Proof: By Euclid
(related to Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements)
- Let any multitude whatsoever of numbers, $B$, $C$, $D$, $E$, be in continued proportion, (starting) from the unit $A$.
- I say that, for $B$, $C$, $D$, $E$, the least (number), $B$, measures $E$ according to some (one) of $C$, $D$.
- For since as the unit $A$ is to $B$, so $D$ (is) to $E$, the unit $A$ thus measures the number $B$ the same number of times as $D$ (measures) $E$.
- Thus, alternately, the unit $A$ measures $D$ the same number of times as $B$ (measures) $E$ [Prop. 7.15].
- And the unit $A$ measures $D$ according to the units in it.
- Thus, $B$ also measures $E$ according to the units in $D$.
- Hence, the lesser (number) $B$ measures the greater $E$ according to some existing number among the proportional numbers (namely, $D$).
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"