Has Euclid stated Cauchy's theorem?

History of Science and Mathematics Asked on September 2, 2021

Cauchy’s Rigidity theorem says that if the corresponding faces of two convex polytopes are isometric (congruent)
then the polytopes are related by a (proper or improper) motion.

Cauchy’s biography (by Bruno Belhoste, Springer 1991) says that this statement
is in Euclid book “9, Theorem 11”. I checked: Proposition 11 in book 9 is not related to this. In general, book 9 is not about stereometry. English and French Wikipedia also say it is in Euclid, but do not say where exactly.

Can anyone tell where in Euclid can this (or similar, or related) statement be found?

One Answer

I found it. It is Definition 10 in Book XI.

Euclid takes the assumption of Cauchy's theorem as the definition of equality of polytopes! Evidently he did not see the difficulty which is involved here. (Cauchy's theorem is highly non-trivial, and far from evident).

Answered by Alexandre Eremenko on September 2, 2021

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