Proof: By Euclid
(related to Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements)
- For as many as A, B, C, D are in multitude, let so many of the least numbers, G, H, K, L, having the same ratio as A, B, C, D, have been taken [Prop. 7.33].
- Thus, the outermost of them, G and L, are prime to one another [Prop. 8.3].
- And since A, B, C, D are in the same ratio as G, H, K, L, and the multitude of A, B, C, D is equal to the multitude of G, H, K, L, thus, via equality, as A is to B, so G (is) to L [Prop. 7.14].
- And as A (is) to B, so E (is) to F.
- And thus as G (is) to L, so E (is) to F.
- And G and L (are) prime (to one another).
- And (numbers) prime (to one another are) also the least (numbers having the same ratio as them) [Prop. 7.21].
- And the least numbers measure those (numbers) having the same ratio (as them) an equal number of times, the greater (measuring) the greater, and the lesser the lesser - that is to say, the leading (measuring) the leading, and the following the following [Prop. 7.20].
- Thus, G measures E the same number of times as L (measures) F.
- So as many times as G measures E, so many times let H, K also measure M, N, respectively.
- Thus, G, H, K, L measure E, M, N, F (respectively) an equal number of times.
- Thus, G, H, K, L are in the same ratio as E, M, N, F [Def. 7.20] .
- But, G, H, K, L are in the same ratio as A, C, D, B.
- Thus, A, C, D, B are also in the same ratio as E, M, N, F.
- And A, C, D, B are in continued proportion.
- Thus, E, M, N, F are also in continued proportion.
- Thus, as many numbers as have fallen in between A and B in continued proportion, so many numbers have also fallen in between E and F in continued proportion.
- (Which is) the very thing it was required to show.
∎
Thank you to the contributors under CC BY-SA 4.0!

- Github:
-

- non-Github:
- @Fitzpatrick
References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"