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Thales understood similar triangles and right triangles, and what is more, used that knowledge in practical ways. The story is told in DL (loc. cit.) that he measured the height of the pyramids by their shadows at the moment when his own shadow was equal to his height. A right triangle with two equal legs is a 45-degree right triangle, all of which are similar. The length of the pyramid's shadow measured from the center of the pyramid at that moment must have been equal to its height.

-- Wikipedia

Apparently Thales's measurement had to wait for that special moment when his shadow was equal to his height because was not able to generalize that whatever the ratio of his own height to his own shadow length, the ratio of pyramid height to pyramid shadow length would be equal; because he knew that isosceles right triangles were similar but he did not know of the more general AAA triangle similarity.

Right?

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    $\begingroup$ Wikipedia is not always a reliable source, but in this case it specifically addresses the Reliability of sources. Thales is a semi-legendary figure, and the pyramid story is most likely a late fable. In any case, the earliest sources that mention him are from several centuries after, so we have no real idea of what he knew or did not know. $\endgroup$ – Conifold Feb 4 at 23:14
  • $\begingroup$ Thales probably knew lots more, see e.g. Robert Hahn, The Metaphysics of the Pythagorean Theorem: Thales, Pythagoras (2017). The method would actually work for the pyramid only on a particular day when the shadow is symmetrical; otherwise one has to find distance from its center to a point of the edge on ground level. $\endgroup$ – sand1 Feb 5 at 15:03
  • $\begingroup$ @sand1 You mean that the sun would have to arc over a perpendicular bisector or angle bisector, in order to cast a symmetrical shadow? I don't see why that's needed. It seems that in that case or in any other case the relevant measurement is from the shadow of the apex to a point on the ground directly below the actual apex, the same point from which pyramid height is measured. $\endgroup$ – Chaim Feb 5 at 19:34
  • $\begingroup$ @sand1 But anyhow I don't see how this answers my question. We could solve using any sun shadows, regardless of a 45 degree sun angle. Why did Thales wait for a 45 degree sun angle? $\endgroup$ – Chaim Feb 5 at 19:34
  • $\begingroup$ Did he? So says Diogenes. Plutarch says "having made two triangles by the sun's rays, he showed that the ratio of the pyramid to the stick is the same as the ratio of the respective shadow". Since both of them lived 8 centuries after Thales, and had only talltales to rely on, this might reflect their knowledge of geometry more than his. Or perhaps, Diogenes thought that getting the number without a calculation sounded more impressive. Btw, Egyptians used similarity in these sorts of measurements long before Thales, the method was called seked. $\endgroup$ – Conifold Feb 5 at 21:30

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