<i>Newton's</i> Principia<i>: The Central Argument</i>

Densmore, Dana; Brackenridge, J. Bruce

doi:10.1063/1.881594

Cited by 11 publications

(5 citation statements)

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“…Outstanding among these authors are the astronomers Sir George Airy in 1834 and Sir John Herschel (son of William, the discoverer of Uranus) in 1849, as well as Henry Lord Brougham (formerly Lord Chancellor) and E. J. Routh, who wrote a very useful Analytical View of Sir Isaac Newton's Principia in 1855. Two recent books make a valiant effort to introduce the reader to the unfamiliar methods of the Principia: Brackenridge (1995) and Densmore (1995) cover only the proofs of Kepler's laws whereas Chandrasekhar (1995) discusses both Books I and III; see the essay review by Westfall in Isis (1996).…”

Section: B Philosophiae Naturalis Principia Mathematicamentioning

confidence: 99%

Moon-Earth-Sun: The oldest three-body problem

1998

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The daily motion of the Moon through the sky has many unusual features that a careful observer can discover without the help of instruments. The three different frequencies for the three degrees of freedom have been known very accurately for 3000 years, and the geometric explanation of the Greek astronomers was basically correct. Whereas Kepler's laws are sufficient for describing the motion of the planets around the Sun, even the most obvious facts about the lunar motion cannot be understood without the gravitational attraction of both the Earth and the Sun. Newton discussed this problem at great length, and with mixed success; it was the only testing ground for his Universal Gravitation. This background for today's many-body theory is discussed in some detail because all the guiding principles for our understanding can be traced to the earliest developments of astronomy. They are the oldest results of scientific inquiry, and they were the first ones to be confirmed by the great physicist-mathematicians of the 18th century. By a variety of methods, Laplace was able to claim complete agreement of celestial mechanics with the astronomical observations. Lagrange initiated a new trend wherein the mathematical problems of mechanics could all be solved by the same uniform process; canonical transformations eventually won the field. They were used for the first time on a large scale by Delaunay to find the ultimate solution of the lunar problem by perturbing the solution of the two-body Earth-Moon problem. Hill then treated the lunar trajectory as a displacement from a periodic orbit that is an exact solution of a restricted three-body problem. Newton's difficultly in explaining the motion of the lunar perigee was finally resolved, and the Moon's orbit was computed by a new method that became the universal standard until after WW II. Poincaré opened the 20th century with his analysis of trajectories in phase space, his insistence on investigating periodic orbits even in ergodic systems, and his critique of perturbation theory, particularly in the case of the Moon's motion. Space exploration, astrophysics, and the landing of the astronauts on the Moon led to a new flowering of celestial mechanics. Lunar theory now has to confront many new data beyond the simple three-body problem in order to improve its accuracy below the precision of 1 arcsecond; the computer dominates all the theoretical advances. This review is intended as a case study of the many stages that characterize the slow development of a problem in physics from simple observations through many forms of explanation to a high-precision fit with the data. [S0034-6861(98)00802-2] CONTENTS I. Introduction 590 A. The Moon as the first object of pure science 590 B. Plan of this review 591 II. Coordinates in the Sky 591 A. The geometry of the solar system 591 B. Azimuth and altitude-declination and hour angle 592 C. Right ascension-longitude and latitude 592 D. The vernal equinox 593 E. All kinds of corrections 593 F. The measurement of time 593 G. The Earth's rota...

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Section: B Philosophiae Naturalis Principia Mathematicamentioning

confidence: 99%

Moon-Earth-Sun: The oldest three-body problem

1998

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“…On the basis of this law a small object like the earth must move around a large object like the sun, rather than vice versa. It is surprisingly dif cult to nd a clear statement that the earth is a planet in Newton's Principia (Densmore 1995), although Newton repeats an argument from Kepler, that the area law works with respect to the Sun, but not with respect to the earth (Newton [1729(Newton [ ]1934. With Newton, then, Copernicanism becomes the thesis that the planets, including the earth, are bound to the sun by gravitational forces which oblige them to move on elliptical paths through free space.…”

Section: Our Copernicusmentioning

confidence: 96%

Constructing Copernicus

Barker

2002

Perspectives on Science

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“…It is clear that to understand Newton's procedure one has to consult Lemma 3 which is given by him as justification for his limit arguments. Remarkably, however, neither Whiteside (MP 6: footnote 19) (Whiteside 1991) nor Aiton (Aiton 1989) commented on this important Lemma, which was also neglected by one of their critics (Erlichson 1992), and by other recent commentators of the Principia (Brackenridge 1995) (Chandrasekhar 1995) (Cohen 1999) (Densmore 1995). In Lemmas 2 and 3 and its corrolaries, Newton described how the area bounded by a given curve and a line and the length of the curve can be approximated by a sequence of parallelograms.…”

Section: Introductionmentioning

confidence: 99%

“…Cohen warmly endorsed Whiteside's analysis (Cohen 1999) while N. Guicciardini in his new book Reading the Principia questioned whether Newton's limit arguments in Prop.1 are well grounded, acknowledging, however, dissenting views (Guicciardini 1999). Other recent authors discussing the Principia either neglected to examine the validity of Newton's limit arguments (Brackenridge 1995) (Chandrasekhar 1995), or failed to understand them (Densmore 1995).…”

Section: Introductionmentioning

confidence: 99%

Kepler's area law in the Principia: filling in some details in Newton's proof of Proposition 1

Nauenberg

2003

Historia Mathematica

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During the past 25 years there has been a controversy regarding the adequacy of Newton's proof of Prop. 1 in Book 1 of the Principia. This proposition is of central importance because its proof of Kepler's area law allowed Newton to introduce a geometric measure for time to solve problems in orbital dynamics in the Principia. It is shown here that the critics of Prop. 1 have misunderstood Newton's fundamental limit argument by neglecting to consider the justification for this limit which he gave in Lemma 3. We clarify the proof of Prop. 1 by filling in some details left out by Newton which show that his proof of this proposition was adequate and well grounded.Rigor merely sanctions the conquests of sound intuition, Jacques Hadamard

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Newton's Principia: The Central Argument

Cited by 11 publications

References 0 publications

Moon-Earth-Sun: The oldest three-body problem

Moon-Earth-Sun: The oldest three-body problem

Constructing Copernicus

Kepler's area law in the Principia: filling in some details in Newton's proof of Proposition 1

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