Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes

(Proposition 3 from Book 8 of Euclid's “Elements”)

If there are any multitude whatsoever of numbers in continued proportion (which are) the least of those (numbers) having the same ratio as them then the outermost of them are prime to one another.

Modern Formulation

If $\frac AB=\frac BC=\frac CD$ are a geometric progression and the least of these numbers, then $A$ and $D$ are co-prime.

Proofs: 1

Proofs: 1 2 3

Thank you to the contributors under CC BY-SA 4.0!



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