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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line ''ℓ'' and a point ''A'', which is not on ''ℓ'', there is exactly one line through ''A'' that does not intersect ''ℓ''. In hyperbolic geometry, by contrast, there are infinitely many lines through ''A'' not intersecting ''ℓ'', while in elliptic geometry, any line through ''A'' intersects ''ℓ''. Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line: *In Euclidean geometry the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels. *In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. *In elliptic geometry the lines "curve toward" each other and intersect. ==History== ===Early history=== While Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century. The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid's work ''Elements'' was written. In the ''Elements'', Euclid began with a limited number of assumptions (23 definitions, five common notions, and five postulates) and sought to prove all the other results (propositions) in the work. The most notorious of the postulates is often referred to as "Euclid's Fifth Postulate," or simply the "parallel postulate", which in Euclid's original formulation is: If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclid's other postulates:
For at least a thousand years, geometers were troubled by the disparate complexity of the fifth postulate, and believed it could be proved as a theorem from the other four. Many attempted to find a proof by contradiction, including Ibn al-Haytham (Alhazen, 11th century), Omar Khayyám (12th century), Nasīr al-Dīn al-Tūsī (13th century), and 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 of the hyperbolic and the elliptic geometries." These theorems along with their alternative postulates, such as Playfair's axiom, played an important role in the later development of non-Euclidean geometry. These early attempts at challenging the fifth postulate had a considerable influence on its development among later European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis and Saccheri.〔Boris A. Rosenfeld & Adolf P. Youschkevitch, "Geometry", p. 470, in Roshdi Rashed & Régis Morelon (1996), ''Encyclopedia of the History of Arabic Science'', Vol. 2, pp. 447–494, Routledge, London and New York: 〕 All of these early attempts made at trying to formulate non-Euclidean geometry however provided flawed proofs of the parallel postulate, containing assumptions that were essentially equivalent to the parallel postulate. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Khayyam, for example, tried 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 & Adolf P. Youschkevitch (1996), "Geometry", p. 467, in Roshdi Rashed & Régis Morelon (1996), ''Encyclopedia of the History of Arabic Science'', Vol. 2, pp. 447–494, 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 which he didn't realize was equivalent to his own postulate. Another example is al-Tusi's son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), who wrote a book on the subject in 1298, based on al-Tusi's later thoughts, which presented another hypothesis equivalent to the parallel postulate. "He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the ''Elements''."〔〔Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., ''Encyclopedia of the History of Arabic Science'', Vol. 2, p. 447–494 (), Routledge, London and New York: 〕 His work was published in Rome in 1594 and was studied by European geometers, including Saccheri〔Victor J. Katz (1998), ''History of Mathematics: An Introduction'', p. 270–271, Addison–Wesley, ISBN 0-321-01618-1: "But in a manuscript probably written by his son Sadr al-Din in 1298, based on Nasir al-Din's later thoughts on the subject, there is a new argument based on another hypothesis, also equivalent to Euclid's, () The importance of this latter work is that it was published in Rome in 1594 and was studied by European geometers. In particular, it became the starting point for the work of Saccheri and ultimately for the discovery of non-Euclidean geometry."〕 who criticised this work as well as that of Wallis.〔(MacTutor's Giovanni Girolamo Saccheri )〕 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「Non-Euclidean geometry」の詳細全文を読む スポンサード リンク
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