Proposition: Prop. 9.17: Last Element of Geometric Progression with Co-prime Extremes has no Integer Proportional as First to Second

(Proposition 17 from Book 9 of Euclid's “Elements”)

If any multitude whatsoever of numbers is in continued proportion, and the outermost of them are prime to one another, then as the first (is) to the second, so the last will not be to some other (number).

Let $A$, $B$, $C$, $D$ be any multitude whatsoever of numbers in continued proportion.
And let the outermost of them, $A$ and $D$, be prime to one another.
I say that as $A$ is to $B$, so $D$ (is) not to some other (number).

fig17e