Figures real, imagined, and missing in Poncelet, Plücker, and Gergonne

Abstract

In the early nineteenth century debate over geometric methodology, Jean-Victor Poncelet characterized pure geometry as reasoning in which the figure is never lost from view. Whether illustrated, described or constructed, Poncelet presented the figure as the primary form of geometrical evidence, a means of justification based in sensory perception. In Poncelet's pure geometry, the objects of geometry were emphatically representational and tangible. By contrast, though classified as analytic geometry, Julius Plücker's contemporary research treated coordinate equations as visual geometric objects—evidence—by focusing on their form and endeavouring to avoid calculations. Working from Poncelet's division between pure and analytic geometries we focus on five versions by three different geometers, of a single conic section construction written between 1817 and 1826. Despite the similarity of their results, Poncelet, Plücker, and Joseph Diaz Gergonne each addressed the problem from contrasting methodological perspectives. We examine how the figure-based distinction materialized in contemporary geometric practices, and what constituted geometric evidence when the figure was lost from view.