In Thm. 4, Prop. 4 of Galileo's 'Two New Sciences' (pg. 187, Crew Translation), Galileo says the following: "From a single point $B$ draw the planes $BA$ and $BC$, having the same length but different inclinations; let $AE$ and $CD$ be horizontal lines drawn to meet the perpendicular $BD$; and let $BE$ represent the height of the plane $AB$, and $BD$ the height of $BC$; also let $BI$ be a mean proportional to $BD$ and $BE$; then the ratio of $BD$ to $BI$ is equal to the square root of the ratio of $BD$ to $BE$." (See figure)figure 5.4, Two New Sciences
But I don't see how this could be the case. If I use two right triangles, one of height $\frac{1}{2}$ and the other of height $\frac{\sqrt{3}}{2}$, I don't get the result Galileo did. Could it be that he erred? Help me clear up my confusion lads
As $BI$ is mean proportional to $BD$ and $BE$.
\begin{array}{l} \Rightarrow \frac{B D}{B I}=\frac{B I}{B E} \\ \Rightarrow \frac{B D}{B I} \times B D=\frac{B I}{B E} \times B D \\ \Rightarrow \frac{B D^{2}}{B I}=\frac{B I \times B D}{B E} \\ \Rightarrow \quad \frac{B D^{2}}{B I^{2}}=\frac{B D}{B E} \end{array}
$Q.E.D$