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In mathematics, Cramer's paradox or the Cramer–Euler paradox〔Weisstein, Eric W. "Cramér-Euler Paradox." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Cramer-EulerParadox.html〕 is the statement that the number of points of intersection of two higher-order curves in the plane can be greater than the number of arbitrary points that are usually needed to define one such curve. It is named after the Swiss mathematician Gabriel Cramer. This paradox is the result of a naive understanding or a misapplication of two theorems: * Bézout's theorem (the number of points of intersection of two algebraic curves is equal to the product of their degrees, provided that certain necessary conditions are met). * Cramer's theorem (a curve of degree ''n'' is determined by ''n''(''n'' + 3)/2 points, again assuming that certain conditions hold). Observe that for all ''n'' ≥ 3, ''n''2 ≥ ''n''(''n'' + 3)/2, so it would naively appear that for degree three or higher there could be enough points shared by each of two curves that those points should determine either of the curves uniquely. The resolution of the paradox is that in certain degenerate cases ''n''(''n'' + 3) / 2 points are not enough to determine a curve uniquely. ==History== The paradox was first published by Maclaurin. Cramer and Euler corresponded on the paradox in letters of 1744 and 1745 and Euler explained the problem to Cramer.〔 It has become known as ''Cramer's paradox'' after featuring in his 1750 book ''Introduction à l'analyse des lignes courbes algébriques'', although Cramer quoted Maclaurin as the source of the statement. At about the same time, Euler published examples showing a cubic curve which was not uniquely defined by 9 points〔Euler, L. "Sur une contradiction apparente dans la doctrine des lignes courbes." Mémoires de l'Academie des Sciences de Berlin 4, 219-233, 1750〕 and discussed the problem in his book Introductio in analysin infinitorum. The result was publicized by James Stirling and explained by Julius Plücker.〔 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「Cramer's paradox」の詳細全文を読む スポンサード リンク
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