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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 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 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 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. ==History== Saccheri quadrilaterals were first considered by Omar Khayyam (1048-1131) 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 of course Saccheri), Khayyam was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle): :Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge.〔Boris A Rosenfeld and Adolf P Youschkevitch (1996), ''Geometry'', p.467 in Roshdi Rashed, Régis Morelon (1996), ''Encyclopedia of the history of Arabic science'', Routledge, ISBN 0-415-12411-5.〕 Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he (correctly) refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid. It was not until 600 years later that Giordano Vitale made an advance on Khayyam in his book ''Euclide restituo'' (1680, 1686), when he used the quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. Saccheri himself based the whole of his long, heroic, and ultimately flawed proof of the parallel postulate around the quadrilateral and its three cases, proving many theorems about its properties along the way. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Saccheri quadrilateral」の詳細全文を読む スポンサード リンク
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