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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Birkhoff's axioms ： ウィキペディア英語版 Birkhoff's axioms In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the approach is similar to a model-based introduction to Euclidean geometry. Birkhoff's axiom system was utilized in the secondary-school text book by Birkhoff and Beatley These axioms were also modified by the School Mathematics Study Group to provide a new standard for teaching high school geometry, known as (SMSG axioms ). A few other textbooks in the foundations of geometry use variants of Birkhoff's axioms. ==Postulates== Postulate I: Postulate of Line Measure. A set of points on any line can be put into a 1:1 correspondence with the real numbers so that |''b'' − ''a''| = ''d''(''A, B'') for all points ''A'' and ''B''. Postulate II: Point-Line Postulate. There is one and only one line, ''ℓ'', that contains any two given distinct points ''P'' and ''Q''. Postulate III: Postulate of Angle Measure. A set of rays through any point ''O'' can be put into 1:1 correspondence with the real numbers ''a'' (mod 2''π'') so that if ''A'' and ''B'' are points (not equal to ''O'') of ''ℓ'' and ''m'', respectively, the difference ''a''''m'' − ''a''''ℓ'' (mod 2π) of the numbers associated with the lines ''ℓ'' and ''m'' is $\backslash angle$''AOB''. Furthermore, if the point ''B'' on ''m'' varies continuously in a line ''r'' not containing the vertex ''O'', the number ''a''''m'' varies continuously also. Postulate IV: Postulate of Similarity. Given two triangles ''ABC'' and ''A'B'C' '' and some constant ''k'' > 0, ''d''(''A', B' '') = ''kd''(''A, B''), ''d''(''A', C' '') = ''kd''(''A, C'') and $\backslash angle$''B'A'C' '' = ±$\backslash angle$''BAC'', then ''d''(''B', C' '') = ''kd''(''B, C''), $\backslash angle$''C'B'A' '' = ±$\backslash angle$''CBA'', and $\backslash angle$''A'C'B' '' = ±$\backslash angle$''ACB''. 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「Birkhoff's axioms」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.