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The conventional view of the prehistory of Newton's synthesis in the Principia of his predecessors' work in planetary theory and terrestrial gravitation is still not seriously changed from that which Newton himself chose to impose on his contemporaries at the end of his life. In his own words:
‘… the same year ‘1666’ I began to think of gravity extending to ye orb of the Moon & having found out how to estimate the force wth wch [a] globe revolving within a sphere presses the surface of the sphere from Keplers rule of the periodical times of the Planets being in a sesquialterate proportion of their distances from the centers of their Orbs, & having deduced that the forces wch keep the Planets in their orbs must [be] reciprocally as the squares of their distances from the centers about wch they revolve: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly …’
^{1} I refer particularly to Rigaud, S. P., Historical Essay on the First Publication of Sir Isaac Newton's Principia, Oxford, 1838; Brewster, D., Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton, Edinburgh, 1855, i, 289 ff.; Ball, W. W. Rouse, An Essay on Newton's ‘Principia’, London, 1893; and More, L. T., Isaac Newton, New York, 1934 (re-issued 1962), 288 ff. See also Koyré, Alexandre, ‘La gravitation universelle de Kepler à Newton’, Archives internationales d'histoire des sciences, 1951, iv, 638–653; and Lohne, J., ‘Hooke versus Newton: An Analysis of the Documents in the Case on Free Fall and Planetary Motion’, Centaurus, 1960, vii, 6–52, for recent variants on the traditional account.
^{2} ULC. Add. 3968.41, 85^{r}. This now celebrated passage, a fuller version of which (not very accurately transcribed) was first printed in A Catalogue of the Portsmouth Collection of Books and Papers written by or belonging to Sir Isaac Newton, Cambridge, 1888, p. xviii, and many times since, is extracted from a cancelled draft of a letter which Newton wrote to Des Maizeaux in the summer of 1718 when the latter was gathering material for his Recueil, London, 1720.
^{3} ULC. Add. 3968.9, 101^{r}, quoted more fully in Lohne's ‘Hooke versus Newton’ (note 1), 48–49. The original manuscript is an English draft preface for an abortive second revision of the Principia which Newton was planning towards the end of 1714. For an excellent account of the Newton-Hooke correspondence (and a full bibliography up to 1950), see Koyré's, A. ‘An Unpublished Letter of Robert Hooke to Isaac Newton’, Isis, 1952, xliii, 312–337.
^{4} Most authorities accept the month as August, but Herivel, J. W., in his ‘Halley‘s First Visit to Newton’, Arch. int. d‘hist. sci., 1960, xiii, 63–65, argues for May.
^{5} First printed by Rigaud in his Essay (note 1), Appendix, 1–19, from the registered copy in the Royal Society archives (Register Book 6, 218), and later from a collation of the two variant Newton autograph/amanuensis-copied originals (Add. 3965.7, 40^{r}–54^{r}/55^{r}–62 bis ^{r}) in the University Library, Cambridge, by Rouse Ball in his Essay (note 1), 35–56, and by A. R. and Hall, M. B. in their Unpublished Scientific Papers of Isaac Newton, Cambridge, 1962, 243–267. Though I cannot at all accept his many hypothetical arguments, I may note also that J. W. Herivel has recently sought to identify the original researches of Newton in late 1679 in a Newton autograph (ULC. Add. 3965.1) first published in Latin translation by Whiston in 1710. (See Herivel's, ‘The Originals of the two Propositions Discovered by Newton in December 1679?’ and ‘Newtonian Studies IV’, Arch. int. d'hist. sci., 1960, xiv, 23–33; 1962, xvi, 13–22. Also compare Whiston, W., Prælectiones Physico-Mathematicæ, Cambridge, 1710 (re-issued 1726), Lect. XIV/XV (for 5/19 February 1704/5), where (p. 137) Whiston notes that the tract he gives is published ‘qualem nempe eam è Charta MS Ipsius Newtoni olim acceperam’.)
^{6} In further confirmation, in a second English draft preface to his abortive second revision of the Principia, Newton remarked about the end of 1714 (ULC. 3968.9, 106^{r}; cf. Brewster's, Memoirs (note 1), i, 471) that of its propositions ‘about ten or twelve … were composed before [December 1684], viz^{t} the 1^{st} & 11^{th} [in answer to Hooke's challenge] in December 1679, the 6^{th} 7^{th} 8^{th} 9^{th} 10^{th} 12^{th} 13^{th} 17^{th} Lib. I & the 1, 2, 3 & 4 of Lib. II [for the De Motu] in June & July 1684’.
^{7} Essay (note 1), pp. 7 ff.
^{8} In his A View of Sir Isaac Newton's Philosophy, London, 1728, Preface, [a1^{v}], quoted by Rigaud, in his Essay (note 1), Appendix, pp. 49–51.
^{9} Cajori, Florian, ‘Newton's Twenty Years' Delay in Announcing the Law of Gravitation’, in Sir Isaac Newton 1727–1927: A Bicentenary Evaluation of His Work, Baltimore, 1928, pp. 127–188, especially 128.
^{10} ‘La gravitation universelle de Kepler à Newton’ (note 1), p. 648.
^{11} Memoirs of the Life of Mr. William Whiston by Himself, London, 1749, i, 38. De Moivre, in the still unpublished memorandum he gave to John Conduitt in November 1727 (now in private possession in New York), adds independent confirmation of Whiston's story, for he relates that after Newton had found wide disagreement in his first moon-test ‘he entertained a notion that with the force of gravity, there might be a mixture of that force which the Moon would have [if] it was carried along in a vortex, but when the Tract of Picard's of the measure of the earth came out, he began his calculation anew, & found it perfectly agreeable to the Theory …’ (I quote from the late nineteenth-century copy (Luard's) in ULC. Add. 4007, 707^{r}). We need not smile at the anti-Cartesian Newton still being willing to accept as a practical physical hypothesis the possible disturbing action of a terrestrial vortex on the moon's orbit. In autograph notes made about 1670 on the rear endpapers of his copy (Trinity College, Cambridge, NQ. 18.36) of Wing's, VincentAstronomia Britannica, London, 1669, Newton explains the disturbance of the moon's orbit from its theoretical elliptical shape through the action of the solar vortex (which ‘compresses’ the terrestrial one bearing the moon by about 4^{1}3 of its width).
^{12} ULC. Add. 3958.2, 45^{r} and 3958.5, 87^{r}–88r. Professor Turnbull first announced his discovery on p. 4 of the Manchester Guardian for Saturday, 3 10 1953, in an article (‘Isaac Newton's Letters’) reprinted in the Manchester Guardian Weekly for Thursday, 8 10 1953, p. 11. The earlier of these documents (Add. 3958.2, 45^{r}) was briefly discussed by Hall, A. R. at the end of ‘Newton and the Calculation of Central Forces’, Annals of Science, 1957, xiii, 62–71, but more adequately described (in relation to a photocopy of the document) by Herivel, J. W. in his ‘Interpretation of an Early Newton Manuscript’, Isis, 1961, lii, 410–416, and independently by Turnbull himself in his edition of The Correspondence of Isaac Newton, Cambridge, 1961, iii, 46–54. The latter was incompletely transcribed and discussed by Hall in his ‘Newton and the Calculation of Central Forces’ and given in full by Turnbull in Newton's, Correspondence, 1959, i, 297–303.
^{13} See his articles, ‘Newton's Discovery of the Law of Centrifugal Force’, Isis, 1962, liii, 546–554, and ‘Sur les premières recherches de Newton en dynamique’, Révue d'histoire des sciences, 1962, xv, 105–140. It is now certain, for example, that in his 1666 moon-test he would have taken a value of 3,500 (Italian) miles for the earth's radius (much too low), that (though, in ignorance of the experiments written up by Riccioli in his 1651 Almagestum Novum, he at first accepted from Galileo's 1638 Discorsi the very low estimate of 8 ft./sec.^{2}) he soon determined an accurate measure of the force of terrestrial gravitation at the earth's surface (about 15½ ft./ sec.^{2}) from his own experiments with conical and vertical pendulums, and that by late 1665 he had arrived at a developed quantitative theory of centrifugal force. On the other hand, there seems now little hope of finding direct autograph evidence of any of Newton's moon-tests, while ULC. Add. 3958.5, 87^{r} (apparently the document to which Newton referred back in later life in support of his claim to have deduced an inverse-square solar gravitational field from Kepler's third law) deduces only that in circularly-orbiting planets the centrifugal forces from the sun are as their corresponding inverse squared distances from it and we must assume that at the time of its composition (c. 1670?) Newton was still unwilling to accept an exact balance between (apparent) planetary centrifugal force and that of solar gravity.
^{14} Astronomia Nova ΑΙΤΙΟΛΟΓΗΤΟΣ, seu Physica Cælestis, tradita commentariis de Motibus Stellæ Martis, Ex observationibus G. V. Tychonis Brahe, Prague, 1609, now available in two excellent editions by Caspar, Max (in Latin in Johannes Kepler: Gesammelte Werke, Munich, 1937, iii; and in German in Die Neue Astronomie, Munich, 1929). No acceptable secondary account of Kepler's derivation of his first two laws exists, but see Small's, RobertAn Account of the Astronomical Discoveries of Kepler, London, 1804 (re-issued Madison, Wisconsin, 1963) and pp. 172–281 of Koyré's, A.La révolution astronomique, Paris, 1961.
^{15} Johannis Kepler's, Harmonices Mundi Libri V …, Linz, 1619, re-edited most recently by Caspar, Max in Kepler's, Gesammelte Werke, Munich, 1939, vi. Compare Koyré's, La révolution astronomique (note 14), pp. 340–343.
^{16} The elliptical orbit was not accepted—though stated—by Riccioli in his 1651 Almagestum Novum, nor by Huygens until he dramatically changed his mind upon reading Newton's Principia on 14 December 1688 (Œuvres complètes de Christiaan Huygens, The Hague, 1944, xxi, 143). The practising astronomer Cassini, of course, worried many theorists in the decade from 1693 with his announcement that the observed planetary paths seemed to be Cassini-ovals (of approximate eccentricity √2).
^{17} Newton's statement in his Principia (London, 1687, Book III, p. 404, and equivalently in all later editions) that the ‘Propositio est Astronomis notissima’ thus becomes one more item of damning evidence of his relative unfamiliarity with contemporary astronomical literature in 1685 (when he apparently first wrote it down).
^{18} I would overweight the present article by giving detailed reference to this mass of observationally accurate theory, all destroyed at one blow by the appearance of Newton's Principia (where it is subsumed into a few closing paragraphs in Book 1, Prop. 31, Scholium). Unfortunately no modern secondary text begins to consider them generally, let alone in detail: indeed the outdated and far from comprehensive discussion by Delambre, J.-B. in his Histoire de l'Astronomie Moderne, 2 vols., Paris, 1821, stands alone. For the present let me merely record the names of Kepler (1609), Cavalieri (1632), Boulliau (1645 and 1657), Ward (1654 and 1656), Pagan (1657), Streete (1661), N. Mercator (1664, 1670 and 1676), Wing (1669), Cassini (1669 and 1693), Newton (1670 and 1679), Halley (1676) and Huygens (1681).
^{19} Streete, Thomas, Astronomia Carolina. A New Theorie of the Cælestial Motions. Composed according to the Best Observations and most Rational Grounds of Art. Yet farre more Easie, Expedite and Perspicuous then any before Extant, London, 1661 (re-issued with a new title-page in 1663).
^{20} Streete in fact bases his account on Ward's, SethAstronomia Geometrica: ubi Methodus proponitur qua Primariorum Planetarum Astronomia sive Elliptica [sive] Circularis possit Geometricè absolvi, London, 1656, Liber primus rather than on the Pagan's, Comte deTheorie des Planétes, Paris, 1657. (Ward had first formulated the hypothesis, by making explicit the consequences of Boulliau's planetary hypothesis as elaborated in the latter's Astronomia Philolaīca, Paris, 1645, in his brief pamphlet In Ismaelis Bullialdi Astronomiæ Philolaicæ Fundamenta, Inquisitio Brevis, Oxford, 1654 (title-page dated 1653), Cap. I.
^{21} Ismaelis Bullialdi Astronomiæ Philolaicæ Fundamenta clarius explicata, & asserta. Adversus Clarissimi Viri Sethi Wardi Oxoniensis Professoris impugnationem, Paris, 1657, particularly Caput III, 16–17.
^{22} ULC. Add. 4004, 1191^{r}. This stray sheet, clearly taken by Newton from the end of the book and inserted with his other optical papers, was diligently, if insensitively, returned to its now outlandish place in the volume by the nineteenth-century cataloguers of the Portsmouth Collection.
^{23} That is, where the upper focus angle is Ψ and mean motion T, then Ψ = T. Newton's text reads: ‘[sit] ed = … distantiæ solis a planetâ.… ab[e] – medio motui … ab aphelio. af = … diametro maximo Ellipseos = be + ed.… bd = … distantiæ focorum.’ We may note that in this paper Newton derived the polar defining equation of the ellipse referred to a focus as origin: that is, if we suppose af = 2a, bd = 2ea (or the ellipse's eccentricity to be e), ed = r and the ellipse's polar equation becomes (In common with seventeenth-century astronomical practice but contrary to modern custom we measure all angles in the figure from aphelion.) I will not stress the importance of Newton's being able to recognize the polar equation of a conic as such: taken in conjunction with the powerful Prop. 41 of Principia's Book 1, that in itself is sufficient argument against those who claim that the inverse problem of gravitation was beyond Newton's powers. In any case the formula had already appeared, in somewhat confused manner admittedly, in Mercator's, NicolausHypothesis Astronomica Nova, et Consensus ejus cum Observationibus, London, 1664, p. 4 (repeated without change in Appendix, pp. 162–184 of his Institutionum Astronomicarum Libri Duo, de Motu Astrorum communi & proprio, secundum Hypotheses Veterum & Recentiorum præcipuas; deque Hypotheseon ex observatis constructione, London, 1676).
^{24} ULC. Add. 3996, 27^{v} ff., especially 30^{r}.
^{25} Whether Newton knew it or not, Boulliau had devised his refinement merely from an empirical examination of Tycho's tables for Mars. Theoretically, however, and using the same analytical equivalents as before the correction yields or, to O(e ^{4}), Ψ = T + ¼e ^{2} sin 2T. In the second part of the scholium to Book 1, Prop. 31 of his Principia, Newton was later to attack the general theory of the upper focus equant: he there deduces from Kepler's first and second laws together that (I neglect a slight confusion in his statement of this result in the first edition.) Clearly Boulliau's correction, as also the similar ones variously formulated by Nicolaus Mercator, Vincent Wing, Huygens and (as we shall see) by Newton himself, is correct to within the cube of the ellipse's eccentricity: in practical terms, taking the limits of contemporary observational accuracy to be about I′, we may say that it is true within those limits. John Machrin (in his Laws of the Moon's Motion according to Gravity, set in appendix to Motte's, AndrewThe Mathematical Principles of Natural Philosophy. By Sir Isaac Newton. Translated into English, London, ii, Appendix, 40 ff., 1729) was, I think, the first to produce an equant hypothesis which incorporated the second correcting term sin^{3}T.
^{26} Almagestum Novum, Bologna, 1661: see Tom. I, Pars Prior, Liber VII, Sectio II, Caput V, Prop. 3, 535.
^{27} Christopher Wren, however, had stated the areal law in a brief, confused passage which introduces his construction of Kepler's problem by a curtate cycloid: ‘Asseruit Keplerus, ex causis physicis, planetas ita ferri circa solem in Orbitâ Ellipticâ, ut velocitas planetæ sit ubique distantiæ ejusdem à Sole reciproce proportionalis; unde sequentem Hypothesin ingeniose commentus est. Secat scilicet aream Ellipseos Planetariæ lineis à sole ductis in infinita Triangula Mixtilinea …: per has autem positiones ponit Planetam æqualibus temporibus ferri’ (Wallis', JohnTractatus de Cycloide, Oxford, 1659, p. 80 of the first pagination). Newton certainly made a deep study at an early age of Wren's following geometrical tract (the gist of which he repeated in Principia, Book 1, Prop. 31) and it is just possible his attention was caught by the fleeting reference to the areal law.
^{28} ULC. Add. 3996, 29^{r}.
^{29} Wing, Vincent, Astronomia Britannica: in qua per Novam, Concinnioremque Methodum hi quinque Tractatus traduntur:… Logistica Astronomica … Trigonometria … Doctrina Spherica … Theoria Planetarum … Tabulæ Novæ Astronomicæ … Cui accessit Observationum Astronomicarum Synopsis Compendaria …, London, 1669. (Newton's copy is now in Trinity College, Cambridge, NQ. 18.36.) For all its inadequacies, Wing's work was a sincere attempt to make freely available in England a sound body of doctrine relating to the ‘new’ astronomy, and was widely read in its day.
^{30} The law was not accepted by Wing as an axiom in his ‘Hypothesis Copernicana’ (pp. 120/121), but Newton added it in an interesting paragraph in his notes: ‘Est equidem regula Kepleriana quod cubi diametrorum (maximarum scilicet) sunt ut quadrata temporum revolutionis … In [Saturno, Martio] ac [Terra] ubi Author [sc. Wing] satis appropinquat has proportiones tabulæ optime consentiunt observationibus, melius consentirent in [Venere] si ejus orbita ad hanc proportionem reduceretur. Orbita [Mercurij] per refractiones ampliatur, et ob eandem rationem Veneris orbita fortasse nonnihil reducitur. An Jovis orbita ad hanc analogiam reduci potest haud scio, id vero suspicor sed hæ ejus tabulæ non satis bene conveniunt cum observationibus, et ut plurimum ex hac emendatione melius convenirent.’ The argument is interesting confirmation of Newton's willingness to modify observational fact to fit a preferred theoretical rule.
^{31} For our present purpose the significant portion of Wing's ‘Copernican hypothesis’ is his first axiom (p. 120) ‘Quòd omnia Planetarum corpora in Ellipsi circa SOLEM moventur, ita ut SOL uno Foco ejus positus est, & medius motus in altero, circa quam fermè æqualiter Planeta volvitur, æquales angulos in temporibus æqualibus describens.’ In his theories of the Earth and Mars (pp. 129 ff., 158) Wing himself implicitly produced an observationally accurate correction of this simple Wardian theory which adds an ‘equation’ of ¼e ^{2} sin 2ω to the central anomaly ω: this yields, to O(e ^{3}), the correct equation at the upper focus of ¼e ^{2} sin 2T (and to O(e ^{4}) is slightly better than Boulliau's). Newton himself did not comprehend the accuracy of Wing's correction for in his brief reference to it (‘In singulis … Planetis maximæ; mediorum motuum correctiones debent esse ut Epicyclorum diametri divisæ per maximas orbitarum diametros’) he erroneously inferred a maximum equation of e (and not ¼e ^{2}).
^{32} As in the following pages this is my translation from Newton's Latin: I have also added the centre O of orbit and the orbit itself (in broken line) to Newton's figure. Since only the shape of the figure is at issue, suppose for simplicity the main diameter AP = 2 units, the (Keplerian) eccentricity SO = e, the angle of mean motion and, following Newton's recommendation, SD = 1, so that (where, as before, we suppose the solar distance SP = r and the true longitude ). Let us. moreover, fix Newton's undetermined constants by taking the equant distance and EP/ED = λ. On the supposition that Newton's construction yields both true longitude … and solar distance r = 1 + e cos T + e ^{2}T + … in terms of mean motion T correctly to O(e ^{3}), we may with some manipulation find three bounds on E, k, l and λ, viz.: where a is some parameter.
^{33} That is, following Kepler, SC = 2 × SO.
^{34} In the notation previously used Newton suggests setting α = − ¼ (or SC = E = 2e - ¼e ^{2}) for best results. Unfortunately this clashes with his further assumption of The best way is to assume α = O (or C to be exactly at the upper focus of orbit) and that the first two deductions alone of note 32 are true: this yields, in agreement with Newton, E = 2e and values which loosely satisfy the third condition The practical effect of this supposition is to make true longitude correct to O(e ^{3}), while constructing (true for an ellipse of eccentricity The empirical origin of Newton's construction in a numerical induction from Wing's tables is the obvious conclusion to draw from this.
^{35} I do not understand this last remark, for the effect of altering the ratio EP: ED is to alter the ellipse's eccentricity and this, as Newton correctly says in the preceding sentence, will in turn alter the true longitude of the planet, making it no longer correct to O(e ^{3}). Clearly the remark is an observational induction rather than an inference from any theoretical considerations.
^{36} ‘Luna defertur in Ellipsi æquabili motu circa Centrum [medii motus], nisi quod … per compressionem vorticis impellitur versus tangentem orbis magni …:… debes potiùs … ad id referre lunares irregularitates quas Reflectionem et Evectionem vocant.’ Cf. note 11 above.
^{37} ULC. Add. 3963.1, 1^{r}/1^{v}. In this unpublished piece, which bears the subheading ‘Problemata in Lectionibus meis [sc. Lucasianis] sic resolvo syntheticè’, Newton gave formal synthetic proof of several geometrical problems (notably 1, 3, 4, 6, 9, 10, 13 and 32) which he had introduced into his Cambridge mathematical lectures in the late 1670's. In particular, the penultimate one in the manuscript (whose enunciation, ‘Prob. De inventione distantiæ Cometæ in Systemate Copernicæa’, alone is there stated) is undoubtedly a first draft of what, in the copy of those lectures he later deposited in the University archives (ULC. Dd. 9.68, especially pp. 122–124, later still to be set in print by Whiston as the Arithmetica Universalis, Cambridge, 1707, especially pp. 205–207), he claimed to have discoursed on in his second lecture of the Michaelmas term of 1680 as ‘Prob. 52. E Cometæ motu uniformi rectilineo per Coelum trajicientis locis quatuor observatis, distantiam a terra, motusque determinationem, in Hypothesi Copernicæa colligere.’ A firm antedate for the composition of this cometary proposition (which depends narrowly on material published by Hooke in his 1678 Cometa (see Gunther, R. T., Early Science in Oxford, Oxford, 1931, viii, 257–258) and by Wallis in the appendix ‘De Cometarum Distantiis Investigandis’ he added to the second edition of his Jeremiæ Horroccii Opera Posthuma, London, 1678) is late 1678, and so also, we conclude, for the following planetary one. I cannot accept a date later than December 1679 for it, when Newton's correspondence with Hooke reached its crucial phase.
^{38} Presumably, as in the preceding proposition ‘in systemate Copernicæa’, that is, presupposing a Keplerian elliptical orbit for the planet. I have added such an orbit (in broken line)—and the centre C—to Newton's figure.
^{39} Newton here cancelled ‘aphelion’. In fact, a complementary theory of equant motion may be constructed in the case where the point A lies beyond C in the aphelion distance SG such that AS remains The case is interesting historically, for it has close affiliation both with Kepler's ‘hypothesis vicaria verior’ (Astronomia Nova (note 14), 257 ff. = Gesammelte Werke, iii, 313) and that expounded in 1664 by Nicolaus Mercator in 1664 in his Hypothesis Astronomica Nova (note 23), both presumably developed on much the same sort of observational evidence as that available to Newton in the present case. It is interesting to note that Kepler supposed the ratio AS/CS (= in theory) to be about while Mercator—partly on theophysical grounds—took it as
^{40} Newton assumes as given, in fact, the shape of the planetary orbit and three corresponding pairs of values of the true longitude and mean motion obviously these latter determine corresponding values of (their difference) and so of so that—if, that is, such a point does exist—two pairs will fix the point A in the plane and the third confirm it in position.
^{41} If, following the notation established in preceding pages, we suppose we deduce immediately that sin [k(T—φ)] = E sin [T—k(T—φ)]. In the elliptical hypothesis this gives, the true planetary longitude in terms of mean motion T correctly to O(e ^{3}) if we take, as Newton suggests, and also in other words, if we set and then the model constructs
^{42} In this context it is interesting to recall Newton's remark to Halley on 20 June 1686 (Correspondence of Isaac Newton (ed. Turnbull, H. W.), Cambridge, 1960, ii, 436) that ‘Kepler knew y^{e} Orb to be not circular but oval & guest it to be Elliptical’. For Huygen's preferred curve, see note 16. Kepler described his ‘via buccosa’ in his Astronomia Nova (note 14), Caput LVIII, 283 ff., having wrestled with it for several months in the early summer of 1605. No one, I think, has pointed out that his ultimate rejection of it (p. 284) for the perfect ellipse (which, he argues, both fits observational fact and yet determines a planet's true anomaly by an amount differing by up to 5½′ from the equivalent anomaly determined in the ‘buccosa’ hypothesis) is fallacious. Kepler, in fact, there forgot to allow for the variation which (by the areal law) the non-symmetrical shape of the ‘buccosa’ imposes on the mean motion from that in the corresponding ellipse: the true maximum difference in true anomaly between the two hypotheses is only about ¾′, an amount observationally negligible in Kepler's day with respect to instrumental error.
^{43} We will remember that in his letter of 20 June 1686 to Halley, he stressed the straight forwardness of the mathematical deduction of the inverse-square field from Kepler's third law after ‘Hugenius [in his 1673 Horologium Oscillatorium] had told how to find y^{e} [centrifugal] force in all cases of circular motion’ (Correspondence (note 42), ii, 438).
^{44} Kepler himself had in his Astronomia Nova postulated a force of gravity decreasing linearly with distance from the sun which, in conjunction with one of magnetism, modified a pristine circular path into the observable elliptical orbit. Ismael Boulliau, however, though in his Astronomia Philolaica of 1645 he had (on the now well-known analogy of gravity to the dispersal of light from a point-source) substituted for Kepler's linear decrease one varying as the inverse-square of the solar distance, could yet in 1657 voice a popular attitude when he wrote that ‘nollem Kepleri famæ detrahere; cui Mathematicarum artium studiosi, præcipuè vero Astronomi, multum debent. Ipse enim mira sagacitate viam Planetæ Ellipticam esse primus invenit, adeoque rationem veram determinandi motus cœlestes tradidit. Coniecturis autem Physicis minus tribuisse virum illum vellem’ (Astronomiæ Philolaicæ Fundamenta clariùs explicata (note 21), 45). In the theory of terrestrial gravity—not yet, of course, identified at large with any celestial attraction—Beaugrand in his Geostatique (Paris, 1636) and Roberval in his Aristarchi Samii de Mundi Systemate … Libellus (Paris, 1644) had expounded qualitative theories, savagely criticized by Descartes in correspondence with Mersenne, which suggested a linear decrease of gravity with distance from the earth's centre (above its surface at least, for Roberval hinted that below the variation would be reversed). Descartes' letters on the subject were printed in the first volume of Clerselier's posthumous edition (in French in 1657) whose 1668 Latin version Newton knew well. (Compare ULC. Add. 4003, printed as ‘De Gravitatione et Æquipendio Fluidorum’ in the Halls' Unpublished Scientific Papers (note 5), 89–121, especially 113, 1. 5.) Moreover, Descartes' criticism that in a force-field round a finite centre not varying directly as the distance the centre of gravity is not stable (originally made by him in his letter of 13 July 1638 to Mersenne) was inserted with Johann Hudde's attempted amplification in Liber V, Sectio XXX of van Schooten's, FransExercitationum Mathematicarum Libri Quinque (Leiden, 1657, pp. 515–516), a work studied minutely by Newton as an undergraduate—indeed Definition 1 ‘Of Gravity’ in his October 1666 fluxional tract (Unpublished Scientific Papers, p. 58) is seemingly a direct reference to Descartes' point.
^{45} See Koyré's, A. able study, ‘A Documentary History of the Problem of [free] Fall from Kepler to Newton. De Motu Gravium Naturaliter Cadentium in Hypothesi Terræ Motæ’, Transactions of the American Philosophical Society, 1955, xlv, 329–395. A complex line of research took its lead from Galileo's remark (Dialogo dei due Massimi Sistemi del Mondo, 1632, 145 ff.)that a body falling freely with, initially, the earth's angular velocity under a constant gravitational tendency to its centre would travel in a semicircle to that centre. In the hypothesis that the centre is indefinitely distant the path would, of course, be a parabola, where the body traverses a horizontal distance proportional to the time of the earth's rotation and falls through a vertical distance instantaneously proportional to the square of the time. Supposing only that the horizontal path is bent into a circle arc round the earth's centre when it is at a finite distance (so that the verticals become radii vectores through that centre) Fermat, in 1636, ingeniously deduced a spiral orbit r = R(1 — φ^{2}/α^{2})^{2}. Generalizing Galileo in a similar way, Mersenne had, about 1635, determined that the line of uniform oblique descent, a straight Une when the earth's centre is supposed infinitely distant, is a logarithmic spiral when the verticals are supposed convergent to a finite centre (Harmonie Universelle, 1636, Tome i, Livre ii, Prop. 8, pp. 113–118: compare Descartes' letters to him of 13 July and especially 12 September 1638 = Clerselier, 1657, i, 327–354). The supposition common to both that the angular velocity of a falling body round the earth's centre will remain constant is, of course, erroneous and one which reveals yet again how little mathematicians in the mid-seventeenth century were willing to avail themselves of Kepler's areal law (or even an approximate equivalent). Before Newton, only Borelli had any glimmering of this necessary application. (Compare the tentative correction of the conventional view in his De Vi Percussionis, Bologna, , 1667, pp. 108 ff., and its more forceful statement in his Risposta … alle Considerazioni fatte sopra alcuni luoghi del suo Libro della Forza della Percossa del R.P.F.… Angeli, Messina, 1668, pp. 16 ff., where Borelli implicitly uses the Keplerian approximation to the areal law v = k/r.)
^{46} In his Theoricæ Mediceorum Planetarum a Causis Physicis Deductæ, Florence, 1666, Borelli suggested that the elliptical motion of Jupiter's satellites (and so, by implication, of the solar planets) could be explained through the interaction of a Huygenian centrifugal force and a constant centripetal gravitational tendency, instantaneously in disequilibrium, whose total effect was thus to modify a circular force-free path (sustained round the force-centre by the continued action of a transverse ‘impetus’). (See Koyré, A., La révolution astronomique (note 14), pp. 474–)Leibniz, in his ‘Tentamen de Motuum Cœlestium Causis’, Acta Eruditorum, 1689, pp. 81–96, was to add the two refinements which made the Borellian theory viable: he assumed that the centrifugal force c ^{2}/^{3}, where is the instantaneous distance from the force-centre, was of itself (in his dynamical system) capable of sustaining motion in a straight line, and, secondly, that the gravitational force is a generally variable function of the radial distance, deducing in his §15 that for elliptical motion round a force-centre situated ata focus the central attraction must be as the inverse-square of that distance. It is an immediate corollary (as in Newton's theory) that the central force in any orbit is measured by c ^{2}/r ^{3}—r¨, where is the radial acceleration in the orbit. (Compare Aiton's, E. J. recent articles, ‘The Celestial Mechanics of Leibniz’ and ‘The Celestial Mechanics of Leibniz in the Light of Newtonian Criticism’, Annals of Science, 1960, xvi, 65–82, and 1962, xviii, 31–42.)
^{47} Compare Patterson, L. D., ‘Hooke's Gravitation Theory and its Influence on Newton’, Isis, 1949, xl, 327–341, and 1950, xli, 32–45; Hall, A. R., ‘Two Unpublished Lectures of Robert Hooke’, Isis, 1951, xlii, 219–230; Koyré, A., ‘An Unpublished Letter …’ (note 3), especially pp. 317–319; and Lohne, J. A., ‘Hooke versus Newton’ (note 1), especially pp. 10–18.
^{48} In England in the 1670's Nicolaus Mercator was its chief popularizer, printing its correct enunciation both in an article, ‘Some Considerations … concerning the Geometrick and direct Method of Signor Cassini for finding the Apogees, Excentricities and Anomalies of the Planets’, Philosophical Transactions, 1670, v, no. 57, 1168–1175 (especially p. 1174: ‘Keplerus … lineam veri motûs Planetæ æqualibus temporibus æquales areas Ellipticas verrere professus est’) and in Caput XX ‘De Hypothesi Kepleri’ of his excellent 1676 compendium, Institutionum Astronomicarum Libri Duo (note 23) (especially p. 145: ‘…areæ, quas radius vector à Sole ad Planetam extensus verrit, cresc[u]nt æqualiter æqualibus temporis momentis’). Mercator was on terms of some familiarity with Newton, having corresponded with him in the mid-1670's on the moon's motion. (Compare Institutionum Astronomicarum Libri Duo, pp. 286–287; and Newton's Principia, Lib. 3, Prop. 17.) Newton's own well-thumbed copy of the Institutiones (Trinity College, Cambridge, NQ. 10.152), though not annotated at the passages where Mercator discusses planetary hypotheses, shows clear signs throughout of having been read continuously.
^{49} Newton to Halley, 20 June 1686: ‘[Hooke] has done nothing & yet written in such a way as if he knew & had sufficiently hinted all but what remained to be determined by y^{e} drudgery of calculations & observations, excusing himself from that labour by reason of his other business: whereas he should rather have excused himself by reason of his inability. For tis plain by his words that he knew not how to go about it. Now is not this very fine? Mathematicians that find out, settle & do all the business must content themselves with being nothing but dry calculators & drudges & another that does nothing but pretend & grasp at all things must carry away all the invention …’ I would not wish to assert that Hooke was devoid of mathematical talent (cf. Patterson, L. D. in Isis, 1950, xli, 35–38), but merely that his mathematical genius was more well-informed than brilliant. Thus, for example, the autograph notes in his copy of Fermat's Opera Varia (Toulouse, 1679) reveal a surface comprehension rather than an intimate understanding.
^{50} From the autograph draft in Trinity College, Cambridge (O. 11a.1^{22}), reproduced in Newton's Correspondence (note 42), ii, 297. Hooke had already announced this fundamental assumption in print in his Attempt to prove the motion of the Earth by Observation, London, 1674, pp. 27 ff., likewise without specifying the quantitative variation of the central attraction with the radial distance.
^{51} Newton to Hooke, 28 November 1679, from the autograph in Trinity College, Cambridge (R. 4.48.1) = Correspondence, ii, 300–301.
^{52} Newton's figure, in which ADEC is the line of free fall, is drawn from the viewpoint of an observer rotating with the earth, and this has misled several historians. H. W. Turnbull, for example, takes Hooke to task for suggesting in his reply that Newton had proposed the falling body would spiral round the earth's centre several times before reaching it: ‘According to Newton's figure there is one revolution [only]’ (Correspondence, ii, 307, n. 12). If, with Hooke, we take the position of a stationary observer, the path will indeed be a spiral of more than one revolution (and I have inserted its possible representation as the thick curve in the present reproduction of Newton's diagram). As J. A. Lohne has pointed out, Newton's figure is extremely badly reproduced in the standard texts: from a close study of the original (whose photocopy Lohne inserted in his ‘Hooke versus Newton’ (note 1), p. 9) it is clear both that the path ADE was drawn tangent to ABC (as it should be) and that Newton did not continue his spiral quite all the way to the earth's centre. It is worthwhile to notice, too, that Newton here rejected implicitly the Fermatian hypothesis of unchanged uniform angular rotation (which would make the path ADE fall wholly in the line ABC).
^{53} Hooke, to Newton, , 9 12 1679 (Correspondence, ii, 305–306). Koyré, who there first published the letter, gives its photocopy on pp. 328/330 of his Unpublished Letter … (note 3). The original is but a poor amanuensis (dictated?) copy and I have not hesitated to alter its orthography for clarity's sake.
^{54} This hypothesis of a perpetual imbalance between a variable centrifugal force and a constant gravitational attraction which together instantaneously modify a circular path (of zero ‘ascent & descent’) is, of course, exactly Borelli's. It is hard to resist the impression that Newton, caught off balance, had looked up the details of his theory of compounded forces immediately on receiving Hooke's letter with its repeated insistence that the path of free fall be explained by compounding motions.
^{55} Newton, to Hooke, , 13 12 1679 (Correspondence, ii, 307–308). Newton (with Borelli) here makes no explicit assumption that the vis centrifuga will by itself carry the falling body out of its circular ‘inertial’ path into a linear ‘gravity-free’ one traversed with uniform motion. (He had indeed in January 1665, in notes entered in his Waste Book, written ‘[Axiome] 2 A quańtity will always move on in y^{e} same straight line (not changing y^{e} determination nor celerity of its motion) unlesse some externall cause divert it’ (ULC. Add. 4004, 10^{v}), but I find this early statement of a linear inertial line impossible to reconcile with his present implicit acceptance of Borelli's imbalance of forces with its corollary of a circular force-free path.) If, however, we accept this assumption we may straightforwardly apply a ‘Newtonian’ analysis to derive the polar equation of the path of free fall as
where the minimum value of the radius vector aC (or the initial velocity at A in the direction AM is and (Compare Pelseneer, J., ‘Une lettre inédite de Newton’, Isis, 1929, xii, 237–254, especially 250 ff.; and Lohne's ‘Hooke versus Newton’ (note 1), p. 43.) Clearly
and is a maximum ) for λ = O or R = p. (Compare Principia, Lib. 1, Prop. 45, Example 1) Newton's autograph sketch, in which p is rather greater than ½R (cf. the photocopy reproduced by Lohne on p. 27 of his ‘Hooke versus Newton’), agrees not too well with this theoretical curve, which in one libration swings through a central angle somewhat less than Newton's. But it is clear that Newton at this time had no such exact theory, perhaps only the intuitive knowledge that as p decreases from p = R to p = O the angle decreases from π/√3 to
^{56} We see very clearly the great stumbling-block to further development implicit in Hooke's continued acceptance of the approximate form of Kepler's areal law, v = k/r (which therefore varies as , where f(r) α 1/r ^{2} is the ‘Attraction’). Compare Koyré's ‘Unpublished Letter …’ (note 3), p. 336, n. 118.
^{57} Hooke, to Newton, , 6 01 1679/1680 (Correspondence, ii, 309).
^{58} Hooke, to Newton, , 17 01 1679/1680 (Correspondence, ii, 313).
^{59} Newton, to Halley, , 20 06 1686: ‘[I] never answered his third [letter]’, Correspondence, ii, 436.
^{60} ‘De motu corporum in gyrum’, ULC. Add. 3965.7, 55^{r}-62 bis ^{r}. The figure I give is an accurate reproduction of that accompanying Prob. 2 (57^{r}) of this autograph text, though for uniformity I have rotated it through a right angle: note particularly that RQ. is in line with QS and not drawn (as later) parallel to SP.
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