Did Galileo use an erroneous geometrical result in 'Two New Sciences'?

Question

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)
The claim struck me as odd and I experimented many times with various geometrical figures to see if I could reproduce this result in at least one instance, but I couldn't. What's going on here?

HiterDean · Answer

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 frac{B D^{2}}{B I^{2}}=frac{B D}{B E}
end{array}
$Q.E.D$

Did Galileo use an erroneous geometrical result in 'Two New Sciences'?

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