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Newton's Early Thoughts on Planetary Motion: AFresh Look
Derek T. Whiteside
The British Journal for the History of Science / Volume 2 / Issue 02 / December 1964, pp 117 -137DOI: 10.1017/S000708740000203X, Published online: 05 January 2009
Link to this article: http://journals.cambridge.org/abstract_S000708740000203X
How to cite this article:Derek T. Whiteside (1964). Newton's Early Thoughts on Planetary Motion: A FreshLook. The British Journal for the History of Science, 2, pp 117-137 doi:10.1017/S000708740000203X
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NEWTON'S EARLY THOUGHTS ON PLANETARYMOTION: A FRESH LOOK
By DEREK T. WHITESIDE
'. . . This Curve truly Calculated will shew the error of thosemany lame shifts made use of by astronomers to approach thetrue motions of the planets with their tables.'
Hooke writing to Newton on 6 January i6yg\8o.
T H E conventional view1 of the prehistory of Newton's synthesis in thePrincipia of his predecessors' work in planetary theory and terrestrialgravitation is still not seriously changed from that which Newton himselfchose to impose on his contemporaries at the end of his life. In his ownwords:
'. . . the same year [1666] I began to think of gravity extending to ye orbof the Moon & having found out how to estimate the force wa w°h [a] globerevolving within a sphere presses the surface of the sphere from Keplers ruleof the periodical times of the Planets being in a sesquialterate proportion oftheir distances from the centers of their Orbs, & having deduced that theforces w°h keep the Planets in their orbs must [be] reciprocally as the squaresof their distances from the centers about wch they revolve: & thereby comparedthe force requisite to keep the Moon in her Orb with the force of gravity atthe surface of the earth, & found them answer pretty nearly . . .'2
'In the end of the year 1679 m answer to a Letter from Dr Hook thenSecretary of the R.S. I wrote that whereas it had been objected against thediurnal motion of the earth that it would cause bodies to fall to the west,the contrary was true. For bodies in falling would keep the motion whichthey had from west to east before they began to fall . . . Dr Hook replied soonafter that they would do so under the Equator but in our latitude they wouldfall not exactly to the east but decline from the east a little to the south. . . .And he added that they would not fall down to the center of the earth butrise up again & describe an Oval as the Planets do in their orbs. WhereuponI computed what would be the Orb described by the Planets. For I had foundbefore by the sesquialterate proportion of the tempora periodica of the Planetswith respect to their distances from the Sun, that the forces wch kept themin their Orbs about the Sun were as the squares of their mean distances fromthe Sun reciprocally: & I found now that whatsoever was the law of the forces
1 I refer particularly to S. P. Rigaud, Historical Essay on the First Publication of Sir IsaacNewton's Principia, Oxford, 1838; D. Brewster, Memoirs of the Life, Writings, and Discoveries ofSir Isaac Newton, Edinburgh, 1855, i, 289 ff.; W. W. Rouse Ball, An Essay on Newton's 'Principia',London, 1893; and L. T. More, Isaac Newton, New York, 1934 (re-issued 1962), 288 ff. See alsoAlexandre Koyre, 'La gravitation universelle de Kepler a Newton', Archives internationalesd'histoire des sciences, 1951, iv, 638-653; and J. Lohne, 'Hooke versus Newton: An Analysis ofthe Documents in the Case on Free Fall and Planetary Motion', Centaurus, i960, vii, 6-52, forrecent variants on the traditional account.
1 ULC. Add. 3968.41, 85r. This now celebrated passage, a fuller version of which (notvery accurately transcribed) was first printed in A Catalogue of the Portsmouth Collection of Booksand Papers written by or belonging to Sir Isaac Newton, Cambridge, 1888, p. xviii, and many timessince, is extracted from a cancelled draft of a letter which Newton wrote to Des Maizeauxin the summer of 1718 when the latter was gathering material for his Recueil, London, 1720.
THE BRITISH JOURNAL FOR THE HISTORY OF SCIENCE VOL. a NO. 6 (1964)2B
u 8 DEREK T. WHITESIDE
w°h kept the Planets in their Orbs, the areas described by a Radius drawnfrom them to the Sun would be proportional to the times in -w°h they weredescribed. And by the help of these two propositions I found that their Orbswould be such Ellipses [i.e. with the sun as force-centre set in the lower focus]as Kepler had described.'3
The sequel is well known and in its main lines factually incontrovertible.Some time in the summer of 16844 Halley went to see Newton in Cam-bridge to talk with him about planetary paths and gravitational theoryand, pleasantly surprised to hear of Newton's researches of four yearsbefore, encouraged him to expand his first results into a short tract,the De Motu,5 which was registered at the Royal Society the followingFebruary. In little more than another eighteen months, in the summer of1686, the final version of the Philosophic Naturalis Principia Mathematicaas we know it today, was ready for the press.6
What has perhaps most worried historians in this traditionalchronology is the apparently unexplained decade and a half gap betweenthe 1666 moon-test and the corresponding deduction at about the sametime from Kepler's third law which together gave satisfactory obser-vational justification for an inverse-square gravitational field round theearth and sun, and Newton's application of that induction, underHooke's encouragement in December 1679, to explain planetary motionin ellipses and his still later generalization of it as his principle of universalgravitation. More than a century ago Rigaud,7 summarizing such earlierviews as those of Henry Pemberton,8 put forward strong evidence for
3 ULC. Add. 3968.9, ioir, 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 ofthe Principia which Newton was planning towards the end of 1714. For an excellent accountof the Newton-Hooke correspondence (and a full bibliography up to 1950), see A. Koyre's'An Unpublished Letter of Robert Hooke to Isaac Newton', his, 1952, xliii, 312-337.
< Most authorities accept the month as August, but J. W. Herivel, in his 'Halley's FirstVisit to Newton', Arch. int. d'hist. sci., i960, xiii, 63-65, argues for May.
5 First printed by Rigaud in his Essay (note 1), Appendix, 1-19, from the registered copyin the Royal Society archives (Register Book 6, 218), and later from a collation of the twovariant Newton autograph/amanuensis-copied originals (Add. 3965.7, 4Or-54r/55r-62 bisT)in the University Library, Cambridge, by Rouse Ball in his Essay (note 1), 35-56, and byA. R. and M. B. Hall 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 thatJ. W. Herivel has recently sought to identify the original researches of Newton in late 1679in a Newton autograph (ULC. Add. 3965.1) first published in Latin translation by Whistonin 1710. (See Herivel's 'The Originals of the two Propositions Discovered by Newton inDecember 1679?' and 'Newtonian Studies IV, Arch. int. d'hist. sci., i960, xiv, 23-33; '96a,xvi, 13-22. Also compare W. Whiston, Prelectiones Physico-Mathematice, Cambridge, 1710(re-issued 1726), Lect. XIV/XV (for 5/19 February 1704/5), where (p. 137) Whiston notesthat the tract he gives is published 'qualem nempe earn e charta MS Ipsius Newtoni olimacceperam'.)
6 In further confirmation, in a second English draft preface to his abortive second revisionof the Principia, Newton remarked about the end of 1714 (ULC. 3968.9, 106*; cf. Brewster'sMemoirs (note 1), i, 471) that of its propositions 'about ten or twelve . . . were composed before[December 1684], viz4 the 1st & nth [m answer to Hooke's challenge] in December 1679, the6th 7th 8th gth iOth i2th I3th i7th Lib. I & the I, 2, 3 & 4 of Lib. II [for the De Motu] in June&July 1684'.
1 Essay (note 1), pp. 7 ff.8 In his A View of Sir Isaac Newton's Philosophy, London, 1728, Preface, [aiv], quoted by
Rigaud in his Essay (note 1), Appendix, pp. 49-51.
Newton's Early Thoughts on Planetary Motion 119
the hypothesis that a significant cause of the delay was Newton's lackof trust in the empirical results of his 1666 moon-test, spoilt initiallyby the erroneous value (much too small) which he took for the earth'sradius. More recently Cajori9 has developed the viewpoint, originallypresented by Adams and Glaisher, that 'the real cause of the delay wasthe question how a sphere attracts an outside particle . . . [one] notclearfed] up until sometime in 1685', while Koyrero has rightly urgedthe obvious point that Newton (no more than Wren, Hooke and Halley)was not able before 1679 to deduce mathematically that a (closed)orbit in an inverse-square gravitational field must be elliptical, addingalso that Newton in 1666 could not have known from available printedsources an approximately true value for the gravitational accelerationat the earth's surface. Certain other inconsistencies in the printed litera-ture seem to have been ignored: for example, Whiston's remark" thatwhen Newton had been disappointed in his first moon-test his failuremade him 'suspect that this Power [which deflects the moon from itsinertial path] was partly that of Gravity, and partly that of Cartesius'sVortices, [so that] he threw aside the Paper of his Calculation, and wentto other Studies'. Only in the past decade has a systematic search of themass of Newton's unpublished papers been made for unpublished materialwhich may shed a new light on the matter. Since 1953, in particular, ourknowledge has been considerably enriched by H. W. Turnbull's discoveryof two important autograph documents12 in the Cambridge University
9 Florian Cajori, '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 a 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 inprivate possession in New York), adds independent confirmation of Whiston's story, for herelates that after Newton had found wide disagreement in his first moon-test 'he entertaineda notion that with the force of gravity, there might be a mixture of that force which the Moonwould have [if] it was carried along in a vortex, but when the Tract of Picard's of the measureof the earth came out, he began his calculation anew, & found it perfectly agreeable to theTheory . . . " (I quote from the late nineteenth-century copy (Luard's) in ULC. Add. 4007, 7O7r).We need not smile at the anti-Cartesian Newton still being willing to accept as a practicalphysical 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 Vincent Wing's Astronomia Britannica, London, 1669, Newtonexplains the disturbance of the moon's orbit from its theoretical elliptical shape through theaction of the solar vortex (which 'compresses' the terrestrial one bearing the moon by aboutis of its width).
11 ULC. Add. 3958.2, 45r and 3958.5, 87r-88r. Professor Turnbull first announced hisdiscovery on p. 4 of the Manchester Guardian for Saturday, 3 October 1953, in an article ('IsaacNewton's Letters') reprinted in the Manchester Guardian Weekly for Thursday, 8 October 1953,p. 11. The earlier of these documents (Add. 3958.2, 45r) was briefly discussed by A. R. Hallat 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 J. W. Herivelin his 'Interpretation of an Early Newton Manuscript', Isis, 1961, lii, 410-416, and independentlyby 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 theCalculation of Central Forces' and given in full by Turnbull in Newton's Correspondence, 1959,i, 297-303.
120 DEREK T. WHITESIDE
Library: these considerably illuminate if somewhat further confuse ourinsight into Newton's early dynamical and astronomical thought,especially when taken in conjunction with Herivel's investigation ofstill earlier dynamical papers in Newton's Waste Book.13 However,certain basic assumptions continue to be made unquestioningly oraccepted unconsciously by Newton's historians, most importantly thatfrom the first Newton both knew and accepted Kepler's three hypothesesof planetary motion and that (as later in the Principia) he always dis-tinguished centrifugal from centripetal accelerations in the planetaryorbits merely by supposing the former to be only the apparent effects ofquantitatively equal ones of the latter. What, by an examination ofNewton's hitherto unpublished early astronomical writings, I wish toshow in the present article is that at various times between early 1664and the winter of 1679/80 (and possibly even later) each of these assump-tions requires serious modification and that we must henceforward settleon late 1679 (and not 1665 or 1666) as the crucial formative period inthe development of those astronomical ideas which were in the middle1680's, together with his earlier Cartesian researches into terrestrialmechanics, to be synthesized in the Principia.
Let us begin by investigating Newton's knowledge and appreciationof Kepler's three planetary hypotheses. The first (that planets move infixed elliptical orbits round the sun placed at a focus and, in Kepler'sextension, that the sun is the centre of a force of gravity which attractsthe planets to it into those orbits from out a defined gravity-free path)and the second (that the radii vectores joining the planets to the sun sweepout areas proportional to the time in which they traverse correspondingorbital arcs) had both been thrown out at the reader as semi-empirical
'3 See his articles, 'Newton's Discovery of the Law of Centrifugal Force', Isis, 1962,Hii, 546-554, and 'Sur les premieres recherches de Newton en dynamique', Rdvue d'histoire dessciences, 1962, xv, 105-140. It is now certain, for example, that in his 1666 moon-test he wouldhave 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 atfirst accepted from Galileo's 1638 Discorsi the very low estimate of 8 ft./sec.2) he soon determinedan accurate measure of the force of terrestrial gravitation at the earth's surface (about 15J ft./sec.2) from his own experiments with conical and vertical pendulums, and that by late 1665he had arrived at a developed quantitative theory of centrifugal force. On the other hand, thereseems now little hope of finding direct autograph evidence of any of Newton's moon-tests,while ULC. Add. 3958.5, 87r (apparently the document to which Newton referred back inlater life in support of his claim to have deduced an inverse-square solar gravitational fieldfrom Kepler's third law) deduces only that in circularly-orbiting planets the centrifugal forcesfrom the sun are as their corresponding inverse squared distances from it and we must assumethat at the time of its composition (c. 1670?) Newton was still unwilling to accept an exactbalance between (apparent) planetary centrifugal force and that of solar gravity.
Newton's Early Thoughts on Planetary Motion 121
guesses—particularly so the latter—in Kepler's Astronomia Nova of i6oo,,I+
while the third (that the squares of the periodical times of orbit of theplanets are proportional to the cubes of their mean distances from thesun) had been much more carefully and directly established by anuncluttered examination of observed orbital times and their collationwith corresponding planetary distances in his Harmonice Mundi15 of tenyears later. Though the fact was reasonably well known to some olderhistorians of astronomy, it has come recently as a surprise to many thatin the interval 1609 to 1687 only the third of Kepler's hypotheses camenear to widespread acceptance by his contemporaries. Of the two othersthe first was, if at all,16 tentatively accepted as a plausible orbit whichneeded further factual confirmation before it could be accepted unreser-vedly, while the second, virtually unjustifiable in a crucial way by obser-vation and empirical inference, was seemingly firmly accepted by no oneand even its formal enunciation but rarely stated in the period.17 Inthe latter's place to fix the position of a planet in its orbit in time therewas developed, in modification of Ptolemy's simple bissextile hypothesisfor regulating motion in his posited eccentric circle orbit, an elaborate,variegated, loosely consistent set of theories which presupposed that theangle (which I will call the 'upper focus angle') made by a planet withthe aphelion at a point (the equant) in line with the sun on the maindiameter of the planetary orbit and at an equal distance from the centrein the opposite direction was, approximately at least, proportional to thetimes of orbit and hence could be used to represent the mean motionof the planet. Suitable further modification would then produce a moreaccurate theory, and indeed those introduced historically in the periodbefore 1687 gave rise to a variety of sophisticated equant theories forregulating planetary motion, many of which rendered the maximumtheoretical error less than 1' of arc and so observationally negligible incomparison with the errors implicit in contemporary observational
'* Astronomia Nova AITIOAOFHTOS, seu Physica Calestis, tradita commentariis de MotibusStelU Martis, Ex observationibus G.V. Tychonis Brake, Prague, 1609, now available in two excellenteditions by Max Caspar (in Latin in Johannes Kepler: Cesammelte Werke, Munich, 1937, iii;and in German in Die Neue Astronomic, Munich, 1929). No acceptable secondary account ofKepler's derivation of his first two laws exists, but see Robert Small's An Account of the AstronomicalDiscoveries of Kepler, London, 1804 (re-issued Madison, Wisconsin, 1963) and pp. 172-281 ofA. Koyrf's La revolution astronomique, Paris, 1961.
"5 Johannis Keppleri, Harmonices Mundi Libri V . . ., Linz, 1619, re-edited most recently byMax Caspar in Kepler's Gesammelte Werke, Munich, 1939, vi. Compare Koyre's La revolutionastronomique (note 14), pp. 340-343.
16 The elliptical orbit was not accepted—though stated—by Riccioli in his 1651 AlmagestumNovum, nor by Huygens until he dramatically changed his mind upon reading Newton'sPrincipia on 14 December 1688 (CEuvres computes de Christiaan Huygens, The Hague, 1944, xxi, 143).The practising astronomer Cassini, of course, worried many theorists in the decade from 1693with his announcement that the observed planetary paths seemed to be Cassini-ovals (ofapproximate eccentricity \ / 2 ) -
•' Newton's statement in his Principia (London, 1687, Book III, p. 404, and equivalentlyin all later editions) that the 'Propositio est Astronomis notissima' thus becomes one moreitem of damning evidence of his relative unfamiliarity with contemporary astronomicalliterature in 1685 (when he apparently first wrote it down).
122 DEREK T. WHITESIDE
techniques.18 We will recognize the fundamental block to theoreticaladvance implicit in this universal repudiation not only of the theoreticalbut even of the physical validity of the areal law. And we ask immediately:what was the extent of Newton's knowledge of this complex of theoryand observation developed by his predecessors in his early years beforehe developed the dynamical structure laid out in the Principia?
Newton's earliest known notes on astronomical theory are thosehe made as an undergraduate, perhaps towards the end of 1664, onStreete's Astronomia Carolina.19 Having accepted Kepler's first law, indiscussing the 'Theory of the Primary Planets Motions' the latterhad there elaborated the simple theory ('formerly published by theCount Pagan, Dr. Seth Ward and others'20) which equates the upperfocus angle exactly with that of mean motion of the planet in its ellipticalorbit, and had then, 'because it is found by diligent observations of thePlanets true places . . . that the Elliptick ^Equation here used requirescorrection', proceeded (pp. 18 ff.) to expound an elegant refinementof the Wardian hypothesis which is historically due to Boulliau" (butthe 'demonstration whereof Streete misleadingly borrows from a certainRobert Anderson). In his Waste Book22 Newton in an early fragmentmade the simple hypothesis of mean motion basic in an elegant techniquehe developed for finding the eccentricity of the earth's elliptical orbitfrom the observed apparent diameters of the sun at quadrant, but it issufficient for our present purpose to note the elements of his scheme:in the accompanying figure (much simplified from Newton's), the planete is set to traverse an elliptical orbit eaf of centre k, foci b and d, andmain diameter qf round the sun d set at a focus so that its position in
18 I would overweight the present article by giving detailed reference to this mass ofobservationally accurate theory, all destroyed at one blow by the appearance of Newton'sPrincipia (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 J.-B. Delambre in his Histoire deI'Astronomie Moderne, 2 vols., Paris, 1821, stands alone. For the present let me merely record thenames 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).
•9 Thomas Streete, Astronomia Carolina. A New Theorie qf the Celestial Motions. Composedaccording to the Best Observations and most Rational Grounds qf Art. Yet fane more Easie, Expedite andPerspicuous then any before Extant, London, 1661 (re-issued with a new title-page in 1663).
20 Streete in fact bases his account on Seth Ward's Astronomia Geometrica: ubi Methodusproponitur qua Primariorum Planetarum Astronomia sive EUiptica [sive] Circularis possit Geometriciabsolvi, London, 1656, Liber primus rather than on the Comte de Pagan's Theorie des Planites,Paris, 1657. (Ward had first formulated the hypothesis, by making explicit the consequencesof Boulliau's planetary hypothesis as elaborated in the latter's Astronomia Philolaica, Paris, 1645,in his brief pamphlet In Ismaelis Bullialdi Astronomia Philolaica Fundamenta, Inquisitio Brevis,Oxford, 1654 (title-page dated 1653), Cap. I.
31 Ismaelis Bullialdi Astronomue Philolaicte Fundamenta clarius explicata, & asserta. AdversusClarissimi Viri Sethi Wardi Oxoniensis Professoris impugnationem, Paris, 1657, particularly Caput III ,16-17.
" ULC. Add. 4004, 1 i9ir. This stray sheet, clearly taken by Newton from the end of thebook and inserted with his other optical papers, was diligently, if insensitively, returned to itsnow outlandish place in the volume by the nineteenth-century cataloguers of the PortsmouthCollection.
Newton's Early Thoughts on Planetary Motion 123
orbit at any given time is fixed by taking the second (here upper) focus bas equant (or abe measures the mean motion).23 Boulliau's refinedhypothesis he wrote up about the same time in explicit notes on Streeteentered in a small undergraduate notebook:24
'Suppose bpd ye Planetary Ellipsis, bed ye circ*mscribed circle. f,s ye
foci, s = O • kfs ye meane Anomaly or angle of middle motion, ca = a linedrawne through ye point g perpendicular to bd, from whos intersection wth y6
circle draw cf to ye focus f, & from (p) its intersection wth ye Ellipsis draw(ps) to ye O • yn is p ye place of ye Planet, yn < bfp is ye corrected Anomaly,
b
'3 That is, where the upper focus angle (abe) is tp and mean motion T, then ift = T.Newton's text reads: '[sit] ed = . . . distantise solis a planeta. . . . ab[e] = medio motui . . .ab aphelio. of = . . . diametro maximo Ellipseos = be+ed. . . . bd = . . . distantise focorum.'We may note that in this paper Newton derived the polar defining equation of the ellipsereferred to a focus as origin: that is, if we suppose of = 2a, bd = 2ea (or the ellipse's eccentricity
to be «), ed = r and ode = <f>, the ellipse's polar equation becomes — = -r. (In common
with seventeenth-century astronomical practice but contrary to modern custom we measureall angles in the figure from aphelion.) I will not stress the importance of Newton's being ableto recognize the polar equation of a conic as such: taken in conjunction with the powerfulProp. 41 of Principia's Book 1, that in itself is sufficient argument against those who claim thatthe inverse problem of gravitation was beyond Newton's powers. In any case the formula hadalready appeared, in somewhat confused manner admittedly, in Nicolaus Mercator's HypothesisAstronomica Nova, et Consensus ejits cum Observationibus, London, 1664, p . 4 (repeated withoutchange in Appendix, pp. 162-184 of his Institutionum Astronomicarum Libri Duo, de Motu Astrorumcommuni & proprio, secundum Hypotheses Veterum & Recentiorum pmcipuas; deque Hypotheseon exobservatis constructione, London, 1676).
** ULC. Add. 3996, 27V ff., especially 3Or.
124 DEREK T. WHITESIDE
< <>fgT variation [<]fps ye Elliptick asquation or Prosthaphaeresis. & drawingPrll gf> y° <spr is ye absolute equation. [<] rsp ye true Anomaly.'25
In neither case, we will note, does Newton make any mention of Kepler'sareal law, but instead uses an equant theory which yields its approximateequivalent. Indeed, perhaps the only way Newton at this time could haveknown even its enunciation was by reading Kepler's Astronomia Novafor himself or by consulting the few pages on the subject in Riccioli'svast, disordered compendium26 where it is mentioned, but there is noevidence to show he did either as a young man.27 With Kepler's thirdlaw we are on surer ground, for Newton unhesitantly followed Streetein accepting its validity as a codification of planetary behaviour: as hewrote in his notebook28 'ye meane distances of ye primary Planets fromye Sunne are in sesquialter proportion to the periods of their revolutionsin time' and in proof he repeated Streete's parallel columns of figureswhich relate the squares of periodic times to the cubes of planetarydistances from the sun for each of Saturn, Jupiter, Mars, Earth, Venusand Mercury.
A few years later, in notes he made on the endpapers of his copyof Wing's Astronomia Britannica,19 the same rejection of the areal law for
a5 Whether Newton knew it or not, Boulliau had devised his refinement merely from anempirical examination of Tycho's tables for Mars. Theoretically, however, and using the sameanalytical equivalents (bd = 2a, fs — lea, bfp = tfi and bfg = T) as before the correction yields
?p = — or, to O(e4), ib = T+Je2 sin 2T. In the second part of the scholium totan 1 •\J l~e*Book 1, Prop. 31 of his Principia, Newton was later to attack the general theory of the upperfocus equant: he there deduces from Kepler's first and second laws together thatiff = 7*+Je2 sin 2T+fe3 sin3 T-\-O(ei). (I neglect a slight confusion in his statement of thisresult in the first edition.) Clearly Boulliau's correction, as also the similar ones variouslyformulated by Nicolaus Mercator, Vincent Wing, Huygens and (as we shall see) by Newtonhimself, is correct to within the cube of the ellipse's eccentricity: in practical terms, taking thelimits of contemporary observational accuracy to be about i', we may say that it is true withinthose limits. John Machrin (in his Laws of the Moon's Motion according to Gravity, set in appendixto Andrew Motte's The Mathematical Principles of Natural Philosophy. By Sir Isaac Newton.Translated into English, London, ii, Appendix, 40 £f., 1729) was, I think, the first to producean equant hypothesis which incorporated the second correcting term f«3 sin3 T.
26 Almagestum Novum, Bologna, 1661: see Tom. I, Pars Prior, Liber VII, Sectio II, Caput V,Prop. 3, 535.
J7 Christopher Wren, however, had stated the areal law in a brief, confused passage whichintroduces his construction of Kepler's problem by a curtate cycloid: 'Asseruit Keplerus, excausis physicis, planetas ita ferri circa solem in Orbita Elliptica, ut velocitas planetee sit ubiquedistantiae ejusdem a Sole reciproce proportionalis; unde sequentem Hypothesin ingeniosecommentus est. Secat scilicet aream Ellipseos Planetariae lineis a sole ductis in infinita TriangulaMixtilinea . . .: per has autem positiones ponit Planetam xqualibus temporibus ferri' (JohnWallis' Tractatus de Cycloide, Oxford, 1659, p. 80 of the first pagination). Newton certainly madea deep study at an early age of Wren's following geometrical tract (the gist of which he repeatedin Principia, Book 1, Prop. 31) and it is just possible his attention was caught by the fleetingreference to the areal law.
28 ULC. Add. 3996, 2gr.39 Vincent Wing, Astronomia Britannica: in qua per Novam, Concinnioremque Methodum hi
quinque Tractatus traduntur: . . . Logistica Astronomica . . . Trigonometria . . . Doctrina Spherica . . .Theoria Planetarum . . . Tabule Nova Astronomies . . . Cui accessit Observationum AstronomicarumSynopsis Compendaria . . ., London, 1669. (Newton's copy is now in Trinity College, Cambridge,NO_. 18.36.) For all its inadequacies, Wing's work was a sincere attempt to make freely availablein England a sound body of doctrine relating to the 'new' astronomy, and was widely read inits day.
NewtorCs Early Thoughts on Planetary Motion 125
an observationally equivalent upper focus equant theory allied with adegree of confidence in Kepler's third law30 are each evident. However,a new element enters: Newton, though loosely willing to accept Wing'srestatement of Kepler's hypothesis of an elliptical planetary orbit,3*preferred to forego an exact ellipse of unit eccentricity in favour of arather unwieldy apparatus of pivoting lines which from the mean motiongiven constructs not merely the true planetary longitude but also theplanetary orbit itself:
'But that [Elliptical] Hypothesis of planetary motion may best be presentedin far neater form and one more suited to computation. Let S be the sun, Athe aphelion and B the perihelion of the planet, C the centre of its mean motion,the angle ACG that mean motion, CG a line of given length. And in the triangleCSG on account of CS, CG and the angle GCS being given the angle CGS isgiven also. Again, let the angle CES be taken in given ratio to the angle CGSand taken from the angle ACG, there will remain the angle ASE which will bethe heliocentric longitude of the planet from aphelion. Further the triangleCES will give SE, and if moreover you take SD of a given length, say half thediameter AB, and between SE and SD take SP the first (or perhaps second
\\
3° 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 equidemregula Kepleriana quod cubi diametrorum (maximarum scilicet) sunt ut quadrata temporumrevolutionis . . . In [Saturno, Martio] ac [Terra] ubi Author [sc. Wing] satis appropinquathas proportiones tabulae 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 hancanalogiam reduci potest haud scio, id vero suspicor sed hse ejus tabulae non satis bene conveniuntcum observationibus, et ut plurimum ex hac emendatione melius convenirent.' The argumentis interesting confirmation of Newton's willingness to modify observational fact to fit a preferredtheoretical rule.
31 For our present purpose the significant portion of Wing's 'Copernican hypothesis' ishis first axiom (p. 120) 'Qudd omnia Planetarum corpora in Ellipsi circa SOLEM moventur,ita ut SOL uno Foco ejus positus est, & medius motus in altero, circa quam ferme aequaliterPlaneta volvitur, squales angulos in temporibus aequalibus describens.' In his theories of theEarth and Mars (pp. 129 ff., 158) Wing himself implicitly produced an observationally accuratecorrection of this simple Wardian theory which adds an 'equation' of Je2 sin 2&> to the centralanomaly w: this yields, to 0{ei), the correct equation at the upper focus of J«2 sin 2T (and to0(e4) is slightly better than Boulliau's). Newton himself did not comprehend the accuracyof Wing's correction for in his brief reference to it ('In singulis . . . Planetis maxima; mediorummotuum correctiones debent esse ut Epicyclorum diametri divisae per maximas orbitarumdiametros') he erroneously inferred a maximum equation of e (and not J«2).
126 DEREK T. WHITESIDE
or third &c) of some number of arithmetical means . . ., then will P be theplace of the planet.32
'Now by applying this hypothesis to the determination of planetarymovements which shall accord with observation, in the case of each planet Ifind GC = | AB, SD = \AB, PE = \AD and angle SEC = f X angle SGC(or angle GSE = f X angle SGE), and that the eccentricity SC ought to bea little less in respect of the diameter AB than is assigned in the ellipticalhypothesis33—specifically, as the diameter to the eccentricity as commonlyassigned, so thereabouts ought to be a fourth part of the epicyclic diameterto the length by which the elliptical eccentricity here ought to be diminished.'34
Drawing on Wing's empirically-based planetary tables, Newton thenlists approximate numerical values (in terms of the earth's diameter asunit) of the diameter AB, 'excentricity' CS and the lengths GC and SD( = \AB in all cases) for each of Saturn, Jupiter, Mars, Earth, Venusand Mercury: he notes too the exact procedure for passing on his modelfrom mean motion to corresponding true planetary longitude and soon to the allied solar distance of the planet. We need not follow himin detail, but may note his concluding remark:
'. . . if the dimensions here assigned do not agree well with observationand if the error be detected in the determination of the heliocentric longitude,the eccentricity SC ought to be increased or decreased slightly. But if it bedetected in the magnitude of the orbit, then all lines should be increased ordiminished proportionately, unless you wish to move the sun S somewhattowards A or B (by taking EP = £ or £ of ED, say).'35
32 As in the following pages this is my translation from Newton's Latin: I have also addedthe centre 0 of orbit and the orbit itself (in broken line) to Newton's figure. Since only the shapeof the figure is at issue, suppose for simplicity the^main diameter AP = 2 units, the (Keplerian)eccentricity SO = e, the angle of mean motion ACG = 7" and, following Newton's recommenda-tion, SD = i, so that CES = T—<f> (where, as before, we suppose the solar distance SP = rand the true longitude ASP = (f>). Let us. moreover, fix Newton's undetermined constantsby taking the equant distance SC = E, $GC = k (T—<t>), CG = / and EPjED = A. On thesupposition that Newton's construction yields both true longitude <j> = T—2e sin 7~+ Je3 sin 2 T—. . . and solar distance r = i + « cos 7*+e3 sin3 7~+ . . . in terms of mean motion T correctlyto O(e3), we may with some manipulation find three bounds on E, k, I and A, viz.:
IE = 2e+ae* cos T, k = f + i o = -. = i/(a+ae cos T) (i — A) and 3 = 2(1 —A)3
where a is some parameter.33 T h a t is, following Kep le r , SC = 2 X SO.34 I n t h e no t a t i on previously used N e w t o n suggests sett ing o = — J (or SC = E = 2e—J«2)
for best results. Unfortunately this clashes with his further assumption of A = f (since k = i + J a ) .The best way is to assume a = O (or C to be exactly at the upper focus of orbit) and that thefirst two deductions alone of note 32 are true: this yields, in agreement with Newton, E = 2eand/ = jjk = 2(1 — A) = f, values which loosely satisfy the third condition 3 = 2(1 — A)!(i + 2&3).The practical effect of this supposition is to make true longitude correct to 0(e3), whileconstructing SP = i + e cos 7~+J§«2 sin2 T + . . . (true for an ellipse of eccentricity 'y i.ie).The empirical origin of Newton's construction in a numerical induction from Wing's tablesis 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 toalter 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 0(e3). Clearlythe remark is an observational induction rather than an inference from any theoretical con-siderations.
Newton's Early Thoughts on Planetary Motion 127
On a later page of his notes on Wing, Newton proposed a like theoryof the moon's motion round the earth in an ellipse, the basis of whichis again mean motion round an upper focus equant though the earthitself is not now set at the second focus. However, as we have seen, thisprimitive elliptical orbit is complicated by Newton's further suppositionthat it is traversed under the disturbing effect of the solar vortex on theterrestrial one in which the moon is borne.36
Even as late as about the spring of 1679 when Newton came todraft a short piece, 'De Problematum resolutione Synthetica',37 hepremised yet a further equant theory of planetary motion, one which(most satisfactorily for those who in his day disliked allotting a funda-mental theoretical role to a point which had no apparent physical signifi-cance) placed the equant not at the upper focus, but firmly at the sunitself. Newton begins:
'Problem. To find the aphelia and true motions [longitudes] of planets.38
'Let S be the sun, T the planet, A some given point between the sun andperihelion,39 the angle [G]SB its mean motion and SB a given line. Suppose
3 6 'Luna defertur in Ellipsi aequabili motu circa Centrum [medii motus], nisi quod . . .per compressionem vorticis impellitur versus tangentem orbis magni . . . : . . . debes potiiis . . .ad id referre lunares irregularitates quas Reflectionem et Evectionem vocant.' Cf. note 11 above.
37 ULC. Add. 3963.1, ir/iv. Jn this unpublished piece, which bears the subheading'Problemata in Lectionibus meis [sc. Lucasianis] sic resolvo synthetice', Newton gave formalsynthetic proof of several geometrical problems (notably I, 3, 4, 6, 9, 10, 13 and 32) whichhe 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 distantieeCometae in Systemate Coperniaea', 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 secondlecture of the Michaelmas term of 1680 as 'Prob. 52. E Cometae motu uniformi rectilineo perCoelum trajicientis locis quatuor observatis, distantiam a terra, motusque determinationem,in Hypothesi Copernicaea colligere.' A firm antedate for the composition of this cometaryproposition (which depends narrowly on material published by Hooke in his 1678 Cometa(see R. T. Gunther, Early Science in Oxford, Oxford, 1931, viii, 257-258) and by Wallis in theappendix 'De Cometarum Distantiis Investigandis' he added to the second edition of hisJeremia Horroccii Opera Posthuma, London, 1678) is late 1678, and so also, we conclude, for thefollowing planetary one. I cannot accept a date later than December 1679 for it, when Newton'scorrespondence with Hooke reached its crucial phase.
3s Presumably, as in the preceding proposition 'in systemate Copernicaea', that is, pre-supposing a Keplerian elliptical orbit for the planet. I have added such an orbit (in brokenline)—and the centre C—to Newton's figure.
39 Newton here cancelled 'aphelion'. In fact, a complementary theory of equant motionmay be constructed in the case where the point A lies beyond C in the aphelion distance SGsuch that AS remains £e. The case is interesting historically, for it has close affiliation both withKepler'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 AstronomicaNova (note 23), both presumably developed on much the same sort of observational evidenceas that available to Newton in the present case. It is interesting to note that Kepler supposedthe ratio ASjCS ( = £ in theory) to be about ViW ^ I -22, while Mercator—partly ontheophysical grounds—took it as y/f)—\ *=a 1-24.
128 DEREK T. WHITESIDE
that the angle BST be in a given ratio, say as 8 to 5, to the angle SBA such that[G]ST be the true longitude of the pianet from aphelion . . .'
H
We need go no further with Newton's development of the proposition.40"In those few words he proposed a most elegant and accurate4* hypothesisof planetary motion which in simplicity excelled those of his predecessors.and contemporaries.
We have said enough to justify our general contention that in the-autumn of 1679 Newton was, if indeed he at all consciously then recog-nized its existence, still unwilling to allow Kepler's crucially importantsecond (areal) law even an empirical place among the axioms of hisastronomical thought. Elliptical planetary orbits he allowed as beingnear enough to their observed paths for purposes of computation butnowhere insisted rigidly on their theoretical accuracy. (He would not,I think, have fought with Huygens if he had known the latter's preferenceat this time for an egg-shaped orbit, much like Kepler's 'buccosa' andlike it both fatter towards aphelion and virtually indistinguishable
4° Newton assumes as given, in fact, the shape of the planetary orbit and three correspondingpairs of values of the true longitude GST and mean motion GSB: obviously these latter deter-mine corresponding values of TSB (their difference) and so of SBA, so that—if, that is, such apoint does exist—two pairs will fix the point A in the plane and the third confirm it in position.
4' If, following the notation established in preceding pages, we suppose GH = 2a, CS = ea,SA = Ea, SB = a, CST = <f>, GSB = T and SBA = kx"fSB = k(T—<j>), we deduceimmediately that sin [k(T—<j>)] = £ sin [T—k(T—$)]. In the elliptical hypothesis this gives,the true planetary longitude GST in terms of mean motion T correctly to O(e3) if we take,as Newton suggests, k = § and also E = \e: in other words, if we set SA = %xCS andSB = i X GH then the model constructs </> as T— 2e sin T+ £e2 sin 2 T . . .
Newton's Early Thoughts on Planetary Motion 129
observationally from a true ellipse.)42 Kepler's third law alone wasaccepted unequivocally by Newton, and it was his good fortune thathe knew it as a trustworthy empirical law at a time when he had indepen-dently developed a workable theory of centrifugal force and its measure,for without either his theoretical deduction in the 166o's of the inverse-square gravitational field round the sun would have been impossible.43
How then could it come about that in a few short months he was tochange the basis of his astronomical thought into its classical 'Newtonian'form? What elements in that new synthesis did he borrow from hiscontemporaries and what were the products of his own insight ?
In the decade between 1666 and 1679, °f course, astronomicalthought had not stood still. Once more, after a long period in whichsuch explanatory schemes were largely rejected, cosmologists werebeginning to return to gravitational theories of planetary motion,44
4* In this context it is interesting to recall Newton's remark to Halley on 20 June 1686(Correspondence of Isaac Newton (ed. H. W. Turnbull), Cambridge, i960, ii, 436) that 'Keplerknew ye Orb to be not circular but oval & guest it to be Elliptical'. For Huygen's preferredcurve, see note 16. Kepler described his 'via buccosa' in his Astronomia Nova (note 14), CaputLVIII, 283 ff., having wrestled with it for several months in the early summer of 1605. Noone, 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 anomalyby an amount differing by up to 5J' from the equivalent anomaly determined in the 'buccosa'hypothesis) is fallacious. Kepler, in fact, there forgot to allow for the variation which (by theareal law) the non-symmetrical shape of the 'buccosa' imposes on the mean motion from thatin the corresponding ellipse: the true maximum difference in true anomaly between the twohypotheses is only about J', an amount observationally negligible in Kepler's day with respectto 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 lawafter 'Hugenius [in his 1673 Horologium OsciUatoriuni] had told how to find ye [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 decreasinglinearly with distance from the sun which, in conjunction with one of magnetism, modifieda pristine circular path into the observable elliptical orbit. Ismael Boulliau, however, thoughin his Astronomia Philolaica of 1645 he had (on the now well-known analogy of gravity to thedispersal of light from a point-source) substituted for Kepler's linear decrease one varying asthe inverse-square of the solar distance, could yet in 1657 voice a popular attitude when hewrote that 'nollem Kepleri famae detrahere; cui Mathematicarum artium studiosi, praecipuevero Astronomi, multum debent. Ipse enim mira sagacitate viam Planetae Ellipticam esseprimus invenit, adeoque rationem veram determinandi motus ccelestes tradidit. Coniecturisautem Physicis minus tribuisse virum ilium vellem' (Astronomia Philolaica Fundamenla clariiisexplicata (note 21), 45). In the theory of terrestrial gravity—not yet, of course, identified atlarge with any celestial attraction—Beaugrand in his Geostatique (Paris, 1636) and Robervalin his Aristarchi Samii de Mundi Systemate . . . Libellus (Paris, 1644) had expounded qualitativetheories, savagely criticized by Descartes in correspondence with Mersenne, which suggesteda 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 subjectwere printed in the first volume of Clerselier's posthumous edition (in French in 1657) whose1668 Latin version Newton knew well. (Compare ULC. Add. 4003, printed as 'De Gravitationeet jEquipendio Fluidorum' in the Halls' Unpublished Scientific Papers (note 5), 89-121, especially113, 1. 5.) Moreover, Descartes' criticism that in a force-field round a finite centre not varyingdirectly as the distance the centre of gravity is not stable (originally made by him in his letterof 13 July 1638 to Mersenne) was inserted with Johann Hudde's attempted amplification inLiber V, Sectio XXX of Frans van Schooten's Exercitationum Mathematicarum Libri Quinque(Leiden, 1657, pp. 515-516), a work studied minutely by Newton as an undergraduate—indeedDefinition 1 'Of Gravity' in his October 1666 fluxional tract (Unpublished Scientific Papers, p. 58)is seemingly a direct reference to Descartes' point.
130 DEREK T. WHITESIDE
while interest in the paths of bodies falling under terrestrial gravity hadbeen continuous since the early 1630's.45 The Italian Borelli, in particular,had in 1666 proposed a qualitative theory of planetary motion whichwas later, with some modification, to be represented in a mathematicallyexact form by Leibniz.46 In England, supremely, Robert Hooke47 hadover twenty years developed his researches into a loose gathering of allthe constituents (elliptical planetary orbits round the sun at a focus,deflection of the planet into its orbit from a linear inertial path by agravitational pull to the sun varying instantaneously as the inverse-square of the solar distance) with the exception of Kepler's true areal law—and knowledge, if not acceptance, of that too was becoming increas-
es See A. Koyre's able study, 'A Documentary History of the Problem of [free] Fall fromKepler to Newton. De Motu Gravium Naturaliter Cadentium in Hypothesi Terra; Motae',Transactions of the American Philosophical Society, 1955, xlv, 329-395. A complex line of researchtook 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 gravita-tional tendency to its centre would travel in a semicircle to that centre. In the hypothesis thatthe centre is indefinitely distant the path would, of course, be a parabola, where the bodytraverses a horizontal distance proportional to the time of the earth's rotation and falls througha vertical distance instantaneously proportional to the square of the time. Supposing only thatthe 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, ingeniouslydeduced a spiral orbit r = R(i — $*/a1)3. Generalizing Galileo in a similar way, Mersenne had,about 1635, determined that the line of uniform oblique descent, a straight line when the earth'scentre is supposed infinitely distant, is a logarithmic spiral when the verticals are supposedconvergent 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,'657, i, 327-354). The supposition common to both that the angular velocity of a falling bodyround the earth's centre will remain constant is, of course, erroneous and one which revealsyet again how little mathematicians in the mid-seventeenth century were willing to availthemselves of Kepler's areal law (or even an approximate equivalent). Before Newton, onlyBorelli had any glimmering of this necessary application. (Compare the tentative correctionof the conventional view in his De Vi Percussionis. Bologna, 1667, pp. 108 ff., and its more forcefulstatement in his Risposta . . . alle Considerazionifatte sopra alcuni luoghi del suo Libra della Forza deltaPercossa del R.P.F.. . . Angeli, Messina, 1668, pp. 16 ff., where Borelli implicitly uses the Keplerianapproximation to the areal law v = kjr.)
4' In his Theoricte Mediceorum Planetarum a Causis Physicis Deducts, Florence, 1666, Borellisuggested that the elliptical motion of Jupiter's satellites (and so, by implication, of the solarplanets) could be explained through the interaction of a Huygenian centrifugal force and aconstant centripetal gravitational tendency, instantaneously in disequilibrium, whose totaleffect was thus to modify a circular force-free path (sustained round the force-centre by thecontinued action of a transverse 'impetus'). (See A. Koyr£, La revolution astronomique (note 14),pp. 474-506.) Leibniz, in his 'Tentamen de Motuum Ccelestium Causis', Ada Eruditorum,1689, pp. 81-96, was to add the two refinements which made the Borellian theory viable:he assumed that the centrifugal force c2/r3, where r is the instantaneous distance from theforce-centre, was of itself (in his dynamical system) capable of sustaining motion in a straightline, and, secondly, that the gravitational force is a generally variable function of the radialdistance, deducing in his §15 that for elliptical motion round a force-centre situated at a focusthe 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 c2/r3—f, where r isthe radial acceleration in the orbit. (Compare E. J. Aiton's recent articles, 'The CelestialMechanics of Leibniz' and 'The Celestial Mechanics of Leibniz in the Light of NewtonianCriticism', Annals of Science, i960, xvi, 65-82, and 1962, xviii, 31-42.)
47 Compare L. D. Patterson, 'Hooke's Gravitation Theory and its Influence on Newton',Isis, 1949, xl, 327-341, and 1950, xli, 32-45; A. R. Hall, 'Two Unpublished Lectures of RobertHooke', Isis, 1951, xlii, 219-230; A. Koyr6, 'An Unpublished Letter . . .' (note 3), especiallypp. 317-319; and J. A. Lohne, 'Hooke versus Newton' (note 1), especially pp. 10-18.
Newton's Early Thoughts on Planetary Motion 131
ingly widespread.48 Unfortunately, too, for Hooke the ultimate fusionneeded a mathematical insight surpassing his moderate talents in thatfield—a point judiciously if heavily made by Newton when he laterrecalled his own triumph for Halley.49
In the event Hooke himself was the immediate catalyst in excitingthe fundamental change through which Newton's astronomical thoughtwent in the winter of 1679/1680. On 24 November 1679, in the attemptto re-establish a working relationship with Newton after the fierce disputesof a few years before on the nature of light, he wrote to Cambridge:
'I hope . . . that you will please to continue your former favours to theSociety by communicating what shall occur to you that is philosophicall . . .For my own part I shall take it as a great favour if you shall please tocommunicate by Letter your objections against any hypothesis or opinionof mine. And particularly if you will let me know your thoughts of that ofcompounding the celestiall motions of the planetts of a direct motion by thetangent & an attractive motion towards the central body . . ..'5°
The letter came out of the blue to Newton (still recovering from hismother's death in the summer) and he was obviously more than a littleunwilling to take up Hooke's suggestion of analysing planetary pathsin terms of compounded motions. Instead he sought (as he thought)to change the subject:
'I did not before ye receipt of your last letter, so much as heare (y* Iremember) of your Hypotheses] of compounding ye celestial motions of y*
4s In England in the 1670's Nicolaus Mercator was its chief popularizer, printing itscorrect enunciation both in an article, 'Some Considerations . . . concerning the Geometrickand direct Method of Signor Cassini for finding the Apogees, Excentricities and Anomaliesof the Planets', Philosophical Transactions, 1670, v, no. 57, 1168-1175 (especially p. 1174:'Keplerus . . . lineam veri motus Planetee aequalibus temporibus zequales areas Ellipticas verrereprofessus est') and in Caput XX 'De Hypothesi Kepleri' of his excellent 1676 compendium,Inslitutionum Astronomicarum Libri Duo (note 23) (especially p. 145: '. . . areae, quas radius vectora Sole ad Planetam extensus verrit, cresc[u]nt aequaliter aequalibus temporis momentis').Mercator was on terms of some familiarity with Newton, having corresponded with him inthe mid-1670's on the moon's motion. (Compare Institutionum Astronomicarum Libri Duo,pp. 286-287; a n d Newton's Principia, Lib. 3, Prop. 17.) Newton's own well-thumbed copy ofthe Institutiones (Trinity College, Cambridge, NQ,. 10.152), though not annotated at the passageswhere Mercator discusses planetary hypotheses, shows clear signs throughout of having beenread continuously.
49 Newton to Halley, 20 June 1686: '[Hooke] has done nothing & yet written in such away as if he knew & had sufficiently hinted all but what remained to be determined by yedrudgery of calculations & observations, excusing himself from that labour by reason of hisother business: whereas he should rather have excused himself by reason of his inability. Fortis plain by his words that he knew not how to go about it. Now is not this very fine? Mathe-maticians that find out, settle & do all the business must content themselves with being nothingbut dry calculators & drudges & another that does nothing but pretend & grasp at all thingsmust carry away all the invention . . . " I would not wish to assert that Hooke was devoid ofmathematical talent (cf. L. D. Patterson in Isis, 1950, xli, 35-38), but merely that his mathe-matical genius was more well-informed than brilliant. Thus, for example, the autograph notesin his copy of Fermat's Opera Varia (Toulouse, 1679) reveal a surface comprehension ratherthan an intimate understanding.
5° From the autograph draft in Trinity College, Cambridge (O. na . i " ) , reproduced inNewton's Correspondence (note 42), ii, 297. Hooke had already announced this fundamentalassumption 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 withthe radial distance.
132 DEREK T. WHITESIDE
Planets, of a direct motion by the tang* to ye curve . . . I shall communicateto you a fansy of my own about discovering ye Earth's diurnal motion . . .'5I
Newton then proposed his spiral line of free fall to the east of the radiusvector ABC rotating uniformly with the earth from west to east.52
Swiftly pouncing on Newton's attempted evasion of the problemof planetary orbits, Hooke was quick to recall attention to it by insistingthat the free fall path is but the same problem posed in another guise:
'. . . as to the curve Line which you seem to suppose [a heavy body]to Des[c]end by . . . Viz* a kind of spirall which after sume few revolutionsleave it in the Center of the Earth my theory of circular motion makes mesuppose it would be very differing and nothing at all akin to a spirall butrather a kind [of] Elleptueid. At least if [the body in falling towards the earth'scentre suffered no resistance but] had the motion of the superficial! partsof the earth from whence it was let fall impressed on it, I conceive the line[AFGH] in which this body would move would resemble an Ellipse [and]the body A would never approach neerer the Center C than G . . .. But w[h]erethe Medium through which it moves has a power of impeding and destroyingits motion the curve in W"* it would move would be some what like the LineAIKLMNOP &c and after many resolutions would terminate in the Center C
51 Newton to Hooke, 28 November 1679, from the autograph in Trinity College, Cam-bridge (R. 4.48.1) = Correspondence, ii, 300-301.
5> Newton's figure, in which ADEC is the line of free fall, is drawn from the viewpointof 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 thefalling body would spiral round the earth's centre several times before reaching it: 'Accordingto 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 onerevolution (and I have inserted its possible representation as the thick curve in the presentreproduction of Newton's diagram). AsJ. A. Lohne has pointed out, Newton's figure is extremelybadly reproduced in the standard texts: from a close study of the original (whose photocopyLohne inserted in his 'Hooke versus Newton' (note 1), p. 9) it is clear both that the path ADEwas drawn tangent to ABC (as it should be) and that Newton did not continue his spiral quiteall the way to the earth's centre. It is worthwhile to notice, too, that Newton here rejectedimplicitly the Fermatian hypothesis of unchanged uniform angular rotation (which wouldmake the path ADE fall wholly in the line ABC).
Newton's Early Thoughts on Planetary Motion 133
. . . I could adde many other considerations which are consonant to myTheory of Circular motions compounded by a Direct motion and an attractiveone to a Center . . ,.'53
Hooke had dared once more to challenge Newton's opinion and thematter could no longer be treated lightly. Newton answered on 13 Decem-ber with a carefully-phrased reply:
'I agree wth you y* . . . if its gravity be supposed uniform [ye body] willnot descend in a spiral to ye very center but circulate wth an alternate ascent& descent made by it's vis centrifuga & gravity alternately overballancing oneanother.54 Yet I imagin ye body will not describe an Ellipsoeid but rathersuch a figure as is represented by AFOGHIKL &c . . . For ye motion of ye
body at G is compounded of ye motion it had at A towards M & of all ye
innumerable converging motions successively generated by ye impresses ofgravity in every moment of it's passage . . . ( . . . I here consider motion
53 Hooke to Newton, 9 December 1679 {Correspondence, ii, 305-306). Koyr£, who therefirst 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 itsorthography for clarity's sake.
54 This hypothesis of a perpetual imbalance between a variable centrifugal force anda 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 thatNewton, caught off balance, had looked up the details of his theory of compounded forcesimmediately on receiving Hooke's letter with its repeated insistence that the path of free fallbe explained by compounding motions.
134 DEREK T. WHITESIDE
according to ye method of indivisibles) . . . Thus I conceive it would be ifgravity were ye same at all distances from ye center. But if it bee supposedgreater neerer ye center . . . the increase of gravity in ye descent may besupposed such y* ye body shall by an infinite number of spiral revolutionsdescend continually till it cross ye center by motion transcendently swift . . .Your acute Letter having put me upon considering thus far ye species of thiscurve, I might add something about its description by points quam proxime. Butthe thing being of no great moment I rather beg your pardon for havingtroubled you thus far wth this second scribble . . .'55
Newton, evidently, was still not prepared either to accept that an ellipsemight be traversed under some suitable law of gravitational decreasewith distance (which should be inverse-square if his deduction fromKepler's third law was accurate) or even to believe that the problemwas of scientific importance.
Hooke knew better and again he returned, forcefully urging bothpoints:
'Your calculation of the Curve by a body attracted by an aequall powerat all distances from the center . . . is right and the two auges will not uniteby about a third of a Revolution. But my supposition is that the Attractionalways is in a duplicate proportion to the Distance from the Center Reciprocall,and consequently that the Velocity will be in a subduplicate proportion tothe Attraction, and Consequently as Kepler supposes reciprocall to the
55 Newton to Hooke, 13 December 1679 (Correspondence, ii, 307-308). Newton (withBorelli) here makes no explicit assumption that the vis centrifuga will by itself carry the fallingbody out of its circular 'inertial' path into a linear 'gravity-free' one traversed with uniformmotion. (He had indeed in January 1665, in notes entered in his Waste Book, written '[Axiorne] 2A quantity will always move on in ye same straight line (not changing ye determination norcelerity of its motion) unlesse some externall cause divert it' (ULC. Add. 4004, iov), but I findthis early statement of a linear inertial line impossible to reconcile with his present implicitacceptance 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' analysisto derive the polar equation of the path of free fall as
1 CR Rp /V1 /
T = J r V^H^TRS+7)+firi " dr =J 8 N/i + A sin 8
</8,3 + A sin 8
where AC = R, aC = r, ACa = <j>, QC = p the minimum value of the radius vector aC (or thef~^\ R-p r(R+p)-aRp.
initial velocity at A in the direction AM is / 1, A = and sin 8 =V R+pf R+P r{R-p)
(Compare J. Pelseneer, 'Une lettre inedite de Newton', Isis, 1929, xii, 237-254, especially250 ff.; and Lohne's 'Hooke versus Newton' (note 1), p. 43.) Clearly
, 1 + A sin 8ACO = / • d& a
J - T T / ' V 3"3+A sin 8and is a maximum ("/'V 3) ^or ^ = 0 or /? = p. (Compare Principia, Lib. 1, Prop. 45, Example 1.)Newton's autograph sketch, in which p is rather greater than \R (cf. the photocopy reproducedby Lohne on p. 27 of his 'Hooke versus Newton'), agrees not too well with this theoreticalcurve, which in one libration swings through a central angle ACH somewhat less than Newton's.But it is clear that Newton at this time had no such exact theory, perhaps only the intuitiveknowledge that as p decreases from p = R to p = O the angle ACO decreases from •nl'Sj 3 to(ACF =)ir/a.
Newton's Early Thoughts on Planetary Motion 135
Distance.56 And that with such an attraction the auges will unite in the samepart of the Circle and that the neerest point of accesse to the center will beopposite to the farthest Distant. Which I conceive doth very Intelligibly andtruly make out all the Appearances of the Heavens. And therefore (thoughin truth I agree with You that the Explicating the Curve in which a bodyDescending to the Center of the Earth, would circumgyrate were a Speculationof noe Use yet) the finding out the proprietys of a Curve made by two suchprinciples will be of greate Concerne to Mankind, because the Invention ofthe Longitude by the Heavens is a necessary Consequence of it . . . What Imentioned in my last concerning the Descent within ye body of the Earth wasBut upon the Supposall of such an attraction, not that I beleive there reallyis such an attraction to the very Center of the Earth, but on the contrary Irather Conceive that the more the body approaches the center, the lesse willit be urged by the attraction . . .'57
And yet once more a few days later:'. . . It now remaines to know the proprietys of a curve Line (not circular
nor concentricall) made by a central attractive power which makes the velocitysof Descent from the tangent Line or equall straight motion at all Distancesin a Duplicate proportion to the Distances reciprocally taken. I doubt notbut that by your excellent method you will easily find out what that Curvemust be, and its proprietys, and suggest a physicall Reason of this pro-portion . . .'58
Newton never replied,59 but the problem had been squarely,unambiguously put to him: Does the central force which, directed toa focus, deflects a body uniformly travelling in a straight line into anelliptical path vary as the inverse-square of its instantaneous distancefrom that focus? To repeat Newton's already quoted words, 'I foundnow that whatsoever was the law of the forces wch kept the Planets intheir Orbs, the areas described by a Radius drawn from them to the Sunwould be proportional to the times in wch they were described. And. . . that their Orbs would be such Ellipses as Kepler had described [when]the forces wch kept them in their Orbs about the Sun were as the squaresof their . . . distances from the Sun reciprocally . . .' We cannot reasonablydeny Newton's claim and may suppose that the proofs he concocteddiffered little in essence from those given by him in the first draft60 ofhis De Motu four years later. In somewhat modernized form Hooke'sproblem, supposing that the body P travelling in the elliptical orbit APof focus S moves instantaneously to Q, by traversing the tangent PR and
i6 We see very clearly the great stumbling-block to further development implicit in Hooke'scontinued acceptance of the approximate form of Kepler's areal law, v = k\r (which thereforevaries as /\J(r), wheref(r) a ijr2 is the 'Attraction'). Compare Koyre's 'Unpublished Letter. . .' (note 3), p. 336, n. 118.
57 Hooke to Newton, 6 January 1679/80 (Correspondence, ii, 309).5s Hooke to Newton, 17 January 1679/80 (Correspondence, ii, 313).59 N e w t o n to H a l l e y , 20 J u n e 1686: ' [ I ] neve r a n s w e r e d his t h i r d [ l e t t e r ] ' , Correspondence,
ii, 436 .60 'De motu corporum in gyrum', ULC. Add. 3965.7, 55r-62 bisT. The figure I give is
an accurate reproduction of that accompanying Prob. 2 (57r) of this autograph text, thoughfor uniformity I have rotated it through a right angle: note particularly that i?Q. is in line withQ_S and not drawn (as later) parallel to SP.
136 DEREK T. WHITESIDE
being simultaneously deflected towards S from R to Q,, is to show that the
quantity of deflection RQcx r ^ X {dt)2, where dt is the indefinitely small
time in which P passes to Q. The mathematical break-through—a notover-difficult piece of conical geometry, as Newton showed—is to noticethat in this limiting case (where RQ and QT both vanish with PQJ)RQ^ = QT2jL, L being the ellipse's (constant) latus rectum. We needthen only show that SP X QT a dt: which is exactly Kepler's areal lawsince, in that limit, SP X QT = 2 X ellipse-sector (SPQ)! Now convincedfor the first time of the theoretical accuracy of that law, Newton had,to complete his analysis, only to demonstrate that the areal law is equi-valent to the assumption of a linear inertial path in a general theory ofcentral forces: that he did, we may assume, by the method he developedin Prop. 1 of his De Motu and from which (though we may now criticizeit as not wholly rigorous) he never varied in the three editions of thePrincipia whose publication he supervised.
At long last, by courtesy of Hooke, Newton had a sound basis onwhich to build the world-system which, when provoked by Halley, hesketched out in his De Motu and then developed with all the vigour onervous excitement in his fundamental Philosophic JVaturalis PrincipiaMathematica. In January 1680 he still had to reduce Borelli's centrifugalforce to physical impotence by equating it with the radial accelerationin the instantaneous inertial path, still to justify the inverse-square lawof attraction as an exact force working above the earth's surface andbetween any two massy disjoint bodies, still to worry at subtle problemsin equinox precession, the moon's motion and many other things. Butfor the first time all the astronomical problems which baffled his con-temporaries became capable of theoretical solution within a frameworkwhich more and more as its intricacies were explored seemed accuratelyto mirror the fundamental structure of observable reality. If I have shownthe minuteness of the part which Newton played in formulating the
Newton's Early Thoughts on Planetary Motion 137
separate axioms of his world-system, I cannot stress too forcefully themagnitude of the intellectual insight needed to mould them to a smoothconsistency and to assign to each its due role, theoretical or empirical,in that synthesis—and in the controlled power of his mathematicalthought Newton surpassed all his contemporaries.
POSTSCRIPT
In the previous issue of this Journal (1964, ii, 1-24) Father J. L. Russellhas published the text of an October 1963 talk—which I heard—where hedisplayed the fruits of his researches into the acceptance and appreciation ofKepler's laws of planetary motion in the period 1609-1666. His able andinformative article should now, in correction and partial refutation of thebleak comment I there allowed myself, be added to my footnote 18 above.The main burden of Russell's paper is to stress the inadequacy of our previousknowledge of the dissemination of Kepler's astronomical thought in thepre-Newtonian period, laying particular emphasis on the extent of con-temporary acquaintance with the first and especially third laws and notingsome half dozen printed references to the areal law before 1666. It will beclear that Russell and I differ appreciably in our interpretations of the signi-ficance of that dissemination: I, in particular, would wish to draw a carefulline between parrot-like quotation and comprehension of those laws, butmust leave the development of my viewpoint for another time. Let me, too,take the opportunity to note that I. B. Cohen has announced imminentpublication of his The New Astronomy: from Kepler to Newton. The script ofthis I have not seen, but it must now surely supersede Delambre as an up-to-dateaccount of mid-17th century astronomical thought.
