Newton's superb theorem: An elementary geometric proof

Newton's "superb theorem" for the gravitational inverse-square-law force states that a spherically symmetric mass distribution attracts a body outside as if the entire mass were concentrated at the center. This theorem is crucial for Newton's comparison of the Moon's orbit with terrestrial gravity (the fall of an apple), which is evidence for the inverse-square-law. Newton's geometric proof in the Principia "must have left its readers in helpless wonder" according to S. Chandrasekhar and J.E. Littlewood. In this paper we give an elementary geometric proof, which is much simpler than Newton's geometric proof and more elementary than proofs using calculus.