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Proposition: Prop. 8.10: Product of Geometric Progressions from One

(Proposition 10 from Book 8 of Euclid's “Elements”)

If (some) numbers fall between each of two numbers and a unit in continued proportion then, as many (numbers) as fall between each of the (two numbers) and the unit in continued proportion, so many (numbers) will also fall in between the (two numbers) themselves in continued proportion.

• For let the numbers $D$, $E$ and $F$, $G$ fall between the numbers $A$ and $B$ (respectively) and the unit $C$ in continued proportion.
• I say that, as many numbers as have fallen between each of $A$ and $B$ and the unit $C$ in continued proportion, so many will also fall in between $A$ and $B$ in continued proportion.

Modern Formulation

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Table of Contents

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