Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals
(Proposition 12 from Book 8 of Euclid's “Elements”)
There exist two numbers in mean proportion to two (given) cube numbers. And (one) cube (number) has to the (other) cube (number) a cubed ratio with respect to (that) the side (of the former has) to the side (of the latter).
- Let $A$ and $B$ be cube numbers, and let $C$ be the side of $A$, and $D$ (the side) of $B$.
- I say that there exist two numbers in mean proportion to $A$ and $B$, and that $A$ has to $B$ a cubed ratio with respect to (that) $C$ (has) to $D$.
- So I say that $A$ also has to $B$ a cubed ratio with respect to (that) $C$ (has) to $D$.
![fig12e](https://github.com/bookofproofs/bookofproofs.github.io/blob/main/_sources/_assets/images/euclid/Book08/fig12e.png?raw=true)
Modern Formulation
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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