Newton’s early computational method for dynamics. (English) Zbl 0801.01005

On the margin of a letter to Hooke of December 13, 1679, Newton drew a diagram for the orbit of a body in a constant central field. Since the first publication of this letter in 1929, several Newton scholars have pointed out that the diagram contains a considerable error, which has long been attributed to some failure in Newton’s method of computation. In this paper, it is demonstrated with an iterative method based on Newton’s mathematical development of the concept of curvature, that Newton’s computation of the orbit was correct: he made an error while drawing the figure representing the orbit, but not in calculating it.


01A45 History of mathematics in the 17th century
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[1] The Correspondence of Isaac Newton vol. II, 1676–1687, edited by H. W.Turnbull (Cambridge University Press, Cambridge, 1960), pp. 304–306.
[2] Reference [1], pp. 307–308.
[3] The termindivisibles was used byBonaventura Cavaliere in the title of his bookGeometria Indivisibilibus Continuorum published in 1635, and later byJohn Wallis in hisArithmetica Infinitorum (Oxon, 1656) which is devoted to the integration of curves bythe method of indivisibles.Newton’s early mathematical studies were also influenced by the work ofWallis, seeThe Mathematical Work of John Wallis byJ. F. Scott, (London, 1938)
[4] J.Pelseneer,Une lettre inédite de Newton, Isis12, 237–254 (1929). · JFM 55.0012.11
[5] J.Lohne,Hooke versus Newton, Centaurus7, 6–52 (1960).Newton’s diagram, Fig. 1, was first published in this paper. · Zbl 0213.00402
[6] R. S.Westfall,Force in Newton’s Physics, (American Elsevier, NY, 1974) pp. 427–429.
[7] The Mathematical Papers of Isaac Newton vol. VI, 1684–1691, ed.D. T. Whiteside (Cambridge University Press, 1974) pp. 12, footnote (36).
[8] H.Erlichson,Newton’s 1679/80 solution of the constant gravity problem, American Journal of Physics59, 728–733 (1990).
[9] A.Koyré,An Unpublished Letter of Robert Hooke to Isaac Newton, Isis43, 312–337 (1952). · Zbl 0048.24301
[10] V. I. Arnol’d,Huygens & Barrow, Newton & Hooke (Birkhaüser, 1990) p. 19.
[11] Ref. [7], pp. 149–153;D. T. Whiteside,The Preliminary Manuscripts for Newton’s 1687Principia, 1684–1685 (Cambridge University Press, 1989), pp. 89–91. In a revised treatise ofDe Motu, written probably in the winter or early spring of 1684–1685,Newton discussed again the problem of orbital motion ”should the centripetal force act uniformly at all distances ...” in the Scholium (which is crossed out and did not appear in this form in the Principia) following proposition XII, problem VII. He now gives explicitly the magnitude of the angle between perigee and apogee to be about 110{\(\deg\)}, in close agreement with the angle 107{\(\deg\)} which I obtained (see section III) from his diagram of 1679, before knowing the existence of this manuscript.Newton also gives the corresponding angle for the case of a force which varies ”reciprocally proportional to the distance from the center” as ”something like 136{\(\deg\)} or 140{\(\deg\)}”. This is a rough numerical approximation, since in this case the maximum angle is 180{\(\deg\)}/27.3{\(\deg\)}. For discrete steps, the curvature method will tend to give a somewhat larger value of this angle, because the radius of curvature increases monotonically until the minimum distance to the center of force is reached. He continues with the statement And universally, if the centripetal force were to decrease in less than the doubled ratio of the distance from the centre, the body would return to its auge [apogee] before it could complete a circle, while if that force were to decrease in a greater than doubled but less than tripled ratio, the body would complete a circle before it could return to its auge. If, however, the same force were to decrease in the tripled or more than tripled ratio of the distance from the centre, and the body should begin to move in a curve which at the start of motion intersected the radius AS at right angles, this, were it once to begin to descend, would continue to descent right to the centre, while if it were once to begin to ascend, it would go off to infinity. In this Scholium,Newton describes the radial dependence of the forces which he had not revealed toHooke in 1679. The only other significant change is that he now refers to force as ”centripetal” rather than ”centrifugal”. His computational results for the case of constant force give almost the same result for the angle between perigee and apogee as in the diagram of 1679. AgainNewton does not reveal in this treatise what computational method he applied to obtain these results, nor has a diagram been found for the orbit in a constant central force field, but there is no reason to suppose that these are not the same as the calculations he sent toHooke in 1679, applying, as we suggest, the curvature method.
[12] In Corollary II of Proposition XLIV, Theorem XIV, Book I of thePrincipia,Newton shows that the central forcef acting on a body moving on the ”revolving ellipse” \(Y_{CH_4 }\) (39) is given by \(f = \frac{{F^2 }}{{r^2 }} + \frac{{L\left( {G^2 - F^2 } \right)}}{{r^3 }}\) (40) wherev=F/G. In this case the angle between perigee and apogee is 180{\(\deg\)}/v. This result gives a powerful approach to obtain approximate orbits for general attractive central forces when the orbit is nearly circular,i.e. {\(\epsilon\)}. For example, the forcef’=c/rn can be approximated by Eq. (40) by settingf’=f anddf’/dr=df/dr arr=L. This givesn=3 2, a result whichNewton derives in Proposition XLV, Problem XXXI. For a constant forcen=0,v=, and the angle between perigee and apogee in this case is 180{\(\deg\)}/03{\(\deg\)}55’23”. For a force proportional to the inverse of the distancen=1,v=, and the corresponding angle is 180{\(\deg\)}/27{\(\deg\)}16’45”. The approximate numerical values for the angles are those given byNewton.
[13] The Mathematical Papers of Isaac Newton vol. I, 1664–1666, ed.D. T. Whiteside (Cambridge University Press, 1967) pp. 252–255.
[14] J. G. Yoder, Unrolling Time (Cambridge University Press, 1988) pp. 71–115.
[15] The Mathematical Papers of Isaac Newton vol. III, 1670–1673, ed.D. T. Whiteside (Cambridge, 1969) pp. 151–159.
[16] D. T.Whiteside,How forceful has a Force Proof to be? Newton’s Principia, Book 1: Corollary 1 to Propositions 11–13 Physis (Rivista Internazionale di storia della scienza) vol.XXVIII 727–749 (1991). · Zbl 0756.01019
[17] J. B. Brackenridge,The critical role of curvature in Newton’s developing dynamics inThe Investigation of difficult things: Essays on Newton and the History of the Exact Sciences in honour of D. T. Whiteside edited byP. M. Harman &Alan E. Shapiro (Cambridge University Press, 1992) pp. 231–260.
[18] B.Pourciau,Radical Principia, Archive for History of Exact Sciences (1992)44, pp. 331–363. · Zbl 0763.01010
[19] J. Herivel,The Background to Newton’s Principia, A Study of Newton’s Dynamical Researches in the Years 1664–84, p. 130 (Oxford, 1965). · Zbl 0134.00402
[20] Christian Huygens,De Vi Centrifuga, in Oeuvres Complètes de Christiaan Huygens XVI, 253–301 (The Hauge, 1929).
[21] D. T.Whiteside,The Prehistory of the Principia from 1664 to 1668, Notes Rec. R. Soc. Lond.45, pg. 15 (1991). · Zbl 0716.01010
[22] Corollary III of Proposition VI, Theorem V, Book I of the third edition of thePrincipia ends with the statement that For PV isQP2/QR. In the diagram in thePrincipia corresponding to this proposition,PV=2{\(\rho\)}sin({\(\alpha\)})QP=v{\(\delta\)}t andQR=(1/2)a{\(\delta\)}t2, where {\(\delta\)}tP{\(\times\)}QT in accordance withKepler’s law of areas.Newton does not provide any proof for this result, which corresponds to Eq. (2) in the form {\(\rho\)}sin({\(\alpha\)})=v2a (14) It should be pointed out that this expression does not depend on an application ofKepler’s law of areas, because the time interval{\(\delta\)}t cancels. Corollary III is absent from the first edition of thePrincipia.
[23] In Corollary III of Proposition VI, Theorem V, book I of the third edition of thePrincipia,Newton introduced a second measure for force (acceleration), using explicitly the conservation of angular momentuml, stating that ”...the centripetal force will be inversely as the solidSY2PV...”. In our notation.SY=rsin({\(\alpha\)})=l/v is the component of the position vector along the curvature vector, andPV=2{\(\rho\)}sin({\(\alpha\)}) is the component of the curvature vector along the position vector,i.e. the chord of the osculating circle which passes through the center of force. Hence,Newton’s second measure of force corresponds (apart from a factor 2l 2) toa=l 2/(r2{\(\rho\)}sin3({\(\alpha\)})), which is the same as Eq. (22).
[24] A.Armitage,Borell’s Hypothesis and the Rise of Celestial Mechanics, Annals of Science,5, 342–351 (1950).
[25] D. T.Whiteside,Newton’s early thoughts on planetary motion: a fresh look, The British Journal for the History of Science,2, 118 (1964). · Zbl 0138.24904
[26] The Mathematical Papers of Isaac Newton, vol. III, 1670–1673, ed.D. T. Whiteside (Cambridge University Press, 1969), pp. 169–173.
[27] Reference [25], p. 123, footnote 23. · Zbl 0138.24904
[28] The Mathematical Papers of Isaac Newton, vol. II, 1667–1670, ed. byD. T. Whiteside (Cambridge University Press, 1968) p. 207.Newton starts his treatise entitledAnalysis by Equations unlimited in the number of terms with his Rule 1: Ifax m/n =y, then will [na/(m+n)]x (m+n)n equal the area ...
[29] I. B. Cohen,Introduction to Newton’s Principia (Cambridge 1971) pg. 295. This quotation is taken from a draft of a letter toDes Maizeaux, written in 1720 or somewhat earlier. Other statements ofNewton make it plausible thatNewton had antedated his discovery, possibly because of his dispute withLeibniz on the development of the calculus. I believe it is significant that in this letterNewton does not refer toKepler’s law of areas, which he demonstrated only after his correspondence withHooke. Historians of science have dismissedNewton’s claim that his first proofs were analytic, based on his fluxions (calculus), primarily because there is no direct documentary evidence, but also because they have not been able to figure out what technique he might have used. However, the analysis presented here, based on the mathematical and physical ideas whichNewton had developed by 1671, makeNewton’s claim entirely plausible.
[30] Reference [29] p. 291. HereNewton states that ... At length in the winter between the years 1676 & 1677 I found the Proposition that by a centrifugal force reciprocally as the square of the distance a Planet must revolve in an Ellipsis about the center of the force placed in the lower umbilicus of the Ellipsis & with a radius drawn to that center describe areas proportional to the times. However, the dates 1676 & 1677 contradictNewton’s statement in several letters that he had discoveredKepler’s law at the time of his correspondence withHooke, which was two years later.
[31] I. B. Cohen,The Newtonian Revolution (Cambridge University Press, 1980), pp. 248–249.
[32] The Mathematical Papers of Isaac Newton, vol. VI, 1684–1691, ed. byD. T. Whiteside (Cambridge University Press, 1974), p. 12, footnote 36.
[33] Thedirect problem in dynamics, given an orbit in polar coordinates where the origin is the center of force to obtain the acceleration (force)a towards this origin, can be readily solved by quadratures using the curvature method. One finds that \(\frac{a}{{a_0 }} = \frac{{w_0 }}{w}\exp \left[ { - 2\int\limits_{r_0 }^r {{{dr'} \mathord{\left/{\vphantom {{dr'} {w\left( {r'} \right)}}} \right.\kern-\nulldelimiterspace} {w\left( {r'} \right)}}} } \right]\) (42) wherew = {\(\rho\)}sin({\(\alpha\)}), andw 0,a 0 andr 0 are the initial values of the corresponding variables. For example, for a logarithmic spiral,w=r, Eq. (19), one obtainsa/a 0=(r 0/r)3, while for a conic section,w=r(2/A), Eq. (16),a/a 0 = (r 0/r)2.
[34] Reference [13] pp. 370–378.
[35] For general bound motion,E<0, in an inverse cube forcec r3, the general solution isr=r0/cosh(b{\(\theta\)}), withE = b2/2(1+b2)r 0 2 , and angular momentuml=c 1/2/(1+b 2)1/2.
[36] In the original tractDe Motu Newton added a comment in aScholium to Problem 1,Whiteside, (1684–1691: VI, p. 45), stating that ’In a spiral which cuts all its radii at a given angle [logarithmic spiral], the centripetal force tending to the spiral’ a pole is reciprocal in the tripled ratio of the distance’. I think that it is significant that at that timeNewton did not given any proof or discuss the method by which he had obtained this result. A geometrical proof based on an elegant self-similarity argument appeared later in the revisedDe Motu as Proposition VIII, Problem III, with thisScholium deleted,Whiteside (1684–1691: VI, p. 137), and in the same form in Proposition IX, Problem VI, book I of thePrincipia.
[37] D. T.Whiteside,The Prehistory of the Principia from 1664 to 1686 Notes Rec. R. Soc. Lond.45, 11–61 (1991). · Zbl 0716.01010
[38] J. Faulkner,Curvature of the ellipse and dynamical consequences (to be published). I thankJohn Faulkner for calling this result to my attention.
[39] The problem of determining the orbital parameter from initial conditions was included byNewton in an extension to Corollary I of Proposition XIII Problem VIII in section III, book I, added in the second edition of thePrincipia. However,Newton does not explain how to obtain these parameters which depends on knowing his unpublished mathematical results of 1671 on curvature. For the focus, the point of contact, and the position of the tangent being given, a conic section may be described, which at that point shall have a given curvature. The ”position of the tangent” is given by sin ({\(\alpha\)}) for a conic section in Eq. (20), and the curvature is obtained from this equation and Eq. (16) \(\rho = \frac{1}{{\sqrt L }}\left[ {2r - \frac{{r^2 }}{A}} \right]^{{3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}}\) (43) These equations can be solved to determine the parametersL andA in terms of given or initial values r0, sin({\(\alpha\)}0) and {\(\rho\)}0.L = {\(\rho\)}0sin3({\(\alpha\)}0) (44) and \(A = \frac{{r_0^2 }}{{2r_0 - \rho _0 \sin \left( {\alpha _0 } \right)}}\) (45)Newton then continues But the curvature is given from the centripetal force and velocity of the body being given; This statement refers to the initial value of {\(\rho\)}0 given by the fundamental dynamical curvature equation, Eq. (12), in the form \(\rho _0 = \frac{{v_0^2 }}{{a_0 \sin \left( {\alpha _0 } \right)}}\) (46) where the magnitudea 0 of the initial acceleration is taken to be equal to the initial value of the ”centripetal force”. It is clearNewton means that not only the magnitude, but also the direction of the initial velocity at the position vector r0 must be given, since only then can one evaluate sin({\(\alpha\)}0) from \(\cos \left( {\alpha _0 } \right) = \frac{{\vec r_0 \cdot \vec v_0 }}{{r_0 v_0 }}\) (47) Hence, substituting the value of{\(\rho\)} 0 obtained from the basic equation of the curvature method, Eq. (46), in Eq. (44) and Eq. (45) one can obtain, explicitly, the orbital parametersL andA in terms of the initial values ofr 0,v 0, sin({\(\alpha\)}0) anda 0, \(\frac{1}{A} = \frac{2}{{r_0 }} - \frac{{v_0^2 }}{c} = \frac{{ - 2E}}{c}\) (48) and \(L = \frac{{v_0^2 r_0^2 \sin ^2 \left( {\alpha _0 } \right)}}{c} = \frac{{l^2 }}{c}\) (49) The geometrical construction in Proposition 1 justifies the uniqueness theorem at the end of this Corollary: and two [different] orbits, touching one the other, cannot be described by the same centripetal force and the same velocity. This completesNewton’s proof that conic sections are the only possible orbits for an inverse square force, as he announced at the start of this Corollary. The uniqueness theorem is also demonstrated by Proposition XLI, book I of thePrincipia. However, the solution of the orbital integral in Proposition XLI for 1/r2 was not mentioned byNewton in any of his three editions of thePrincipia. A direct solution was first published byJohann Bernoulli in theMemoire de l’Academie Royale des Sciences 1710, pp 519–533. Recently there has been a lively debate in the literature, initiated byR. Weinstock, concerning the question whetherNewton actually proved in thePrincipia that given a central force 1/r 2, the only orbits are conic sections.R. Weinstock,Dismantling a Centuries-Old Myth: Newton’s Principia and Inverse Square Orbits, American Journal of Physics50, 610–617 (1982). For further references, and a careful study of this question seeG. H. Pourciau,On Newton’s Proof that Inverse-Square Orbits Must be Conies, Annals of Science48, 159–172 (1991);Newton’s Solution of the One-Body Problem, Archive for History of Exact Sciences, vol.44, 125–146 (1992).M. Nauenberg,The Mathematical Principles Underlying the Principia Revisited (to be published in the College Mathematics Journal).
[40] Reference [19] pp. 246–256. · Zbl 1184.93079
[41] R. S.Westfall,A note on Newton’s demonstrations of motion in ellipses, Archives Internationales d’Histories des Sciences,22, 52–60 (1969). For a recent discussion see,J. B. Brackenridge,The Locke/Newton Manuscripts revisited: conjugates, curvatures, & conjectures (to be published in Archives Internationales d’Histories des Sciences).
[42] D. T. Whiteside,The Preliminary Manuscripts for Newton’s 1687 Principia, 1684–1685 (Cambridge University Press, 1989) p. xv. According to a memorandum whichConduitt had fromDeMoivre in November 1727, ...in 1684 Dr. Halley made Sir Isaac a visit at Cambridge and there in a conversation the Dr. asked him what he thought the Curve would be that would be described by the Planets supposing the force of attraction towards the Sun to be reciprocal to the square of their distances from it. Sir Isaac replied immediately that it would be an Ellipsis. The Doctor struck with joy and amazement asked him how he knew it. Why saith he I have calculated it. Whereupon Dr. Halley asked him for his calculation without any further delay. Sir Isaac looked among his papers but could not find it, but he promised him to renew it, and then to send it to him... A couple of months laterNewton sentHalley the manuscriptDe Motu. PartlyHalley’s urging that he publish his results,Newton then spent the next two years in further work which culminated in his monumentalPrincipia.
[43] Reference [1] pp. 433–434.
[44] Reference [1] pp. 300–303.
[45] Reference [19] pp 304–326.
[46] D. T. Whiteside,The Preliminary Manuscripts for Isaac Newton’s 1687 Principia, 1684–1685 (Cambridge University Press, 1989) pp. 3–11.
[47] M. Nauenberg,Hooke, Orbital Motion and the Principia (to be published in the American Journal of Physics).
[48] Reference [1] pp. 297–298. · JFM 51.0465.10
[49] P. J. Pugliese,Robert Hooke and the dynamics of motion in a curved path, inRobert Hooke, New Studies, edited byM. Hunter andS. Schaffer, (Boydell Press, 1989) pp. 181–205
[50] To avoid this lack of convergence,Newton could have applied an adaptive algorithm to iterations in the impulse method. For example, the time step{\(\delta\)}t could be taken proportional to the radial distance from the center.
[51] The impulse equations correspond to a canonical or simplectic mapping of the coordinates \(\vec r\) and \(\vec v\) . This is the consequence of the fact that these equations are the exact solutions of a Hamiltonian with impulsive forces. For a discussion of simplectic transformations in Hamiltonian mechanics, see, for example,V. I. Arnol’d,Mathematical Methods of Classical Mechanics (Springer-Verlag, 1984).
[52] In Theorem 4 ofDe Motu Newton proves that Supposing that the centripetal force be reciprocally proportional to the square of the distance from the centre, the squares of the periodic times in ellipses are as the cubes of their transverse axes.
[53] The first treatise on calculus was published by 1’Hospital, based on lectures by his tutor,Johann Bernoulli, about a decade after the publication of thePrincipia, entitledAnalyse des Infiniment Petits, Pour l’intelligence des lignes courbes (A Paris, de l’Imprimerie Royale, MD-CXCVI). An unpublished translation of this book into English (which I found amongHooke’s manuscripts in the Royal Society) was provided byCharles Hayes who subsequently incorporated it into a book entitledA treatise on fluxions (London 1704) (D. T. Whiteside, private communication). An exact translation of 1’Hospital’s book was published in 1730 byE. Stone under the titleThe Method of Fluxions both Direct and Inverse (The Former being a translation from ..., and the later supply’d by the Translator).Newton’s original treatise of 1671 was first published byJohn Colson in English in 1736, in a book entitledThe Method of Fluxions and Infinite Series with its Application to the Geometry of Curve-Lines, By the Inventor Sir Isaac Newton, Kt, Late president of the Royal Society. Translated from theAuthor’s Latin Original not yet made publick. London, Printed by Henry Woodfall; M.DCC.XXXVI.
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