The Principia

The Principia: Mathematical Principles of Natural Philosophy

I. Bernard Cohen
Anne Whitman
assisted by Julia Budenz
Preceded by I. Bernard Cohen
Copyright Date: 1999
Edition: 1
Pages: 991
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  • Book Info
    The Principia
    Book Description:

    In his monumental 1687 workPhilosophiae Naturalis Principia Mathematica, known familiarly as thePrincipia, 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 completely new translation, the first in 270 years, is based on the third (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, thePrincipiaalso 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 illuminating Guide to thePrincipiaby I. Bernard Cohen, along with his and Anne Whitman's translation, will make this preeminent work truly accessible for today's scientists, scholars, and students.

    eISBN: 978-0-520-96091-6
    Subjects: General Science

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Table of Contents
    (pp. ix-x)
  3. Preface
    (pp. xi-xviii)
  4. A Guide to Newton’s Principia
    • Contents of the Guide
      (pp. 3-8)
    • Abbreviations
      (pp. 9-10)
    • CHAPTER ONE A Brief History of the Principia
      (pp. 11-25)
      I. Bernard Cohen

      Isaac Newton’sPrincipiawas published in 1687. The full title isPhilosophiae Naturalis Principia Mathematica,orMathematical 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,¹ “dynamics” provides an appropriate designation of the subject matter of thePrincipia,since “force” is a primary concept of that work. Indeed, thePrincipiacan...

    • CHAPTER TWO Translating the Principia
      (pp. 26-42)
      I. Bernard Cohen

      Newton’sPrincipiahas been translated in full, in large part, or in close paraphrase into many languages, including Chinese, Dutch, English, French, German, Italian, Japanese, Mongolian, Portuguese, Romanian, Russian, Spanish, and Swedish.¹ The whole treatise has been more or less continuously available in an English version, from 1729 to the present, in a translation made by Andrew Motte and modernized more than a half-century ago. In addition, a new version of book 1 by Thomas Thorp was published in 1777.²

      Motte’s translation of 1729 was reprinted in London in 1803 “carefully revised and corrected by W. Davis” and with “a...

    • CHAPTER THREE Some General Aspects of the Principia
      (pp. 43-84)
      I. Bernard Cohen

      When Newton enlarged thePrincipiafrom two books to three, he changed the title fromDe Motu Corporumto the more grandiose titlePhilosophiae Natu ralis Principia Mathematica.In the first edition, the title page displays the wordsPhilosophiaeandPrincipiain large capital letters, stressing these two nouns at the expense of the modifiersNaturalisandMathematica(see figs. 3.1, 3.2). We don’t know for certain whether this emphasis was initially the result of a conscious decision by Halley or whether it came about through the printer’s design, merely stressing the nouns at the expense of the adjectives. But...

    • CHAPTER FOUR Some Fundamental Concepts of the Principia
      (pp. 85-108)
      I. Bernard Cohen

      In a scholium to the Definitions (discussed in §4.11), Newton says that he has undertaken “to explain the senses in which less familiar words are to be taken in this treatise.” These do not include “time, space, place, and motion,” which are “familiar to everyone.” It is important, however, he notes, to take into account that “these quantities are popularly conceived solely with reference to the objects of sense perception.” Newton says that he will “distinguish these quantities into absolute and relative, true and apparent, mathematical and common.”

      Preliminary to any such discussion is the set of eight definitions of...

    • CHAPTER FIVE Axioms, or the Laws of Motion
      (pp. 109-127)
      I. Bernard Cohen

      Newton’s three laws of motion were set forth under the general heading “Axiomata, sive Leges Motus,” or the “Axioms, or the Laws of Motion.” As mentioned (§3.1 above), it is difficult to believe he was not (even if unconsciously) making a direct improvement on the laws announced by Descartes in hisPrincipiaas “Laws of Nature” or “Regulae quaedam sive Leges Naturae.” On at least one occasion, in a letter to Roger Cotes during the preparation of the second edition of thePrincipia,Newton inadvertently called one of his laws of motion by Descartes’s name, “Law of nature.”

      Newtonians generally...

    • CHAPTER SIX The Structure of Book 1
      (pp. 128-160)
      I. Bernard Cohen

      ThePrincipia,in its final form, consists of four prefaces, a set of “Definitions,” “Axioms, or the Laws of Motion,” books 1 and 2 on “The Motion of Bodies,” book 3 on “The System of the World,” and a concluding “General Scholium.” The subject of book 1 is motion in free spaces, that is, in spaces devoid of resistance, while book 2 deals with motion under several different kinds of resistance. Here Newton’s subject is extended to include many other topics of natural philosophy, such as the principles of wave motion, along with some aspects of the general theory of...

    • CHAPTER SEVEN The Structure of Book 2
      (pp. 161-194)
      I. Bernard Cohen

      Book 2 of thePrincipiadiffers from books 1 and 3 in a variety of ways. One of the most striking of these is the fact that its major contents, the theoretical and experimental study of the forces of resistance to motion in various types of fluids, is usually not discussed and often not even mentioned by historians of science or even by some historians of mechanics, such as Ernst Mach. Those historians who have studied book 2 have been generally concerned with certain special topics and not with the main theme of book 2: the theoretical and experimental investigation...

    • CHAPTER EIGHT The Structure of Book 3
      (pp. 195-273)
      I. Bernard Cohen

      Book 3 is composed of six distinct parts. The first contains a set of rules (“regulae”) for proceeding in natural philosophy; the second the “phenomena” on which the exposition of the system of the world is to be based. Next comes the application of mathematical principles (primarily as developed in book 1) to explain the motion of planets and their satellites by the action of universal gravity. The fourth part sets forth Newton’s gravitational theory of the tides. The fifth part of book 3 (props. 22 to 30) contains an analysis of the motion of the moon, at once—as...

    • CHAPTER NINE The Concluding General Scholium
      (pp. 274-292)
      I. Bernard Cohen

      The first edition of thePrincipiahad no proper conclusion, since Newton suppressed his draft “Conclusio.” In the second edition, he planned to have a final discussion about “the attraction of the small particles of bodies,” as he wrote to Cotes on 2 March 1712/13,¹ when most of the text had already been printed off. On further reflection, however, he abandoned the temptation to expose his theories of the forces, interactions, structure, and other aspects of particulate matter, and instead composed the concluding General Scholium, in which, he said, he had included a “short Paragraph about that part of Philosophy.”...

    • CHAPTER TEN How to Read the Principa
      (pp. 293-368)
      I. Bernard Cohen

      The literature concerning Isaac Newton and hisPrincipiais vast and ever increasing. Among those many works there are several that can be especially recommended as first guides to anyone who wishes to study the mathematical and technical structure of thePrincipia.These are D. T. Whiteside’s “Before thePrincipia:The Maturing of Newton’s Thoughts on Dynamical Astronomy” and “The Mathematical Principles Underlying Newton’sPrincipia” (§1.2, n. 9 above) and Curtis Wilson’s “The Newtonian Achievement” (cited in Abbreviations, pp. 9-10 above). The latter, in a brief compass, gives a splendid overall view of the development of Newton’s ideas on dynamics...

    • CHAPTER ELEVEN Conclusion
      (pp. 369-370)
      I. Bernard Cohen

      Newton’sPrincipiais a book of mathematical principles applied to nature insofar as nature is revealed by experiment and observation. As such, it is a treatise based on evidence. Never before had a treatise on natural philosophy so depended on an examination of numerical predictions and numerical evidence. The significance of Newton’s numbers, in the context of the mission of thePrincipia,was not only to provide convincing evidence of the correctness of the mathematical principles being applied to natural philosophy, but to explore the theoretical significance of possible conflicts between simple—perhaps overly simple—theory and the evidential universe....

  5. The Principia (Mathematical Principles of Natural Philosophy)
    • [Introduction]
      (pp. 371-378)

      Newton’sPrincipiais a book of mathematical principles applied to nature insofar as nature is revealed by experiment and observation. As such, it is a treatise based on evidence. Never before had a treatise on natural philosophy so depended on an examination of numerical predictions and numerical evidence. The significance of Newton’s numbers, in the context of the mission of thePrincipia,was not only to provide convincing evidence of the correctness of the mathematical principles being applied to natural philosophy, but to explore the theoretical significance of possible conflicts between simple—perhaps overly simple—theory and the evidential universe....

    • Halley’s Ode to Newton
      (pp. 379-380)
      Edm. Halley

      Behold the pattern of the heavens, and the balances of the divine structure; Behold Jove’s calculation and the laws

      That the creator of all things, while he was setting the beginnings of the world, would not violate;

      Behold the foundations he gave to his works.

      Heaven has been conquered and its innermost secrets are revealed;

      The force that turns the outermost orbs around is no longer hidden.

      The Sun sitting on his throne commands all things

      To tend downward toward himself, and does not allow the chariots of the heavenly bodies to move

      Through the immense void in a straight...

    • Newton’s Preface to the First Edition
      (pp. 381-383)
      Is. Newton

      SINCE THE ANCIENTS (according to Pappus) considered mechanics to be of the greatest importance in the investigation of nature and science and since the moderns—rejecting substantial forms and occult qualities—have undertaken to reduce the phenomena of nature to mathematical laws, it has seemed best in this treatise to concentrate onmathematicsas it relates to natural philosophy. The ancients divided mechanics into two parts: the rational, which proceeds rigorously through demonstrations, and thepractical? Practical mechanicsis the subject that comprises all the manual arts, from which the subject ofmechanicsas a whole has adopted its name. But since...

    • Newton’s Preface to the Second Edition
      (pp. 384-384)
      Is. Newton

      IN THIS SECOND EDITION of thePrinciples,many emendations have been made here and there, and some new things have been added. In sec. 2 of book 1, the finding of forces by which bodies could revolve in given orbits has been made easier and has been enlarged. In sec. 7 of book 2, the theory of the resistance of fluids is investigated more accurately and confirmed by new experiments. In book 3 the theory of the moon and the precession of the equinoxes are deduced more fully from their principles; and the theory of comets is confirmed by more...

    • Cotes’s Preface to the Second Edition
      (pp. 385-399)
      Rogerv Cotes

      THELONG-AWAITED NEW EDITION of Newton'sPrinciples of Natural Philosophyis presented to you, kind reader, with many corrections and additions. The main topics of this celebrated work are listed in the table of contents and the index prepared for this edition. The major additions or changes are indicated in the author's preface. Now something must be said about the method of this philosophy.

      Those who have undertaken the study of natural science can be divided into roughly three classes. There have been those who have endowed the individual species of things with specific occult qualities, on which—they have then...

    • Newton’s Preface to the Third Edition
      (pp. 400-402)
      Is. Newton

      IN THIS THIRD EDITION, supervised by Henry Pemberton, M.D., a man greatly skilled in these matters, some things in the second book concerning the resistance of mediums are explained a little more fully than previously, and new experiments are added concerning the resistance of heavy bodies falling in air. In the third book, the argument proving that the moon is kept in its orbit by gravity is presented a little more fully; and new observations, made by Mr. Pound, on the proportion of the diameters of Jupiter to each other have been added. There are also added some observations of...

    • Definitions
      (pp. 403-415)

      lf the density of air is doubled in a space that is also doubled, there is times as much air, and there is six times as much if the space is tripled.bcase is the same for snow and powders condensed by compression or liquefaction, and also for all bodies that are condensed in various ways by any causes whatsoever. For the present, I am not taking into account any medium, there should be any, freely pervading the interstices between the parts of

      bodies. Furthermore, I mean this quantity whenever I use the term “body” or “mass” in the following...

    • Axioms, or the Laws of Motion
      (pp. 416-430)

      Every body perseveres in its state of being at rest or of movingauniformly straight forward,aexcept insofar asbitbis compelled to changecitscstate by forces impressed.

      Projectiles persevere in their motions, except insofar as they are retarded by the resistance of the air and are impelled downward by the force of gravity. A spinning hoop,dwhich has parts that by their cohesion continually draw one another back from rectilinear motions, does not cease to rotate, except insofar as it is retarded by the air. And larger bodies—planets and comets—preserve for a longer time both...

    • Book 1: The Motion of Bodies
      (pp. 431-630)

      The method of first and ultimate ratios, for use in demonstrating what follows

      Quantities, and also ratios of quantities, which inaany finite timeaconstantly tend to equality, and which before the end of that time approach so close to one another that their difference is less than any given quantity, become ultimately equal.

      If you deny this,blet them become ultimately unequal, andblet their ultimate difference be D. Then they cannot approach so close to equality that their difference is less than the given difference D, contrary to the hypothesis.

      If in any figureAacE,comprehended by the...

    • Book 2: The Motion of Bodies
      (pp. 631-790)

      The motion of bodies that are resisted in proportion to their velocity

      If a body is resisted in proportion to its velocity, the motion lost as a result of the resistance is as the space described in moving.

      For since the motion lost in each of the equal particles of time is as the velocity, that is, as a particle of the path described, then, by composition [or componendo], the motion lost in the whole time will be as the whole path. Q.E.D.

      Corollary. Therefore, if a body, devoid of all gravity, moves in free spaces by its inherent force...

    • Book 3: The System of the World
      (pp. 791-938)

      In the preceding books I have presented principles of philosophy that are not, however, philosophical but strictly mathematical—that is, those on which the study of philosophy can be based. These principles are the laws and conditions of motions and of forces, which especially relate to philosophy. But in order to prevent these principles from seeming sterile, I have illustrated them with some philosophical scholiums [i.e., scholiums dealing with natural philosophy], treating topics that are general and that seem to be the most fundamental for philosophy, such as the density and resistance of bodies, spaces void of bodies, and the...

    • General Scholium
      (pp. 939-946)

      The hypothesis of vortices is beset with many difficulties. If, by a radius drawn to the sun, each and every planet is to describe areas proportional to the time, the periodic times of the parts of the vortex must be as the squares of the distances from the sun. If the periodic times of the planets are to be as the 3/2 powers of the distances from the sun, the periodic times of the parts of the vortex must be as the 3/2 powers of the distances. If the smaller vortices revolving about Saturn, Jupiter, and the other planets are...

  6. Index
    (pp. 947-966)
  7. Back Matter
    (pp. 967-967)