Saccheri Quadrilateral

A Saccheri quadrilateral ( 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 first known consideration of the Saccheri quadrilateral was by Omar Khayyam in the late 11th century, and it may occasionally be referred to as the Khayyam-Saccheri quadrilateral.

For a Saccheri quadrilateral ABCD, the sides AD and BC (also called legs) are equal in length and perpendicular to the base AB. The top CD is called the summit or upper base and the angles at C and D are called the summit angles.

The advantage of using Saccheri quardrilaterals 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 they are acute, this quadrilateral leads to hyperbolic geometry, and when they are obtuse, the quadrilateral leads to elliptical geometry (provided that 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 this in the obtuse case, but failed to properly handle the acute case.