To find a third (straight line) proportional to two given straight lines. * Let $BA$ and $AC$ be the [two] given [straight lines], and let them be laid down encompassing a random angle. * So it is required to find a third (straight line) proportional to $BA$ and $AC$.
Given two segments $\overline{AB}$, $\overline{AC}$ with the ratio $$\frac{|\overline{AB}|}{|\overline{AC}|},$$ it is possible to construct a third segment $\overline{CE}$ with a squared ratio. $$\frac{|\overline{AB}|}{|\overline{AC}|}=\frac{|\overline{AC}|}{|\overline{CE}|}.$$
This construction can be repeated $2$ times to construct a segment with a cubed ratio or an arbitrary number $n$ of times to construct a segment with the $n-1$-th continued proportion.
Proofs: 1
