Prove that in a Kepler elliptic orbit with small eccentricity e the angular motion of a particle as viewed from the empty focus of the ellipse is uniform (the empty focus is the focus that is not the center of attraction) to first order in e. It is this theorem that enables the Ptolemaic picture of planetary motion to be a reasonably accurate approximation. On this picture the Sun is assumed to move uniformly on a circle whose center is shifted from Earth by a distance called the equant. If the equant is taken as the distance between the two foci of the correct elliptical orbit, then the angular motion is thus described by the Ptolemaic picture accurately to first order in e.