Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square

(Proposition 16 from Book 8 of Euclid's “Elements”)

If a square number does not measure a(nother) square number then the side (of the former) will not measure the side (of the latter) either. And if the side (of a square number) does not measure the side (of another square number) then the (former) square (number) will not measure the (latter) square (number) either. * Let $A$ and $B$ be square numbers, and let $C$ and $D$ be their sides (respectively). * And let $A$ not measure $B$. * I say that $C$ does not measure $D$ either. * So, again, let $C$ not measure $D$. * I say that $A$ will not measure $B$ either.


Modern Formulation

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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