Polymodal Style in Euclidean Proof

Abstract

A mathematical proof can be presented using, sequentially, various expressive modalities; we will say that in such a case it uses a polymodal style. Euclid uses this resource, but what could be understood as the underlying motivation? Such a question asks for a possible logic, not for the historically correct hypothesis —although this question may, in some cases, be useful to the historian. An initially plausible answer could point to aesthetic and pedagogical reasons. However, the methodological exploitation of the contrast between structure and expression shows the limitations of such a hypothesis: the expressive modality, in certain circumstances, is imposed by the structure of the proof. The main purpose of this article is therefore to provide an initial understanding of the polymodal style in Euclid.