Contents of the Principia: Book I on motion, Book II on resisted motion and fluids, Book III on the System of the World, and the famous Rules of Reasoning in Philosophy.

What Is in Newton’s Principia Mathematica?

The Philosophiae Naturalis Principia Mathematica — the “Mathematical Principles of Natural Philosophy,” usually called simply the Principia — was published in London in 1687. The book is the central work of Isaac Newton’s career, and one of the most consequential single works in this history. In three books, it set out the mathematical framework of classical mechanics, derived the three laws of motion and the law of universal gravitation, and demonstrated that the same laws governed the motion of the planets, the Moon, the tides, and a falling apple. The book is the completion of the Scientific Revolution: the unification of celestial and terrestrial mechanics that Copernicus, Kepler, and Galileo had all worked toward. This page looks at the contents of the Principia in some detail. The biographical and contextual notes are on the Newton page. The story of how Newton arrived at universal gravitation is told in How Did Newton Discover Gravity?. The empirical background of the laws is treated in What Are Kepler’s Three Laws of Planetary Motion? and in The Laws of Motion and Gravity.

The Publication

The Principia was published on 5 July 1687. The book was financed by Edmund Halley, the Astronomer Royal, who had been the catalyst for the work. Halley had visited Newton at Cambridge in 1684, asking what curve a planet would trace if it were attracted to the Sun by a force varying inversely with the square of the distance. Newton had replied that the curve was an ellipse. Halley had urged Newton to publish the argument. Newton had at first refused, then accepted, and the result was the Principia. The book was published by the Royal Society, of which Newton was a fellow, and at the Society’s expense. The original printing was about 250 copies. It sold quickly, and a second edition, with corrections, was published in 1713, edited by Roger Cotes. A third edition, edited by Henry Pemberton, appeared in 1726, the year before Newton’s death. The book was widely read across Europe. It was translated into English (by Andrew Motte, in 1729), into French (by the Marquise du Châtelet, in 1759), and into other languages. It became, by the early eighteenth century, the standard reference in theoretical natural philosophy.

Book I: The Motion of Bodies

The first book of the Principia is the most mathematically demanding. It sets out the framework of classical mechanics, in a series of propositions proved by the methods of synthetic geometry that Newton had learned from Euclid and Apollonius. The book is divided into fourteen sections. The first section, “On the Method of First and Last Ratios,” introduces the method of fluxions, Newton’s version of the calculus, in a geometric form. The method is used to find the velocities of bodies at given instants, by considering the limit of the average velocities over smaller and smaller intervals of time. Newton, in a famous passage, compares the method to the “geometry of the ancients,” in which curves are treated as the limits of polygons. The method is the foundation of the analytical approach of the rest of the book. The second section, “On the Determination of Centripetal Forces,” lays out the central problem: given a body moving in a curve around a fixed center, find the force directed toward that center. The section contains the propositions on the relationship between the force and the curvature of the path, the area law (Kepler’s second law), and the motion of a body in a non-circular orbit. The third and fourth sections develop the motion of bodies in conic sections — ellipses, parabolas, and hyperbolas — and the conditions under which a body will move in each. The famous Proposition 11, that a body moving under the action of a central force varying inversely with the square of the distance moves in a conic section, is the heart of the book. The proposition is the inverse of Kepler’s first law: Kepler’s first law says that the planetary orbits are ellipses; Newton’s proposition says that if the force is inverse-square, then the orbit is a conic section. The remaining sections of Book I treat the motion of bodies in non-resisting media of various kinds, the motion of pendulums, the motion of bodies under the action of mutual forces, and the motion of bodies in orbits that are not closed. The book ends with a series of propositions on the motion of the Moon, including the famous “lunar theory,” in which Newton works out the perturbations of the Moon’s orbit by the Sun.

Book II: The Motion of Bodies in Resisting Media

The second book of the Principia is the most surprising. It is a long, detailed treatment of the motion of bodies in resisting media, including fluids, pendulums, and projectiles. The book is also, the most controversial. Newton used the book to argue, against Descartes, that the planetary system could not be explained by the action of vortices in a subtle fluid. The vortex theory, which held that the planets were carried around the Sun by huge whirlpools of subtle matter, was the leading alternative to the gravitational theory in the late seventeenth century. Newton showed, by a series of careful arguments, that the vortex theory could not account for the observed motions of the planets and comets. The structure of Book II is roughly as follows. The first section treats the motion of bodies in fluids of various kinds, with different ratios of resistance to velocity. The second section treats the motion of pendulums in resisting media, with applications to the measurement of the resistance. The third section treats the motion of fluids themselves, including the propagation of waves and the resistance of fluids to the motion of bodies through them. The fourth and fifth sections continue the analysis of fluids, with applications to the figure of the Earth and the motion of comets. The most important result of Book II, for the long-term development of physics, is the theory of wave propagation in fluids. Newton, in Propositions 41–50, worked out the speed of sound in air, on the assumption that sound is a wave of compression and rarefaction in the air. The theoretical speed turned out to be about 9 percent too low, a discrepancy that Newton was aware of and that he attributed to the “crassitude” of the air particles. The discrepancy was eventually resolved, in the early nineteenth century, by Laplace, who showed that the discrepancy is due to the heating of the air during compression, a phenomenon that Newton had not considered. The climax of Book II is the critique of the vortex theory. The vortex theory, as Descartes had developed it, was a serious alternative to the gravitational theory, and it was widely accepted in France and the Low Countries. Newton argued, by a series of careful arguments, that the vortex theory could not account for the observed motions of the planets, the comets, and the satellites of the planets. The argument was, decisive: by the early eighteenth century, the vortex theory had few defenders, and the gravitational theory had become the consensus view of European natural philosophy.

Book III: The System of the World

The third book of the Principia is the most accessible and the most famous. It applies the mathematical framework of Books I and II to the actual solar system. The book is the answer to the question that Halley had asked in 1684: what curve does a planet trace if it is attracted to the Sun by a force varying inversely with the square of the distance? The book is also, in a broader sense, the synthesis of the new physics: the unification of celestial and terrestrial mechanics, the demonstration that the same laws govern the heavens and the Earth. The book opens with the famous “Rules of Reasoning in Philosophy,” four methodological rules that Newton proposed to govern the practice of natural philosophy:

“We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.”
“Therefore to the same natural effects we must, as far as possible, assign the same causes.”
“The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.”
“In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.” These four rules are the methodology of the Principia. They are empirical, conservative, and explicit. They were a working codification of the new mechanical philosophy, and they were widely cited in the eighteenth century. The book continues with a series of “phenomena” — observed regularities in the motions of the planets, the satellites, and the comets — and then “propositions” — demonstrations of the regularities from the laws of motion and the law of universal gravitation. The most important propositions are the demonstrations of Kepler’s three laws as consequences of the inverse-square law. Proposition 13 shows that the equatorial plane of a planet in an elliptical orbit sweeps out equal areas in equal times (Kepler’s second law). Propositions 14 and 15 show that the elliptical orbits of the planets, and the harmonic law relating their periods and sizes, follow from the inverse-square law. The book also contains the famous “lunar theory,” a long and difficult analysis of the perturbations of the Moon’s orbit by the Sun. The lunar theory was one of the great triumphs of the Principia: it showed that the motion of the Moon, which had been the bane of gravitational theory for centuries, could be accounted for in detail by the inverse-square law. The theory was not, in 1687, fully correct, and Newton knew it. The lunar theory would be improved, over the next two centuries, by a long line of mathematicians including Euler, Clairaut, d’Alembert, and Laplace. The book ends with a treatment of comets. Newton showed, by a careful analysis of the data on the comet of 1680, that comets move in conic sections around the Sun, on orbits that are, in general, very elongated ellipses or near-parabolic orbits. The analysis was a tour de force of mathematical astronomy, and it showed that comets are a normal part of the solar system, not (as they had often been thought to be) atmospheric phenomena or supernatural visitations.

The Argument of the Whole Work

The argument of the Principia is, in brief, the following. The planets move in conic sections around the Sun, with the Sun at one focus; the area law holds for each planet; the periods of the planets are related to the sizes of their orbits by the harmonic law; the Moon moves around the Earth under the action of the same force that makes a stone fall; the tides are caused by the gravitational pull of the Sun and the Moon on the oceans of the Earth; the figure of the Earth is the equilibrium shape under the action of the centrifugal force of its rotation and the gravitational pull of its parts; the comets move around the Sun on elongated orbits. All of these phenomena, Newton argued, are consequences of a single principle: that every particle of matter in the universe attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. The argument was, in the eighteenth and nineteenth centuries, the central argument of the new physics. The Principia was, for two hundred years, the dominant text in theoretical physics, and the framework it established — the three laws of motion, the law of universal gravitation, the calculus, the analytical methods — was the framework within which most physical science was done. The unification of celestial and terrestrial mechanics that the Principia accomplished was the completion of the central project of the Scientific Revolution. The story of how Newton arrived at this unification is told in How Did Newton Discover Gravity?.

The Reception of the Principia

The Principia was, from the start, a controversial book. The Cartesians on the Continent rejected the gravitational theory as an “occult quality” — a return to the scholastic philosophy that Descartes had been trying to replace. The Leibnizians quarreled with the Newtonians about the priority of the calculus. The Jesuits, who ran the leading astronomy programs in Catholic Europe, were slow to adopt the new framework. The English Newtonians, including Halley and Cotes, worked hard to promote the book and to defend it against its critics. however, the Principia won. By the early eighteenth century, the gravitational theory had become the consensus view of European natural philosophy. The book was translated into English, French, German, and other languages, and it became the standard reference in theoretical natural philosophy. The eighteenth century was the Age of Newton, just as the seventeenth century had been the Age of Galileo and Descartes. The full reception of the book is one of the great stories in this history, and it is the central narrative of the Newton page and of the Laws of Motion and Gravity section.

What Is in Newton's Principia Mathematica?