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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)¹ cube numbers. And (one) cube (number) has to the (other) cube (number) a cubed² 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 $D$.

Modern Formulation

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Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016

Footnotes