Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube

(Proposition 23 from Book 8 of Euclid's “Elements”)

If four numbers are in continued proportion, and the first is cube, then the fourth will also be cube. * Let $A$, $B$, $C$, $D$ be four numbers in continued proportion, and let $A$ be cube. * I say that $D$ is also cube.


Modern Formulation

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Proofs: 1

Proofs: 1 2 3 4 5 6 7

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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",, 2016