Book: Mathematical Principles of Natural Philosophy

Overview

Isaac Newton’s Mathematical Principles of Natural Philosophy (1687) recasts the study of nature as a deductive, quantitative science. Using precise definitions, geometrical lemmas, and axioms of motion, Newton derives the behavior of terrestrial and celestial bodies within a single framework. The treatise unifies Galileo’s kinematics, Kepler’s astronomy, and contemporary experimental knowledge by demonstrating that the same forces governing falling bodies on Earth also determine planetary and cometary motions.

Structure and Aims

The book is divided into three parts. Book I develops the mathematical theory of motion under central forces in a vacuum, establishing the consequences of different force laws. Book II treats motion in resisting media and uses experiments and analysis to argue against Cartesian vortex theories. Book III, titled “The System of the World, ” applies the established principles to the solar system, showing that universal gravitation explains the phenomena cataloged by astronomers: planetary orbits, satellite periods, comet paths, tides, and terrestrial variations.

Mathematical Foundations

Newton begins with definitions of quantity of matter (mass), quantity of motion (momentum), and centripetal force, followed by a scholium distinguishing absolute and relative, true and apparent, time, space, place, and motion. He then introduces lemmas on “first and last ratios, ” a geometric limit method anticipating differential calculus, to compute areas, tangents, curvature, and orbital relations. This synthetic geometry allows him to prove general results without algebraic calculus notation, linking motion to force through carefully crafted limit arguments.

Laws of Motion and Dynamics

At the core stand the three laws of motion: bodies persist in uniform straight-line motion unless acted upon; change of motion is proportional to impressed force and along its line; and action equals reaction. From these, with corollaries, Newton derives conservation of momentum for isolated systems and the collective motion of centers of mass. He analyzes centripetal forces that constantly deflect bodies from straight paths, showing that a central force implies equal areas swept in equal times, and that an inverse-square central force produces conic-section orbits with the focus at the center of force.

Universal Gravitation

Book I’s theorems set the stage for Book III’s identification of gravity as an attractive force between all masses, proportional to their quantities of matter and inversely proportional to the square of their separation. Newton argues from terrestrial gravity and the Moon’s orbital fall that the same inverse-square dependence governs both, establishing universality. He computes that such a law yields Kepler’s area and harmonic relations, explains the precession of apsides, and predicts orbital parameters from observed periods and distances.

Applications and Tests

Book III derives the masses of the Sun, planets, and their satellites from orbital data, explains the tides by superposed lunar and solar attractions, and accounts for the precession of the equinoxes by the torque on Earth’s oblate figure, itself deduced from rotational flattening. Comets, observed to move on highly inclined and elongated paths, are fitted as conic trajectories deflected by solar gravity, contradicting vortex drag. Book II’s analysis of resistance shows that planetary motions could not persist in dense vortical media, reinforcing the vacuum model required by astronomical regularities and comet behavior.

Method and Legacy

Newton frames Book III on stated assumptions and “phenomena” distilled from observation, then deduces consequences that can be checked against data, emphasizing mathematical demonstration over speculative mechanisms. He declines to assign a physical cause for gravity’s transmission while insisting on its measurable laws and predictive power. The Principia thereby refounds natural philosophy on mathematical principles, forging classical mechanics, validating universal gravitation, and providing a template for linking theory, calculation, and observation across the natural sciences.