This article starts with a cognitive analysis of an imaginary debate – reconstructed with excerpts from historical sources – concerning the shapes of particular sections of a right cone and of a right cylinder. Our analysis, based on the theory of figural concepts (Fischbein, 1993), suggests the following hypothesis: Where conic sections are concerned, a rupture between the figural and the conceptual aspects is to be expected and is not easily compounded. We carried out an exploratory study with expert university students in order to validate this hypothesis, exploring how students could compund the rupture and restore the unity between figural and conceptual components. In particular, what kind of geometrical concepts (conceptualisation), if any, these students were able to mobilise. After reporting the findings of the study, we shift to the didactic plane and suggest which tools of semiotic mediation could be introduced in order to enable the studen