notebooks that he composed in Saratov on the banks of the Volga, during his captivity as a prisoner of war, following the Russian campaign of the Napoleonic army to which he took part as a lieutenant of engineers in 1812. After commenting upon the first articles that Poncelet wrote still as a student at the École polytechnique, these Notebooks are here analysed and studied in detail; and it is showed how the main ideas, techniques, and results, later expounded in the Traité, developed in his mind. A particular attention is given to the “principle of continuity”, that Poncelet put at the basis of his geometrical thought, but distinguishing it from Carnot’s principle of correlation; and to the “principles of projection”, by which he formalised the projection method already used by Brianchon. To develop this programme, aiming to confer to synthetic geometry the same generality that algebra offered to the analytic one, Poncelet introduced two fundamental concepts, namely the ones of “ideal secant” and of “ideal common chord” of two or more conic sections, by means of which he was able to manage imaginary elements without using algebra. All this is discussed at length, and it occupies the first one third of the chapter. Then, the articles that are considerevole are the ones that Poncelet published in Gergonne’s Annales after his return to France, to forward his ideas and methods. As it is seen in this chapter, also the correspondence he had with Terquem, Servois, and Brianchon, that Poncelet published in 1864 at the end of his life, helps to understand how the project of the Traité matured in the years 1818–1820. As he later showed in his treatise, Poncelet refused the slightest hint of analytic geometry in favour of a purely synthetic presentation. In fact, his principal aim, already emerged in Saratov, was the study of the graphical properties of figures, which he defined as those properties that do not involve distances and angles (which he regarded as contingent properties), whereas the graphical ones remain unchanged by the operations of projection and section. This led him to contrast Gergonne, defender of the analytic approach, on the roles that synthetic and algebraic techniques had to play in geometry. To present and to comment upon Poncelet’s works in the years 1815–1820 is the theme of the central part of the chapter. The last one third of the chapter is devoted to illustrate and to comment upon the contents of the Traité. In particular, the salient topics of the treatise are discussed, as his foundational general principles, the theory of reciprocal pole and polars, the centres of similitude and homology, the ideal common chords and the theory of double contacts, and the discovery of cyclic points. Finally, his theory of polygons inscribed in a conic and circumscribed about another, which led him to state the porism today known with his name, is commented at some length.