The Other Focus

Tom Wieting
Department of Mathematics, Reed College

Abstract: In 1609 ad, Johannes Kepler proposed a new model for planetary motion:
  • Each Planet follows an elliptical orbit for which the Sun occupies one focus.
  • The line joining the Sun and the Planet sweeps out equal areas in equal times.
In consequence, the new model required for the location of the Planet upon its orbit as a function of time, the solution of a transcendental equation: The Equation of Kepler. Most astronomers, notably Kepler and Nicholas Mercator, believed the solution to be non-constructible and they acknowledged the circumstance with regret. Enter Seth Ward (1656 ad), who proposed an elegant constructive scheme for locating the Planet upon its orbit as a function of time, by exploiting the unoccupied focus of the ellipse. He restored, so he said, geometry to astronomy. The object of this lecture is to set the foregoing issues in historical context and to describe and evaluate Ward's scheme.

Prerequisite: Trigonometry.