Historical reconstruction of the upward anthyphairetic step in ancient incommensurability proofs

Determine whether the ancient proofs of the incommensurability of the diameter to the side of a square and of the mean-and-extreme ratio employed an upward anthyphairetic step to the whole, as conjectured from the surviving ancient sources.

Background

The paper compares two geometric cases of Pythagorean incommensurability: the diameter-to-side ratio of a square and the mean-and-extreme ratio. In both cases, the author identifies a transformation from a dyad to a larger whole that preserves the relevant ratio and interprets this transformation as an “upward anthyphairetic step to the whole.”

The author presents this as a historical conjecture about the methods used in ancient proofs, rather than as a deduction established from an extant proof. The problem is therefore to determine whether the ancient demonstrations of these two incommensurabilities actually followed a common upward-step procedure.