Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube

(Proposition 15 from Book 8 of Euclid's “Elements”)

If a cube number measures a(nother) cube number then the side (of the former) will also measure the side (of the latter). And if the side (of a cube number) measures the side (of another cube number) then the (former) cube (number) will also measure the (latter) cube (number) . * For let the cube number $A$ measure the cube (number) $B$, and let $C$ be the side of $A$, and $D$ (the side) of $B$. * I say that $C$ measures $D$. * And so let $C$ measure $D$. * I say that $A$ will also measure $B$.


Modern Formulation

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Proofs: 1

Proofs: 1

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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