Cramér-Euler Paradox

A curve of order n is generally determined by n(n+3)/2 points. So a conic section is determined by five points and a cubic curve should require nine. But the Maclaurin-Bézout theorem says that two curves of degree n intersect in n^2 points, so two cubics intersect in nine points. This means that n(n+3)/2 points do not always uniquely determine a single curve of order n. The paradox was publicized by Stirling, and explained by Plücker.

See also

Cubic Curve, Maclaurin-Bézout Theorem

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Euler, L. "Sur une contradiction apparente dans la doctrine des lignes courbes." Mémoires de l'Academie des Sciences de Berlin 4, 219-233, 1750 Reprinted in Opera Omnia, Series Prima, Vol. 26. Boston: Birkhäuser, pp. 33-45, 1992.Sandifer, E. "How Euler Did It: CramerÕs Paradox." Aug. 2004.

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Cramér-Euler Paradox

Cite this as:

Weisstein, Eric W. "Cramér-Euler Paradox." From MathWorld--A Wolfram Web Resource.

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