The Principia: The Authoritative Translation: Mathematical Principles of Natural Philosophy

The Principia: The Authoritative Translation: Mathematical Principles of Natural Philosophy

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In his monumental 1687 work, Philosophiae Naturalis Principia Mathematica, known familiarly as the Principia, Isaac Newton laid out in mathematical terms the principles of time, force, and motion that have guided the development of modern physical science. Even after more than three centuries and the revolutions of Einsteinian relativity and quantum mechanics, Newtonian physics continues to account for many of the phenomena of the observed world, and Newtonian celestial dynamics is used to determine the orbits of our space vehicles.

This authoritative, modern translation by I. Bernard Cohen and Anne Whitman, the first in more than 285 years, is based on the 1726 edition, the final revised version approved by Newton; it includes extracts from the earlier editions, corrects errors found in earlier versions, and replaces archaic English with contemporary prose and up-to-date mathematical forms.

Newton's principles describe acceleration, deceleration, and inertial movement; fluid dynamics; and the motions of the earth, moon, planets, and comets. A great work in itself, the Principia also revolutionized the methods of scientific investigation. It set forth the fundamental three laws of motion and the law of universal gravity, the physical principles that account for the Copernican system of the world as emended by Kepler, thus effectively ending controversy concerning the Copernican planetary system.
The translation-only edition of this preeminent work is truly accessible for today's scientists, scholars, and students.

Product Details

ISBN-13: 9780520290747
Publisher: University of California Press
Publication date: 02/05/2016
Edition description: First Edition
Pages: 616
Sales rank: 150,505
Product dimensions: 6.90(w) x 10.00(h) x 1.20(d)

About the Author

I. Bernard Cohen (1914–2003) was Victor S. Thomas Professor of the History of Science at Harvard University. He was the author of Benjamin Franklin's ScienceInteractions, and Science and the Founding Fathers
Anne Whitman (1937–1984) was coeditor (with I. Bernard Cohen and Alexander Koyré) of the Latin edition, with variant readings, of the Principia
Julia Budenz, author of From the Gardens of Flora Baum, is a multilingual classicist and poet.

Read an Excerpt

The Principia

Mathematical Principles of Natural Philosophy

By Isaac Newton, I. Bernard Cohen, Anne Whitman


Copyright © 1999 The Regents of the University of California
All rights reserved.
ISBN: 978-0-520-96478-5


A Brief History of the Principia

The Origins of the Principia

Isaac Newton's Principia was published in 1687. The full title is Philosophiae Naturalis Principia Mathematica, or Mathematical Principles of Natural Philosophy. A revised edition appeared in 1713, followed by a third edition in 1726, just one year before the author's death in 1727. The subject of this work, to use the name assigned by Newton in the first preface, is "rational mechanics." Later on, Leibniz introduced the name "dynamics." Although Newton objected to this name, "dynamics" provides an appropriate designation of the subject matter of the Principia, since "force" is a primary concept of that work. Indeed, the Principia can quite properly be described as a study of a variety of forces and the different kinds of motions they produce. Newton's eventual goal, achieved in the third of the three "books" of which the Principia is composed, was to apply the results of the prior study to the system of the world, to the motions of the heavenly bodies. This subject is generally known today by the name used a century or so later by Laplace, "celestial mechanics."

The history of how the Principia came into being has been told and retold. In the summer of 1684, the astronomer Edmond Halley visited Newton in order to find out whether he could solve a problem that had baffled Christopher Wren, Robert Hooke, and himself: to find the planetary orbit that would be produced by an inverse-square central force. Newton knew the answer to be an ellipse. He had solved the problem of elliptical orbits earlier, apparently in the period 1679–1680 during the course of an exchange of letters with Hooke. When Halley heard Newton's reply, he urged him to write up his results. With Halley's prodding and encouragement, Newton produced a short tract which exists in several versions and will be referred to as De Motu (On Motion), the common beginning of all the titles Newton gave to the several versions. Once started, Newton could not restrain the creative force of his genius, and the end product was the Principia. In his progress from the early versions of De Motu to the Principia, Newton's conception of what could be achieved by an empirically based mathematical science had become enlarged by several orders of magnitude.

As first conceived, the Principia consisted of two "books" and bore the simple title De Motu Corporum (On the Motion of Bodies). This manuscript begins, as does the Principia, with a series of Definitions and Laws of Motion, followed by a book 1 whose subject matter more or less corresponds to book 1 of the Principia. The subject matter of book 2 of this early draft is much the same as that of book 3 of the Principia. In revising this text for the Principia, Newton limited book 1 to the subject of forces and motion in free spaces, that is, in spaces devoid of any resistance. Book 2 of the Principia contains an expanded version of the analysis of motion in resisting mediums, plus discussions of pendulums, of wave motion, and of the physics of vortices. In the Principia, the system of the world became the subject of what is there book 3, incorporating much that had been in the older book 2 but generally recast in a new form. As Newton explained in the final Principia, while introducing book 3, he had originally presented this subject in a popular manner, but then decided to recast it in a more mathematical form so that it would not be read by anyone who had not first mastered the principles of rational mechanics. Even so, whole paragraphs of the new book 3 were copied word for word from the old book 2.

Steps Leading to the Composition and Publication of the Principia

The history of the development of Newton's ideas concerning mechanics, more specifically dynamics, has been explored by many scholars and is still the subject of active research and study. The details of the early development of Newton's ideas about force and motion, however interesting in their own right, are not directly related to the present assignment, which is to provide a reader's guide to the Principia. Nevertheless, some aspects of this prehistory should be of interest to every prospective reader of the Principia. In the scholium to book 1, prop. 4, Newton refers to his independent discovery (in the 1660s) of the v2/r rule for the force in uniform circular motion (at speed v along a circle of radius r), a discovery usually attributed to Christiaan Huygens, who formally announced it to the world in his Horologium Oscillatorium of 1673. It requires only the minimum skill in algebraic manipulation to combine the v2/r rule with Kepler's third law in order to determine that in a system of bodies in uniform circular motion the force is proportional to 1/r2 or is inversely proportional to the square of the distance. Of course, this computation does not of itself specify anything about the nature of the force, whether it is a centripetal or a centrifugal force or whether it is a force in the sense of the later Newtonian dynamics or merely a Cartesian "conatus," or endeavor. In a manuscript note Newton later claimed that at an early date, in the 1660s, he had actually applied the v2/r rule to the moon's motion, much as he does later on in book 3, prop. 4, of the Principia, in order to confirm his idea of the force of "gravity extending to the Moon." In this way he could counter Hooke's allegation that he had learned the concept of an inverse-square force of gravity from Hooke.

A careful reading of the documents in question shows that sometime in the 1660s, Newton made a series of computations, one of which was aimed at proving that what was later known as the outward or centrifugal force arising from the earth's rotation is less than the earth's gravity, as it must be for the Copernican system to be possible. He then computed a series of forces. Cartesian vortical endeavors are not the kind of forces that, in the Principia, are exerted by the sun on the planets to keep them in a curved path or the similar force exerted by the earth on the moon. At this time, and for some years to come, Newton was deeply enmeshed in the Cartesian doctrine of vortices. He had no concept of a "force of gravity" acting on the moon in anything like the later sense of the dynamics of the Principia. These Cartesian "endeavors" (Newton used Descartes's own technical term, "conatus") are the magnitude of the planets' endeavors to fly out of their orbits. Newton concludes that since the cubes of the distances of the planets from the sun are "reciprocally as the squared numbers of their revolutions in a given time," their "conatus to recede from the Sun will be reciprocally as the squares of their distances from the Sun."

Newton also made computations to show that the endeavor or "conatus" of receding from the earth's surface (caused by the earth's daily rotation) is 12½ times greater than the orbital endeavor of the moon to recede from the earth. He concludes that the force of receding at the earth's surface is "4000 and more times greater than the endeavor of the Moon to recede from the Earth."

In other words, "Newton had discovered an interesting mathematical correlation within the solar vortex," but he plainly had not as yet invented the radically new concept of a centripetal dynamical force, an attraction that draws the planets toward the sun and the moon toward the earth. There was no "twenty years' delay" (from the mid-1660s to the mid-1680s) in Newton's publication of the theory of universal gravity, as was alleged by Florian Cajori.

In 1679/80, Hooke initiated an exchange of correspondence with Newton on scientific topics. In the course of this epistolary interchange, Hooke suggested to Newton a "hypothesis" of his own devising which would account for curved orbital motion by a combination of two motions: an inertial or uniform linear component along the tangent to the curve and a motion of falling inward toward a center. Newton told Hooke that he had never heard of this "hypothesis." In the course of their letters, Hooke urged Newton to explore the consequences of his hypothesis, advancing the opinion or guess that in combination with the supposition of an inverse-square law of solar-planetary force, it would lead to the true planetary motions. Hooke also wrote that the inverse-square law would lead to a rule for orbital speed being inversely proportional to the distance of a planet from the sun. Stimulated by Hooke, Newton apparently then proved that the solar-planetary force is as the inverse square of the distance, a first step toward the eventual Principia.

We cannot be absolutely certain of exactly how Newton proceeded to solve the problem of motion in elliptical orbits, but most scholars agree that he more or less followed the path set forth in the tract De Motu which he wrote after Halley's visit a few years later in 1684. Essentially, this is the path from props. 1 and 2 of book 1 to prop. 4, through prop. 6, to props. 10 and 11. Being secretive by nature, Newton didn't tell Hooke of his achievement. In any event, he would hardly have announced so major a discovery to a jealous professional rival, nor in a private letter. What may seem astonishing, in retrospect, is not that Newton did not reveal his discovery to Hooke, but that Newton was not at once galvanized into expanding his discovery into the eventual Principia.

Several aspects of the Hooke-Newton exchange deserve to be noted. First, Hooke was unable to solve the problem that arose from his guess or his intuition; he simply did not have sufficient skill in mathematics to be able to find the orbit produced by an inverse-square force. A few years later, Wren and Halley were equally baffled by this problem. Newton's solution was, as Westfall has noted, to invert the problem, to assume the path to be an ellipse and find the force rather than "investigating the path in an inverse-square force field." Second, there is no certainty that the tract De Motu actually represents the line of Newton's thought after corresponding with Hooke; Westfall, for one, has argued that a better candidate would be an essay in English which Newton sent later to John Locke, a position he maintains in his biography of Newton. A third point is that Newton was quite frank in admitting (in private memoranda) that the correspondence with Hooke provided the occasion for his investigations of orbital motion that eventually led to the Principia. Fourth, as we shall see in §3.4 below, the encounter with Hooke was associated with a radical reorientation of Newton's philosophy of nature that is indissolubly linked with the Principia. Fifth, despite Newton's success in proving that an elliptical orbit implies an inverse-square force, he was not at that time stimulated — as he would be some four years later — to move ahead and to create modern rational mechanics. Sixth, Newton's solution of the problems of planetary force depended on both his own new concept of a dynamical measure of force (as in book 1, prop. 6) and his recognition of the importance of Kepler's law of areas. A final point to be made is that most scholarly analyses of Newton's thoughts during this crucial period concentrate on conceptual formulations and analytical solutions, whereas we know that both Hooke and Newton made important use of graphical methods, a point rightly stressed by Curtis Wilson. Newton, in fact, in an early letter to Hooke, wrote of Hooke's "acute Letter having put me upon considering ... the species of this curve," saying he might go on to "add something about its description quam proxime," or by graphic methods. The final proposition in the Principia (book 3, prop. 42) declared its subject (in the first edition) to be: "To correct a comet's trajectory found graphically." In the second and third editions, the text of the demonstration was not appreciably altered, but the statement of the proposition now reads: "To correct a comet's trajectory that has been found."

When Newton wrote up his results for Halley (in the tract De Motu) and proved (in the equivalent of book 1, prop. 11) that an elliptical orbit implies an inverse-square central force, he included in his text the joyous conclusion: "Therefore the major planets revolve in ellipses having a focus in the center of the sun; and the radii to the sun describe areas proportional to the times, exactly ["omnino"] as Kepler supposed." But after some reflection, Newton recognized that he had been considering a rather artificial situation in which a body moves about a mathematical center of force. In nature, bodies move about other bodies, not about mathematical points. When he began to consider such a two-body system, he came to recognize that in this case each body must act on the other. If this is true for one such pair of bodies, as for the sun-earth system, then it must be so in all such systems. In this way he concluded that the sun (like all the planets) is a body on which the force acts and also a body that gives rise to the force. It follows at once that each planet must exert a perturbing force on every other planet in the solar system. The consequence must be, as Newton recognized almost at once, that "the displacement of the sun from the center of gravity" may have the effect that "the centripetal force does not always tend to" an "immobile center" and that "the planets neither revolve exactly in ellipses nor revolve twice in the same orbit." In other words, "Each time a planet revolves it traces a fresh orbit, as happens also with the motion of the moon, and each orbit is dependent upon the combined motions of all the planets, not to mention their actions upon each other." This led him to the melancholy conclusion: "Unless I am much mistaken, it would exceed the force of human wit to consider so many causes of motion at the same time, and to define the motions by exact laws which would allow of any easy calculation."

We don't know exactly how Newton reached this conclusion, but a major factor may have been the recognition of the need to take account of the third law, that to every action there must be an equal and opposite reaction. Yet, in the texts of De Motu, the third law does not appear explicitly among the "laws" or "hypotheses." We do have good evidence, however, that Newton was aware of the third law long before writing De Motu. In any event, the recognition that there must be interplanetary perturbations was clearly an essential step on the road to universal gravity and the Principia.

In reviewing this pre-Principia development of Newton's dynamics, we should take note that by and large, Newton has been considering exclusively the motion of a particle, of unit mass. Indeed, a careful reading of the Principia will show that even though mass is the subject of the first definition at the beginning of the Principia, mass is not a primary variable in Newton's mode of developing his dynamics in book 1. In fact, most of book 1 deals exclusively with particles. Physical bodies with significant dimensions or shapes do not appear until sec. 12, "The attractive forces of spherical bodies."

Newton's concept of mass is one of the most original concepts of the Principia. Newton began thinking about mass some years before Halley's visit. Yet, in a series of definitions which he wrote out some time after De Motu and before composing the Principia, mass does not appear as a primary entry. We do not have documents that allow us to trace the development of Newton's concept of mass with any precision. We know, however, that two events must have been important, even though we cannot tell whether they initiated Newton's thinking about mass or reinforced ideas that were being developed by Newton. One of these was the report of the Richer expedition, with evidence that indicated that weight is a variable quantity, depending on the terrestrial latitude. Hence weight is a "local" property and cannot be used as a universal measure of a body's quantity of matter. Another was Newton's study of the comet of 1680. After he recognized that the comet turned around the sun and after he concluded that the sun's action on the comet cannot be magnetic, he came to believe that Jupiter must also exert an influence on the comet. Clearly, this influence must derive from the matter in Jupiter, Jupiter acting on the comet just as it does on its satellites.


Excerpted from The Principia by Isaac Newton, I. Bernard Cohen, Anne Whitman. Copyright © 1999 The Regents of the University of California. Excerpted by permission of UNIVERSITY OF CALIFORNIA PRESS.
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Table of Contents


Publisher's Note,
Preface to the 1999 Edition,
A Brief History of the Principia,
The Principia (Mathematical Principles of Natural Philosophy),
Halley's Ode to Newton,
Newton's Preface to the First Edition,
Newton's Preface to the Second Edition,
Cotes's Preface to the Second Edition,
Newton's Preface to the Third Edition,
Axioms, or the Laws of Motion,
Book 1: The Motion of Bodies,
Book 2: The Motion of Bodies,
Book 3: The System of The World,
General Scholium,
Notes Added in Second Printing,

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