Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second
(Proposition 7 from Book 8 of Euclid's “Elements”)
If there are any multitude whatsoever of [continuously] [proportional]bookofproofs$2328 [numbers]bookofproofs$2315, and the first measures the last, then (the first) will also measure the second.
* Let $A$, $B$, $C$, $D$ be any number whatsoever of numbers in continued proportion.
* And let $A$ measure $D$.
* I say that $A$ also measures $B$.
Modern Formulation
If $A,B,C,D,$ etc. is a geometric progression, then there exist a positive integer $n$ with $B=An,$ $C=An^2,$ $D=An^3,$ etc.
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Fitzpatrick
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