Division of Other curves than circles The coordinates of an arc of a circle of length $\frac{2pi}{p}$ are an algebraic number, and when $p$ is a Fermat prime you can find it in terms of square roots.
Gauss said that the method applied to a lot more curves than the circle. Will you please tell if you know any worked examples of this (finding the algebraic points on other curves)?

To flesh out the comment I gave: Prasolov and Solovyev mention an example due to Euler and Serret: consider the plane curve with complex parametrization
$$z=\frac{(t-a)^{n+2}}{(t-\bar{a})^n (t+i)^2}$$
where $a=\frac{\sqrt{n(n+2)}}{n+1}-\frac{i}{n+1}$ and $n$ is a positive rational number.
The arclength function for this curve is $s=\frac{2\sqrt{n(n+2)}}{n+1}\arctan\,t$; since
$$\arctan\,u+\arctan\,v=\arctan\frac{u+v}{1-u v}$$
the division of an arc of this curve can be done algebraically (with straightedge and compass for special values).
Here are plots of these curves for various values of $n$:

Serret also considered curves whose arclengths can be expressed in terms of the incomplete elliptic integral of the first kind $F(\phi|m)$; I'll write about those later once I figure out how to plot these... (but see the Prasolov/Solovyev book for details)
