Méthodes, Figures Et Pratiques De La Géométrie Au Début Du Dix

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Méthodes, Figures Et Pratiques De La Géométrie Au Début Du Dix THÈSE DE DOCTORAT Discipline : Mathématiques présentée par Jemma Lorenat pour obtenir le grade de Docteur de l’université Pierre et Marie Curie et de Doctor of Philosophy of Simon Fraser University “Die Freude an der Gestalt ”: méthodes, figures et pratiques de la géométrie au début du dix-neuvième siècle Thèse dirigée par Thomas Archibald et Catherine Goldstein présentée et soutenue publiquement le 10 avril 2015 devant un jury composé de Mme Nilima Nigam, professor, Simon Fraser University (Burnaby, Canada), présidente du jury M. Christian Gilain, professeur émérite, Université Pierre et Marie Curie (Paris, France), examinateur M. Dirk Schlimm, associate professor, Mac Gill University (Montréal, Canada), rapporteur M. Thomas Archibald, professor, Simon Fraser University (Burnaby, Canada), directeur Mme Catherine Goldstein, directrice de recherche au CNRS, Institut de mathématiques de Jussieu-Paris Gauche (Paris, France), directrice Rapporteur non présent à la soutenance : M. Philippe Nabonnand, pro- fesseur à l’université de Lorraine (Nancy, France) c Simon Fraser University (Canada) et Université Pierre et Marie Curie (France) ii ACKNOWLEDGEMENTS First, this research would not have been possible without the encouragements, critiques, and administrative feats of my dissertation directors, Tom Archibald and Catherine Goldstein. They continually inspire me with their dedication, understanding, exactitude, generosity, and often uncanny ability to succinctly state what I am trying to express in so many words. I am grateful for all the reporters and members of my committee, in particular to Nilima Nigam who joined at the very beginning and has offered valuable support and recommendations over the past four years. My family has been a source of comfort and wisdom. Bobby has shown me how to be an artist, my sister has allowed me to feel like a role model, my mother has been a rock of optimism in a sea of doubts, and my father has never stopped telling me to ask good questions. Thank you. iii ABSTRACT As recounted by later historians, modern geometry began with Jean Victor Poncelet, whose contributions then spread to Germany alongside an opposition between geometric methods that came to be exemplified by the antagonism of Julius Plücker, an analytic geometer, and Jakob Steiner, a synthetic geometer. To determine the participants, arguments, and qualities of this perceived divide, we drew upon historical accounts from the late nineteenth and early twentieth centuries. Several themes emerged from the historical perspective, which we investigated within the original sources. Our questions centred on how geometers distinguished methods, when op- position arose, in what ways geometry disseminated from Poncelet to Plücker and Steiner, and whether this geometry was “modern” as claimed. Our search for methodological debates led to Poncelet’s proposal that within pure geometry the figure was never lost from view, while it could be obscured by the calculations of algebra. We examined his argument through a case study that revealed visual attention within constructive problem solving, regardless of method. Further, geometers manipulated and represented figures through textual descriptions and coor- dinate equations. In these same texts, Poncelet and Joseph-Diez Gergonne instigated a debate on the principle of duality. Rather than dismiss their priority dispute as external to mathemat- ics, we consider the texts involved as a medium for communicating geometry in which Poncelet and Gergonne developed strategies for introducing new geometry to a conservative audience. This conservative audience did not include Plücker and Steiner, who adapted new vocabulary, techniques and objects. Through comparing their common research, we found they differenti- ated methods based on personal considerations. Plücker practiced a “pure analytic geometry” that avoided calculation. Steiner admired “synthetic geometry” because of its organic unity. These qualities contradicted descriptions of analytic geometry as computational or synthetic geometry as ad-hoc. Finally, we turned to claims for novelty in the context of contemporary French books on geometry. Most of these books point to a pedagogical orientation, where the methodological divide was grounded in student prerequisites and “modern” implied the use of algebra in geometry. By contrast, research publications exhibited evolving forms of geometry that evaded dichotomous categorization. Keywords: geometry, nineteenth century, analysis and synthesis, visualization, Jean Victor Poncelet, Joseph-Diez Gergonne, Julius Plücker, Jakob Steiner iv RÉSUMÉ Die Freude an der Gestalt: méthodes, figures et pratiques dans la géométrie au début du dix-neuvième siècle L’histoire standard de la géométrie projective souligne une opposition entre les méthodes ana- lytiques et synthétiques. Selon les historiens de la fin du dix-neuvième siècle, la géométrie mod- erne a commencé avec le Traité des propriétés projectives de Jean Victor Poncelet en 1817, puis, pendant le premier tiers du siècle, les contributions de Poncelet se répandirent en Allemagne, ainsi qu’une opposition entre différentes approches géométriques dont l’exemple toujours cité est l’antagonisme entre Julius Plücker, un géométre analytique, et Jakob Steiner, un géomètre synthétique. Ce n’est qu’à partir des années 1870, selon ces récits, que les géomètres mirent fin à une distinction qui avait cessé d’être pertinente. Pour déterminer les participants, les arguments et les qualités de cette apparente division méthodologique, nous avons puisé dans les récits historiques écrits à la fin du dix-neuvième siècle et au début du vingtième siècles. Même s’ils insistent pour résumer la situation dans l’opposition globale et binaire que nous avons décrite, leurs écrits suggèrent déjà qu’il y avait plutôt une multitude d’oppositions à plus petite échelle, qui résultaient en particulier de ce que des découvertes multiples, presque simultanées, étaient faites par un petit groupe de géomètres. Plusieurs thèmes principaux émergeaient de cette perspective historiographique, qui forme le premier chapitre de cette thèse, et nous avons décidé de les approfondir en les confrontant à une étude détaillée des textes originaux : des textes de Poncelet, Plücker, Steiner et Joseph-Diez Gergonne sur la géométrie et la méthodologie écrits pendant le premiers tiers du dix-neuvième siècle. Nos questions sont centrées sur la manière dont ces géomètres ont distingué leurs pro- pres méthodes géométriques de celles des autres mathématiciens contemporains, quand une opposition surgissait en géométrie, sur la manière dont à la fois la géométrie et ces opposi- tions se sont transmises de Poncelet à Plücker et Steiner, et dans quelle mesure cette géométrie était “moderne” et nouvelle comme le clamaient ses praticiens, et plus tard les historiens de la géométrie. Dans le deuxième chapitre, nous examinons donc en détail un problème de mathématiques concernant l’inscription de figures dans des coniques, problème qui a été discuté entre plusieurs mathématiciens (Gergonne, Poncelet, Plücker, etc.) et a déclenché des polémiques sur les méthodes de solution. J’ai étudié en détail les arguments, les techniques, etc. et les oppositions, j’ai aussi montré le rôle des nouveaux moyens de diffusion que sont les journaux mathématiques. Notre recherche sur ce débat méthodologique nous a conduit à un échange épistémologique entre Poncelet et Gergonne sur l’utilisation de l’analyse algébrique en géométrie. En distinguant ce qu’il appelle la pure géométrie de la géométrie analytique, Poncelet insistait sur le role central v de la figure. Il suggérait que tant dans l’ancienne géométrie pure, celle héritée d’Euclide et de l’Antiquité, que dans la géométrie pure moderne qu’il incarnait, la figure n’était jamais perdue de vue, et qu’elle pourrait être obscurcie par les calculs de l’algèbre appliquée à la géométrie. En géométrie synthétique, les objets de la géométrie étaient representationnels et tangibles, ces qualités étant encore mises en avant même lorsque les géomètres introduisirent des points imaginaires ou à l’infini. Nous examinons ici l’argument de Poncelet en action à travers l’étude de plusieurs solutions à un même problème de construction géométrique qui a été discuté et résolu dans un ensemble de publications étroitement connectées, mais écrites par plusieurs auteurs différents. Il s’agit de construire une courbe du second ordre ayant un contact d’ordre trois avec une courbe plane donnée, dont cinq solutions paraissent entre 1817 et 1826. En étudiant comment les auteurs invoquaient les figures et quelles formes alternatives leur servaient à représenter les objets géométriques, nous avons trouvé que l’attention visuelle est au coeur de la résolution de ces problèmes de construction, indépendamment de la méthode suivie, mais qu’elle n’est pas non plus réservée aux figures. Se fondant sur ce qu’il décrit comme le manque d’élégance des calculs analytiques, Poncelet perçoit cet exemple de construction comme particulièrement adapté à une approche de géométrie pure et développe en particulier le concept de corde idéale pour généraliser les cas où les points de tangence sont imaginaires. Figure 1: Illustration selon Poncelet de la corde idéale commune à l’infini entre deux cercles concentriques, Traité des propriétés projectives, 1822. Mais Poncelet et Plücker ont aussi développé des stratégies pour manipuler et visualiser d’autres types de représentations du problème. Plücker par exemple a développé de nouvelles formes d’équations
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