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In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate): In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point. It is equivalent to Euclid's parallel postulate in the context of Euclidean geometry〔more precisely, in the context of absolute geometry.〕 and was named after the Scottish mathematician John Playfair. The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists. The statement is often written with the phrase, "there is one and only one parallel". In Euclid's Elements, two lines are said to be parallel if they never meet and other characterizations of parallel lines are not used.〔(Euclid's elements, Book I, definition 23 )〕 This axiom is used not only in Euclidean geometry but also in the broader study of affine geometry where the concept of parallelism is central. In the affine geometry setting, the stronger form of Playfair's axiom (where "at most" is replaced by "one and only one") is needed since the axioms of neutral geometry are not present to provide a proof of existence. Playfair's version of the axiom has become so popular that it is often referred to as ''Euclid's parallel axiom'',〔for instance, Rafael Artzy (1965) ''Linear Geometry'', page 202, Addison-Wesley〕 even though it was not Euclid's version of the axiom. ==History== Proclus (410–485 A.D.) clearly makes the statement in his commentary on Euclid I.31 (Book I, Proposition 31) In 1785 William Ludlam expressed the parallel axiom as follows:〔William Ludlam (1785) ''The Rudiments of Mathematics'', p. 145, Cambridge〕 :Two straight lines, meeting at a point, are not both parallel to a third line. This brief expression of Euclidean parallelism was adopted by John Playfair in his textbook ''Elements of Geometry'' (1795) that was republished often. He wrote〔John Playfair (1836) (Elements of Geometry ) from Google books, page 22 for parallel axiom.〕 :Two straight lines, which intersect one another, cannot be both parallel to the same straight line. Playfair acknowledged Ludlam and others for simplifying the Euclidean assertion. In later developments the point of intersection of the two lines came first, and the denial of two parallels became expressed as a unique parallel through the given point. In 1883 Arthur Cayley was president of the British Association and expressed this opinion in his address to the Association:〔William Barrett Frankland (1910) ''Theories of Parallelism: A Historic Critique'', page 31, Cambridge University Press〕 :My own view is that Euclid's Twelfth Axiom in Playfair's form of it, does not need demonstration, but is part of our notion of space, of the physical space of our experience, which is the representation lying at the bottom of all external experience. When David Hilbert wrote his book, Foundations of Geometry (1899), providing a new set of axioms for Euclidean geometry, he used Playfair's form of the axiom instead of the original Euclidean version for discussing parallel lines. 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「Playfair's axiom」の詳細全文を読む スポンサード リンク
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