In geometry, the parallel postulate, also called Euclid's fifth postulate since it is the fifth postulate in Euclid's Elements, is a distinctive axiom in what is now called Euclidean geometry. It states that:
If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. A geometry where the parallel postulate cannot hold is known as a non-euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the first four postulates) is known as absolute geometry (or, in other places known as neutral geometry).
Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from elliptic geometry. The Elements contains the proof of an equivalent statement (Book I, Proposition 17): Any two angles of a triangle are together less than two right angles. The proof depends on an earlier proposition: In a triangle ABC, the exterior angle at C is greater than either of the interior angles A or B. This in turn depends on Euclid's unstated assumption that two straight lines meet in only one point, a statement not true of elliptic geometry.
In other words, the converse of the fifth postulate follows from Euclid's axioms minus the fifth postulate, plus an axiom stating that two distinct non-parallel straight lines meet in only one point.
Exactly one line can be drawn through any point not on a given line parallel to the given line in a plane.
Many other equivalent statements have been suggested, some of them appearing at first to be unrelated to parallelism, and some seeming so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates.
However, the alternatives which employ the word "parallel" cease appearing so simple when one is obliged to explain which of the three common definitions of "parallel" is meant - constant separation, never meeting or same angles where crossed by a third line - since the equivalence of these three is itself one of the unconsciously obvious assumptions equivalent to Euclid's fifth postulate.
Omar Khayyám (1050-1123) recognized that three possibilities arose from omitting Euclid's Fifth; if two perpendiculars to one line cross another line, judicious choice of the last can make the internal angles where it meets the two perpendiculars equal (it is then parallel to the first line). If those equal internal angles are right angles, we get Euclid's Fifth; otherwise, they must be either acute or obtuse. He persuaded himself that the acute and obtuse cases lead to contradiction, but had made a tacit assumption equivalent to the fifth to get there.
Girolamo Saccheri (1667-1733) pursued the same line of reasoning more thoroughly, correctly obtaining absurdity from the obtuse case (proceeding, like Euclid, from the implicit assumption that lines can be extended indefinitely and have infinite length), but failing to debunk the acute case (although he managed to wrongly persuade himself that he had).
Where Saccheri and Khayyám had attempted to prove Euclid's fifth by disproving the only possible alternatives, the nineteenth century finally saw mathematicians exploring those alternatives and discovering the logically consistent geometries which result.
In 1829, Nikolai Ivanovich Lobachevsky published an account of acute geometry in an obscure Russian journal (later re-published in 1840 in German).
In 1831, János Bolyai included, in a book by his father, an appendix describing acute geometry, which, doubtlessly, he had developed independently of Lobachevsky.
Carl Friedrich Gauss had actually studied the problem before that, but he didn't have a courage to publish any of his results.