An archimedean proof of heron’s fonnula for the area of a triangle: Heuristics reconstructed

Abstract

I believe, as did al-Bīrūnī, that Archimedes invented and proved Heron’s formula for the area of a triangle. But I also believe that Archimedes would not multiply one rectangle by another, so he must have had a another way of stating and proving the theorem. It is possible to “save” Heron’s received text by inventing a geometrical counterpart to the un-Archimedean passage and inserting that before it, and to consider the troubling passage as Archimedes’ own translation into terms of measurement. My invention is based on a reconstruction of the heuristics that led to the proof I prove a crucial lemma: If there are five magnitudes of the same kind, a, b, c, d, m, and m is the mean proportional between a and b, and a: C = d: B, then m is also the mean proportional between c and d.