J. Bruce Brackenridge is Alice G. Chapman Professor of Physics at Lawrence University.

The Key to Newton’s Dynamics

By J. Bruce Brackenridge

University of California Press

Copyright © 1996 J. Bruce Brackenridge
All right reserved.
ISBN: 9780520202177
One
A Simplified Solution:
The Area Law, the Linear Dynamics Ratio, and the Law of Gravitation

Isaac Newton’s Philosophiae Naturalis Principia Mathematica (The mathematical principles of natural philosophy), hereafter referred to as the Principia , 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 the Principia are 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 of a classic as a work that everyone wants to have read but that nobody wants to read. The essential core of the Principia , however, does not lie beyond the reach of any interested and open-minded individual who is willing to make a reasonable effort.

In 1693, Richard Bentley, a young cleric who was later to become Master of Newton’s college, wrote to ask Newton for advice on how to master the work. Newton suggested a short list of background materials, and then, concerning the Principia itself, advised Bentley to read only the first three sections in Book One (i.e., the first sixty pages of the four hundred pages that make up the first edition). These sections provide the theoretical background for the astronomical applications that Newton presented in Book Three and regarded as of popular scientific interest. In the introduction to Book Three, Newton repeated the advice that he had given to Bentley:

I had composed the third book in a popular method so that it might be read by many. But since those who had not sufficiently entered into the principles could not easily discern the strength of the consequences nor put aside long-held prejudices, I chose to rework the substance of that book into the form of propositions in the mathematical way, so that they might be read only by

those who had first mastered the principles. Nevertheless, I do not want to suggest that anyone should read all of these propositionswhich appear there in great numbersince they could present too great an obstacle even for readers skilled in mathematics. It would be sufficient for someone to read carefully the definitions, laws of motion, and the first three sections of the first book; then let [the reader] skip to this [third] book.1

Newton’s sage advice to the general reader to concentrate on the first three sections of Book One of the Principia appeared in the first edition of 1687 and remained unchanged in the two revised editions published in 1713 and 1726, all during Newton’s lifetime. It is the third and final edition that has been reproduced in many subsequent editions and translated into many other languages. Because this third edition is readily available and because it is seen to represent Newton’s most fully developed views, it is almost exclusively taken as a basis for the study of Newton’s dynamics. The general reader, however, should not begin with this final edition and its many additions and revisions, but rather with the first edition and its relatively straightforward presentation.

In 1684, Newton sent to London a tract entitled On the Motion of Bodies in Orbit (On Motion) that was to serve as the foundation for the first edition of the Principia of 1687. This comparatively short tract presents in a clean and uncluttered fashion the basic core of Newton’s dynamics and its application to the central problem of elliptical motion. The brief set of definitions that appeared in On Motion was expanded in the Principia into a much larger set of definitions, laws, and corollaries. Further, the first four theorems and four problems in On Motion were expanded into fourteen lemmas and seventeen propositions in the Principia . (Theorem 1 of On Motion is Proposition 1 of the Principia but Problem 4 of On Motion is Proposition 17 of the Principia ). The expanded framework of numbered propositions by itself, however, does not tell the entire story. Even more troublesome for the general reader is Newton’s practice of adding new material to the old framework. Having established the expanded set of propositions and lemmas in the early draft of the first edition, Newton elected to hold to that framework as he inserted additional material in his published revised editions. Even in the preface to the first edition, Newton apologized to his readers for such insertions.

Some things found out after the rest, I chose to insert in places less suitable, rather than to change the number of the propositions as well as the citations. I heartily beg that what I have done here may be read with patience.2

After the publication of the first edition, Newton began work on a grand radical revision of the Principia in which many of the propositions would have been renumbered and retitled. In contrast to the single method of the first edition, Newton clearly presented three alternate methods of dy-

namic analysis in this projected revised scheme, each method set forth in a new proposition. Unfortunately, Newton never implemented this new scheme of renumbering the propositions and lemmas in the published revisions. If the challenge of renumbering the propositions and correcting the cross-references was too much in the limited first edition, then it was apparently overwhelming in the expanded revised editions. The new material added to the published revised editions simply was inserted into the old structure of the first edition. The third method of dynamic analysis, so clearly differentiated in the projected revision, was distributed throughout the theorems and problem solutions of the second and third sections of the published revisions. The reader of On Motion and, to a lesser extent, of the first edition is not faced with this difficulty. In those works, Newton clearly explicates his analysis with a single method applied uniformly to several problems; until the reader understands his original method and his unpublished restructuring, however, Newton’s additions to the much studied revised third edition appear as distractions rather than enrichments.

A Simplified Solution

The story of Isaac Newton and the apple is a familiar one. We have all seen the portrayal of an English gentleman who is sitting under a tree and is struck on the head by a falling apple. In a flash, he leaps to his feet and runs off shouting about the theory of universal gravitation. The story has its foundation in Newton’s own telling and is attested by a number of memoranda written by those close to him in his later years. The setting is the garden of his country home, the time is 1666, and Newton, a young man of twenty-four, is home after a few years at university. The apple tree that provides his inspiration stands in his front garden, and the fruit it bears is a yellow-green cooking apple called the Flower of Kent. One version of the story, told by Newton in his later years and recorded by an associate, John Conduitt, includes the following statement:

Whilst he was musing in a garden it came into his thought that the power of gravity (which brought an apple from the tree to the ground) was not limited to a certain distance from the earth but that this power must extend much farther than was usually thought. Why not as high as the moon said he to himself and if so that must influence her motion and perhaps retain her in her orbit, where upon he fell to calculating what would be the effect of that supposition.3

There is evidence that Newton made a calculation comparing the moon’s centrifugal force, a celestial event, with the local force of gravity, a terrestrial event. Since it is a calculation that could have been inspired by any falling object, why not an apple? That early calculation of 1666 did not

supply the mathematical basis for the general demonstration that the force necessary to maintain a planet in an elliptical orbit about the sun located at a focus of the ellipse is inversely proportional to the square of the distance between the sun and the planet (i.e., the law of universal gravitation). It was late in 1684, after Edmund Halley’s famous visit to Newton’s rooms at Cambridge University, before Newton gave anyone a copy of such a proofa proof which Newton claimed to have produced in 1679. The inspiration of the falling apple of 1666 required more than a decade to reach its final goal.

In 1684, Newton sent Halley a solution to the problem of planetary motion in the tract On Motion . That solution is expressed neither in the mathematics of classical geometry nor in the mathematics of contemporary differential and integral calculus. As such it is a challenge to the modern physicist as well as to the classical scholar. The outline of the solution, however, is not complicated; it is the details that provide the challenge. Newton adapted the linear kinematics of Galileo to the inertial dynamics of Descartes and determined the nature of the force necessary to maintain planetary motion as described by Kepler. If a constant linear acceleration A is acting on a body of mass m , then its displacement D is proportional to the constant acceleration A and the square of the time t (i.e., D = (1/2) At 2 ). If one adds to this simple kinematic relationship the dynamic relationship that the acceleration A is proportional to the force F (i.e., F = mA ), then the force F is directly proportional to the displacement D and inversely proportional to the square of the time t (i.e., F = (2m ) D / t 2 ). Newton’s genius manifests itself in adapting this simple proportional relationship of constant rectilinear force, distance, and time to the more complex problem of the nature of the planetary force, which is not constant. Newton’s unique contribution was the assumption that the variable force could be considered to be approximately constant over a very short period of time. The three elements that Newton generated to produce the solution can be set forth quite simply: first, the relationship that expresses the time in terms of the area; second, the relationship that expresses the force in terms of the displacement and the time (and hence in terms of the area); and finally, the relationship that expresses the force necessary for planetary motion in terms of distance (i.e., the demonstration that the gravitational force is inversely proportional to the square of the distance).

Theorem 1

The first element is the law of equal areas in equal times, demonstrated in figure 1.1. If the force acting on a body is always directed to a fixed point S , then the time required to travel from point P to point Q is proportional to the shaded area SPQ . If successive areas are generated in equal

Figure 1.1
If a body moves from point P to point Q under a centripetal
force directed toward the fixed point S , then the shaded area
SPQ is proportional to the time.

Figure 1.2
The force F required to maintain any orbit APQ about a
center of force S is proportional to the displacement QR and
inversely proportional to the area SQP .

times, then the areas swept out by the line from the body to the center of force are equal. This relationship was first recognized by the astronomer Johannes Kepler in 1609, but it was not until after 1679 that Newton demonstrated its general application to any motion under any force directed toward a fixed center. The area law is the link that was missing in Newton’s earlier analysis of motion and it is the key element in his celestial dynamics; it appears as Theorem 1 in the 1684 tract On Motion and as Proposition 1 in the 1687 Principia (see chapter 4 for details).

Theorem 3

The second element is the basic relationship that I have elected to call the “linear dynamics ratio.” Figure 1.2 is similar to Newton’s diagram for Theorem 3 in On Motion and for Proposition 6 in the Principia . The line RPZ is the tangent to the curve APQ . If no force acted on a moving body at

point P , then the body would continue in a given time interval along the tangent line to the point R . Because a force does act continuously on the body, however, it moves instead to the point Q . The displacement QR represents the deviation of the body from the tangential path PR due to the action of the force. Galileo had demonstrated in his experiments with inclined planes that for motion under a given constant force, the distance traveled is proportional to the square of the time. Newton assumes that as the point Q shrinks back to the point P , then the force can be treated as if it were constant. Thus, the distance QR is proportional to the square of the time and to the magnitude of the force at point P , or what is equivalent, the force is directly proportional to the distance QR and inversely proportional to the square of the time. From Theorem 1, the time is proportional to the triangular area SPQ and thus can be expressed in terms of the altitude QT and the base SP . The result is that the force F at point P can be expressed as follows:

The challenge is to express the ratio QR/QT 2 in terms of SP and constants of the orbital figure, and hence to express the linear dynamics ratio QR /(QT 2 W SP 2 ), and thus the force, in terms of the radial distance SP (see chapter 2 for a review and chapter 4 for a detailed discussion of this theorem).

Problem 3

The third element is a demonstration by Newton of a relationship between portions of an ellipse. Figure 1.3 is a drawing of a planetary ellipse APQ with a focus at point S . The line LSL drawn through the focus S and perpendicular to the major diameter of the ellipse is called the latus rectum L . Newton demonstrates in Problem 3 and in Proposition 11 that as the point Q shrinks back to the point P , the ratio QR / QT 2 becomes equal to the reciprocal of the latus rectum L , which is a constant for a given ellipse. Thus, the force can be obtained quite simply from the linear dynamics ratio above:

This result states that the force required to maintain a planet in an elliptical orbit about the sun located at a focus of the ellipse is proportional to the inverse square of the distance between the planet P and the sun S . Thus is demonstrated the mathematical basis for the law of universal gravitation, the essence of celestial interactions, which Newton provides for future astronomers and physicists (see chapter 2 for a review and chapter 5 for a detailed discussion of this problem).

Figure 1.3
As point Q shrinks back to point P , the ratio QT 2 /QR becomes equal to the
line LSL , which is a constant (the latus rectum ) for a given ellipse.

The details of the demonstrations of the relationships above are more demanding than is evident in this verbal gloss. Taken step by step, however, the analysis will become clear to the reader. At times Newton makes analytical leaps that for him are obvious and it is then my duty to supply the intervening steps. Thus, it is the number of steps rather than the size of any single step that offers the challenge. The reward for the patient reader is an insight into the solution of the problem of planetary motion, a problem that challenged astronomers for millennia. That solution is now universally held to have provided a major turning point in astronomy and natural philosophy in the late seventeenth century.

The Reception

Professional scholars, however, did not greet the publication of the 1687 Principia with unreserved praise. The dominant figure in seventeenth-century natural philosophy was the French scholar Reni Descartes, whose mechanical description of planets carried in a swirling vortex of celestial ether provided the model for many other natural philosophers. Two other outstanding figures in European mathematics and natural philosophy at the time of the publication of the first edition of the Principia were the Dutch scholar Christiaan Huygens and the German scholar Wilhelm Gottfried Leibniz. Both felt that Newton’s description of the mathematical

nature of gravitational force had failed to address the fundamental question of the physical cause of the force. It would appear that Huygens accepted the inverse-square law as a genuine discovery, although he believed that its cause remained to be investigated. Leibniz initially praised the 1687 Principia as one of the most important works of its kind since Descartes. He criticized Newton, however, for his rejection of Cartesian vortices and for his failure to provide an alternate physical cause for the gravitational attraction. In England, the astronomer and mathematician Edmund Halley served as the editor of the 1687 Principia and it was published under the imprimatur of the Royal Society. Even with this auspicious beginning, it was not without controversy that the Principia was finally published. The English scientist Robert Hooke claimed priority for the discovery of the inverse square nature of the gravitational force, a claim that Newton vehemently rejected. Despite individual reservations, the overall reception by the scholarly community was positive, and Newton established himself as one of the leading mathematicians of Britain and Europe. As one modern scholar of Leibniz’s work put it, “Already in . . . 1695, Leibniz had abandoned the project of presenting a theory capable of competing with Newton’s. Despite his subtle philosophical and theological objections, in the eighteenth century Leibniz had left Newton master of celestial mechanics.”4

As the scholarly reputation of the Principia grew, even those who professed little or no mathematical ability came to pay homage. The English philosopher John Locke, in exile in Holland at the time of the book’s publication, obtained assurance from Huygens that the mathematical propositions of the Principia were valid and then applied himself to understanding Newton’s conclusions. Locke eventually referred to “the incomparable Mr. Newton” in the preface to his An Essay Concerning Human Understanding . The French writer and philosopher Frangois Voltaire waxed even more eloquent when he drew the following comparison between Newton and the German astronomer Johannes Kepler: “Before Kepler, all men were blind. Kepler had one eye, Newton had two.” The English poet Alexander Pope’s often quoted heroic couplet, published shortly after Newton’s death, revealed even more forcefully the popular view that Newton and the Principia opened doors that had long been closed: “Nature, and Nature’s Laws lay hid in night:/God said, Let Newton be! and all was light.” In the dedicatory poem to the first edition of the Principia , Edmund Halley reflected on Newton’s “unlocking the treasury of hidden truth” and concluded that “nearer to the Gods no mortal may approach.”

As the eighteenth century drew to a close, not everyone continued to praise the new world that appeared in Newton’s work. Figure 1.4 is an early nineteenth-century caricature of Newton by the philosopher-poetartist William Blake, who, putting imagination above reason, reacted negatively to the eighteenth-century veneration of Newton. In the portrait,

Figure 1.4
William Blake’s portrait of Isaac Newton.
Courtesy Tate Gallery, London.

triangles abound as the symbol of the geometrical and mathematical mentality that Blake opposed. Newton holds a triangular compass as he draws a triangular figure on the parchment, his fingers make triangles with the object in his hand, his legs form triangles with each other and with the rock on which he sits, the muscles of his body take on geometric forms that defy anatomical description, and triangular eyes scheme as they look down a triangular nose at geometric plans that triangular hands create below. For Blake, Newton symbolized the eighteenth-century regard for human reason that placed God above and separate from women and men, while Blake regarded human imagination as the essential divine quality by which God was made manifest.5

Neither Newton nor his Principia deserves the extreme judgments of Pope and Blake; the work is ranked as one of the major intellectual achievements of Western culture. The Enlightenment of the eighteenth century and the Romanticism of the nineteenth century both have their roots in the acceptance or rejection of the new worldview that paid homage to Newton’s scientific writings. Just as his Optics provided a model for the experimental method, so his Principia laid the foundations for the theoretical method. It was, in fact, the wave of the future.

Continues…
Excerpted from The Key to Newton’s Dynamicsby J. Bruce Brackenridge Copyright © 1996 by J. Bruce Brackenridge. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.