An Archimedean Proof of Heron’s Fonnula for the Area of a Triangle: Heuristics Reconstructed

Christian Marinus Taisbak

A chapter in From Alexandria, Through Baghdad, 2014, pp 189-198 from Springer

Abstract: 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, 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.

Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-36736-6_9

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DOI: 10.1007/978-3-642-36736-6_9

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