A Saccheri quadrilateral (also known as a Khayyam–Saccheri quadrilateral) is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his book Euclides ab omni naevo vindicatus (literally Euclid Freed of Every Flaw) first published in 1733, an attempt to prove the parallel postulate using the method Reductio ad absurdum. The Saccheri quadrilateral may occasionally be referred to as the Khayyam–Saccheri quadrilateral, in reference to the 11th century Persian scholar Omar Khayyam. For a Saccheri quadrilateral ABCD, the sides AD and BC (also called the legs) are equal in length, and also perpendicular to the base AB. The top CD is the summit or upper base and the angles at C and D are called the summit angles. The advantage of using Saccheri quadrilaterals when considering the parallel postulate is that they place the mutually exclusive options in very clear terms: Are the summit angles right angles, obtuse angles, or acute angles? As it turns out: when the summit angles are right angles, the existence of this quadrilateral is equivalent to the statement expounded by Euclid's fifth postulate. When the summit angles are acute, this quadrilateral leads to hyperbolic geometry, and when the summit angles are obtuse, the quadrilateral leads to elliptical or spherical geometry (provided that also some other modifications are made to the postulates). Saccheri himself, however, thought that both the obtuse and acute cases could be shown to be contradictory. He did show that the obtuse case was contradictory, but failed to properly handle the acute case. History While the quadrilaterals are named for Giovanni Gerolamo Saccheri, they were considered in the works of earlier mathematicians. Saccheri's first proposition states that if two equal lines AC and BC form equal angles with the line AB, the angles at CD will equal each other; a version of this statement appears in the works of the ninth century scholar Thabit ibn Qurra. Rectifying the Curved, a 14th century treatise written in Castile, builds off the work of Thabit ibn Qurra and also contains descriptions of Saccheri quadrilaterals. Omar Khayyam (1048-1131) described them in the late 11th century in Book I of his Explanations of the Difficulties in the Postulates of Euclid. Unlike many commentators on Euclid before and after him (including of course Saccheri), Khayyam was not trying to prove the parallel postulate as such but to