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.

Read more about Saccheri QuadrilateralHistory, Properties, A Formula, See Also

Other articles related to "saccheri quadrilateral, quadrilaterals, saccheri, quadrilateral":

Saccheri Quadrilateral - Properties
... Let ABCD be a Saccheri quadrilateral having AB as base, CA and DB the equal sides that are perpendicular to the base and CD the summit ... The following properties are valid in any Saccheri quadrilateral in hyperbolic geometry ... The line segment joining the midpoint of the base and the midpoint of the summit divides the Saccheri quadrilateral into two Lambert quadrilaterals ...
Non-Euclidean Geometry - History - Early History
... and the Italian mathematician Giovanni Girolamo Saccheri (18th century) ... The theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were "the first few theorems ... among later European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis and Saccheri ...
Parallel Postulate - History
... He formulated the Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral", and his attempted ... The Saccheri quadrilateral was also first considered by Omar Khayyám in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid ... Unlike many commentators on Euclid before and after him (including Giovanni Girolamo Saccheri), Khayyám was not trying to prove the parallel postulate as such but to derive it from ...
Lambert Quadrilateral
... In geometry, a Lambert quadrilateral, named after Johann Heinrich Lambert, is a quadrilateral three of whose angles are right angles ... Historically, the fourth angle of a Lambert quadrilateral was of considerable interest since if it could be shown to be a right angle, then the Euclidean parallel postulate could be proved as a theorem ... It is now known that the type of the fourth angle depends upon the geometry in which the quadrilateral lives ...