Why did mathematicians fight the fifth postulate for 2,000 years?

Euclid's fifth postulate, the parallel postulate, says that if a line crosses two other lines and the interior angles on one side add up to less than two right angles, then the two lines, extended far enough, meet on that side. Unlike Euclid's first four postulates, it is not self-evident, and that is why people spent more than two thousand years trying to prove it from them.

Euclid himself seems to have felt it. If the order in which the postulates were listed is significant, it indicates that he included this one only when he realised he could not prove it or proceed without it.

The attempts form a long line of near misses. Ptolemy produced a false proof, Proclus pointed it out, then offered a false proof of his own. Over and over the mistake was the same: assuming some obvious-looking property that turned out to be equivalent to the fifth postulate itself. One famous equivalent is Playfair's axiom: given a line and a point not on it, at most one line parallel to the first can be drawn through the point. Another is that the angles of every triangle add up to 180 degrees. Each looks innocent, each secretly smuggles the postulate in.

The way out came in the nineteenth century. Instead of trying to refute the alternatives, mathematicians explored them, and discovered logically consistent geometries. Lobachevsky published one in 1829, and János Bolyai, independently, another in an appendix to his father's book, dated 1832 in most accounts and 1831 in some. Gauss had studied the problem too, but published nothing. In 1868 Eugenio Beltrami finally demonstrated that the fifth postulate is independent of Euclid's other axioms: it can neither be proved nor disproved from them.