◀ ▲ ▶Branches / Geometry / Elements-euclid / Book--8-continued-proportion / Proof: By Euclid

Proof: By Euclid

(related to Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square)

• 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.

• For if $C$ measures $D$ then $A$ will also measure $B$ [Prop. 8.14].
• And $A$ does not measure $B$.
• Thus, $C$ will not measure $D$ either.
• So, again, let $C$ not measure $D$.
• I say that $A$ will not measure $B$ either.
• For if $A$ measures $B$ then $C$ will also measure $D$ [Prop. 8.14].
• And $C$ does not measure $D$.
• Thus, $A$ will not measure $B$ either.
• (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"