Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square

(Proposition 22 from Book 8 of Euclid's “Elements”)

If three numbers are in continued proportion, and the first is square, then the third will also be square. * Let $A$, $B$, $C$ be three numbers in continued proportion, and let the first $A$ be square. * I say that the third $C$ is also square.


Modern Formulation

(not yet contributed)

Proofs: 1

Proofs: 1 2 3 4

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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