An approach to orbital mechanics, which is accessible to beginning physics students and presupposes no knowledge of calculus, is presented. A theory of orbits is developed for the inverse-square central force law which differs considerably from the usual deductive approach. This document begins with qualitative aspects of solutions, and leads to a number of geometrically realizable physical invariants of the orbits. Consequently, most of the theorems rely only on simple geometrical relationships. Despite its simplicity, this planetary geometry is powerful enough to treat a wide range of perturbations with relative ease. It is felt that this treatment provides a better view of "what doing physics is really like" than the standard route via algebraic manipulations. The document concludes with suggestions for further research into the geometry of planetary orbits. (MP)