# Were there impossibility proofs for constructions in Greek geometry?

Greek geometry was confronted with problems such as squaring the circle. Straightedge and compass constructions were unable to provide a solution, but other mechanical curves, such as the quadratix, were introduced to deal with these difficult problems.

(a) Was there a general consensus/awareness that it is impossible (as opposed to: “not yet knowing how to”) to solve certain geometrical problems with straightedge and compass constructions?

(b) Who was the first person in history to give an "impossibility proof" that some geometrical problems cannot be solved with such-and-such construction principle. I know Gauss found a proof for the constructibility of regular polygons, but I am looking for earlier mathematical contributions.

As in most such cases, nobody was "first". One can present already the Pythagorean proof of incommensurability of the side and the diagonal of a square as an "impossibility proof". It amounts to saying that it is impossible to split the side into a whole number of equal segments so that the diagonal can be assembled from a whole number of segments of the same length. The proof (in Euclid) is by reductio, as are some other proofs that can be similarly presented. And Gauss did not prove the impossibility part, it was Wantzel in 1837, probably the first geometric impossibility proof recognized as "proper" by modern standards. But by then there were "flawed" 17th century arguments, Euler's proof of irrationality of $$e$$, Lambert's of $$\pi$$, and Abel's of insolvability of the quintic in radicals.