Robert Hooke and celestial mechanics

Time was when Robert Hooke was best known as a historical figure for his disputes with the eminent Isaac Newton. Remnants of that attitude still remain, but since the 1980s there has been a steady flow of studies which reveal a figure distinguished in his own right. There is a Robert Hooke compendium which covers much of the immense scope of Hooke's output. One area that is not included on that site is Hooke's contribution to celestial mechanics. This is the area around which the most enduring controversy arose. The strength and endurance of feeling that is aroused is perhaps because the concepts are so widely taught at elementary level, and their significance is so highly estimated. For example, the eminent Newtonian scholar I Bernard Cohen introduced his paper on ‘Newton’s Discovery of Gravity’ with:

The high point of the Scientific Revolution was Isaac Newton's discovery of the law of universal gravitation: All objects attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of their separation. By subsuming under a single mathematical law the chief physical phenomena of the observable universe Newton demonstrated that terrestrial physics and celestial physics are one and the same. In one stroke the concept of universal gravitation revealed the physical significance of Johannes Kepler's three laws of planetary motion, solved the thorny problem of the origin of the tides and accounted for Galileo Galilei's curious and unexplained observation that the descent of a free-falling object is independent of its weight. ¹ I B Cohen ‘Newton’s Discovery of Gravity’, Scientific American, Vol. 244, No. 3, March 1981, p. 169

This passage illustrates the great esteem attached to the discovery of the concept of universal gravitation, and also some of the points of contention in the controversy around it. This is because that concept was presented by Hooke to Newton in a correspondance between them ² Newton, I. (1960). The Correspondence of Isaac Newton (H. Turnball, Ed.). Cambridge: Cambridge University p.297 Letters 235 to 241 doi:10.1017/9781108627375 , in which Newton is reluctantly persuaded to consider that terrestrial and celestial mechanics are one and the same, and are governed by the same gravity. Hooke urges him to verify by calculation that the elliptical orbits of the planets and moons can be derived from their mechanical motions when the gravitational attractions between them are inversely proportional to the square of their separation.

In the context of this correspondence it might appear that Cohen's introduction is distinctly misleading, but it is not a simple matter to summarise the issues. In Cohen's view, Hooke's correspondence showed that he “could not make the leap from intuitive hunch and guesswork to exact science”, and that his concepts were both limited and inexact. This assessment of Hooke's efforts is fairly prevalent, for example all the four relevant entries in Wikipedia ³ Controversy with Hooke, Hooke on gravitation, Newton's "de Motu", Newton's law of universal gravitation. include the oft quoted 18th century aphorism “The example of Hook shows what a distance there is between a glimpsed truth and a proven truth” ⁴ "Explication abregée du systême du monde, et explication des principaux phénomenes astronomiques tirée des Principes de M. Newton" (1759), at Introduction (section IX), page 6: "Il ne faut pas croire que cette idée ... de Hook diminue la gloire de M. Newton", [and] "L'exemple de Hook" [serves] "à faire voir quelle distance il y a entre une vérité entrevue & une vérité démontrée". . The reality behind this turn of phrase is assessed by an examination of the correspondence, and other works by Hooke, which is found in the next section below.

Whatever the outcome of the detailed debate, even his severest critics admit that Hooke's ideas were sound, although not backed up by mathematical proof. It is the ideas and their inception, however, that capture the popular imagination. So, for example, on the Wikipedia page on celestial mechanics:

Isaac Newton (25 December 1642–31 March 1727) is credited with introducing the idea that the motion of objects in the heavens, such as planets, the Sun, and the Moon, and the motion of objects on the ground, like cannon balls and falling apples, could be described by the same set of physical laws. In this sense he unified celestial and terrestrial dynamics. ⁵ Celestial Mechanics

Where popular presentation is concerned, a truth glimpsed has more impact than a truth proved. Unfortunately for Hooke's reputation, although he came up with the ideas, and Newton's stature is supposed to rest on the precision and depth of his analysis, it is Newton who is given all of the credit.

The debate on Hooke's contribution is referred to in one of the Wikipedia entries, but the verdict is pronounced against Hooke:

There has been scholarly controversy over exactly what if anything Newton really gained from Hooke, apart from the stimulus that Newton acknowledged.

In the sections below a range of contributions to both sides of the debate are examined, with the conclusion that they do not support the verdict as pronounced.

Guesses and glimpses?

A good starting point is a passage from Hooke's 1670 Cutlerian Lecture Attempt to Prove the Motion of the Earth, published in 1674. The passage is widely quoted ⁶ Robert Weinstock, Problem in two unknowns: Robert Hooke and a worm in Newton’s apple, The Physics Teacher 30, 282 (1992); DOI: 10.1119/1.2343543
Ofer Gal, Meanest Foundations and Nobler Superstructures: Hooke, Newton and the "Compounding of the Celestiall Motions of the Planetts" Springer-Science. 2002.
Michael Nauenberg, Robert Hooke's Seminal Contribution to Orbital Dynamics, Physics in perspective, 2005, Vol.7 (1), p.4 Springer DOI: 10.1007/s00016-004-0226-y as a demonstration of the success and originality of Hooke's analysis, years before Newton adopted it.

I shall explain a system of the world, differing in many particulars from any yet known, answering in all things to the common rules of mechanical motions. This depends upon 3 suppositions; first, that all celestial bodies have an attractive of gravitating power towards their own centres, whereby they attract not only their own parts, and keep them from flying from them, as we ‘observe the Earth to do, but that they do also attract all the other ‘celestial bodies that are within the sphere of their activity, and ‘consequently that not only the Sun and the Moon have a influence upon the body and motion of the Earth, and the Earth upon them, but that Mercury also, Venus, Mars, Saturn and Jupiter, by their attractive powers have a considerable influence upon its motion, as, in the same manner, the corresponding attractive power of the Earth has a considerable influence upon every one of their motions also. The second supposition is this, that all bodies, that are put into direct and simple motion will continue to move forwards in a straight line, until they are by some other effectual powers deflected and bent into a motion describing a circle, ellipsis, or some other uncompounded curve line. The third supposition is, that these attractive powers are ‘so much the more powerful in operating, by how much nearer the body wrought upon is to their own centres

Hooke's three suppositions comprise a universal gravitational attraction between celestial bodies acting just as the earth's gravity does, the law of motion itemised as the first in Newton's Principia, and a proposal that gravitation decreases with increasing separation. Taken separately, each is state of the art, but not a great breakthrough. But, as Ofer Gal has pointed out, it is the assertion that the celestial motions might be calculated by applying just these common rules of mechanic motions that is the true breakthrough. Previous theories of the celestial motions had supposed that they are sustained by a naturally rotating system, such as a vortex driven round by the sun at its centre. How could lumps of matter flying around independently like so many balls in the air, all interacting through the gravitational attractions between them, sustain this paradigm of stability and regularity? Surely there had to be some grand apparatus for keeping it all in order.

This system of the world was the product of a period of work marked by Hooke's paper presented in 1666 ⁷ Thomas Birch: The History of the Royal Society of London for improving of Natural Knowledge, from its first rise. Vol. II, London, 1756, p90. to the Royal Society proposing that the

cause of inflecting a direct motion into a curve may be from an attractive property of the body placed in the center whereby it continually endeavors to attract or draw it to itself. For if such a principle be supposed, all the phenomena of the planets seem possible to be explained by the common principle of mechanical motions

The presentation was followed by a demonstration using an arrangement of pendulums analogous to the earth and moon showing their orbit about a common centre. Hooke made it clear that the attraction in the case of a pendulum was not supposed to be of the same degree as that of a planet.

The advance that Hooke's thinking represents is demonstrated in his written correspondence with Newton, launched by Hooke in 1679 as an employee of the Royal Society ⁸ Newton, I. (1960). The Correspondence of Isaac Newton vol 2 (H. Turnball, Ed.). Cambridge: Cambridge University Press. doi: 10.1017/9781108627375 . It is clear from the correspondence that Newton has no prior grasp of Hooke's approach, and indeed has to be convinced of it despite his reservations. The first letter, with considerable diplomacy, includes a request for Newton's opinion on Hooke's System of the World:

Sr, Finding by our Registers that you were pleasd to correspond wth Mr Oldenburg and having also had the happinesse of Receiving some Letters from you my self, make me presume to trouble you wth this present scribble. [...] I am not ignorant that both heretofore and not long since also there have been some who have indeavourd to misrepresent me to you and possibly they or others have not been wanting to doe the like to me, but Difference in opinion if such there be (especially in Philosophicall matters where Interest hath little concerne) me thinks shoud not be the occasion of Enmity - tis not wth me I am sure. For my own part I shall take it as a great favour if you shall please to communicate by Letter your objections against any hypothesis or opinion of mine, And particularly if you will let me know your thoughts of that of compounding the celestiall motions of the planetts of a direct motion by the tangent & an attractive motion towards the centrall body, Or what objections you have against my hypothesis ofthe lawes or causes of Springinesse. I have lately Received from Paris a new Hypothesis invented by Monsieur Mallement De Messange, Dr of the Sorbon....

Newton replies within a few days, explaining that he is

...almost wholy unacquainted wth what Philosophers at London or abroad have of late been imployed about. And perhaps you will incline ye more to beleive me when I tell you yt I did not before ye receipt of your last letter, so much as heare (yt I remember) of your Hypotheses of compounding ye celestial motions of ye Planets, of a direct motion by the tangt to ye curve & of ye laws & causes of springyness, though these no doubt are well known to ye PhilosophIcal world. ...

and he goes on to present a proposal of his own to demonstrate experimentally the daily rotation of the earth:

In requital of this advertisement I shall communicate to you a fansy of my own about discovering the earth's diurnal motion. In order thereto I will consider ye Earth's diurnal motion alone without ye annual, that having little influence on ye experimt I shall here propound.

Suppose then BDG represents the Globe of ye Earth carried round once a day about its center C from west to east according to ye order of ye letters BDG; & let A be a heavy body suspended in the Air & moving round with the earth so as perpetually to hang over ye same point thereof B. Then imagin this body B let fall & it's gravity will give it a new motion towards ye center of ye Earth without diminishing ye old one from west to east. Whence the motion of this body from west to east, by reason that before it fell it was more distant from ye center of ye earth then the parts of ye earth at wch it arrives in its fall, will be greater then the motion from west to east of ye parts of ye earth at wch ye body arrives in it's fall: & therefore it will not descend in ye perpendicular AC, but outrunning ye parts of ye earth will shoot forward to ye east side of the perpendicular describing in it's fall a spiral line ADEC, quite contrary to ye opinion of ye vulgar who think that if ye earth moved, heavy bodies in falling would be outrun by its parts & fall on the west side ofye perpendicular. The advance of ye body from ye perpendicular eastward will in a descent of but 20 or 30 yards be very small & yet I am apt to think it may be enough to determin the matter of fact. ...

Newton goes on to give many details of how such an experiment might be set up. On his consideration of Hooke's System he says

If I were not so unhappy as to be unacquainted with your Hypotheses abovementioned (as I am wth almost all things which have of late been done or attempted in Philosophy) I should so far comply wth your desire as to send you what Objections I could think of against them if I could think of any. And on ye other hand I could with pleasure heare & answer any Objections made against any notions of mine in a transient discourse for a divertisment. But yet my affection to Philosophy being worn out, so that I am almost as little concerned about it as one tradesman uses to be about another man's trade or a country man about learning, I must acknowledge my self avers from spending that time in writing about it wch I think I can spend otherwise more to my own content & ye good of others:

Hooke, who is famed for his experiments, has attempted to put his theories to Newton, who is famed for his theoretical powers. Newton counters with offering Hooke some instruction in experiment. But Newton has added to the physical path of the body AD an imagined path, as though it could pass onward in some manner, to the centre of the earth at C. This shows a view very different from Hooke's system. The imagined path is of a body moving under the influence of a central force, and gives an example in which the body will end at rest at the centre of the force. This does not address Hooke's proposal that such motions could account for the eternal and regular motion of the celestial bodies.

In his reply, Hooke sympathises with Newton, saying that philosophy should not be forced, but has to be inspired, and says that Newton's rejection of the vulgar opinion on his proposed experiment is certainly right and true. Hooke goes on to take up Newton's imagined line of descent of the falling body through the earth to its centre, and use it to direct his thinking to the workings of the System of the World:

But as to the curve Line which you seem to suppose it to Desend by (though that was not then at all Discoursed of) Vizt a kind of spirall which after sume few revolutions Leave it in the Center of the Earth my theory of circular motion makes me suppose it would be very differing and nothing att all akin to a spirall but rather a kind Elleptueid. At least if the falling Body were supposed in the plaine of the equinoxciale supposing then ye earth were cast into two half globes in the plaine of the equinox and those sides separated at a yard Distance or the lilke to make Vacuity for the Desending Body and that the gravitation to the former Center remained as before and that the globe of the earth were supposed to move with a Diurnall motion on its axis and that the falling body had the motion of the superficiall parts of the earth from whence it was Let fall Impressed on it, I conceive the line in which this body would move would resemble An Elleipse: for Instance Let ABDE represent the plaine of the equinox litmited by the superficies of the earth: C the Center therof to which the lines of Gravitation doe all tend. Let A represent the heavy Body let fall at A and attracted towards C but Moved also by the Diurnall Revolution of the earth from A towards BDE &c. I conceive the curve that will be described by this descending body A will be AFGH and that the body A would never approach neerer the Center C then G were it not for the Impediment of the medium as Air or the like but would continually proceed to move round in the Line AFGHAFG &c. But w[h]ere the Medium through which it moves has a power of impeding and destroying its motion the curve in wch it would move would be some what like the Line AIKLMNOP &c and after then many resolutions would terminate in the Center C.

Hooke has made Newton's thought experiment somewhat better defined, and chosen a case to illustrate sustained orbital motion. In the absence of an impeding or deflecting force a body travels in a straight line; in the presence of a deflecting force its path is curved. Without impediment, the body must be expected to oscillate for ever in some pattern around the centre between extremes of distance from the centre. This pattern might take the elliptical shape of a planetary orbit.

Newton's reply accepts the strength of Hooke's argument, but is still sceptical that a body travelling under a central force could produce anything like an ellipse:

I agree wth you yt ye body in our latitude will fall more to ye south then east if ye height it falls from be any thing great. And also that if its gravity be supposed uniform it will not descend in a spiral to ye very center but circulate wth an alternate ascent & descent made by it's vis centrifuga & gravity alternately overballancing one another. Yet I imagin ye body will not describe an Ellipsoeid but rather such a figure as is represented by AFOGHIKL &c. Suppose A ye body, C ye center of ye earth, ABDE quartered wth perpendicular diameters AD, BE, wch cut ye said curve inF & G; AM ye tangent in wch ye body moved before it began to fall & GN a line drawn parallel to yt tangent. When ye body descending through ye earth (supposed pervious) arrives at G, the determination of its motion shall not be towards N but towards ye east between N & D.

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 of gravity in every moment of it's passage from A to G: The motion from A to M being in a parallel to GN inclines not ye body to verge from ye line GN. The innumerable & infinitly little motions (for I here consider motion according to ye method of indivisibles) continually generated by gravity in its passage from A to F incline it to verge from GN towards D, & ye like motions generated in its passage from F to G incline it to verge from GN towards C. But these motions are proportional to ye time they are generated in, & the time of passing from A to F (by reason ofye longer journey & slower motion) is greater then ye time of passing from F to G. And therefore ye motions generated in AF shall exceed those generated in FG & so make ye body verge from GN to some coast between N & D. The nearest approach therefore of ye body to ye center is not at G but somewhere between G & F as at O. And indeed the point O, according to ye various proportions of gravity to the impetus of ye body at A towards M, may fall any where in ye angle BCD in a certain curve wch touches ye line BC at C & passes thence to D. Thus I conceive it would be if gravity were ye same at all distances from ye center. But ifit be supposed greater nearer ye center ye point O may fall in ye line CD or in ye angle DCE or in other angles yt follow, or even no where. For the increase of gravity in ye descent may be supposed such yt ye body shall by an infinite number of spiral revolutions descend continually till it cross ye center by motion transcendently swift.

Your acute Letter having put me upon considering thus far ye species of this curve, I might add something about its description by points quam proxime. But the thing being of no great moment I rather beg your pardon for having troubled you thus far wth this second scribble wherin if you meet wth any thing inept or erroneous I hope you will pardon ye former & ye latter I submit & leave to your correction remaining Sr...

Newton has chosen to look at a case where the force of gravity is constant, which makes the analysis more tractable. He is still not seeing these mechanical motions as a model for the celestial ones. His statement that the thing is of no great moment reinforces that impression, but it may just be a polite way to round off the communication. Newton has in earlier work found as a consequence of Kepler's third law of planetary motion that the outward endeavours of the planets in their orbits are inversely proportional to the square of the distance from the sun ⁹ J. Herivel, The Background to Newton’s Principia: A Study of Newton’s Dynamical Researches in the Years 1664-84 (Oxford U.P., London, 1996), p197 , but he has not connected this finding with any attractions between them ¹⁰ Curtis Wilson "The Newtonian achievement in astronomy" in R Taton & C Wilson (eds) (1989) The General History of Astronomy (Cambridge UP 1989), Volume, 2A, page 237-8. . Newton has instead calculated the forces that would have to be sustained by a centrally governed rotational system to keep the planets in their allotted orbits. He compares these endeavours to gravity, using the latter term only to refer to terrestrial weight, with no suggestion that this phenomenon might extend further.

Hooke's reply confirms his understanding of the mechanics of the celestial motions, and firmly asserts the significance of the problem. It also clearly states the inverse square law of attraction as the key to a successful analysis, and that although the gravity of the bodies acts from their full extent it can be considered as if acting from their centres.

Your Calculation of the Curve by a body attracted by an equal 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. But my supposition is that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall, and Consequently that the Velocity will be in a subduplicate proportion to the Attraction and Consequently as Kepler Supposes Reciprocall to the Distance. And that with Such an attraction the auges will unite in the same part of the Circle and that the neerest point of accesse to the center will be opposite to the furthest Distant. Which I conceive doth very Intelligibly and truly make out all the Appearances of the Heavens. And therefore (though in truth I agree with You that the Explicating the Curve in which a body Descending to the Center of the Earth, would circumgyrate were a Speculation of noe Use yet) the finding out the proprietys of a Curve made by two such principles will be of great Concerne to Mankind. because the Invention of the Longitude by the Heavens is a necessary Consequence of it: for the composition of two such motions I conceive will make out that of the moon. What I mentioned in my last concerning the Descent within ye body of the Earth was But upon the Supposall of such an attraction, not that I beleive there really is such an attraction to the very Center of the Earth, but on the Contrary I rather Conceive that the more the body approaches, the Center, the lesse will it be Urged by the somewhat like the Gravitation on a pendulum or a body moved in a Concave Sphrere where the power Continually Decreases the neerer the body inclines to a horizontall motion, which it hath when perpendicular under the point of suspension, or in the Lowest point, and there the auges are almost opposite, and the nearest approach to the Center is at about a quarter of a Revolution. But in the Celestiall Motions the Sun Earth or Centrall body are the cause of the Attraction, and though they cannot be supposed mathematicall points yet they may be Conceived as physicall and the attraction at a Considerable Distance may be computed according to the former proportion as from the very Center. This Curve truly Calculated will shew the error of those many lame shifts made use of by astronomers to approach the true motions of the planets with their tables.

Hooke goes on to describe his investigation of the experiment proposed by Newton in his first letter. There seems to have been no reply to this letter – after about ten days Hooke again writes:

I gave you an account by my Last of the 6th Instant that by the tryalls I had made without Doors your Expt succeeded very well. I can now assure you that by two tryalls since made in two severall places wthin doors it succeeded. Alsoe that I am now perswaded the Experiment is very certaine, and that It will prove a demonstration of the Diurnall motion of the earth as you have very happily intimated. It now remaines to know the proprietys of a curve Line (not circular nor concentricall) made by a centrall attractive power which makes the Velocitys of Descent from the tangent Line or equall straight motion at all Distances in a Duplicate proportion to the Distances Reciprocally taken. I doubt not but that by your excellent method you will easily find out what that Curve must be, and its proprietys, and suggest a physicall Reason of this proportion. If you have had any time to consider of this matter, a word or two of your Thoughts of it will be very gratefull to the Society (where it has been debated)

Newton's only further reply to Hooke mentions his report on the experimental investigation, but not Hooke's proposal for applying his system of the world. The problem of demonstrating that Hooke's system would confirm Kepler's assertions on celestial orbits remained unsolved despite the efforts of Hooke, Edmund Halley, and Sir Christopher Wren. However, when Halley visited Newton in August 1684, he learnt that Newton had proved that for a body moving in an ellipse under an attractive force directed to a focus the law of force was that of the inverse square ^8a Newton, I. (1960). The Correspondence of Isaac Newton vol 2 (H. Turnball, Ed.). Cambridge: Cambridge University Press. doi: 10.1017/9781108627375 . It seems evident that, when the penny finally dropped, Newton's “worn out affection to Philosophy” was fast mended and he took up the analysis of Hooke's system, but without the courtesy of any acknowledgement or response to Hooke. Newton's manuscript De motu corporum in gyrum, produced later in the same year in response to Halley's visit is an extended study of Hooke's direct motion by the tangent & an attractive motion towards the centrall body. This manuscript, revised and extended, forms the opening sections of Principia Mathematica.

The case and controversy

It is no surprise that Hooke sought recognition for his contribution in Principia when it was published. His case ¹¹ R.T. Gunther, Early Science in Oxford, 10: 57-60 was a summary of the pendulum demonstrations and the letters reviewed above. The controversy is visible in letters between Halley and Newton from 1686, starting:

Your Incomparable treatise intituled Philosophiæ Naturalis Principia Mathematica, was by Dr Vincent presented to the R. Society on the 28th past [...] There is one thing more that I ought to informe you viz,that Mr Hook has some pretensions upon the invention of ye rule of the decrease of Gravity being reciprocally as the squares of the distances from the Center. He sais you had the notion from him, though he owns the Demonstration of the Curves generated therby to be wholly your own. ¹² Newton, I. (1960). The Correspondence of Isaac Newton p431: #285 Halley to Newton 22 May 1686 vol 2 (H. Turnball, Ed.). Cambridge: Cambridge University Press. doi:10.1017/9781108627375.

Newton replies that “In the papers in your hands there is noe one proposition to which he can pretend, & soe I had noe proper occasion of mentioning him there”. In subsequent letters to Halley he becomes more dismissive of Hooke's contribution, and asserts that he had previously communicated an explanation of orbits to the Royal Society that implied an inverse square decrease of gravity, that Hooke had made a mistaken assertion about gravity within the earth, that Hooke had plagiarised the work of others on the inverse square relation, that he himself had calculated the forces of the planets as being as the inverse square of their distance from the sun, and that Hooke's suppositions were just guesses without mathematical proof. Halley, in his replies, says that Hooke had promised to produce a proof of the elliptical orbit but that he had failed to do so, and that opinion in the society was that Newton should be considered as the inventor, and that “I must now again beg you, not to let your resentments run so high, as to deprive us of your third book”.

The controversy has continued, on and off, since then. The presentations in the relevant Wikipedia entries include some account of Hooke's case, but echo many of the counterclaims of Newton. The pages make frequent reference to a comprehensive chapter in The General History of Astronomy by Curtis Wilson ¹³ Curtis Wilson, "The Newtonian achievement in astronomy", in R Taton & C Wilson (eds) The General History of Astronomy (Cambridge UP 1989) Volume, 2A, chapter 13, pages 233–274 : "The Newtonian achievement in astronomy". Wilson does not adjudicate on the controversy, but simply presents the historical account, introducing it at p233:

In brief, Newton before 1679 was so far from having entertained the idea of universal gravitation that he had not yet considered the orbital paths produced by different laws of centripetal attraction as a mathematical problem. This problem was posed to him by Robert Hooke (1635-1703) in 1679, Still in the five years following, there is no sign that Newton regarded universal gravitation as the obvious generalization it has since been taken to be.

Several of Newton's claims are disposed of on subsequent pages: the significance of his earlier work on inverse square attractions on page 237, his claim of plagiarism against Hooke on page 239, and his claim of Hooke's mistake on gravity within the earth on page 245. Pages 240-246 on the correspondence about orbital motion begin with:

Of seminal importance, however, will be Hooke’ proposal of the idea of obtaining the orbital motions of planets by combining an attraction to a centre with inertial motion along the tangent... In a correspondence that Hooke initiated with Newton in 1679, he so insisted on this way of conceiving orbital motion as to goad Newton into discovering a major consequence of the idea, namely the areal rule; and this consequence was to provide the basis for all Newton’s future work in planetary dynamics.

There is clearly enough here to justify Hooke's expectation of a special acknowledgement in Newton's subsequent publications.

Glimpses and proofs

There remains the issue of the soundness of Hooke's grasp of celestial mechanics, and the status of his suppositions: are they guesses or brilliant insights? The latter opinion depends on the former: if Hooke does not understand the mechanics his suppositions are likely to be stabs in the dark. For Cohen, Hooke's statement

the Velocity will be in a subduplicate proportion to the Attraction and Consequently as Kepler Supposes Reciprocall to the Distance

is “a fundamental error that must have convinced Newton that Hooke did not entirely understand what he was talking about” (although Newton does not seem to have expressed that view). Actually, Kepler's statement at that time was as Hooke described, and is a good approximation for planetary orbits, even the more eccentric Mars orbit studied by Kepler. Wilson describes Hooke's assumption as an inexactitude, not a fundamental error.

Wilson remarks(p245) about Hooke's penultimate letter:

This passage is remarkable for the correctness of a number of its guesses. Hooke rightly supposes that the force of gravity beneath the Earth's surface decreases with approach to the centre,... His conception is that of an attraction resulting from the attractions of all the parts of which the attracting body is made up.

Wilson perhaps unconsciously echoes Newton's term for Hooke's conceptions. A less improbable interpretation of this passage is that they are not guesses, but are based on a sound mechanical understanding. The latter conception corresponds exactly to Newton's theorem on gravitational attraction. It seems that Hooke's case for recognition has been accepted by current historical authority, but is not well represented in prevalent popular views.

What, then, is the real distance between a glimpsed truth and a proven one? In practice, the former precedes the latter, but the latter is inaccessible without the former. In between the two there is a process of finding a way forward, with some turnings leading to dead ends and others illuminating and advancing the investigation. This process is evident in Newton's own work, where effects such as the tides and perturbations of the lunar orbit are explored in an approximate manner, and the calculations do not produce results that match observation. Although his analysis of tides is sound in principle, the calculated value of tidal motion is about one tenth of that observed. Only when the analysis is extended to include additional effects does the calculation become realistic. This does not diminish the value of Newton's laying out the framework for the analysis; nor is the value of Hooke's system of the world diminished by its being a framework not completely analysed.

It should be made clear that whatever the outcome of the controversy, Newton remains a towering figure in the development of celestial mechanics. To illustrate this I have rewritten the passage from Cohen's paper above so as to render a statement of Newton's contribution which follows the account given by Curtis Wilson:

A turning point of the Scientific Revolution was Isaac Newton's development of analytical dynamics, and its application to celestial mechanics. When a model of celestial motions as bodies freely moving under gravitational forces, just as bodies on earth move under the force of gravity, these forces being inversely proportional to the square of the distance between them, was put to him, he was able not only to confirm that the motions observed by Kepler conform to this model, but in a series of brilliant proofs extend the analysis to derive the modification of Kepler's third law required because the planet attracts the Sun as well as the Sun the planet; to prove that spheres attract as though their masses are concentrated at their centres; to account for the chief perturbations in the motions of the Moon; to explain the precession of the equinoxes on the assumption that the Earth a spheroid flattened at the poles; and to fit parabolic trajectories to the observations of comets. The latter result demonstrated that even the extraordinary behaviour of comets required no modification to the universal association of gravity with weight.

The power and completeness of this analysis, based on such simple conceptions, changed the the world's view of what could be achieved by a mathematical description of nature.

Hooke's solution

There has been some speculation as to how far Hooke progressed in his attempts to analyse orbital motion:

Beyond this farseeing but admittedly qualitative understanding of the problem of planetary motion, however, we mostly have to guess what Hooke meant by his claim to have solved (or “perfected”) it, as he did in 1679, or by the promise in his lecture on light of 1681 of a solution “with Geometrical Certainty and Exactness.” ¹⁴ Robert D. Purrington, The First Professional Scientist: Robert Hooke and the Royal Society of London, Science Networks, Historical Studies, Volume 39 2009, p169

Hooke's claim seems uncharacteristically far fetched in view of the advanced and complex nature of the subsequent analysis by Newton:

From a modern perspective, and taking into account Hooke’s abilities, we know that a full solution, the deduction of Kepler’s Laws of planetary motion from a theory of attraction between the Sun and a planet, which is what Newton would do, was far more difficult than Hooke could have imagined, and, in fact, beyond him. ¹⁵ Purrington, p176

A possible solution to this puzzle is that Hooke was not taking the same approach to the problem as Newton, but was employing an earlier principle from mechanics:

From several of Hooke's writings we know that he had formulated for himself the following principle, which he believed to be generally true: “the comparative velocities of any body moved are in subduplicate proportion to [that is, as the square root of] the aggregates or sums of the powers by which it is moved.”... In fact, Hooke's principle turns out to yield the right result whenever his “aggregates or sums of the powers” can be interpreted as work or potential energy. ¹⁶ Wilson p244

An indication that this was the approach that Hooke considered is found in the Classified papers of the Royal Society: CLP/20/349 is a single sheet with notes beginning:

Since the velocitys are as the Roots of the Reciprocalls of ye Distances the times are as the Roots of the Distances or the square of ye times as the Distances. for the weights or powers of gravity are as the Reciprocalls of the squares of the Distances. therefore the aggregates of those increasing powers are as the Reciprocalls of the Distances. [then a passage with many corrections] The planetary elliptical motion is compounded of this and equall [?]

This demonstrates that Hooke has the correct expression for the “aggregates or sums of the powers” of an inverse square attraction, being as the reciprocal of the distance. Unfortunately for him, he is still attempting to pass directly from there to deduce the time, as he did in his unsuccessful analysis in “of Spring” in 1678. What he is failing to employ is the idea of instantaneous velocity; this seems an elementary deficiency in the present day, when we have experience of strobe lighting and slow motion shots, but for terrestrial mechanics in those days, the only way to verify an analysis was by calculating the aggregate times and distances and measuring how they vary with experiment. For celestial mechanics, however, the motions are so slow and regular that an effective instant in the motion can be a measurable timescale, as Kepler's observations show.

Actually, the deduction of Kepler’s Laws of planetary motion from a theory of attraction between the Sun and a planet is not so difficult as has been supposed, if Hooke's principle is employed instead of Newton's procedures. The first step is to check the plausibility: the principle predicts that the velocity is highest at the point of closest approach and lowest at the most distant, as observed. The task is to demonstrate that the changes in velocity between these extremes are just right to produce an elliptical orbit. For this we need the understanding mentioned in one of Newton's waste books from about 1664:

If the body b moved in an Ellipsis that its force in each point (if its motion in that point bee given) [will ?] bee found by a tangent circle of Equall crookednesse with that point of the Ellipsis. ¹⁷ Herivel p130

This can then be combined with Huygens' formulation for circular motion and Hooke's understanding of orbital motion as sustained by a central force, to show that the task is to confirm that the normal component of the force at each point matches that required by the velocity and radius of curvature at that point. A proof, in contemporary style follows:

C is the centre of curvature at the point P, and so PC is the normal at P. The focal property of the ellipse is that angle EPC = angle CPF, let that angle be α. The ellipse can be constructed as the locus of points such that the sum of PE and PF is constant.

The gravitational attraction A acts along PE and we suppose that A is as 1 / PE², the inverse square.

To sustain orbital motion at speed v around this curve at point P the component of A along PC is required to be as v² / PC (Huygens).

Combing these we require v² is as PC.cos(α) / PE². The component of A along the tangent at P serves to change the speed of the body: the problem is to show that the resulting changes in speed around the ellipse are just right to sustain the orbit. We use Hooke's principle that the change in v² is as the aggregate of A with respect to the distance along PE, which is is as 1 / PE. So we have to prove that, for the geometry of the ellipse, the change in PC.cos(α) / PE² is as 1 / PE.

Now, for an ellipse, PC.cos(α) is as PE.PF; this is a standard result, but it also follows simply from the geometry of the rotation of the lines EP, FP and CP under the constraint that CP bisects angle FPC, as shown in the red triangles in Figure 1. Then, since PE + PF is constant, we can obtain PC.cos(α) / PE² is as 1 / PE + a constant, and so the change in that quantity is as 1 / PE, as required.

The variation of v² simplifies as PF/PE, from which, using another standard result for an ellipse, the Kepler equal area rule is demonstrated.

The proof by Newton, together with the lemmas, corollaries and propositions it depends on, are shown here. This shows that the distance between Hooke's insights, his approach, and the proven truth is very much less than has been supposed.

Newton's legacy

“Nature and Nature's laws lay hid in night: God said, Let Newton be! and all was light.”

Well, not quite all; Newton's monumental status gave free rein to his baleful jealousy:

Meanwhile, the Newtonian theory, considered as a program to account for the celestial motions, remained nearly at a standstill during the first half of the eighteenth century. The perturbations of the Moon were not yet accurately enough known to permit determination of the longitude at sea. The vagaries of Jupiter and Saturn remained unaccounted for. At least some of the blame for this standstill must be placed upon Newton himself. In his later years he could not bear in any way to be seen to be wrong, nor to acknowledge that the solution of problems in the Principia might be improved upon. ¹⁸ Wilson p273

And in optics this was continued beyond the grave by adopters of his mantle:

Both Young and Brougham were very able men, but whereas Brougham could influence the general reading public through the medium of his paper, Young could only command the ear of the most accomplished scientists of the day and one or two enlightened ministers of the Government. Hence the majority who preferred to obtain their ideas through the more easily assimilable form of the Review condemned Young as a scientist, and considered his Optical discoveries a travesty of Newton’s genius. ¹⁹ Thomas Young, F.R.S. Philosopher and Physician, Frank Oldham, Edward Arnold & Co. 1933, Chapter xiii Contemporary criticism of dr. Young’s work on optics, p117

It appears also that even in dynamics, subsequent advances have framed the problem in terms of energy rather than force:

There are two important alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics. These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, speed and momentum, for describing mechanical systems in generalized coordinates. These are basically mathematical rewriting of Newton's laws, but complicated mechanical problems are much easier to solve in these forms. ²⁰ https://en.wikipedia.org/wiki/Classical_mechanics

Perhaps Hooke was tomorrow's man, rather than yesterday's man