"Die Freude an der Gestalt" : methods, figures and practices in early nineteenth century geometry

L'histoire standard de la géométrie projective souligne l'opposition au 19e siècle entre méthodes analytiques et synthétiques. Nous nous interrogeons sur la manière dont les géomètres du 19e siècle ont vraiment opéré ou non des distinctions entre leurs méthodes et dans quelle mesure cette géométrie était "moderne'' comme le clamaient ses praticiens, et plus tard leurs historiens. Poncelet insistait sur le rôle central de la figure, qui selon lui pourrait être obscurci par les calculs de l'algèbre. Nous étudions son argument en action dans des problèmes de construction résolus par plusieurs auteurs différents -comme la construction d'une courbe du second ordre ayant un contact d'ordre trois avec une courbe plane donnée, dont cinq solutions paraissent entre 1817 et 1826. Nous montrons que l'attention visuelle est au coeur de la résolution, indépendamment de la méthode suivie, qu'elle n'est pas réservée aux figures, et que les débats sont aussi un moyen de signaler de nouvelles zones de recherche à un public en formation. Nous approfondissons ensuite la réception des techniques nouvelles et l'usage des figures dans les travaux de deux mathématiciens décrits d'ordinaire comme opposés, l'un algébriste, Plücker, et l'autre défendant l'approche synthétique, Steiner. Nous examinons enfin les affirmations de modernité dans les manuels français de géométrie publiés pendant le premier tiers du dix-neuvième siècle. Tant Gergonne et Plücker que Steiner ont développé des formes de géométrie qui ne se pliaient pas en fait à une caractérisation dichotomique, mais répondaient de manière spécifique aux pratiques mathématiques et aux modes d'interaction de leur temps.The standard history of nineteenth century geometry began with Jean Victor Poncelet's contributions which then spread to Germany alongside an opposition between Julius Plücker, an analytic geometer, and Jakob Steiner, a synthetic geometer. Our questions centre on how geometers distinguished methods, when opposition arose, in what ways geometry disseminated from Poncelet to Plücker and Steiner, and whether this geometry was "modern'' as claimed.We first examine Poncelet's argument that within pure geometry the figure was never lost from view, while it could be obscured by the calculations of algebra. Our case study reveals visual attention within constructive problem solving, regardless of method. Further, geometers manipulated and represented figures through textual descriptions and coordinate equations. We also consider the debates involved as a medium for communicating geometry in which Poncelet and Gergonne in particular developed strategies for introducing new geometry to a conservative audience. We then turn to Plücker and Steiner. Through comparing their common research, we find that Plücker practiced a "pure analytic geometry'' that avoided calculation, while Steiner admired "synthetic geometry'' because of its organic unity. These qualities contradict usual descriptions of analytic geometry as computational or synthetic geometry as ad-hoc.Finally, we study contemporary French books on geometry and show that their methodological divide was grounded in student prerequisites, where "modern'' implied the use of algebra. By contrast, research publications exhibited evolving forms of geometry that evaded dichotomous categorization.The standard history of nineteenth century geometry began with Jean Victor Poncelet's contributions which then spread to Germany alongside an opposition between Julius Plücker, an analytic geometer, and Jakob Steiner, a synthetic geometer. Our questions centre on how geometers distinguished methods, when opposition arose, in what ways geometry disseminated from Poncelet to Plücker and Steiner, and whether this geometry was "modern'' as claimed.We first examine Poncelet's argument that within pure geometry the figure was never lost from view, while it could be obscured by the calculations of algebra. Our case study reveals visual attention within constructive problem solving, regardless of method. Further, geometers manipulated and represented figures through textual descriptions and coordinate equations. We also consider the debates involved as a medium for communicating geometry in which Poncelet and Gergonne in particular developed strategies for introducing new geometry to a conservative audience. We then turn to Plücker and Steiner. Through comparing their common research, we find that Plücker practiced a "pure analytic geometry'' that avoided calculation, while Steiner admired "synthetic geometry'' because of its organic unity. These qualities contradict usual descriptions of analytic geometry as computational or synthetic geometry as ad-hoc.Finally, we study contemporary French books on geometry and show that their methodological divide was grounded in student prerequisites, where "modern'' implied the use of algebra. By contrast, research publications exhibited evolving forms of geometry that evaded dichotomous categorization