We develop a theory of orbits for the inverse‐square central force law that differs considerably from the usual deductive approach. In particular, we make no explicit use of calculus. By beginning with qualitative aspects of solutions, we are led to a number of geometrically realizable physical invariants of the orbits. Consequently, most of our theorems rely only on simple geometrical relationships. Despite its simplicity, our planetary geometry is powerful enough to treat a wide range of perturbations with relative ease. Furthermore, without introducing any more machinery, we obtain full quantitative results. The paper concludes with suggestions for further research into the geometry of planetary orbits.