Cramer's paradox

The paradox was first published by Colin Maclaurin in 1720. Cramer and Leonhard Euler corresponded on the paradox in letters of 1744 and 1745 and Euler explained the problem to Cramer. At about the same time, Euler published examples showing a cubic curve which was not uniquely defined by 9 points and discussed the problem in his book Introductio in analysin infinitorum. The result was publicized by James Stirling and explained by Julius Plücker. ==No paradox for lines and nondegenerate conics==

For first-order curves (that is, lines) the paradox does not occur, because n=1, so n^2=1. In general, two distinct lines intersect at a single point unless the lines are of equal slope, in which case they do not intersect at all. A single point is not sufficient to define a line (two are needed); through the point of intersection there pass not only the two given lines but an infinite number of other lines as well. Two nondegenerate conics intersect in at most at four finite points in the real plane, which is precisely the number given as a maximum by Bézout's theorem. However, five points are needed to define a nondegenerate conic, so again in this case there is no paradox. ==Cramer's example for cubic curves==

In a letter to Euler, Cramer pointed out that the cubic curves x^3-x=0 and y^3-y=0 intersect in precisely nine points. The first equation defines three vertical lines x=-1, x=0, and x=1, and similarly the second equation defines three horizontal lines; these lines intersect in a grid of nine points. Hence nine points are not sufficient to uniquely determine a cubic curve in degenerate cases such as these. ==Euler's resolution==

A bivariate equation of degree n has 1 + n(n + 3) / 2 coefficients, but the set of points described by the equation is preserved if the equation is divided through by one of the non-zero coefficients, leaving one coefficient equal to 1 and only n(n + 3) / 2 coefficients to characterize the curve. Given n(n + 3) / 2 points (xi, yi), each of these points can be used to create a separate equation by substituting it into the general polynomial equation of degree n, giving n(n + 3) / 2 equations linear in the n(n + 3) / 2 unknown coefficients. If this system is non-degenerate in the sense of having a non-zero determinant, the unknown coefficients are uniquely determined and hence the polynomial equation and its curve are uniquely determined. But if this determinant is zero, the system is degenerate and the points can be on more than one curve of degree n. ==References==