Hooke and Newton: Divining Planetary Motions

The year 2003 marked the 300th anniversary of the death of Robert Hooke, one of the greatest scientists of the 17th century. Hooke's legacy is currently being restored after three centuries of oblivion. It might be expected that his seminal influence on Isaac Newton's development of the theory of planetary motion would be well known and understood by now, if not by physicists then at least by historians and philosophers of science. But that is not the case, as one finds in Ofer Gal's book, Meanest Foundations and Nobler Superstructures: Hooke, Newton and the "Compounding of the Celestiall Motions of the Planetts," which was reviewed recently by George E. Smith (Physics Today, September 2003, page 61).

To appreciate the importance of Hooke's contribution to planetary motion, which he communicated to Newton during a correspondence in the autumn of 1679, one must understand not only Hooke's views, described at length in Gal's book, but also Newton's own knowledge of the subject at the time. Smith repeats the standard science historians' argument that "before his correspondence with Hooke, Newton (along with many others) thought of the planetary orbits as involving equilibrium between, using Newton's phrasing, 'an endeavor to recede from the center' associated with circular motion and some other mechanism." But that explanation is inadequate. Moreover, it is not the one offered by Gal, who makes the unprecedented claim that, before Hooke's intervention, "representing force-driven motion by straight lines or open curves, while reserving the closed orbit to represent force free motion, expressed a common understanding of the relation between force and motion."¹ Gal also avers without justification that the "novelty of De Motu thus encapsulated [Newton's] willingness to represent forced motions by closed curves."² (De Motu was Newton's first draft of the Principia, written five years after his correspondence with Hooke.)

Newton's letter of 13 December 1679 to Hooke clearly shows that these claims are incorrect.³ Unlike his contemporaries, Newton had developed a sophisticated mathematical theory of orbital motion, as is evident in his description of orbital curves under the action of various central forces. In that letter, Newton even included a comment on the special case of a 1/r³ force (not treated in most physics textbooks), which leads to an orbit that rotates toward the center "by an infinite number of spiral revolutions";³ that remarkable result, however, has been ignored.

Considerable evidence exists that Newton's early theory of orbital motion was based on the mathematical description of curvature that he and, independently, Christiaan Huygens had developed. However, Newton's approach, based on a decomposition of motion along the tangent and along the normal to the orbital curve, was missing an essential ingredient. Newton was not yet aware that for central forces, angular momentum is conserved, and thus Kepler's second law (area law) is justified.

Newton later admitted that, in 1679, "in answer to a letter to Dr. Hook . . . I found now that whatsoever was the law of the forces which kept the Planets in their Orbs, the areas described by the Radius drawn from them to the Sun would be proportional to the times in which they were described."⁴ What Hooke had suggested to Newton is that orbital motion could be decomposed into "a direct [inertial] motion by the tangent, and an attractive motion [radial] towards the central body."⁴ For a central impulsive force acting at periodic intervals, this decomposition of motion makes the conservation of angular momentum self-evident, as Newton subsequently showed in De Motu. His proof became a cornerstone of the Principia, because it allowed him to geometrize orbital dynamics by replacing the time variable by the area swept by the radial line drawn to the center of force.

Contrary to Gal's argument that Hooke's scientific style was "radically different from Newton's," and Smith's assertion of a "monumental contrast" between their approaches to science, both Hooke and Newton had a very similar and quite modern approach, based on observations and experiments, to the understanding of natural phenomena. For example, Hooke reached his views of planetary motion by analyzing the motion of a conical pendulum, as he explained in detail in a 1666 lecture given at the Royal Society of London.⁴ In his review, Smith states that "the great value of Gal's book lies in his analysis of how Hooke arrived at his conception [of orbital motion] through his research in optics." But in his Royal Society lecture, Hooke explicitly rejected the optical analogy that the "ending" of the motion of planets into a curve is caused by the "unequal density of the medium." The essential difference in their approaches is that Newton was able to translate physical concepts into mathematical form and solve the resulting equations, while Hooke lagged far behind in that ability. The theory of planetary motions was not "divined" by Newton, but should be recognized as a remarkable joint scientific achievement of Newton and Hooke.

• 1. O. Gal, Meanest Foundations and Nobler Superstructures: Hooke, Newton and the "Compounding of the Celestiall Motions of the Planetts," Kluwer Academic, Boston (2003), p. 184.
• 2. Ref. 1, p. 188.
• 3. M. Nauenberg, Arch. Hist. Exact Sci. 46, 221 (1994); J. B. Brackenridge, M. Nauenberg, in The Cambridge Companion to Newton, I. B. Cohen, G. E. Smith, eds., Cambridge U. Press, New York (2003), p. 85.
• 4. M. Nauenberg, Am. J. Phys. 62, 331 (1994) [INSPEC].
• 3. F. Perrin, Comptes Rendus B 208, 1394 (1939); S. Flügge, Naturwissenschaften 27, 402 (1939); R. Peierls, Proc. Cambridge Philos. Soc. 35, 610 (1939).