A part of Khayyam's commentary on Euclid's Elements deals with the parallel axiom.
The treatise of Khayyam can be considered the first treatment of the axiom not based on petitio principii, but on a more intuitive postulate.
Khayyam refutes the previous attempts by other mathematicians to prove the proposition, mainly on grounds that each of them had postulated something that was by no means easier to admit than the Fifth Postulate itself.
Drawing upon Aristotle's views, he rejects the usage of movement in geometry and therefore dismisses the different attempt by Al-Haytham.
Unsatisfied with the failure of mathematicians to prove Euclid's statement from his other postulates, Omar tried to connect the axiom with the Fourth Postulate, which states that all right angles are equal to one another.

Khayyam was the first to consider the three cases of acute, obtuse, and right angle for the summit angles of a Khayyam-Saccheri quadrilateral, three cases which are exhaustive and pairwise mutually exclusive.
After proving a number of theorems about them, he proved that the Postulate V is a consequence of the right angle hypothesis, and refuted the obtuse and acute cases as self-contradictory.
Khayyam's elaborate attempt to prove the parallel postulate was significant for the further development of geometry, as it clearly shows the possibility of non-Euclidean geometries.
The hypotheses of the acute, the obtuse, and the right angle are now known to lead respectively to the non-Euclidean hyperbolic geometry of Gauss-Bolyai-Lobachevsky, to that of Riemannian geometry, and to Euclidean geometry.

Tusi's commentaries on Khayyam's treatment of parallels made its way to Europe.
John Wallis, professor of geometry at Oxford, translated Tusi's commentary into Latin.
Jesuit geometer Girolamo Saccheri, whose work (euclides ab omni naevo vindicatus, 1733) is generally considered as the first step in the eventual development of non-Euclidean geometry, was familiar with the work of Wallis.
The American historian of mathematics, David Eugene Smith, mentions that Saccheri "used the same lemma as the one of Tusi, even lettering the figure in precisely the same way and using the lemma for the same purpose".
He further says that "Tusi distinctly states that it is due to Omar Khayyam, and from the text, it seems clear that the latter was his inspirer."
A part of Khayyam's commentary on Euclid's Elements deals with the parallel axiom. The treatise of Khayyam can be considered the first treatment of the axiom not based on petitio principii, but on a more intuitive postulate. Khayyam refutes the previous attempts by other mathematicians to prove the proposition, mainly on grounds that each of them had postulated something that was by no means easier to admit than the Fifth Postulate itself. Drawing upon Aristotle's views, he rejects the usage of movement in geometry and therefore dismisses the different attempt by Al-Haytham. Unsatisfied with the failure of mathematicians to prove Euclid's statement from his other postulates, Omar tried to connect the axiom with the Fourth Postulate, which states that all right angles are equal to one another.
Khayyam was the first to consider the three cases of acute, obtuse, and right angle for the summit angles of a Khayyam-Saccheri quadrilateral, three cases which are exhaustive and pairwise mutually exclusive. After proving a number of theorems about them, he proved that the Postulate V is a consequence of the right angle hypothesis, and refuted the obtuse and acute cases as self-contradictory. Khayyam's elaborate attempt to prove the parallel postulate was significant for the further development of geometry, as it clearly shows the possibility of non-Euclidean geometries. The hypotheses of the acute, the obtuse, and the right angle are now known to lead respectively to the non-Euclidean hyperbolic geometry of Gauss-Bolyai-Lobachevsky, to that of Riemannian geometry, and to Euclidean geometry.
Tusi's commentaries on Khayyam's treatment of parallels made its way to Europe. John Wallis, professor of geometry at Oxford, translated Tusi's commentary into Latin. Jesuit geometer Girolamo Saccheri, whose work (euclides ab omni naevo vindicatus, 1733) is generally considered as the first step in the eventual development of non-Euclidean geometry, was familiar with the work of Wallis. The American historian of mathematics, David Eugene Smith, mentions that Saccheri "used the same lemma as the one of Tusi, even lettering the figure in precisely the same way and using the lemma for the same purpose". He further says that "Tusi distinctly states that it is due to Omar Khayyam, and from the text, it seems clear that the latter was his inspirer."
Omar Khayyam