Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers

(Proposition 9 from Book 8 of Euclid's “Elements”)

If two numbers are prime to one another and there fall in between them (some) numbers in continued proportion then, as many numbers as fall in between them in continued proportion, so many (numbers) will also fall between each of them and a unit in continued proportion. * Let $A$ and $B$ be two numbers (which are) prime to one another, and let the (numbers) $C$ and $D$ fall in between them in continued proportion. * And let the unit $E$ be set out. * I say that, as many numbers as have fallen in between $A$ and $B$ in continued proportion, so many (numbers) will also fall between each of $A$ and $B$ and the unit in continued proportion.

Modern Formulation

(not yet contributed)

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