Proposition: 8.06: First Element of Geometric Progression not dividing Second

(Proposition 6 from Book 8 of Euclid's “Elements”)

If there are any multitude whatsoever of numbers in continued proportion, and the first does not measure the second, then no other (number) will measure any other (number) either.


Modern Formulation

If $A,B,C,D,E$ are given with $B=An,$ $C=An^2,$ $D=An^3,$ $E=An^4,$ and $n$ is not a positive integer, then $A$ is not a divisor of $B.$ In this geometric progression, none of the numbers will be a multiple of any of the proceeding numbers.

Proofs: 1

Proofs: 1

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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