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

Area of Science:

• History of Mathematics
• Ancient Greek Geometry

Background:

• Euclid's Elements exhibits undergeneralization due to non-recognition of reflex angles.
• This omission leads to geometric inconsistencies, such as Proclus' four-sided triangles.

Purpose of the Study:

• To dispute the claim that Euclid's undergeneralization stemmed from a lack of awareness.
• To propose and support an alternative explanation rooted in Euclid's commitment to rigor.

Main Methods:

• Analysis of Euclid's definitions and geometric proofs.
• Examination of the role of diagrams (diagrammata) in Greek mathematics, referencing Netz (1998, 2003).
• Evaluation of Euclid's perspective on the reliability of sense perception.

Main Results:

• The non-recognition of reflex angles in Euclid's Elements is attributed to a representational constraint imposed by the use of diagrams.
• This constraint, influenced by the propositional role of diagrams and Euclid's views on perception, guided geometric rigor.
• The careful construction of definitions and exclusion of reflex angles demonstrate adherence to this constraint.

Conclusions:

• Euclid was aware of the limitations of diagrams in geometric representation.
• He intentionally operated within these limitations to maintain mathematical rigor in the Elements.
• The omission of reflex angles was a deliberate choice to preserve the integrity of his geometric system.