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

History of Science and Mathematics Asked by Zachary Candelaria on August 11, 2020

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?

As $$BI$$ is mean proportional to $$BD$$ and $$BE$$.

$$begin{array}{l} Rightarrow frac{B D}{B I}=frac{B I}{B E} \ Rightarrow frac{B D}{B I} times B D=frac{B I}{B E} times B D \ Rightarrow frac{B D^{2}}{B I}=frac{B I times B D}{B E} \ Rightarrow frac{B D^{2}}{B I^{2}}=frac{B D}{B E} end{array}$$

$$Q.E.D$$

Answered by HiterDean on August 11, 2020