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.
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Becker, ?., 1933. Eudoxos-Studien П. Warum haben die Griechen die Existenz der vierten Proportionale angenommen? Quellen und Studien zur Geschichte der Mathematik, Astronomie, und Physik, Abteilung B, 2, 369–387.
Dijksterhuis, EJ., 1956. Archimedes. Ejnar Munksgaard, Copenhagen.
Heath, T.L., 1926. The Thirteen Books of Euclid’s Elements. Cambridge University Press, Cambridge.
Mueller, I., 1981. Philosophy of Mathematics and Deductive Structure in Euclid’s Elements. MIT Press, Cambridge.
Netz, R., 2009. Ludic Proof. Cambridge University Press, Cambridge.
Schöne, H., 1903. Heron von Alexandria: Vermessungslehre und Dioptra, Heronis Alexandrini opera quae supersunt omnia, vol. 3. Teubner, Leipzig.
Taisbak, C .M , 1980. An Archimedean proof of Heron’s Formula for the area of a triangle; Reconstructed. Centauras 24, 110–116.
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Taisbak, C.M. (2014). An Archimedean Proof of Heron’s Fonnula for the Area of a Triangle: Heuristics Reconstructed. In: Sidoli, N., Van Brummelen, G. (eds) From Alexandria, Through Baghdad. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36736-6_9
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