Hippocrates of Chios – His Elements and His Lunes: A critique of circular reasoning

Hippocrates of Chios, active during the later fifth century BCE, is stated in Proclos's ―catalogue of geometers‖ to have been the first writer of elements, and is also known to have worked on the squaring of ―lunes‖, plane figures contained by a convex and a concave circular arc. Customarily this is taken to mean that he wrote a book in the axiomatic style of Euclid's Elements, and a Euclidean reading of the text on lunules has been used to produce a list of such Euclidean propositions as must already have been in Hippocrates's Elements.
The present article, analyzing Hippocrates's procedures closely, makes the observation that little or nothing of what Hippocrates makes use of had not been known and used in practical geometry, in part for more than a millennium, in part for at least a century. It suggests that Hippocrates's argumentation, instead of being rooted in an axiomatic system, was based on the ―locally obvious‖, such knowledge as his audience would be familiar with and could be supposed to accept as evident. His ―elements‖, far from being an axiomatic system, would be a list of such locally obvious techniques and insights.
We know Hippocrates's work on lunes from Simplicios's sixth-century commentary to Aristotle's Physics. Simplicios proposes two versions, first a report of what Alexander of Aphrodisias had told in his commentary to the same work and passage, next (as he claims) the description of the work that Eudemos had offered, adding only Euclidean proofs where he supposes Eudemos has omitted them. Comparison of the two show that they are both genuine, none of them derived from the other. It is suggested that Alexander draws on Hippocrates's teaching, being based either on lecture notes of his or on students' notes; the Eudemos-version may instead go back to what Hippocrates published more officially.