Constructive Euclidean Proofs of the Equivalence Between Keplerian Orbits and Newton's Inverse-Square Law

Abstract

Kepler's first two laws state that a planet moves on an ellipse with the Sun at a focus and sweeps out equal areas in equal times (constant areal speed). In the Principia, Newton showed how these laws connect to universal gravitation. Since then, the equivalence between orbital laws and force laws has remained a central topic in celestial mechanics. We present fully geometric proofs, built from explicit Euclidean straightedge-and-compass constructions, of this equivalence in both directions. The proof system combines finite-step constructions, tangent and triangle geometry, affine transport, local displacement ratios, conic invariants, and several hodograph realizations. Within this broader framework, one contribution is to use the auxiliary circle as the primary hodograph proxy in configuration space rather than the directrix-circle normalization of radius 2a. Our emphasis is a Principia-style argument that avoids differential equations while remaining close to Euclidean methods.