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The Key to Newton's Dynamics: The Kepler Problem and the Principia

J. Bruce Brackenridge
with English translations from the Latin by Mary Ann Rossi
Copyright Date: 1995
Pages: 330
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  • Book Info
    The Key to Newton's Dynamics
    Book Description:

    While much has been written on the ramifications of Newton's dynamics, until now the details of Newton's solution were available only to the physics expert.The Key to Newton's Dynamicsclearly explains the surprisingly simple analytical structure that underlies the determination of the force necessary to maintain ideal planetary motion. J. Bruce Brackenridge sets the problem in historical and conceptual perspective, showing the physicist's debt to the works of both Descartes and Galileo. He tracks Newton's work on the Kepler problem from its early stages at Cambridge before 1669, through the revival of his interest ten years later, to its fruition in the first three sections of the first edition of thePrincipia.

    eISBN: 978-0-520-91685-2
    Subjects: General Science

Table of Contents

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  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
    (pp. vii-x)
    (pp. xi-xiv)

    • ONE A Simplified Solution: The Area Law, the Linear Dynamics Ratio, and the Law of Gravitation
      (pp. 3-11)

      Isaac Newton’sPhilosophise Naturalis Principia Mathernatica(The mathematical principles of natural philosophy), hereafter referred to as thePrincipia, justifiably occupies a position as one of the most influential works in Western culture, but it is a work more revered than read. Three truths concerning thePrincipiaare held to be self-evident: it is the most instrumental, the most difficult, and the least read work in Western science. A young student who passed Newton on the streets of Cambridge is reported to have said, “There goes the man who writ the book that nobody can read.” It fits Mark Twain’s definition...

    • TWO An Overview of Newton’s Dynamics: The Problem of the Planets and the Principia
      (pp. 12-39)

      The authority of thePrincipiastemmed initially from its ability to provide a solution to one of the major challenges that astronomers faced, the problem of the planets. The vast majority of stars maintain a fixed position relative to one another. The planets, however, move against the background of the constellations (the word “planet” comes from the Greek word meaning “wanderer”). Their appearances, retrograde loops, and disappearances have provided a continuous challenge to astronomers for over three thousand years. An early version of the problem of the planets was set for astronomers by Plato, who challenged them to find the...

    • THREE Newton’s Early Dynamics: On Uniform Circular Motion
      (pp. 40-66)

      Isaac Newton was born on 25 December 1642, in the manor house of Woolsthorpe, a very small village seven miles south of the town of Grantham. Fifty miles south of Grantham lies the university town of Cambridge, the center of Newton’s university training and scholarly activities. Seventy miles south of Cambridge lies London, the location of Newton’s later professional life and of his death on 27 March 1727. Newton exerted an influence, however, that extended well beyond the miles between Grantham and London and lasted much longer than the period of eighty-four years between his birth and death; his published...


    • FOUR The Paradigm Constructed: On Motion, Theorems 1, 2, and 3
      (pp. 69-94)

      Newton’s thoughts on dynamics were awakened late in 1679 by a series of letters from Robert Hooke, who had recently become secretary of the Royal Society and was attempting to revive Newton’s interest in contributing to its proceedings. The Royal Society had been founded in London in 1661, and its meetings and publications served to inform the intellectual community of progress in natural philosophy. Newton had been elected to membership in 1672 but had threatened to resign the following year because of criticism of his paper on the theory of colors. Hooke’s letter of conciliation of November 1679 opened with...

    • FIVE The Paradigm Applied: On Motion, Problems 1, 2, and 3
      (pp. 95-118)

      Theorem 3 in Newton’s tractOn Motionprovides the basic paradigm for solutions to direct problems: Given the orbit and the location of the force center, find the force. As Newton put it, “specifically the solidSP² XQT² /QRmust be computed.” He concluded Theorem 3 with the statement, “We shall give examples of this point in the following problems.” In this tract, he elected to solve three examples of direct problems. The most important example was Problem 3, the Kepler problem: find the centripetal force required to maintain planetary elliptical motion about a center of force located at...

    • SIX The Paradigm Extended: On Motion, Theorem 4 and Problem 4
      (pp. 119-138)

      Contemporary textbooks in astronomy or physics attribute three laws governing planetary motion to the work of Kepler: the first law states that a planet moves in an elliptical path about the sun located at a focus of the ellipse; the second law states that a line joining the sun and a planet sweeps out equal areas in equal times; and the third law states that the period of a planet about the sun is proportional to the three-halves power of the transverse axis of the elliptical orbit. In Theorem 1 ofOn Motion,Newton demonstrates that the second law is...


    • SEVEN The Principia and Its Relationship to On Motion: A Reference Guide for the Reader
      (pp. 141-165)

      Just as the seven problems and four theorems of the tractOn Motionfar exceeded Halley’s request in 1684 for a solution to the single problem of elliptical motion about a focal center of force, so the published text of the first edition of thePrincipiain 1687 far exceeded the contents of the original tract of 1684. Newton divided the work into three books: the first book is devoted to the analysis of motion in a nonresistive medium, the second book to the analysis of motion in resistive media, and the third book to the analysis of the data...

    • EIGHT Newton’s Unpublished Proposed Revisions: Two New Methods Revealed
      (pp. 166-181)

      Following the publication of the first edition of thePrincipiain 1687, Newton began to make corrections in his working copy of the text and to propose revisions and additions for a possible second edition. When, twentysix years later, in 1713, the second edition was published, many of these hand-written revisions were incorporated. Several revisions, however, never appeared in printed form. Of particular interest are the unpublished revisions of the fundamental dynamics of Sections 2 and 3 of Book One. These revisions, if published, would have provided a dramatically different format for these fundamental sections. They have been masterfully reconstructed...

    • NINE Newton’s Published Recast Revisions: Two New Methods Concealed
      (pp. 182-210)

      Two more editions of thePrincipiawere published during Newton’s lifetime: the second edition appeared in 1713, twenty-six years after the first edition, and the third edition appeared in 1726, just one year before Newton died. Neither of these revised editions displays the radical restructuring of the propositions and lemmas that Newton had shown to the mathematician David Gregory in 1694. That proposed restructuring was set aside and Newton elected to retain the formal structure of the 1687 edition. All the propositions and lemmas of the revised editions retain their original headings with only minimal verbal changes. Many of the...

    • TEN Newton’s Dynamics in Modern Mathematical Dress: The Orbital Equation and the Dynamics Ratios
      (pp. 211-222)

      Throughout this book I have attempted to view Newton’s creative process in terms of the dynamics and mathematics that preceded his analysis, rather than to view it with hindsight from a modern perspective. In this closing chapter, however, I reverse the procedure and express Newton’s dynamic measures of force in current mathematical notation. Contemporary textbooks in physics present a second-order differential equation called “Newton’s Second Law” in the familiar form ofF= ma. For motion in one dimensionx, this equation has only one component equation:F(x) =m(d²x/dt²). The force functionF(x) is equal to the product of the...

  8. APPENDIX An English Translation of Sections 1, 2, and 3 of Book One from the First (1687) Edition of Newton’s Mathematical Principles of Natural Philosophy
    (pp. 225-268)
  9. NOTES
    (pp. 269-288)
    (pp. 289-292)
    (pp. 293-296)
    (pp. 297-300)
  13. Back Matter
    (pp. 301-301)