Proof: By Euclid
(related to Proposition: Prop. 9.32: Power of Two is Even-Times Even Only)
- For let any multitude of numbers whatsoever, $B$, $C$, $D$, have been (continually) doubled, (starting) from the dyad $A$.
- I say that $B$, $C$, $D$ are even-times-even (numbers) only.
- In fact, (it is) clear that each [of $B$, $C$, $D$] is an even-times-even (number).
- For it is doubled from a dyad [Def. 7.8] .
- I also say that (they are even-times-even numbers) only.
- For let a unit be laid down.
- Therefore, since any multitude of numbers whatsoever are in continued proportion, starting from a unit, and the (number) $A$ after the unit is prime, the greatest of $A$, $B$, $C$, $D$, (namely) $D$, will not be measured by any other (numbers) except $A$, $B$, $C$ [Prop. 9.13].
- And each of $A$, $B$, $C$ is even.
- Thus, $D$ is an even-times-even (number) only [Def. 7.8] .
- So, similarly, we can show that each of $B$, $C$ is [also] an even-times-even (number) only.
- (Which is) the very thing it was required to show.
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"