The Value of Generality in Michel Chasles’s Historiography of Geometry
Résumé
Michel Chasles’s Aperçu historique sur l’origine et le développement des méthodes en géométrie (1837) was composed at the time when the domain of projective geometry took shape. This domain emerged as the outcome of a historical, philosophical and mathematical reflection that developed over some decades at the beginning of the 19th century in France. This reflection explicitly aimed at determining how one could achieve in geometry, by purely geometrical means, as significant a generality as the one that had been achieved by way of analytical methods from the 17th century onwards. The book, written by one of the founders and practitioners of this domain of geometry, can be considered a synthesis of these reflections, both regarding geometry itself and the history of the progressive introduction of generality in the field. Moreover, the book presents the author’s personal ideas on both aspects. It thereby reveals that the founders of the domain, among whom we can reckon Monge, Carnot, Poncelet and Chasles, held different views regarding the means for increasing generality in geometry. The article intends to rely on the analysis developed by Michel Chasles in his book in order to reflect on generality in geometry. Moreover, the article aims at indicating the specificities of Chasles’s reflections in the context of early 19th century France and how they can be correlated with his own mathematical contributions to the field. Among other features, we sketch how Chasles suggested we generalize the idea that lay at the root of Monge and Poncelet’s use of projections. We also show how Chasles reformulated Poncelet’s “principle of continuity”. The latter outcome of Chasles’s philosophical reflection on general methods is important from a historical point of view, since it yielded the “principle of contingent relationships,” which inspired Kummer’s way of introducing ideal elements in number theory and, beyond, Hilbert’s method of ideal elements.