A Critical Historiographical Reexamination of Menelaus's Non- Euclidean Spherical Geometry and Its Alleged Proto-Intuitionism

This paper reexamines Roshdi Rashed’s interpretation of Menelaus’s Spherics as a work of proto-intuitionistic mathematics. I examine its geometrical and logical grounds through the Aristotelian understanding of scientific disciplines and demonstration. Sciences are distinguished first through their subject matters, consisting of the subject of inquiry and the scientific consideration; their distinction through principles follows secondarily. On this basis, I distinguish Menelaus’s spherical geometry from Theodosius’s spherical stereometry and explain the occasional use of stereometry to establish principles of spherical geometry. This distinction also provides grounds for reconsidering the claim that Menelaus retains Euclid’s postulates except the fifth. The distinction between necessarily true knowledge and complete scientific knowledge supplies a reason for preferring demonstration propter quid without rejecting proof by contradiction as invalid. I then examine Proposition I.11 and its converse, Menelaus’s criticism of Theodosius, and al-Harawi’s explanation of that criticism. I argue that the criticism concerns the appropriateness of principles and proofs to the subject matter. Menelaus’s occasional use of Euclidean stereometry likewise supplies no ground for qualifying his alleged intuitionism as proto-intuitionism. The evidence supports a deliberate preference for direct demonstrations and restriction to that method, while leaving unestablished the claim that Menelaus rejected the principle of excluded middle or regarded proof by contradiction as logically invalid.