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Proof: By Euclid

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

• Let $A$, $B$, $C$, $D$ be four numbers in continued proportion, and let $A$ be cube.

• I say that $D$ is also cube.

• For since two numbers, $B$ and $C$, are in mean proportion to $A$ and $D$, $A$ and $D$ are thus similar solid numbers [Prop. 8.21].

• And $A$ (is) cube.
• Thus, $D$ (is) also cube [Def. 7.21] .
• (Which is) the very thing it was required to show.
  ∎

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"