Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional

(Proposition 11 from Book 8 of Euclid's “Elements”)

There exists one number in mean proportion to two (given)1 square numbers. And (one) square (number) has to the (other) square (number) a squared2 ratio with respect to (that) the side (of the former has) to the side (of the latter). * Let $A$ and $B$ be square numbers, and let $C$ be the side of $A$, and $D$ (the side) of $B$. * I say that there exists one number in mean proportion to $A$ and $B$, and that $A$ has to $B$ a squared ratio with respect to (that) $C$ (has) to $D$. * So I say that $A$ also has to $B$ a squared ratio with respect to (that) $C$ (has) to $D$.

Modern Formulation

(not yet contributed)

Proofs: 1

Proofs: 1 2

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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, given two square numbers, their geometric mean is a number (i.e. a positive integer).

2. Literally, "double" (translator's note).