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

and

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

and

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.