Proposition: Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime

Euclid's Formulation

If any multitude whatsoever of numbers is in continued proportion, (starting) from a unit, and the (number) after the unit is prime, then the greatest (number) will be measured by no [other] (numbers) except (numbers) existing among the proportional numbers. * Let any multitude whatsoever of numbers, $A$, $B$, $C$, $D$, be in continued proportion, (starting) from a unit. * And let the (number) after the unit, $A$, be prime. * I say that the greatest of them, $D$, will be measured by no other (numbers) except $A$, $B$, $C$.


Modern Formulation

(not yet contributed)

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