  In Book 1, Proposition 7, Problem 2 of his 1687 Philosophiae Naturalis
Principia Mathematica, Isaac Newton poses and answers the following question:
Let the orbit of a particle moving in a central force field be an off-center
circle. How does the magnitude of the force depend on the position of the
particle onthat circle? In this article, we identify a potential that can
produce such a force, only at zero energy. We further map the zero-energy
orbits in this potential to finite-energy free motion orbits on a sphere; such
a duality is a particular instance of a general result by Goursat, from 1887.
The map itself is an inverse stereographic projection, and this fact explains
the circularity of the zero-energy orbits in the system of interest. Finally,
we identify an additional integral of motion - an analogue of the Runge-Lenz
vector in the Coulomb problem - that is responsible for the closeness of the
zero-energy orbits in our problem.
