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 $B$.
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 $A^n,A^nq,\ldots,A^nq^n.$
Table of Contents
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)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BY-SA 3.0 Sources:
- Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016