On Friday 13 November 2009 05:49:43 Dan Asimov wrote:
<< Stirling (in 1717) prove that 9 points uniquely define a cubic.
McLaurin (in 1720) proved that two cubics intersect in at most nine points. ... Around 1750, Euler and Cramer noticed that these seem to contradict, since different cubics are passing through the same nine points.
Um, what's the apparent contradiction?
Different lines intersect in at most one point. Many lines can pass through that same point. The point being???
1. No one claims that one point uniquely defines a line. 2. Not only do two cubics intersect in at most nine points, but there are examples where they intersect in exactly nine points. So you get two cubics going through the same nine points, but supposedly nine points define a *unique* cubic. (The solution to the "paradox" is that nine points *in general position* uniquely define a cubic, where in this case "in general position" means "outside some horrendous thing of codimension 1".) -- g