Newton's superb theorem: An elementary geometric proof

TL;DR

This paper presents a simpler, elementary geometric proof of Newton's superb theorem, demonstrating that a spherically symmetric mass attracts outside bodies as if all mass were concentrated at the center, crucial for understanding gravitational laws.

Contribution

It provides a more elementary and straightforward geometric proof of Newton's superb theorem, improving upon Newton's original and calculus-based proofs.

Findings

Simpler geometric proof of Newton's superb theorem

Clarifies the gravitational attraction of spherically symmetric bodies

Enhances understanding of inverse-square law implications

Abstract

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.