To quote Goursat:
It is shown in treatises on Analytic Geometry that every unicursal curve of degree n has $\frac{(n-1)(n-2)}{2}$ double points, and, conversely, that every curve of degree n which has this number of double points is unicursal. P222
Would somebody mind developing some intuition for this statement, along with an example or four (if not an intuitive proof), that would help motivate me to pick up classical books on analytical geometry & encourage me to wade through hundreds of pages to get to results like this one? I know so little about topics like these that I'm still trying to figure out the intuition for deriving multiple points, since it seems like authors do it in different ways each time, thanks.
I hope this is helpful; I was reading this proof of the theorem:http://www.jstor.org/stable/2973979?seq=1
It didn't give me much intuition; essentially, it seemed like they found algebraic equations for the double points and then used Bezout's theorem to get exact answers, with an intermediate step that narrows down the number of solutions.
So it seems like Bezout's theorem explains why there are intersections at all.
As for examples, notice that quadratic curves like ellipses have no double points while cubics do (whether real or complex); the cusp in $y^2=x^3$ is a double point.
As for intuition, think of a string wrapped around a pole. The more times it wraps around, the more it must intersect itself. The degree of a unicursal curve can be thought of as measuring how many times the curve "wraps around" itself.