Die Freude an der Gestalt: Methods, Figures, and Practices in Early Nineteenth Century Geometry

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 opposition 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 coordinate 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 mathematics, 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 differentiated 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.