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Proposition: Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number

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

If any multitude whatsoever of numbers is in continued proportion, (starting) from a unit, and the (number) after the unit is square, then all the remaining (numbers) will also be square. And if the (number) after the unit is cube, then all the remaining (numbers) will also be cube.

• Let any multitude whatsoever of numbers, $A$, $B$, $C$, $D$, $E$, $F$, be in continued proportion, (starting) from a unit.
• And let the (number) after the unit, $A$, be square.
• I say that all the remaining (numbers) will also be square.
• And so let $A$ be cube.
• I say that all the remaining (numbers) are also cube.

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