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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 $D$.

fig12e

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

  1. In other words, between two given cube numbers there exist two numbers in continued proportion (translator's note). ↩

  2. Literally, "triple" (translator's note). ↩