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

Part of the Science Networks. Historical Studies book series (SNHS, volume 39)


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.


Centrifugal Force Circular Motion Elliptical Orbit Kepler Problem Planetary Orbit 
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  1. 1).
    Diary I, 18 and 21 October 1679.Google Scholar
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    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
  3. 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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    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
  11. 22).
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    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
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  32. 56).
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    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
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    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
  47. 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
  48. 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
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    Whiteside (1967–90), VI p. 11, n. 32.Google Scholar
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    Whiteside (1991), Kollerstrom (1999), and the earlier works by Lohne (1960) and Patterson (1950).Google Scholar
  51. 76).
    Corresp., II, pp. 431–447; letters 285–291; see n. 35.Google Scholar
  52. 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
  53. 78).
    Corresp., II, p. 297.Google Scholar
  54. 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
  55. 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
  56. 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
  57. 84).
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  58. 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
  59. 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
  60. 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
  61. 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
  62. 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
  63. 94).
    Pugliese (1989).Google Scholar
  64. 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
  65. 98).
    Ibid, fol. 4r. The passage is quoted by Pugliese (1989).Google Scholar
  66. 100).
    Nauenberg (1994a, 2006).Google Scholar
  67. 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
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    See, for example, Nauenberg (2003).Google Scholar
  69. 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
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    Pugliese (1989), p. 204.Google Scholar
  71. 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
  72. 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
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    For example, Cohen (1999), Whiteside (1970, 1989, 1991), Koyré (1955).Google Scholar
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    Whiteside (1991), Wilson (1989).Google Scholar
  75. 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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    Herivel (1965), p. 282, Brackenridge (2001) in Buchwald and Cohen (2001), pp. 105–115.Google Scholar
  77. 113).
    Westfall (1980), pp. 404–408, and all of his Chapter 10.Google Scholar
  78. 115).
    Halley to Newton, 22 May 1686, Corresp., II, 431.Google Scholar
  79. 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

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