Proposition: 8.02: Construction of Geometric Progression in Lowest Terms

(Proposition 2 from Book 8 of Euclid's “Elements”)

To find the least numbers, as many as may be prescribed, (which are) in continued proportion in a given ratio.


Modern Formulation

If $q:=\frac AB$ is a reduced ratio, i.e. $A,B$ are co-prime, then

It follows $\frac CD=\frac DE=\frac AB.$

It follows $\frac FG=\frac GH=\frac HK=\frac AB.$

And so forth with the numbers $A^n,A^nq,\ldots,A^nq^n.$

Proofs: 1 Corollaries: 1

Proofs: 1 2 3 4 5 6 7

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