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.

Proofs: 1

Proofs: 1 2

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