Johann Bernoulli, John Keill and the inverse problem of central forces

References

• A number of papers dealing with Newton's treatment of central force motion have recently appeared Whiteside D.T. The mathematical principles underlying Newton's Principia mathematica Journal for the History of Astronomy 1970 1 116 138 idem, ‘How forceful has a force proof to be? Newton's Principia, Book 1: Corollary 1 to Propositions 11–13’, Physis, 28 (1991), 727–49; H. Erlichson, ‘Newton's polygon model and the second order fallacy’, Centaurus, 35 (1992), 243–58; F. De Gandt, ‘Le probl`eme inverse (prop. 39–41)’, Revue d'Histoire des Sciences, 40 (1987), 281–309; S. Di Sieno and M. Galuzzi, ‘Calculus and geometry in Newton's mathematical work: some remarks’, in Science and Imagination in 18th-Century British Culture, edited by S. Rossi (Milan, 1987), 177–89; M. S. Mahoney, ‘Algebraic vs. geometric techniques in Newton's determination of plantetary orbits’, in Action and Reaction, edited by P. Theerman and A. F. Seeff (Newark, 1993), 183–205; B. Pourciau, ‘Newton's solution of the one-body problem’, Archive for History of Exact Sciences, 44 (1992), 123–46; J. Bruce Brackenridge, ‘Newton's mature dynamics and the Principia: a simplified solution to the Kepler problem’, Historia Mathematica, 16 (1989), 36–45; ‘Newton's unpublished dynamical principles: a study in simplicity’, Annals of Science, 47 (1990), 3–31.
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• A good study of Varignon is Blay Michel La naissance de la mécanique analytique, la science du mouvement au tournant des XVIIe et XVIIIe si`ecles Paris 1992 The reception of Newton's dynamics on the Continent is admirably discussed in chapter 9 of Domenico Bertoloni Meli, Equivalence and Priority: Newton versus Leibniz (Oxford, 1993). Both Bertoloni Meli and Blay stress the importance of the achievements of the Continentals, which are described as more general and more flexible. In this paper I will try to take a different point of view, i.e. I will try to understand the reasons defended by the Newtonians.
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• Keill , John . 1708 . Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum . Philosophical Transactions , 26 : 174 – 188 .
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• Sir Isaac Newton Mathematical Principles of Natural Philosophy, and, System of the World Berkeley1962 48 49 translated into English by Andrew Motte in 1729, revised by Florian Cajori 5th imp. What Newton means here is that one has to make Q approach to P as a limit. All the statements are valid in this limiting situation, when the are PQ is ‘nascent’ or ‘vanishing’. Newton's theory of ‘limits of prime and ultimate ratios’ was developed in Section 1, Book 1.
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• Newton Mathematical Principles of Natural Philosophy, and, System of the World Berkeley1962 35 35 translated into English by Andrew Motte in 1729, revised by Florian Cajori 5th imp.,
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• Newton Mathematical Principles of Natural Philosophy, and, System of the World Berkeley1962 61 61 translated into English by Andrew Motte in 1729, revised by Florian Cajori 5th imp., Newton's original Latin at p. 59 of the third edition of the Principia is ‘Ex tribus novissimis propositionibus consequens est, quod si corpus quodvis P secundum lineam quamvis rectam PR quacunque cum velocitate exeat de loco P, & vi centripeta, quae sit reciproce proportionalis quadrato distantiae locorum a centro, simul agitetur; movebitur hoc corpus in aliqua sectionum conicarum umbilicum habente in centro virium; & contra’.
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• Newton Mathematical Principles of Natural Philosophy, and, System of the World Berkeley1962 65 65 translated into English by Andrew Motte in 1729, revised by Florian Cajori 5th imp.,
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• Keill's paper on central forces Keill John Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum Philosophical Transactions 1708 26 174 188 was presented at the Royal Society in 1708 and published in 1710. Johnann Bernoulli's criticisms (see § 4) were presented at the Académie des Sciences in 1710 and published in 1713 as ‘Extrait de la Réponse de M. Bernoulli `a M. Herman, datée de Basle le 7 Octobre 1710’, Mémoires de l'Académie des Sciences (1710), 521–33.
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• The Correspondence of Isaac Newton et al. Cambridge 1959–77 v 5 6 7 vols I. Newton (note 6), 61. Newton's original Latin at p. 59 of the third edition of the Principia is ‘Nam datis umbilico, & puncto contactus, & positione tangentis, describi potest sectio conica, quae curvaturam datam ad punctum illud habebit. Datur autem curvatura ex data vi centripeta, & velocitate corporis: & orbes duo se mutuo tangentes eadem vi centripeta eademque velocitate describi non possunt’. The logical correctness of this Newtonian procedure has been discussed in a number of papers: R. Weinstock, ‘Dismantling a centuries-old myth: Newton's Principia and inverse-square orbits’, American Journal of Physics, 50 (1982), 610–17; idem, ‘Long-buried dismantling of a centuries-old myth: Newton's Principia and inverse-square orbits’, American Journal of Physics, 57 (1989), 846–9; J. T. Cushing, ‘Kepler's laws and universal gravitation in Newton's Principia’, American Journal of Physics, 50 (1982), 617–28; V. I. Arnol'd, Huygens and Barrow, Newton and Hooke, translated from the Russian by Eric J. F. Primrose (Basle 1990). Careful analyses of this Corollary are given also in the papers by D. T. Whiteside, J. Bruce Brackenridge and Bruce Pourciau cited in note 1. In what follows I am greatly indebted to Pourciau's reconstruction of the logic of Corollary 1.
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• Newton Mathematical Principles of Natural Philosophy, and, System of the World Berkeley1962 130 130 translated into English by Andrew Motte in 1729, revised by Florian Cajori 5th imp., Newton's original Latin at p. 125 of the third edition of the Principia is ‘Posita cujuscunque generis vi centripeta & concessis figurarum curvilinearum quadraturis, requiruntur tum trajectoriae in quibus corpora movebuntur, tum tempora motuum in trajectoriis inventis’.
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• Newton Mathematical Principles of Natural Philosophy, and, System of the World Berkeley1962 130 130 translated into English by Andrew Motte in 1729, revised by Florian Cajori 5th imp.,
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• The Mathematical Papers of Isaac Newton Whiteside D.T. Cambridge1967–81 VI 348n 348n 8 vols iii, 252.
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• Whiteside , D.T. , ed. 1967–81 . The Mathematical Papers of Isaac Newton Vol. VI , 435 – 437 . Cambridge 8 vols
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• The procedure employed by Newton to determine SVX is rather complicated and will not detain us here. For a detailed analysis of Corollary 3 to Proposition 41, see Erlichson H. The visualization of quadratures in the mystery of Corollary 3 to Proposition 41 of Newton's Principia Historia Mathematica 1994 21 148 161
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• For a discussion of this topic, see Whiteside's notes in Newton The Mathematical Papers of Isaac Newton Whiteside D.T. Cambridge 1967–81 VI 345 355 8 vols
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• It has been often repeated that Newton was unable to apply the calculus to dynamics. For instance: ‘One would be wrong in thinking that Newton knew how to formulate and resolve problems through the integration of differential equations’ Costabel P. Newton's and Leibniz's dynamics Texas Quarterly 1967 10 119 126 (125).
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• Herivel , J.W. 1965 . The Background to Newton's Principia Oxford R. S. Westfall, Force in Newton's Physics: The Science of Dynamics in the Seventeenth Century (London, 1971); I. Bernard Cohen, Introduction to Newton's Principia (Cambridge, 1971); D. T. Whiteside, ‘The mathematical principles underlying Newton's Principia Mathematica’, Journal for the History of Astronomy, 1 (1970), 116–38.
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• De Gandt , F. 1986 . Le style mathématique des Principia de Newton . Revue d'Histoire des Sciences , 29 : 195 – 222 .
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• ‘C'est encore une justice dˆuë au sçavant M. Newton, & que M. Leibniz luy a renduë luy-mˆeme: Qu'il avoit aussi trouvé quelque chose de semblable au Calcul différentiel, comme il paroˆit par l'excellent Livre intitulé Philosophiae Naturalis Principia Mathematica, qu'il nous donna en 1687: lequel est presque tout de ce calcul’. de L'Hˆopital Guillaume François Antoine Analyse des infiniment petits Paris 1696 Preface, éiivéiiir .
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• Newton . 1967–81 . The Mathematical Papers of Isaac Newton Edited by: Whiteside , D.T. Vol. VI , 538ff – 538ff . Cambridge 8 vols
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• Newton . 1967–81 . The Mathematical Papers of Isaac Newton Edited by: Whiteside , D.T. 450 – 455 . Cambridge 8 vols
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• Corollary 5 to Proposition 6 in Newton Mathematical Principles of Natural Philosophy, and, System of the World Berkeley 1962 49 49 translated into English by Andrew Motte in 1729, revised by Florian Cajori 5th imp.,
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• Newton The Mathematical Papers of Isaac Newton Whiteside D.T. Cambridge1967–81 III 150ff 150ff 8 vols For a discussion of the role of curvature in Newton's dynamics, see J. Bruce Brackenridge, ‘The critical role of curvature in Newton's developing dynamics’, in The Investigation of Difficult Things: Essays on Newton and the History of Exact Sciences, in Honour of D. T. Whiteside, edited by P. M. Harman and A. E. Sharpiro (Cambridge, 1992), 231–60.
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• Newton . 1967–81 . The Mathematical Papers of Isaac Newton Edited by: Whiteside , D.T. Vol. VII , 122 – 123 . Cambridge 8 vols
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• Newton The Mathematical Papers of Isaac Newton Whiteside D.T. Cambridge1967–81 VI 588 593 8 vols 598–9. Referring to this manuscript, Whiteside observes: ‘our assessment of the handwriting style places it as not much later than this [1692] terminus ante quem non’ (ibid., 597).
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• ‘Si Lex vis centripetae investiganda sit qua corpus in orbe quocunque APE circa centrum quodvis S movebitur: sit AB = x Absicssa & BP = y ordinata ad rectos angulos insistens. Cape BT in ea ratione ad BP quam habet [xdot] ad [ydot] et acta TP curvam tanget in P. Ordinatae BP parallelam age SG tangenti occurrens in G et vis qua corpus in Orbe APE circa centrum S movebitur erit ut (ÿSP)/([xdot][xdot]SG3 )’. Newton The Mathematical Papers of Isaac Newton Whiteside D.T. Cambridge 1967–81 VI 598 598 8 vols
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• Among the various studies devoted to the results on central force motion developed by Leibniz's adherents at the turn of the eighteenth century, Eric Aiton's papers are particularly detailed: Aiton E. The inverse problem of central forces Annals of Science 1964 20 81 99 ‘The contributions of Isaac Newton, Johann Bernoulli and Jakob Hermann to the inverse problem of central forces’, Studia Leibnitiana, Sonderheft, 17 (1987), 47–58. See also D. T. Whiteside's notes in Newton (note 18), vi, 146–9.
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• Varignon , Pierre . 1700 . Mémoires de l'Académie des Sciences 22 – 27 . 83–101, 224–42; (1701), 20–38; (1703), 212–29; (1706), 178–235. On Varignon, see Blay (note 2).
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• Hermann , Jakob . 1710 . “ Metodo di investigare le orbite dei pianeti ” . In Giornale dei Letterati d'Italia 447 – 467 . ‘Extrait d'une lettre de M. Herman `a M. Bernoulli, datée de Padoüe le 12 Juillet 1710’, Mémoires de l'Académie des Sciences (1710), 519–21. Bernoulli (note 14).
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• ‘votre Solution paroˆit faite `a dessein, accommodée `a ce que vous cherchiez, & `a ce que vous connoissiez déja’ Bernoulli Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum Philosophical Transactions 1708 26 521 521
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• ‘De plus il ne suit pas encore de votre Solution particuliere qu'elle ne convienne qu'aux seules Sections Coniques: apr`es la premiere intégration de votre équation differentio-differentielle vous avez oublié d'y ajoˆuter de part ou d'autre une quantité constante; ce qui pourroit laisser quelqu'un en doute, si outre les Sections Coniques, il n'y auroit point encore quelqu'autre genre de Courbes qui satisfist `a votre question’ Bernoulli Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum Philosophical Transactions 1708 26 522 522
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• Jacopo Riccati defended Hermann's solution in Risposta ad alcune opposizioni fatte dal Sig. Giovanni Bernoulli Giornale dei Letterati d'Italia 1714 19 185 210 On this topic, see S. Giuntini, ‘Jacopo Riccati e il problema inverso delle forze centrali’, in I Riccati e la cultura della marca nel Settecento europeo, edited by G. Piaia and M.-L. Soppelsa (Florence, 1992), 127–49.
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• The first one to publish on singular solutions of fluxional equations was Taylor Brook Methodus incrementorum London 1715 A general theory of singular solutions was developed later by Alexis-Claude Clairaut.
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• Bernoulli's two orbits correspond to what I indicate below (in § 5) as case 1 and case 2b. Neither of them was included in Newton's treatment of the inverse problem of inverse cube forces given in Corollary 3 to Proposition 41 of the Principia (see § 2). An attempt to emend this deficiency in the third edition by adding a reference to Proposition 9, Book 1 (i.e. to the logarithmic spiral) was abandoned. See Newton The Mathematical Papers of Isaac Newton Whiteside D.T. Cambridge 1967–81 VI 355n 355n 8 vols
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• ‘J'ai dit ci-devant que M. Newton, apr`es avoir démontré que les forces centrales d'un corps, dirigées par un des foyers d'une Section Conique quelconque décrite par ce corps, sont toujours entr'elles en raison renversée des quarrées des distances de ce mˆeme corps `a ce foyer, suppose l'inverse de cette proposition sans la démontrer […] pour voir encore la necessité de la démonstration que je viens de donner de cette inverse, il n'y a qu'`a considerer que de ce qu'un corps pour se mouvoir sur une spyrale logarithmique, requiert des forces centrales en raison réciproque des cubes des distances au foyer au centre de cette courbe; ce n'est pas une conséquence qu' avec de telles forces il descrivˆit toujours une telle courbe; Puisqu'il est aisé de se convaincre par les formules directes des forces centrales, que ce corps auroit aussi ces forces en cette raison s'il décrivoit une spyrale hyperbolique’ Bernoulli Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum Philosophical Transactions 1708 26 532 533
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• Apr`es avoir trouvé ce théoreme je le montrai `a M. Newton et je me flattois qu'il lui paroˆitroit nouveau, mais M. Newton m'avoit prévenu; il me le fit voir dans les papiers qu'il prépare pour une seconde édition de ses Principia Mathematica: toute la différence qu'il y avoit, c'est qu'au lieu d'exprimer la loi de la force centripete par le moyen du rayon de la concavité, il l'exprimoit par le moyen d'une corde inscrite dans le cercle de la concavité: mais il me dit qu'il valoit mieux l'exprimer par le rayon, comme j'avois fait’. Der mathematische Briefwechsel zwischen Johann I Bernoulli und Abraham de Moivre Verhandlungen der Naturforschenden Gesellschaft in Basel Wollenschläger Karl 1931–2 43 151 317 (214). See also Newton (note 18), vi, 548, where D. T. Whiteside reconstructs the history of equations 16 and 17.
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• Bernoulli . 1708 . Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum . Philosophical Transactions , 26 : 521 – 533 . David Gregory's own proof of De Moivre's result is in ‘Codex E’ in Christ Church, Oxford, MS 346 (manuscript dated 1707). Pierre Varignon (see § 4) had already published an equivalent result in ‘Autre r`egle générale des forces centrales’, Memoires de l'Académie des Sciences (1701), 20–38. It seems that Varignon's theorem was not noticed either by Bernoulli or by the British, when, in 1714–19, Keill and Bernoulli squabbled over who first discovered equation 26. On this, see Whiteside's note 25 in Newton (note 18), vi, 548–9.
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• Keill . 1708 . Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum . Philosophical Transactions , 26 : 174 – 188 . On the part that Keill's declaration—in a short paragraph of a few lines incidental to his main paper—came to play in the Newton-Leibniz controversy, see A. Rupert Hall, Philosophers at War (Cambridge, 1980), chapter 7.
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• The use of pedal coordinates in Newtonian dynamics has been discussed in Pourciau The mathematical principles underlying Newton's Principia mathematica Journal for the History of Astronomy 1970 1 131 134
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• On inserting the constants of proportionality one should write : Euler L. Mechanica St Petersburg 1736 See, for example (= Opera Omnia, ser. 2, i, 196); and C. Maclaurin, Treatise of Fluxions (Edinburgh, 1742), art. 874.
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• ‘data relatione SA[= r] ad SP[= p], facile invenietur lex vis centripetae’ Keill Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum Philosophical Transactions 1708 26 181 181
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• ‘ex data lege vis centripetae, invenire potest relatio SA[= r] ad SP[= p], ac proinde per methodum tangentium inversam, exhiberi potest curva, quae data vi centripeta describi possit’ Keill Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum Philosophical Transactions 1708 26 183 184
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• Posito quod vis centripeta (cujus quantitas absoluta nota est), sit reciproce, ut distantiae quadratum & projiciatur corpus secundum datam rectam cum data velocitate. Invenire curvam in qua movetur corpus’ Keill Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum Philosophical Transactions 1708 26 188 188
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• For a modern proof of the property of conic sections employed by Keill, see Pourciau The mathematical principles underlying Newton's Principia mathematica Journal for the History of Astronomy 1970 1 140 140
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• Keill was proud in which he saw his originality here. He wrote ‘Hoc a me prius ostensum est in actis philosophicis Londiniensibus Anno 1708’ Keill Observationes … de inverso problemate virium centripetarum Philosophical Transactions 1714 29 95 95
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• Bernoulli . 1708 . Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum . Philosophical Transactions , 26 : 174 – 188 . ‘De motu corporum gravium’, Acta eruditorum (1713), 77–95, 115–32.
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• Cotes , R. 1714 . Logometria . Philosophical Transactions , 29 : 5 – 45 . Whiteside informs us that Cotes had achieved this result in 1709 while checking the correctness of Corollary 3 to Proposition 41, Book 1, of the Principia. Newton (note 18), vi, 353n. A reconstruction of Cotes's procedures is in R. Gowing, Roger Cotes: Natural Philosopher (Cambridge, 1983), 51–4, 195–203.
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• Keill . 1714 . Observationes … de inverso problemate virium centripetarum . Philosophical Transactions , 29 : 95 – 95 . In a letter, dated 9 November 1713, in which Keill informed Newton of his coming paper on inverse cube orbits, he wrote ‘I shall be much obliged to you if you will send me your throughts on this matter’. See Newton (note 15), vi, 37–8. The controversy between Keill and Bernoulli continued up to the death, in 1721, of one of the contenders. The close contacts between Newton and Keill during the priority dispute are discussed in Hall (note 50), chapter 10. See also D. T. Whiteside's notes in Newton (note 18), vi, 350.
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• This is how Keill described Bernoulli's analytical techniques applied to the solution of the inverse problem (see § 4). Keill Observationes … de inverso problemate virium centripetarum Philosophical Transactions 1714 29 94 94
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• From equations 42 and 31 the following differential equation is obtained Bernoulli Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum Philosophical Transactions 1708 26 129 129 Cases 2a, c and 3 correspond, by suitably adjusting the constants of integration respectively to: In Cases 2c and 3 the particle reaches the centre of force as 0 → ∞. In cases 2a and c the particle has enough energy to escape to infinity, while in case 3 the trajectory is confined into a bounded region. Bernoulli, in 1713, had given an incomplete classification of inverse cube orbits. He wrote: ‘Docebo his naturam harum curvarum, pro quibus legem virium centripetarum ope theorematis Moyvraeani supra demonstrati quilibet facile inveniet’
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• Keill . 1714 . Observationes … de inverso problemate virium centripetarum . Philosophical Transactions , 29 : 95 – 95 . Bernoulli (note 14).
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• Keill , J. 1716 . Défense du Chevalier Newton . Journal litéraire , 8 : 418 – 433 .
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• Bernoulli . 1708 . Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum . Philosophical Transactions , 26 : 174 – 188 . Keill (note 3); Bernoulli (note 14).
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• Bernoulli , Johann . 1716 . Epistola pro eminente mathematico . Acta eruditorum , : 296 – 315 .
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• Kruse , Johann H. 1718 . Responsio ad Cl. Viri Johannis Keil . Acta eruditorum , : 454 – 466 .
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• Keill , John . 1719 . Lettre…`a M. Jean Bernoulli . Journal litéraire , 10 : 261 – 287 . Whiteside writes that ‘a wealth of Newton's autograph English drafts and extensive corrections of De Moivre's French renderings of Keill's propagandist pieces exists in ULC Add.3968.23, Add.3985.14 and especially Res. 1893.5’ (Newton (note 18), vi, 350n).
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• Je suis surpris de voir que M. Bernoully dans les Actes de Leipsic 1713 pag. 130 dise, qu'il ait été le premier qui a trouvé & demontré le Probl`eme inverse des Forces Centripetes […] Il ya plus de sept ans que je l'ai demontré, comme on le peut voir dans les Transactions Philosophiques pour le mois de Septembre & d'Octobre 1708, o`u je donnai une Solution générale de ce Probl`eme que j'appliquai ensuite au cas particulier. Cette Solution fut imprimée & publiée plus de deux ans avant que M. Bernoully envoyˆat sa Solution `a l'Academie des Sciences, il n'a donc nul droit de vouloir passer pour en ˆetre le premier Inventeur’ Keill Défense du Chevalier Newton Journal litéraire 1716 8 422 422
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• ‘J'ai donc démontré en tr`es peu de lignes, `a l'aide de ma Solution générale, que lors que la force est réciproque au quarré de la Distance, la Courbe décrite doˆit ˆetre une Section Conique; ce que M. Bernoully ne déduit de la sienne qu'`a force de travail: il paroˆit donc que j'ai été le premier & non lui, qui, apr`es M. Newton ait essayé & résolu le Probl`eme des Forces Centripetes’ Défense du Chevalier Newton Journal litéraire 1716 8 424 424
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• ‘Itaque nihil hactenus effecit Keilius ex isto theoremate solutionem problematis inversi centralium virium in casu speciali derivaturus, cum non demonstrarit, tres istos valores non nisi sectionibus conicis competere. Hanc demonstrationem cum nondum dederit, abunde liquet, ipsum non esse primum solutionis problematis inversi virium centralium in casu speciali inventorem, quemadmodum supra contendit’ Kruse Responsio ad Cl. Viri Johannis Keil Acta eruditorum 1718 458 458
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• ‘LaSolution de M. Bernoully ne differant de celle de M. Newton, que dans les Caract`eres ou Symboles’ Keill Défense du Chevalier Newton Journal litéraire 1716 8 418 418
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• ‘vidi Bernoullianam formulam omnino cum Newtoniana coincidere; nec nisi in notatione quantitatum ab ea differre. Nam si pro ab — εF[xdot] ponatur ABGE, pro ac ponatur Q, & x pro A, a pro CX, & [xdot] pro IN […] constat formulam illam non magis a Newtoniana discrepare, quam verba latinis literis expressa differunt ab iisdem verbis scriptis in Graecis characteribus’ Keill Observationes … de inverso problemate virium centripetarum Philosophical Transactions 1714 29 114 114
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• ‘Mon équation fait voir de plus si la trajectoire cherchée est Algebrique ou non dans quelque hypothese que ce soit des forces données. Car si l'intégrale de Bernoulli Epistola ad Clarissimum Virum Edmundum Halleium Geometriae Professorem Savilianum, de Legibus Virium Centripetarum Philosophical Transactions 1708 26 526 526 se trouve réductible `a un arc de cercle dont le rayon soit OA (a) comme nombre `a nombre; la Courbe cherchée sera nécessairement alors Algebrique’
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• Keill to Newton, 9 November 1713, in Newton The Correspondence of Isaac Newton et al. Cambridge 1959–77 VI 37 37 7 vols
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• ‘Mais apr`es tout, M. Newton dans la Page 53 de la seconde Edition de ses Principes, nous a donné en trois lignes une Démonstration de ce cas particulier, au lieu que M. Bernoully y employe 7 `a 8 Pages’ Keill Défense du Chevalier Newton Journal litéraire 1716 8 420 420
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• ‘Vous pouvez tirer cette consequence, si vous pouvez toˆujours décrire une Spirale Logarithmique dans laquelle si un corps se meut, il doive avoir, `a un point donné, une direction donnée, une vitesse donnée & une force centripetale absoluë donnée; qui est le cas dans les Sectiones Coniques; mais parce que cela ne se peut, dans la Spirale Logarithmique, nous en devons tirer une consequence toute contraire’ Keill Lettre…`a M. Jean Bernoulli Journal litéraire 1719 10 276 277
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• ‘Lubens credo quod ais de aucto Corollario 1, Prop. XIII Lib. 1. Operis Tui incomparabilis Princip.Phil.hoc nempe factum esse antequam hae lites coeperunt, neque dubitavi unquam, Tibi esse demonstrationem propositionis inversae quam nude asserueras in prima Operis Editione, aliquid dicebam tantum contra formam illius asserti, atque optabam, ut quis analysin daret, qua inversae veritatem inveniret a priori, ac non supposita directa jam cognita. Hoc vero, quod Te non invito dixerim, a me primo praestitum esse puto, quantum saltem hactenus mihi constat' (Bernoulli to Newton, 10 December 1719, in Newton The Correspondence of Isaac Newton et al. Cambridge 1959–77 VII 76 76 7 vols
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• ‘Negabant enim a Neutono satis esse demonstratum praeter sectiones conicas nullam aliam curvam quaesito satisfacere, quamvis prop. XVII. lib. 1. Princ. Philos. Naturalis hoc satis clare evincere videatur’ Euler Mechanica St Petersburg 1736 221 221
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• ‘En généralisant ensuite ses recherches, ce grand géom`etre fit voir qu'un projectile peut se mouvoir dans une section conique quelconque, en vertu d'une force dirigée vers son foyer, et réciproque au carré des distances: il développa les diverses propriétés du mouvement dans ce genre de courbes: il détermina les conditions nécessaires pour que la courbe soit un cercle, une ellipse, une parabole ou une hyperbole, conditions qui ne dépendent que de la vitesse et de la position primitive du corps. Quelles que soient, cette vitesse, cette position et la direction initiale du mouvement, Newton assigna une section conique que le corps peut décrire, et dans laquelle il doit conséquemment se mouvoir; ce qui répond au reproche que lui fit Jean Bernoulli, de n'avoir pas démontré que les sections coniques sont les seules courbes que puisse décrire un corps sollicité par une force réciproque au carré des distances’. Laplace Pierre-Simon Exposition du syst`eme du monde Paris 1796 I quote from pp. 521–2 of the sixth (1835) edition, revised by the author and published by Fayard in 1984.
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• ‘Objectio fuit quod Corollarium Propositionis XIII Libri primi in editione prima non demonstraverim. Dicunt enim quod corpus P secundum rectam positione datam data cum velocitate a dato loco exiens, vi centripeta cujus lex datur Curvas plures describere posse. Sed hallucinantur. Si mutetur vel positio rectae vel velocitas corporis, mutari potest Curva describenda et ex circulo Ellipsis, ex Ellipsi Parabola vel Hyperbola fieri. Sed positione rectae et velocitate corporis et lege vis centripetae manentibus curva alia atque alia describi non possunt. Ideoque si ex data curva determinatur vis centripeta, vicissim ex data vi centripeta determinabitur curva. In secunda paucis tantum verbis attigi. In utraque constructionem hujus Corollarii in Prop. XVII exhibui qua veritas ejus satis elucesceret, & Problema generaliter solutum in Prop. XLI Lib.I’ Newton The Mathematical Papers of Isaac Newton Whiteside D.T. Cambridge 1967–81 VIII 457 458 8 vols translation from the Latin by D. T. Whiteside).
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• Arnol'd The Correspondence of Isaac Newton et al. Cambridge 1959–77 v 32 32 7 vols For the existence and uniqueness of solutions of differential equations, see I. G. Petrovski, Ordinary Differential Equations (New York, 1973) 4; and A. Dhar, ‘Nonuniqueness in the Solution of Newton's Equation of Motion’, American Journal of Physics, 61 (1993), 58–61.
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• Newton Mathematical Principles of Natural Philosophy, and, System of the World Berkeley1962 61 61 translated into English by Andrew Motte in 1729, revised by Florian Cajori 5th imp.
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• For instance Hermann wrote Jakob Hoc problema primum solutionem accepit a Cel. Newtono Prop. 41 Lib. 1 Princ. Phil. Nat. Math. & postea a Perspicacissimo Geometra Joh. Bernoulli gemino modo’ Phoronomia Amsterdam 1716 73 73
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• Cotes . 1714 . Logometria . Philosophical Transactions , 29 : 5 – 45 . Harmonia mensurarum (Cambridge, 1722); Taylor (note 43); Thomas Simpson, Essays on Several Curious and Useful Subjects (London, 1740); idem, Mathematical Dissertations (London, 1743); idem, Miscellaneous Tracts (London, 1757); William Emerson, The Doctrine of Fluxions (London, 1743); Thomas Simpson, The Doctrine and Application of Fluxions (London, 1750); Nicholas Saunderson, The Method of Fluxions (London, 1756).
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• John Machin's studies of the Moon's motion were added by Henry Pemberton in the third edition of Newton Principia 451 454 and in the Motte's translation of 1729 as The laws of the Moon's motion according to gravity.
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• Walmesley's Charles Théorie de mouvement des apsides Paris1749 main publications are; ‘Two essays’, Philosophical Transactions, 49 (1756), 700–58; ‘Of the irregularities in the motion of a satellite’, Philosophical Transactions, 50 (1758), 809–35; and ‘Of the irregularities in the planetary motions’, Philosophical Transactions, 52 (1761), 275–335.
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• Landen's , John . 1771 . Animadversions on Dr. Stewart's computation of the Sun's distance from the Earth London was written as a criticism of Matthew Stewart, The distance of the Sun from the Earth determined by the theory of gravity (Edinburgh, 1763).
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• Simpson , Thomas . 1757 . Miscellaneous tracts London Preface.
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• Gregory , David . 1702 . Astronomiae physicae & geometricae elementa Oxford For Newton's interest in the use of geometry in dynamics, see Newton (note 18), viii, 442–59.
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• Ubi Propositio aliqua per Analysin quamcunque prodijt, haec non statim in Geometriam admittenda est sed demonstratio ejus prius componi debet. Nam Geometriae vis & laus omins ab ejus certitudine, certitude autem a Demonstrationibus syntheticis dependet. Ea de causa Veteres non solebant Analyticas Propositionum suarum inventionem in lucem dare sed inventas semper demonstrabant synthetice & hujusmodi demonstrationibus munitas edebant. Eadem de causa et nos Propositiones per Analysin nostram inventas demonstravimus synthetice ut in Geometria admitterentur & Philosophia Coelorum quae in Libro tertio habetur fundaretur in Propositionibus geometrice demonstratis’ Newton Astronomiae physicae & geometricae elementa Oxford 1702 444 445 translation from the Latin by D. T. Whiteside).
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• On the empiricist tradition in Scottish mathematics, see Olson R. G. Scottish philosophy and mathematics Journal for the History of Ideas 1971 32 29 42 idem, Scottish philosophy and British physics, 1750–1880 (Princeton, 1975); and E. Sageng, ‘Colin Maclaurin and the Foundations of the Method of Fluxions’, PhD (Princeton University, 1989).
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• Newton . 1967–81 . The Mathematical Papers of Isaac Newton Edited by: Whiteside , D.T. Vol. VIII , 122 – 122 . Cambridge 8 vols
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• Newton . 1967–81 . The Mathematical Papers of Isaac Newton Edited by: Whiteside , D.T. Vol. VIII , 597 – 597 . Cambridge 8 vols
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• There is a huge literature on the controversy originated by Berkeley George The analyst London 1734 Among the recent studies, see D. Sherry, ‘The wake of Berkeley's Analyst: rigor mathematicae?’, Studies in History and Philosophy of Science, 18 (1987), 455–80. See also I. Grattan-Guinness, ‘Berkeley's criticism of the calculus as a study in the theory of limits’, Janus, 56 (1969), 213–27.
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• See John Colson's commentary in Newton Isaac Method of fluxions and infinite series London 1736 270 270 It should be noted that, for Newton, fluxional ‘time’ is just any fluent whose fluxion is constant. Many eighteenth-century ‘Newtonians’ misrepresented Newton on this point, equating fluxional and real time.
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• On Newton's theory of the Moon, see Whiteside D.T. Newton's lunar theory: from high hope to disenchantment Vistas in Astronomy 1976 19 317 328 C. B. Waff, ‘Isaac Newton, the motion of the lunar apogee, and the establishment of the inverse square law’, Vistas in Astronomy, 20 (1976), 99–103; and S. Aoki, ‘The Moon-Test in Newton's Principia: accuracy of the inverse-square law of universal gravitation’, Archive for History of Exact Sciences, 44 (1992), 145–90. On Newton's theory of perturbations, see I. Bernard Cohen, The Newtonian Revolution (Cambridge, 1980); and C. Wilson, ‘The Newtonian achievement in astronomy’, in The General History of Astronomy, edited by René Taton and Curtis Wilson (Cambridge, 1989), ii, part A, 233–74.
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• Newton Mathematical Principles of Natural Philosophy, and, System of the World Berkeley1962 67 67 translated into English by Andrew Motte in 1729, revised by Florian Cajori 5th imp.
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• On Newton's approximation techniques for the computation of orbits, see Nauenberg M. Newton's early computational method for dynamics Archive for History of Exact Sciences 1994 46 221 252
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