Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals
(Proposition 19 from Book 8 of Euclid's “Elements”)
Two numbers fall (between) two similar solid numbers in mean proportion. And a solid (number) has to a similar solid (number) a cubed ratio with respect to (that) a corresponding side (has) to a corresponding side.
- Let A and B be two similar solid numbers, and let C, D, E be the sides of A, and F, G, H (the sides) of B.
- And since similar solid (numbers) are those having proportional sides [Def. 7.21] , thus as C is to D, so F (is) to G, and as D (is) to E, so G (is) to H.
- I say that two numbers fall (between) A and B in mean proportion, and (that) A has to B a cubed ratio with respect to (that) C (has) to F, and D to G, and, further, E to H.

Modern Formulation
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Table of Contents
Proofs: 1
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Proofs: 1 2 3 4 5
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References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BY-SA 3.0 Sources:
- Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016
Footnotes