Schopenhauer discussed various attempts to prove it and claimed it's a "synthetic proposition a priori."
Schopenhauer discussed various attempts to prove it and claimed it's a "synthetic proposition a priori." Klügel's doctoral thesis sought to demonstrate flaws in 28 proofs of it, and Beltrami showed that no proof was possible. A stronger form was developed by Playfair and Brodie that showed it was equivalent to the Pythagorean theorem. Abandoned by Bolyai and Lobachevsky in their independent development of hyperbolic geometry, it was also discarded in an elliptical geometry developed by Riemann. For 10 points,, give this axiom of geometry proposed by Euclid involving non-intersecting lines.
Schopenhauer discussed various attempts to prove it and claimed it's a "synthetic proposition a priori."
{parallel} postulate [or Euclid’s {Fifth} postulate]