# Centripetal Force and Resulting Trajectories

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## Abstract

Chapter 4 applies polar calculus within a comprehensive, self-contained treatment of Newton’s theory of gravitation. Newton’s treatise *Principia Mathematica* had been a miracle. It provided a synthesizing and penetrating solution to a question that had occupied many of humanity’s best minds for about 3000 years: how do the heavens work? Kepler had discovered the three laws of planetary motion with painstaking observations, but Newton came to recognize the deeper reality. All three of Kepler’s laws rest on a combination of mathematical methods, basic laws of motion, and the inverse square law of universal gravitation. The chapter includes a complete analysis of the connection between the magnitude of a centripetal force and the geometry of the trajectory of a point-mass—or a sphere that has its mass radially distributed—on which the force acts.