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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.

• Let $A$, $B$, $C$, $D$, $E$ be any multitude whatsoever of numbers in continued proportion, and let $A$ not measure $B$.
• I say that 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.

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References

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", https://proofwiki.org/wiki/Main_Page, 2016