Newton's mature dynamics: Revolutionary or reactionary?

References

• My first discussion of the ‘alternate dynamics ratio’ was presented in Moscow at the International Conference on ‘Newton and Science’ in October 1987. That presentation provided the basis for the short paper, Brackenridge J. Bruce Newton's Mature Dynamics and the Principia: A Simplified Solution to the Kepler Problem submitted to Historia Mathematica in November, 1987. The present paper provides a general introduction to the subject and extends the analysis of the role played by the alternate ratio in Newton's dynamics. In particular, see the discussion of the manuscript shown in figure 14.
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• Brackenridge , J. Bruce . 1982 . Kepler, Elliptical Orbits, and Celestial Circularity: A Study in the Persistence of Metaphysical Commitment . Annals of Science , 39 : 117 – 142 . Part I Part II, 265–95. Part I is concerned with the role of the circle for Kepler in geometry, music, and astrology. Part II continues with its application to astronomy. It concludes with the demonstration that elliptical orbits do not, for Kepler, destroy the principle of celestial circularity.
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• Newton , Isaac . 1727 . Mathematical Principles of Natural Philosophy , 3rd edn xvii – xvii . London All page references are to the English translation by Motte (London, 1729) and edited by F. Cajori, 2 vols (Berkeley, 1971). I have used this edition where possible, because it is readily available in paperback. As with any translation and edition, there are shortcomings, and other editions have also been consulted and variations noted.
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• Simpson , Thomas K. 1976 . “ Newton and the Liberal Arts ” . In The College , 1 – 11 . Annapolis : St John's College .
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• Simpson , Thomas K. 1976 . “ Newton and the Liberal Arts ” . In The College , 2 – 3 . Annapolis : St John's College .
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• Newton . 1727 . Mathematical Principles of Natural Philosophy , 3rd edn xvii – xvii . London
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• One expands the terms SP, QT, and QR in a power series, as functions of the angle dϑ = angle QSP. From the diagram for my reconstruction of Proposition 6 (see Figure 1) one obtains SP = r, QT = (SQ) sin (dϑ) and QR = PU - PT = PU - (ST - SP), where ST = (SQ) cos (dϑ) and PU = (RU) cot (UPR) = (- QT) cot (SPR). Expanding and retaining terms in the differential angle dϑ to second order, one obtains the orbital equation. See Whiteside The Mathematical Papers of Isaac Newton Cambridge 1967–74 42 43 8 vols Also, note 31, Whiteside (footnote 11), p. 123. There is a difference in a sign between the two versions that is resolved by making the radius an increasing function of angle, as in my Figure 1.
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• Clarke , John . 1730 . A Demonstration of Some of the Principal Sections of Sir Isaac Newton's Principles of Natural Philosophy 164 – 164 . London Plate VII, opposite
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• Herivel , John . 1965 . The Background to Newton's Principia 12 – 12 . Oxford
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• Herivel , John W. 1964 . Newton's First Solution to the Problem of Kepler Motion . British Journal for the History of Science , 2 : 351 – 354 . (p. 352). The choice of 1679 as the date of Newton's adoption of celestial rectilinear inertia has been challenged by some and defended by others. See, for example, note 9 of the review by Whiteside (footnote 14).
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• Whiteside , D.T. 1970 . The Mathematical Principles Underlying Newton's Principia Mathematica . The Journal of the History of Astronomy , 1 : 116 – 138 . (p. 122). ‘Here again we slide over the subtlety that the deviation RQ is not in general a straight line, and may only be considered so when it is—as here, fortunately—assumed to be of second-order infinitesimal magnitude.’ See footnotes 13 and 15 for a further discussion of the deviation RQ as a second-order infinitesimal.
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• Herivel , John W. 1970 . “ Newton's Achievements in Dynamics ” . In The Annus mirabilis of Sir Isaac Newton 1666–1966 Edited by: Palter , R. 117 – 117 . Cambridge, Mass. in As both Aiton and Whiteside have pointed out, however, this first-order infinitesimal parabolic are in Proposition 6 is made up of an infinite sum of infinitesimal straight lines, which must be second-order infinitesimals. See the references in footnotes 11 and 15 and, most recently, the paper by E. J. Aiton, ‘Polygons and Parabolas: Some Problems Concerning the Dynamics of Planetary Orbits in the Seventeenth Century’, presented at the Conference on ‘Leibniz, Newton, and their Disciples’ at Harvard and Tufts Universities, 4–6 March 1988 (to be published in Centaurus). See also E. J. Aiton, The Vortex Theory of the Planetary Motions (London and New York, 1972), pp. 104–5.
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• Whiteside , D.T. 1966 . Newtonian Dynamics . History of Science , 5 : 109 – 110 .
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• Whiteside , D.T. 1966 . Newtonian Dynamics . History of Science , 5 : 112 – 112 . This curved deviation is the first-order infinitesimal are discussed in footnote 13.
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• Newton , Isaac . 1687 . Philosophiae Naturalis Principia Mathematica 44 – 44 . London English translation by Mary Ann Rossi.
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• Newton , Isaac . 1687 . Philosophiae Naturalis Principia Mathematica 32 – 32 . London
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• Newton . 1727 . Mathematical Principles of Natural Philosophy , 3rd edn 21 – 21 . London
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• Whiteside , D.T. 1967–74 . The Mathematical Papers of Isaac Newton 133 – 133 . Cambridge 8 vols VI, note 90
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• See, for example, Grosholz E. Some Uses of Proportion in Newton's Pricipia, Book 1: A Case Study in Applied Mathematics Studies in History and Philosophy of Science 1987 18 209 220 ‘It [QR] is also directly proportional to t 2 by Lemma 10.’ The reference given is to the third edition where, in Proposition 6, Lemma 10 is clearly stated as the second choice.
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• Newton . 1687 . Philosophiae Naturalis Principia Mathematica 48 – 48 . London
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• See Whiteside The Mathematical Papers of Isaac Newton Cambridge 1967–74 III 151 195 8 vols for an extended discussion of Newton's considerable work on curvature.
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• Newton . 1727 . Mathematical Principles of Natural Philosophy , 3rd edn 37 – 37 . London
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• Newton . 1727 . Mathematical Principles of Natural Philosophy , 3rd edn 48 – 48 . London
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• Newton . 1727 . Mathematical Principles of Natural Philosophy , 3rd edn 49 – 49 . London
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• Westfall , Richard S. 1980 . Never at Rest 416 – 416 . Cambridge
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• Westfall , Richard S. 1980 . Never at Rest 106 – 106 . Cambridge note 4
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• Whiteside . The Mathematical Papers of Isaac Newton Vol. VI , 8 vols see in particular note 86, 130 and note 25, pp. 548–50. In note 5 (p. 597), the alternate dynamics ratio is called the ‘“Moiverean” measure of central force’ and provides, in note 10 (p. 598) the basis for a fluxional version of the dynamics ratio.
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• Brougham , H. and Routh , E.J. 1855 . Analytical View of Sir Isaac Newton's Principia London All page references are to the modern reprint (New York, 1972), pp. 43–8.
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• Newton . Mathematical Principles of Natural Philosophy , 3rd edn 49 – 51 . London
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• Brackenridge , J. Bruce . 1985 . “ The defective diagram as an analytical device in Newton's Principia ” . In Religion, Science, and Worldview Edited by: Osler , M.J. and Farber , P.L. 61 – 93 . London in
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• Frost , Percival . 1883 . Newton's Principia, First Book, Sections I., II., III. 95 – 96 . Cambridge All page references are to the fourth edition.
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• Newton did produce a version of the solution that I presented above but it never appeared in print. In what Whiteside calls ‘more radical restructuring’, Newton proposed in the early 1690s to revise the entire logical and expository framework of this dynamical section of the Principia. In all three published editions, Lemma 12 simply makes a statement concerning parallelograms in a given ellipse. In a proposed drastic revision of Lemma 12 (with four corollaries) Newton develops the relationships that I present as step II above. Further, in a new Proposition 10 he applies these results to the general conic. ‘Let a body move in the perimeter of the conic PQ: there is required the centripetal force tending to any given point S.’ See Whiteside The Mathematical Papers of Isaac Newton Cambridge 1967–74 VI 583 589 8 vols The alternative dynamics ratio is the measure of the force and the general solution is applied in Corollary 1 to a force directed to the conic centre and in Corollary 2 to a force directed to the conic focus.
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• Herivel , J. 1964 . Newton's First Solution to the Problem of Kepler Motion . British Journal for the History of Science , 2 : 353 – 353 . note 8
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• Whiteside . 1967–74 . The Mathematical Papers of Isaac Newton Vol. VI , 548 – 548 . Cambridge 8 vols ‘In a preliminary recasting on Add. 3965.6:37r Newton toyed with the notion of making this derived property basic as a variant “Prop.VI.Theor.V.”.’ See also note 1, p. 538. ‘In 1713 he [Newton] also concurrently drafted a number of tentative further improvements which never found their way into print.’
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• Manuscript Add. 3965.6:37r , Cambridge : University Library . Translation by Mary Ann Rossi
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• Ation , Eric J. 1964 . The Inverse Problem of Central Forces . Annals of Science , 20 : 89 – 89 .
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• Whiteside . 1967–74 . The Mathematical Papers of Isaac Newton Vol. VI , 146 – 149 . Cambridge 8 vols note 124
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• Frost . 1883 . Newton's Principia, First Book, Sections I., II., III. 202 – 202 . Cambridge
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• Brougham . 1855 . Analytical View of Sir Isaac Newton's Principia 63 – 63 . London All page references are to the modern reprint, New York, 1972
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