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

In: From Alexandria, Through Baghdad

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  • Christian Marinus Taisbak

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.

Suggested Citation

  • Christian Marinus Taisbak, 2014. "An Archimedean Proof of Heron’s Fonnula for the Area of a Triangle: Heuristics Reconstructed," Springer Books, in: Nathan Sidoli & Glen Van Brummelen (ed.), From Alexandria, Through Baghdad, edition 127, pages 189-198, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-36736-6_9
    DOI: 10.1007/978-3-642-36736-6_9
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