Essay Review: Newtonian Dynamics: The Background to Newton's Principia

To repair two examples of the latter which the reader may find especially confusing, on p. 142 my transcription of paragraph 7—with square brackets denoting restoration of fragmented text—is ‘If two body[s (a and b) movein]g y^e same way towards o, [refle]ct against one ano[th]er (a) overtaking (b) none of theire motion shall bee lost. for (a) press[eth (b) soe much] as (b) presseth (a) and th[eref]ore y^e motion of (b) shall incre[ase soe muc]h as y^t of (a) decreaseth’; while on p. 306, ll. 9–11 the manuscript reading is clearly ‘… exercetur verò proportionaliter mutationi ['allatæ’ cancelled] status et quatenus exercetur dici potest corporis vis exercita conatus et reluctatio. Hujus ['generis' cancelled] una species est vis centrifuga gyrantium'.

I will come later to stress certain particular cases, but let me state here that significant structural inaccuracy is present in the diagrams on pp. 122, 171, 239, 243, 259, 264 and 268, while those on pp. 124, 131, 134, 247, 261, 262, 269 (both figures) and 273 could profitably be improved or allocated variants when two texts are being collated.

Halley in early May had just suffered the simultaneous deaths of his father (apparently by suicide) and of his younger brother. Apart from the emotional upset of this double tragedy he had to settle the estate of his father, who had left no will, in conflict with his fractious stepmother (whom later he took to court). (See MacPike E. F. , Hevelius, Flamsteed and Halley, London, 1937, 46.) The evidence cited by Herivel, on pp. 96–7, in favour of a May visit by Halley is very late and not at all firm. Conduitt's ‘May’ is an unsupported interpolation in an account given him by De Moivre in November 1727 which merely specifies ‘1684’. The two Newtonian extracts quoted (ULC. Add 3968.9, 101^r/106^r) both come from an intended preface to an abortive revision of the Principia's second edition planned about 1715: This is exceedingly roughly written and certainly not to support an objective autobiographical account of the Principia's composition. On f. 106^r, incidentally, a first version of ‘June and July’ reads less precisely ‘in the summer time of the year 1684’.

In the opening phrase of ‘Hyp. 3’ of the Cambridge autograph (p. 246) ‘two’ is a careted insertion absent from the ‘Locke’ version, while the lower figure on p. 247 is less complicated and slightly differently lettered. As presented, of course, Herivel's internal and circumstantial arguments presuppose three mutually exclusive hypotheses (p. 117), though he might well have considered how far any original version of MS VIII, however composed, might have been altered and expanded for Locke's especial benefit. He has also, it seems to me, ignored one piece of documentary evidence (from De Moivre's 1727 memorandum) which supports the hypothesis that such an original was initially conceived as a variant proof of the elliptical orbit's being traversed under an inverse square force when, in the summer of 1684, he could not immediately recover his first proof (of about Christmas 1679?) for Halley: ‘S^r Isaac, in order to make good his promise, fell to work again, but he could not come to that conclusion which he thought he had before examined with care; however, he attempted a new way, which, though longer than the first, brought him again to his former conclusion. Then … he made both his calculations agree together’ (cf. More L. T. , Isaac Newton, New York, 1934, 299).

Add 4004, 1^r = p. 129. In my figure I have drawn a regular inscribed polygon in generalisation of Newton's square, but this corresponds merely to his argument ‘by y^e same proceeding’.

Reproduced, in its Motte-Cajori English version, on p. 11. Herivel developed this present view more fully a few years ago in an article (‘Newton's Discovery of the Law of Centrifugal Force’, Isis 51, 1960, 546–53) which may, of course, no longer accurately represent his standpoint.

Add 3958.5, 87^r = p. 193, some time later than 1670 by its writing, if invented independently of Huygens' Horologium oscillatorium. Here the vis centrifuga is measured straightforwardly by the continuous deviation which it induces (in a simply infinitesimal time) between the rectilinear, tangential path and the resulting circular orbit. I may add that the essential portion of my restoration was conceived several years ago. Subsequently I. B. Cohen has brought to my attention that an equivalent explanation of the Principia scholium was given by Percival Frost in his Newton's Principia, First book, Sections I, II, III with notes and illustrations (London, 1878, 168) though without comment on its logical soundness.

‘Newton's First Solution to the Problem of Kepler Motion’, British journal for the history of science, 2, 1965, 352/4: Compare pp. 16–22 of the present work.

I have elsewhere (‘Newton's Early Thoughts on Planetary Motion: A Fresh Look’, British journal for the history of science, 2, 1964, 117–37) cited certain documentary evidence which at least cast doubts on Newton's undivided belief in a principle of rectilinear inertia on a cosmic scale, referring in particular to his quotation, in a letter to Hooke of 13 December 1679 (p. 243), of Borelli's theory that the planetary orbits are traversed under the joint action of a constant centripetal ‘gravity’ and a variable vis centrifuga. Of course, if one does not accept rectilinear inertia one cannot deduce Kepler's areal law as its theoretical consequence in a central force field and a mainstay to Herivel's interpretation snaps. Newton's early statements of the rectilinear inertial principle (cf. pp. 44–5) are uniquely based on Descartes' Principia philosopiæ, Part II, §§ 37ff, but it is pertinent to add that in the latter's theory of planetary motion (Principia, Part III, §§ 140–4) the principle is allowed only for a freely moving ‘star’ (sidus, exemplified by Descartes as a comet); as soon, however, as the star is caught up in the grip of a (deferent) vortex its primitive motion becomes one of ‘philosophical rest’ (uniform and circular) though as many as five permanent distortions of that circular ‘inertial’ motion are permitted to exist. (Newton was suitably sour in his discussion of this paradox in his [De gravitatione] (p. 221), asking ‘Quid itaque? an hic conatus [recedendi] a quiete Planetarum juxta Cartesium vera et Philosophica, vel potius a motu vulgi et non Philosophico derivandus est?’) Newton's implicit citation of Borelli's theory in his Hooke letter is not, by the way, unique: He makes a second use of the model about April 1681 in accounting for the motion of a comet near perihelion (Correspondence of Isaac Newton, ii (1960) 361). As late as 1684 he is still capable of making the curious observation that ‘Motus … uniformis hic est duplex, progressivus secundum lineam rectam quam corpus centro suo æquabiliter lato describet et circularis circa axem suum quemvis qui vel quiescit vel motu uniformi latus semper manet positionibus suis prioribus parallelus’ (MS Xa, Lex 1 = p. 397, compare p. 86). I should add that Herivel's epitomisation (p. 59) of Borelli's theory is (in common with so many recent accounts, by Koyré and others) fundamentally unsound in its illogical insistence that the centripetal pull of gravity (constant for Borelli as for Newton in his 1679 letter) should balance a variable vis centrifuga.

Hooke to Newton, 9 December 1679 and 17 January 1679/80. I am not persuaded by Herivel's recent assertion (‘Newton's First Solution …’ (note 8), 353–4) that these two assumptions ‘add up to a complete solution of the problem of Kepler Motion’. In the present account of the Newton-Hooke correspondence on the path of free fall down from the earth's surface, Herivel's reproductions of the two Newtonian trajectories (pp. 239/243) are structurally false. While I agree that these letters are ‘chiefly memorable’ for the evidence they provide of the undeveloped state of Newton's thought on the problem and that there is ‘no firm’ indication that he actually attempted a quantitative solution (pp. 244/5), a good deal could have been done, if only negatively, to reinforce that conclusion. For example, Newton's ‘spiral’ fall curve (of 28 November 1679) is highly approximated by ϕ = k ( [ R 2 / r 2 - 1 ] - cos - 1 r / R ) , k ≈ π / 18 and this corresponds to a true fall path (in a stationary reference system) φ = k √ [R²/r² — I]: For low values of r the latter is traversed in an approximately inverse cube field.

In fact, qp will be tangent to the arc PQ at p and the orbital sector (pSQ) equal in area to the triangle pSq. (The latter affords a simple definition of the deviation arc.) Evidently, if QR is drawn parallel to SP the segment (PpQ) is equal to the curvilinear area (QRR′).

Following Varignon, Johann Bernoulli and Hermann, the standard eighteenth century treatment of planetary motion essentially took its lead from Book I, Prop. 41 of the Principia by equating the orbital acceleration in the instantaneous direction of motion to the accelerative component of the central force acting in that direction (d²s/dt² = f(r). dr/ds)—an approach which avoids any direct calculation of the deviation from the instantaneous (rectilinear) inertial path.

Even to the point of omitting an explanatory figure in his excerpt from Newton's letter of June 1674 to Collins (reproduced on p. 237) and of leaving unnoticed (on p. 273) a significant mathematical faux pas on Newton's part.

In the first edition this reads: ‘Ex tribus novissimis Propositionibus [sc. 11, 12 and 13] consequens est, quod si corpus quodvis P, secundum lineam quamvis rectam PR, quacunque cum velocitate exeat de loco P, & vi centripeta quæ sit reciproce proportionalis quadrato distantiæ a centro, simul agitetur; movebitur hoc corpus in aliqua sectionum Conicarum umbilicum habente in centro virium; & contra’. In October 1709 ( Edleston J. , Correspondence of Newton and Professor Cotes (London, 1850) 5) he somewhat clarified this with an addendum: ‘Nam datis umbilico et puncto contactus & positione tangentis, describi potest Sectio conica quæ curvaturam datam ad punctum illud habebit. Datur autem curvatura ex data vi centripeta: Et Orbes duo se mutuo tangentes eadem vi describi non possunt’. He was, however, always careful in later years to assert that ‘The Demonstration of the first Corollary of the nth, 12th and 13th Propositions being very obvious, I omitted it in the first edition and contented myself with adding the 17th Proposition whereby it is proved that a body in going from any place with any velocity will in all cases describe a conic Section; w^ch is that very Corollary’ (ULC. Add 3968.9, 101^v: The last point needs careful justification to be valid).

For, as Newton says in the second edition of his Principia (see previous note) two distinct orbits touching one another at a point cannot have the same curvature and so are not described under the same central force. Newton, of course, was the discoverer (in his 1671 fluxions tract) of the formula which expresses the length of the radius of curvature in a polar coordinate system.