Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements

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

If between two numbers there fall (some) numbers in continued proportion then, as many numbers as fall in between them in continued proportion, so many (numbers) will also fall in between (any two numbers) having the same ratio [as them] in continued proportion.


Modern Formulation

Given some equal continued proportions e.g. $\frac AB=\frac EF=\frac GL$ (which we can also write $B=Aq^n$ , $F=Eq^n$, $L=Gq^n$ for some $q > 0$ and some positive integer $n\ge 3$) there are $n-1$ numbers falling between each of them, e.g, for $n=3$:

Proofs: 1

Proofs: 1 2 3 4 5 6 7 8

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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