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.
* Let the given ratio, (expressed) in the least numbers, be that of $A$ to $B$.
* So it is required to find the least numbers, as many as may be prescribed, (which are) in the ratio of $A$ to $B$.
Modern Formulation
If $q:=\frac AB$ is a reduced ratio, i.e. $A,B$ are coprime, then
 $C=A^2,$
 $D=AB=A^2q,$
 $E=B^2=A^2q^2.$
It follows $\frac CD=\frac DE=\frac AB.$
 $F=A^3,$
 $G=A^2B=A^3q,$
 $H=AB^2=A^3q^2,$
 $K=B^3=A^3q^3.$
It follows $\frac FG=\frac GH=\frac HK=\frac AB.$
And so forth with the numbers $A^n,A^nq,\ldots,A^nq^n.$
Table of Contents
Proofs: 1 Corollaries: 1
Mentioned in:
Proofs: 1 2 3 4 5 6 7
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016