When Nine Points Are Worth But Eight: Euler’s Resolution of Cramer’s Paradox - Intersection of Lines and Curves

Author(s):
Two straight lines in the plane intersect in one point. This statement is always true, except when the two lines are parallel (although the statement is always true in the projective plane). And like the previous question, the problem of intersection can be considered for curves of higher degree. For example, a line can typically intersect an ellipse, parabola or hyperbola in two points. Two conics can intersect in as many as 4 points. By considering various low order cases, many 18th century mathematicians came to the conclusion that two curves of order $m$ and $n$ typically intersect in $mn$ points, and can never intersect in more than $mn$ points. This result is now called Bézout's Theorem, after Etienne Bézout (1730-1783), who gave the first acceptable proof of this result in 1779.
Hold on, now … there's a problem here! Let's suppose $m=n=3.$ Then on the one hand, two curves of order three will typically intersect in 9 points. On the other hand, if $n=3,$ then $\frac{n^2+3n}{2}=9,$ so those same 9 points should have been enough to determine a unique curve of order three! This apparent contradiction goes by the name of Cramer's Paradox, although it was first noticed by Maclaurin and it was resolved at least as successfully by Euler as it was by Cramer.