The First Professional Scientist pp 165-202 | Cite as

# The System of the World: Hooke and Universal Gravitation, the Inverse-square Law, and Planetary Orbits

## Abstract

Of all the problems which occupied Hooke’s fertile mind over the four decades of his scientific career, the most important was without question planetary motion, despite the fact that his contributions are not yet generally acknowledge. It was, of course, a problem which Kepler necessarily left unsolved, and even in the early 1660s one which no one knew how to approach. In the two decades between 1665 and 1685, Hooke may have occasionally lost interest, perhaps because he thought he had already solved it, or because he was simply too busy to give it much attention, but in those years it was almost always on his mind. It is an issue that brought Hooke and Newton into serious conflict, and was, of course, what led Newton to the *Principia*.

## Keywords

Centrifugal Force Circular Motion Elliptical Orbit Kepler Problem Planetary Orbit## Preview

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- 1).
*Diary I*, 18 and 21 October 1679.Google Scholar - 4).
*Birch*, 2, 91.*Register Book*, iii, p. 114. Hooke also speculated that an increase in the density of the interplanetary medium could keep a body from moving off along a tangent, as did Newton somewhat later. See, for example, Pugliese (1989).Google Scholar - 7).Westfall (1980), Chapter 5. We note that Hooke was discussing the problem in terms of “direct motion by the tangent, and of another endeavour tending to the center,” to use Birch’s words, while Newton was, following, or perhaps “paralleling” Huygens — Westfall’s term — in balancing gravity and centrifugal force. Much later Newton coined the term “centripetal” for the tendency toward the center that Hooke was talking about. There is a very real difference between these positions, i.e., whether gravity balanced centrifugal force, or whether it simply provided the centripetal tendency (to use Newton’s term) that disturbed the inertial motion. On 11 July 1667 Hooke further commented that, according to Birch, “he had a theory, which would solve all the unequal motions of the planets...”Google Scholar
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*Gunther*, VIII, pp. 27–28.Google Scholar - 13).
*Corresp*. II, p. 297, 24 November 1679.Google Scholar - 14) “An Attempt to Prove the Motion of the Earth”, p. 28.Google Scholar
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- 20).Newton acknowledged as much in a letter to Halley, written 14 July 1686: “This is true, that his Letters occasioned my finding the method of determining Figures, wch when I tried in ye Ellipsis I threw the calculation by [,] being upon other studies & so it rested for about 5 yeares till upon your request I sought for yt paper ...”
*Corresp*, II, 444.Google Scholar - 22).We have already noted that one reason for this is that his published work, a major exception being the
*Micrographia*, is largely based on lectures, mostly to the Royal Society. This left little room for quantitative development. While the geometrical argument in his “Laws of Circular Motion” (see below) is impressive, one notes with some dismay his comment in his*Diary*on 8 July 1693, that “I masterd Logarithms.” (*Gunther*, X, p. 257).Google Scholar - 26).“Discourse of Comets,”
*PW*, p. 182.Google Scholar - 29).This is reminiscent of the early Newton. See Kollerstrom (1991).Google Scholar
- 32).Especially the comments on gravity in his “Discourse of Comets”,
*PW*, pp. 167–185.Google Scholar - 33).He concluded with “Sr Chrisopher was little satisfied that he could do it, and tho Mr Hook then promised to show it him, I do not yet find that in that particular he has been good as his word.” Halley to Newton, 29 June 1686;
*Corresp*. II, 441–2. The context was Newton’s threat to suppress Book III of the*Principia*in the face of Hooke’s claims that his contribution to Newton’s understanding had not been acknowledged, and Halley’s attempt to smooth Newton’s ruffled feathers. Although Halley attributes to Hooke the term “centripetal” force, it is not at all clear that Hooke actually used it in the conversation in question. Halley disclosed that he had found that Kepler’s Third Law would result from an inverse-square force, or in Halley’s words: “having from the consideration of the sesquialter proportion of Kepler, concluded that the centripetall force descreased in the proportion of the squares of the distances reciprocally ...” Halley’s proof would have been restricted to circular oribits.Google Scholar - 35).Herivel (1965) thought that Halley may have gone earlier, possibly in May, 1684, and that this was the visit which revived Newton’s interest in dynamics. This would conform more readily with the two-month deadline set by Wren in January. Herivel (1965), p. 97.Google Scholar
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- 37).This is how Newton described it: “his [Hooke’s] letters occasioned my finding the method of determining Figures, wch when I had tried in ye Ellipsis, I threw the calculation by being upon other studies & so it rested for about 5 yeares till upon your request I sought for yt paper & not finding it did it again ...” Newton to Halley, 14 July 1686 (
*Corresp.*, II, 444). DeMoivre’s famous description was as follows: “In 1684 Dr. Halley came to visit him at Cambridge ... the Dr. asked him what he thought the Curve would be that would be described by the Planets supposing the force of attraction towards the Sun to be reciprocal to the square of their distances from it. Sir Isaac replied that it would be an Ellipsis, the Doctor struck with joy & amazement asked him how he knew it, why saith he I have calculated it, whereupon Dr. Halley asked him for his calculation without any further delay, Sir Isaac looked among his papers but could not find it, but he promised him to renew it, & then to send it to him...” (MS 1075-7, University of Chicago Library). As we have noted, it is by no means clear that Hooke could*not*have made the same claim that Newton made to Halley. In fact, he essentially did. It merely required combining the knowledge that the planets moved in elliptical orbits with the conviction that gravity was an inverse square force. He might even have claimed that he had “calculated it,” and indeed he made such claims.Google Scholar - 38).Herivel (1965), gives the text of
*De Motu*, and an English translation.Google Scholar - 39).See
*Birch*, 4, 347.Google Scholar - 40).That is, the orbit in the form of the radius vector as a function of
*t*or θ, as we would now require. For example, Proposition XXXI of Book I.Google Scholar - 42).Quoted in Bell (1947), p. 89.Google Scholar
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- 47).J.B. Duhamel,
*Regia scientiarum academiae historia*(Paris, 1698); noted in*CHO*, p. 63, n. 1.Google Scholar - 49).June-July, 1663; see, particularly Shapin and Schaffer (1985), pp. 249–52.Google Scholar
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*Micrographia*, p. 246.Google Scholar - 51).
*PW*, p. 201–2. On the causal connection Huygens saw between centrifugal force and gravity, see, for example, Yoder (1988), chapter 3.Google Scholar - 52).Gal (2003).Google Scholar
- 53).
*Gunther*, VIII, 28.Google Scholar - 54).
*PW*, pp. 149–190 (though p. 149 is labelled “194”). As compiled by Waller, the lectures run to about 35,000 words, which might be delivered in about 5 hours. Thus 5–10 lectures at minimum. The*Journal Book (Birch)*records lectures on the subject for 25 October, 8 and 15 November, 20 December, and 14 February (1682/3). The treatment of gravity is mainly on pages 176–185. Michaelmas, honoring St. Michael, was traditionally September 29, coinciding very nearly with the autumnal equinox, hence an equinox celebration. The Michaelmas term at British universities is, and was, the fall term, beginnng about Michaelmas. The Royal Society would usually resume meetings, after a summer hiatus, around Michaelmas. In 1682 they began again on 25 October and Hooke read the first part of a discourse on comets, prompted by the bright comet of 1680 and 1681.Google Scholar - 55).“Discourse of Comets,”
*PW*, p. 177. Hooke, of course was right; while the small bodies of the solar system are irregular, gravity has moulded the large bodies into approximate spheres.Google Scholar - 56).Should one be reminded here of Aristotle’s arguments about the sphericity of the Earth, it is clear that Hooke’s understanding of the physics of the problem is very different. In “A Discourse of Comets” Hooke advances the argument that if they did not possess gravity, the parts of rotating bodies like the Sun and Jupiter, would be dispersed by their rotation.
*PW*, p. 178.Google Scholar - 57).
*PW*, P. 176.Google Scholar - 58).
*PW*, p. 178.Google Scholar - 59).A fact which Westfall came close to conceding as early as 1967 (Westfall, 1967).Google Scholar
- 60).
*PW*, 178.Google Scholar - 61).
*PW*, p. 178.Google Scholar - 62).Though he does argue that “the perpendicular of Gravity will not always point to the center of the Earth,” because of its rotation (to which an exclamation mark might be added).
*PW*, p. 181.Google Scholar - 64).
*PW*, pp. 185.Google Scholar - 65).
*Corresp.*, II, p. 309. As noted below, Newton thought that Wren knew this is 1677, and if true, Hooke would have as well.Google Scholar - 66).
*PW*, p. 184.Google Scholar - 67).
*Works of Richard Bentley*, London, 1938, Vol. 3, pp. 210–11. Cited in Cajori (1934), p. 633.Google Scholar - 68).Aside from his recollection of 50 years later, the evidence comes mostly from the
*Waste Book*. See Herivel (1965), Chapter II.Google Scholar - 69).
*Corresp.*, II, pp. 433, 435. Newton to Halley, 27 May and 20 June 1686.Google Scholar - 70).Kollerstrom (1999) interprets a Halley letter to Newton (
*Corresp*. II, p. 442) as carrying the implication that Wren did not confirm Newton’s claim. Whether that interpretation is warranted or not, nothing Halley or Wren, or for that matter Hooke wrote, suggests that Wren was ahead of Hooke in arriving at the inversesquare nature of gravity. Yet they exchanged ideas on the subject regularly over coffee.Google Scholar - 71).We should note that in the second edition of the
*Principia*(1713), in the Scholium to Proposition IV, Theorem IV, Newton does acknowledge Hooke, along with Wren and Halley.Google Scholar - 72).The evidence against Newton’s story lies in his treatment of gravitation in the intervening period. See Kollerstom (1991) or Wilson (1989), for example. Though, to be fair, the lack of any evidence in support of Newton’s claims only makes them improbable, and does not rule them out.Google Scholar
- 73).The total of fourteen letters between Hooke and Newton after Oldenburg’s death consist of six from the fall of 1677 to the spring of 1678, and eight between 24 November 1679 and 18 December 1680. Of the latter, the first six constitute the exchange over dynamics: Hooke to Newton, 24 November 1679; Newton to Hooke, 28 November 1679; Hooke to Newton, 9 December 1679; Newton to Hooke, 13 December 1679; Hooke to Newton, 6 January 1679/80; Hooke to Newton, 17 January 1679/80. There are two more letters from almost a year later: Newton to Hooke, 3 December 1680 and Hooke to Newton, 18 December 1690. (
*Corresp*. II. We have alluded to this correspondence in several places. Two of these letters, Newton to Hooke, 28 November 1679, and Hooke to Newton, 17 January 1679/80, were reproduced in facsimile by*Gunther*, X, 52–55. (*Corresp.*, II, letters #235–241, 243). See n. 2, above.Google Scholar - 74).Whiteside (1967–90), VI p. 11, n. 32.Google Scholar
- 75).Whiteside (1991), Kollerstrom (1999), and the earlier works by Lohne (1960) and Patterson (1950).Google Scholar
- 76).
*Corresp.*, II, pp. 431–447; letters 285–291; see n. 35.Google Scholar - 77).Gunther published the 28 November and 17 January letters in facsimile in Vol X of his.
*Early Science in Oxford*(pp. 52–55).Google Scholar - 78).
*Corresp.*, II, p. 297.Google Scholar - 79).He had buried his mother in June of 1679 and had just returned from Lincolnshire when he received Hooke’s first letter. Westfall (1980), pp. 339–340.Google Scholar
- 80).
*Corresp.*, II, p. 300, 302. In evaluating this statement, one should weigh in the balance the relative care with which Newton seems to have used centrifugal force in 1664 (Brackenridge and Nauenberg, 2002, p. 88b ff.)Google Scholar - 82).To wit: “Your Deserting Philosophy in a time when soe many other Eminent freinds have also left her ... Seems a little Unkind yet tis to be hoped her allurements may sometimes make you ... alter your resolutions ...”
*Corresp.*, II, pp. 304–6. Hooke may have later regretted those generous words. Newton’s argument was that since the body was above the surface of the earth when released, it had a greater velocity toward the east than the point directly below on the surface, and thus advanced on the earth, falling to the east. Hooke pointed out to Newton that there was a southerly deflection as well. and Newton acknowledged that that was indeed the case. We now know that if the rotating earth is viewed as an inertial system, there is a centrifugal force which has a southerly component as well as one perpendicular to the earth’s surface. The Coriolis force will deflect the resultant southerly motion to the west (in the northern hemisphere). But the Coriolis force also acts on the much larger velocity component toward the center of the earth, causing a resultant easterly deflection. So the object will fall to the east of the perpendicular, and indeed, to the SE, as Hooke argued. It is unclear how he arrived at this conclusion. See also the discussion by Lohne (1960). It might be further noted that a plumb line suffers a similar deflection due to the centrifugal force. But there remains a very small southerly deflection due to the Coriolis force.Google Scholar - 84).It should be said that not everyone agrees that Newton’s very small figure was anything other than a qualitative solution. Whiteside wrote that “it is evident that we should not lay too great an emphasis on the precise shape of Newton’s roughly sketched fall curve.” Whiteside (1967-80), VI, p. 10, n. 29.Google Scholar
- 85).6 January 1679/80.
*Corresp.*, II, p. 309. For comparison, consult what Wren told Hooke two years earlier, on 20 September 1677;*Diary I*, p. 314.Google Scholar - 87).Newton assumed gravity to be constant, and a ball rolling in a
*circular path*on the interior surface of a cone has a constant centripetel force, which is the radial component of the normal force. As Hooke put it: “Your Calculation of the Curve by a body attracted by an aequall power at all Distances from the center Such as that of a ball Rouling in an inverted Concave cone is right and the two auges will not unite by about a third of a Revolution.”*Corresp.*II, 309.Google Scholar - 89).Though Ofer Gal has written a book which has a very different take on these issues and the same is the case in his contribution to the Hooke 2003 symposium (Gal, 2002, 2003). I do not find all of his arguments unconvincing, in particular details of the discussion of his relationship to Kepler in Gal (2003). See also Brackenridge and Nauenberg (2002).Google Scholar
- 90).Newton to Halley, 20 June and 14 July 1686.
*Corresp.*, II, letters 288 and 290. Among other things, Newton wanted to defend himself against Hooke’s later claim that he had shown his ignorance of the inverse-square nature of gravity when he assumed it constant in the letter of 28 November 1679. Newton heatedly wrote to Halley that “in my answer to his first letter I refused his correspondence, told him I had laid Philosophy aside, sent him only ye experiment of projectiles to sweeten my Anser, expected to heare no further from, could scarce perswade myself to anwer his second letter, did not answer his third, was upon other things, thought no further of philosophical matters when his letters put me upon it, & therefore may be allowed not to have had my thoughts of that kind about me so well at that time.” (*Corresp.*, II, p. 436). Very self-serving, one might say. Halley’s reply of 29 June (*Corresp.*II, pp. 441–3) described the meeting of the Society at which the*Principia*was presented, and Hooke’s reaction.Google Scholar - 91).As we mentioned in Chapter 5, the
*Journal Book (Birch)*entry for the meeting of 10 December 1684 notes that “Mr Halley gave an account, that he had lately seen Mr. Newton at Cambridge, who had shewed him a curious treatise,*De Motu*...”. Hooke may have first seen it somewhere around this time, though some have argued, as did Lohne, that he “only gradually became aware of contents of the work Newton was composing. But from the fall of the year 1686 we can see him [Hooke] feverishly active to assert his claims of priority.” Lohne (1960).Google Scholar - 94).Pugliese (1989).Google Scholar
- 97).Now in the collection of Hooke material in the Wren Library at Trinity College, Cambridge, Ms 0.11a.1/16. Pugliese (1989), p. 201. The manuscript consists of seven pages, and includes a diagram, reproduced in both Pugliese and Nauenberg (1994a); Ms 0.11a folio 6r. Nauenberg (1994a, p. 346) gives a transcription of the text in the figure.Google Scholar
- 98).Ibid, fol. 4r. The passage is quoted by Pugliese (1989).Google Scholar
- 100).Nauenberg (1994a, 2006).Google Scholar
- 101).In Newton’s case, the force on a moving body was taken to be proportional to the distance it would recede from the center if it continued along its tangential path (in some short time interval). Also Hall (1957). An almost identical approach was employed by Huygens as early as 1659. See Yoder (1998).Google Scholar
- 102).See, for example, Nauenberg (2003).Google Scholar
- 104).On the other hand, perhaps it was the failure of such an attempt that explains why no proof ever surfaced and why Wren was unconvinced. It is worth emphasizing that while Nauenberg has shown what Hooke
*could*do, and in fact did do in the case of a force proportional to the distance, the only evidence that Hooke might have done the same for the inverse-square force lies in his repeated claims that he had solved the problem. The views of Ofer Gal on Hooke’s and Newton’s constructions are different from Nauenberg’s. See Gal (2002). Nauenberg is mostly correct, but the issue cannot be further elaborated here.Google Scholar - 105).Pugliese (1989), p. 204.Google Scholar
- 106).Halley’s argument (
*Corresp.*II, 442) was just the opposite, that if Hooke was waiting for others to fail, in order to appreciate the value of his discovery, that no longer applied after*De Motu*.Google Scholar - 108).Although the date on the document is the better part of a year after
*De Motu*was presented to the Society, there are at least two reasons for believing that Hooke had not seen*De Motu*before he offered his “proof,” or at least that he was giving an independent argument. They are, first, that Hooke’s arguments are his own, typical of his reasoning and seemingly not informed by a reading of*De Motu*, and second, that based on the lack of discussion of*De Motu*at Society meetings, it may very well be that its detailed contents were not made available. Hooke was no longer Secretary and may have had no special access to correspondence or other documents. Perhaps trumping these theoretical arguments, however, is that in a letter to Newton on 29 June 1686, Halley wrote, referring to*De Motu*“...since which time is has been entered upon the Register books of the Society as all this past Mr. Hook was acquainted with it...”.*Corresp.*, II, 442. In Problem 2 of*De Motu*, Newton posed the following problem: “Given a body revolving in the ellipse of the ancients, there is requird the law of centripetal force directed to the center of the ellipse.” The result is “directly as the distance.” But the approach is entirely different. Herivel (1965), p. 281.Google Scholar - 109).For example, Cohen (1999), Whiteside (1970, 1989, 1991), Koyré (1955).Google Scholar
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- 111).Nauenberg in “Newton’s early computational method for dynamics” (Nauenberg, 1994a), Brackenridge and Nauenberg (2002), and Nauenberg (2006), have attempted to reconstruct the process by which Newton arrived at the orbits he drew in his second letter to Hooke. Their analysis is very interesting and very possibly correct, but speculative. Nonetheless, the mathematical imperatives are such that there a very few ways for Newton to have arrived at his result, assuming that he had a method, and that the argument wasn’t merely qualitative. As to when Newton first either obtained Kepler’s Second Law or showed that elliptical orbits resulted from inverse-square forces, Cohen calls Newton’s claim, made around 1718, that he had done the latter in 1677 (conveniently before the correspondence with Hooke), as “bogus history.” Cohen (1980), pp. 248–9.Google Scholar
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*Corresp.*, II, 431.Google Scholar - 118).Aubrey to Anthony Wood, 15 September 1689;
*Corresp.*, III, pp. 40–44; especially notes 1–9 pp. 43–44. See also Chapter 12.Google Scholar