For two thousand years, mathematicians tried to prove Euclid's awkward fifth rule
Euclid's first four postulates are almost obviously true. The fifth, about when two lines must eventually meet, is long and clumsy, and generations of brilliant thinkers were sure it could be derived from the rest. Every attempted proof smuggled in the very thing it set out to prove.
In its original wording, the postulate says that if a line crosses two others and the interior angles on one side total less than two right angles, those two lines will meet on that side if extended far enough. It does not mention parallels directly, though Euclid defined parallel lines just before the postulates, in Book I, Definition 23. The best-known restatement is named after the Scottish mathematician John Playfair, who in a 1795 commentary proposed replacing it with a simpler claim: through a point off a given line, at most one parallel can be drawn.
Part of the trouble is how many innocent-looking ideas turn out to be equivalent. Assuming that every triangle's angles sum to 180 degrees, that rectangles exist, that similar but differently sized triangles exist, that triangles can have unlimited area, or even Pythagoras' theorem, all quietly commit you to the fifth postulate. Even the word parallel hides the problem, since treating its four usual meanings as interchangeable is itself equivalent to the postulate.
The attempts stretch across centuries and cultures. Proclus, writing in the fifth century, noted that Ptolemy's proof was flawed and then offered a flawed one of his own. Ibn al-Haytham brought motion and transformation into geometry while trying. Omar Khayyám derived early results of what would become elliptic and hyperbolic geometry and first studied the quadrilateral later named after Saccheri. Nasir al-Din al-Tusi critiqued Khayyám in 1250, and a 1298 book based on his ideas, published in Rome in 1594, influenced European geometers. Girolamo Saccheri disposed of one alternative case but wrongly convinced himself he had eliminated the other.
That surviving case was the key. Denying the postulate does not produce contradictions; it yields perfectly consistent geometries. Where the original form fails, you get hyperbolic geometry; drop its converse and you get elliptic geometry, where on a sphere two lines meet at exactly two points. More than 2,200 years after Euclid, the fifth postulate is still a postulate.
Source: Parallel postulate
