Copernicus's epicycles from Newton's gravitational force law via linear   perturbation theory in geometric algebra
  We derive Copernicus's epicycles from Newton's gravitational force law by assuming that a planet's orbit is a perturbed circular orbit, with the perturbation defined to be co-rotating with the said orbit. We substitute this orbit expression into Newton's gravitation law and showed that the perturbation satisfies the linear part of Hill's oscillator equation for lunar motion. We solve this oscillator equation using an exponential Fourier series and impose the boundary conditions at the aphelion and perihelion to derive Copernicus's formulas for the eccentric, deferent, and epicycle. We show that for small eccetricity, the Copernican orbit expression also leads to Kepler's law of areas for planetary motion. The formalism we use is the Clifford (geometric) algebra Cl_{2,0}. 
Quirino M. Sugon Jr., Sarah Bragais, and Daniel J. McNamara