Yes, I'm familiar with this definition of "algebraic", but I wasn't aware that an algebraic irrational number could be geometrically defined using Euclid's methods of geometry. So I'm excited about that result to be sure.

Ok, I'm aware that $\displaystyle \pi$ and $\displaystyle e$ are both transcendental numbers in terms of having been proven not to satisfy any polynomial equations where the coefficients are rational numbers. But I didn't realize how that spills over to geometric constructions.

So I've learned something new and valuable here. Good thing I posted my "proof".

I should call my proof, "Squaring the Circle Transcendentally"

I've downloaded some proofs that $\displaystyle \pi$ and $\displaystyle e$ are both transcendental numbers but I haven't been able to understand them yet. It appears that they have to do with derivatives and even limits or infinite series or sums. I haven't fully understood them yet. I'd like to get a handle on those particular proofs.