The proof is not that ancient. Pi was proven to be irrational in 1761, and proven to be transcendental in 1882.
For a long time the problem was known as “squaring the circle”: Given a circle in a plane, construct a square with the same area using a compass and straightedge. This was a famous unsolved problem in mathematics from antiquity all the way through the renaissance.
FishFace@piefed.social 1 day ago
Thanks for the correction - misremembered that.
A slight clarification in return: the constructible numbers are a strict subset of the algebraic (i.e. non-transcendental) real numbers.
(The constructible numbers are those numbers resulting from the closure of the rational numbers under square roots.)
This means that although the proof of pi’s transcendentality proved that squaring the circle is impossible, it could have been the case that pi was neither transcendental nor constructible. A simple example of such a number is the cube root of 2.