Euclidean rigor and the curious case of the (missing) reflex angle

Abstract

I examine a known case of undergeneralization in Euclid's Elements arising from Euclid's non-cognizance of the reflex angle. Sir Thomas Heath (1956) attributed the undergeneralization to Euclid's lack of awareness, an assessment that I dispute. Non-recognition of reflex angles also leads to Proclus' four-sided triangles which violate an essential property of triangles. I show that these issues are resolvable. However, the question as to why Euclid did not acknowledge the reflex angle remains. I claim that the best explanation is that Euclid was guided by reasons of rigor. I argue that the propositional role of diagrammata in Greek mathematics as expounded by Netz (1998,2003) and Euclid's view of the reliability of sense perception impose a crucial representational constraint that governs how diagrams could be used in the Elements. I show that the missing reflex angle, and the careful crafting of definitions in the Elements are evidence that the representation constraint was indeed meticulously followed, even at apparent cost. I argue that alternative explanations for the missing reflex angle are not tenable. In sum, Euclid was aware of the limitations of diagrams, and worked assiduously within their limitations to preserve rigor.