Graphical Derivation of the Inverse-Square Law of Gravitation from an Elliptic Orbit and Kepler's Law of Areas

Motivated by the common knowledge that earth satellites move in elliptic paths, a graphical demonstration is presented which shows that if a small mass moves in an elliptic orbit about a large mass at one focus and obeys Kepler's law of areas, the force of attraction on it must vary inversely as the square of the distance between the two masses. The demonstration differs from the usual elementary proofs of the inverse-square law in that it assumes elliptic rather than circular orbits, uses the law of areas instead of the dependence of period on semimajor axis, and requires a minimum of formal mathematics. By using a similar construction it may be shown that if the large mass is at the center of the ellipse, the force varies linearly with distance.