Poncelet, Chasles, and the Early Years of Projective Geometry
Poncelet, faced with growing opposition to his ideas at the Ecole Polytechnique, and despite the enthusiasm with which some geometers adopted it elsewhere in France, eventually abandoned the subject for the mathematical analysis of machines. It was rescued by a younger mathematician, Michel Chasles, who dropped Poncelet’s strange ideas and founded the subject on the idea of the invariance of cross-ratio under projective transformations. This was presented in his historically rich Aperçu historique (1839) and at greater length in his Théorie de géométrie supérieure (1852), and established projective geometry as a legitimate, even rigorous discipline in France. More precisely, it established the theory of plane figures under projective transformations. It often proceeds, as we did in Chapter 3, by using a suitable transformation to reduce a figure to a special case and then arguing metrically. An important illustration of this is his projective definition of a conic section using only the property of cross-ratio, that was also given by Steiner independently.
A final section briefly describes the modern, formal treatment of real projective geometry, which will be taken up later, and notes that there are axiomatic formulations in which, for example, Desargues’ theorem is false.
Extracts are given from Chasles’s Aperçu historique to illustrate his thoughts about Monge, his descriptive geometry, and the school around him.
KeywordsProjective Space Projective Plane Projective Geometry Projective Line Projective Transformation
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