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From Natural Philosophy to Natural Science: the Entrenchment of Newton's Ideal of Empirical Success

From Natural Philosophy to Natural Science: the Entrenchment of Newton's Ideal of Empirical Success

From Natural to Natural : The Entrenchment of Newton's Ideal of Empirical Success

Pierre J. Boulos

Graduate Pro gram in Philosophy

Submitted in partial fulfillment of the requirements for the degree of

Faculty of Graduate Studies The Universly of Western Ontario , Ontario April 1999

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The author retains ownership of the L'auteur conserve la propriete du copyright in this thesis. Neither the droit d'auteur qui protege cette these. thesis nor substantial extracts &om it Ni la these ni des extraits substantieis may be printed or otherwise de celle-ci ne doivent Stre imprimes reproduced without the author's ou autrement reproduits sans son permission. autorisation. ABSTRACT

William Harper has recently proposed that Newton's ideal of empirical success as exempli£ied

in his deductions fiom phenomena informs the transition fiom to . This dissertation examines a number of methodological themes arising fiom the

Principia and that purport to exemplifjr Newton's ideal of empirical success. Among these themes is the method of answering important theoretical questions empirically by f?om phenomena According to Newton's ideal, a successll has its parameters measured by the phenomena it purports to explain. The ideal of empirical success is not limited to prediction, but also generates reliable of the parameters of the phenomena.

Newton's Principiu exempliiies deductions fiom phenomena where a higher level theoretical claim is inferred f?om certain other high level theoretical principles along with phenomena. Newton established that a measure of the rate of orbital could sensitively measure the exponent of the law. In the -test, Newton turned the theoretical question regarding the measure of the force holding the moon m her into an empirical question of measurement of the length of a seconds pendulum on . Two different phenomena are giving agreeing measurements of the same inverse-square centripetal force . The agreement in measured vahres is a higher level which relates the two phenomena in quest Newton's model of evidential reasoning exploits these agreements in such a way that the stakes are raised for rival .

The done on the bar perturbation problem forms the focus of this project. In considering the effect of the sun on the moon m her orbit, Newton was able to account for only half of the observed lunar precession. The solution to this problem came in the late

1740s and early 1750s through the work of Clairaut, d1Alembert. and Euler. The zero precession left over after the effect of the sun on the lunar orbit had been correctly solved removed impediments to the acceptance of Newton's The solution exhibited

Newton's stronger ideal.

Newton's ideal of empirical success did not immediately become part of the practice of natural philosophy. When one traces the developments of Euler, Clairaut, and others working on lunar precession and perturbat ion theory, these developments exhibit a better realization of Newton's ideal. Thus a good candidate for exploring the entrenchment of

Newton's ideal is the solution ofthe lunar perturbation problem. The question is "when did it occur?" We will examine whether Euler had m this stronger ideal after he first denounced the Newtonian system for failing to solve the lunar problem (and the n-body problem in general) arid then accepted it largely based on Clairaut's solution to the problem.

Although this stronger ideal is exemplified in the solution to the lunar perturbation problem, it is not clear that Euler exhibits an understanding and an awareness of the kind of evidential reasoning which drives Newton in his own work. Newton's ideal suggests that a priori metaphysical commitments ought not to drive scientific investigations. Although Euler is willing to bracket an a priuri metaphysical commitment to an aether theory, he nonetheless seems to accept the inverse-square law on hypothetico-deductive grounds.

Keywords: Newton, Principia, Ideal of Empirical Success, Hypothetico-Deduct ivism,

Measurement, Phenomena, , Precession, Perturbation, Universal

Gravitation, Clairaut, dlAlembert, and Euler. Acknowledgements

To claim that this project would not have been realised without the support of a number of people is an understatement. I find myselftrying to put into prose an elegant way of saying thank you to all whose support I have found to be priceless. The point is, though, that at the end of the the best way to show gratitude is with a "thank you." To John Nicholas I owe a debt of gratitude for the helpll comments and LiveIy hallway discussions. Andrea Purvis has been extremely helpfbl in steering me through the various hurdles of graduate studies.

Of course, Bill Harper has spent countless hours challenging me and encouraging me m this dissertation Bill has taught me the value of pursuing a project in which one can truly be excited and energetic. There are not words that can capture my gratitude to Biu.

To my parents, Victor and Yvette, I would Like to express my appreciation for standing by me f?om the start and for instilling in me a love and thirst for Wah and

Laura, my in-laws, have atways shown me encouragement and support. I do not envy all my siblings, parents, and extended family for having to witness my journey in this dissertation.

I do, however, thank them for their unconditional support. Writing a dissertation can take on a life of its own. This one did. Nonetheless, if it were not for the opportunity to witness my children's first steps or to hear their first words, and to share these with a life partner, this dissertation could not have been written. I would like to dedicate this dissertation to Chada&

Am, and Andrea whose love, support, and patience have inspired me and motivated me in achieving my dream This is a dream I will always remember. Certificate of Esarnination ,Abstract Acknowledgements Table of Contents List of -1ppendices Chapter 1 Introduction Chapter 2 Lght and Colours Paper (1672) Espedmenm Crucis The Doctrine of Colour Hooke and Huygens: Challenges to Newton a. The Hooke and Newton Debate over &ht b. The Huygens and Newton Debate over Light Newton and E-xpehentd Phdosophy Chapter 3 The Rules Phenomena The Path to Universal Gravitation Chapter 4 Chapter 5 Newon and the of the Lunar Apse: The Setting of the Problem Chapter 6 Euler, Lunar Theory, and the Vindication of Universal Gravitation Clairaut and dlAlembert Sketch of Clairaut's Solution Chapter 7 Newton's Ideal of Empirical Success and Natural Science Euler on the motion of the Lunar Apse Appendix A Appendix B Appendix C Appendix D Bibliography Vita -1ppendis .i Clairaut Translation: Concerning the Appendix B Clairaut Translation: Demonstration of the Fundamental of bly Lunar Theory .\ppen&x C Detailed Sketch of Clairaut's Proof Appendix D Maple V Workbook: The LUNAR Orbit Chapter 1

Newton the unparaIIelTd,whose Name No Time will wear out of the Book of Fame, Coelestiall Science has promoted more, Than all the Sages that have shone before. compell'd his piercing Mind obeys, And gladly shows him all her secret Ways; 'Gainst she has no Defence, And yields t' experimental Consequence; His Tow'ring Genius, fkom its certain Cause Ev'ry Appearance a pion' draws, And shews th' Almighty Architect's der'd Laws.

Desaguliers (1 729), I3e N~onianSystem of World, the Best Model of Government, An AIegorical Poem

Introduction

It is a widely held view that the natural are empirically oriented, and that the progress we have seen in the natural sciences over the past two to three hundred is due entirely to the decision taken by natural philosophers some centuries back to be concerned strictly with what is known through the senses. One mark of the is that, to this day, science is a leading method by which one understands the world. The revolution in science that took place during the sixteenth and seventeenth centuries facilitated the search for truth in the material world-

Sir (1642472'7) has been characterised by mainstream Anglo-

American philosophers of science as a founding practitioner of what we have come to understand as the methods of natural science, and more often than mt, he has been viewed as an outstanding and unique example of the great natural .' On this and Newton specificdy, are thought to be not at all preoccupied with not known empirically. The scientist, under this view, is one whose interests preclude things that are "transcendent" or "metaphysical." Ln the Prefice to the first edition of his

Philosophiae Naturalis Principia Mathernatica (1 687)- Newton observes that

All the difficulty of philosophy seems to consist in this---fiom the phenomena of to investigate the of natureT and then hmthese forces to demonstrate other phenomena2

Newton goes on to claim that

By the mathematically demonstrated in the former books, we in the third derive from the celestial phenomena the forces of with which bodies tend to the sun and the several planets.3

Not long ago E.A. Burtt penned the following:

Would that in the pages ofsuch a man we might fhd a clear statement of the method used by his powem mind in the accomplishment of his dazzling performances, with perhaps specific and illuminating directions for those less fled; or an exact and consistent logical analysis of the ultimate bearings of the unprecedented intellectual revolution which he carried to such a decisive issue! But what a disappointment as we turn the leaves of his works! Only a handful of general and often vague statements about his method, which have to be laboriously interpreted and supplemented by a painstaking study of his scientific biography.4

Rcichenbach (1 968) Sir Isaac Newtog Rin- Mottr's Tdtion Revised by F. Cajori (BukJe~The UniverSirg of California Prcss, 1934), page 4. Hcrcafkr ref4to as fhk@k 3%~pagc4. Bum (1925), page 208. Butt's disappointment was lessened by the that in comparison to some of Newton's famous predecessors Newton actually hed somewhat better in stating his method.

Nonetheless. says Burtt, a disappointing feature of the period associated with these great is that none of them seemed to know exactly what they were doing. This prompted historians of popular science, like Arthur Koestler, to brandish the famous representatives of the scientific revolution as "sleepwalkers." In the case of Newton, Butt would have us believe that the sleepwalker in Newton is the "scientist" and what Ever awoke was the philosopher.

In scientific discovery and formulation Newton was a marvellous genius; as a philosopher he was uncritical, sketchy, inconsistent, even second-rate.'

Even more recently, Casper Hakfoort (Hakfooa, 1995) examines the debates surrounding the theories of light m the eighteenth century- He admits of portraying a Kuhnian position and the starting point for Hakfoort, like many historians of science, is to treat

'Newtonianism" or the so-called "Newtonian Revolution" as strictly a scientific framework. Implicit in this is that Newton, the scientist and not the philosopher, gave us many new discoveries and a '3vorld view" to which his name is preked.

In contrast to this position, William Harper (Harper' 1991) has recently proposed that Newton's ideal of empirical success as exemplified in his deductions of the phenomena is what illuminates the transition fiom natural philosophy to natural science. What is promising m Harper's analysis is that it recasts the in a new way.

Historians and philosophers of science have not dealt adequately with the question regarding the transition fiom natural philosophy to oatural science. According Harper, this transition involves understanding Newton's philosophical position of what counts as theoretical success and tracing the acceptance of this position in the community of practising philosophers of nature. The revolution in science bearing Newton's name is not a shift &om one to another but is, rather, a transition to a new way of inquiring into nature.

This dissertation will examine a number of methodological themes arising from the

Principia and that purport to exemplify Newton's ideal of empirid success. Among these themes is the method of answering important theoretical questions empirically by measurement &omphenomena. According to Newton's method, which, contrary to Burtt, is much less buried, a successll theory has its parameters measured by the phenomena it purports to explain. The ideal of success is not limited to prediction, but also generates reliable measurements of the parameters by the

Consider how Newton provides resources to answer the theoretical question of the direction of the force deflecting a planet into its orbit by one that is answered empirically by measurement. Theorem 1 of Book I of the Principia asserts:

The areas which revolving bodies describe by radii drawn to an immovable centre of force do lie in the same immovable planes and are proportional to the times in which they are descnid.'

If the force deflecting a body into an orbit is directed at a centre, the orbit will Lie in a plane and the radius will sweep out equal areas m equal times. Further along in Theorem

2 Newton shows tbat ifa body moves in a plaue orbit and satisfies the area law (that isy it sweeps out areas proportional to the times) with respect to a centre, then it is deflected

- -- Hvper (1991). Rz- page 40. Wewill explore this in Chaptus 3 and 4. into that orbit by a force directed at the centre. There are two important corollaries to this last theorem and these concern the situations of increasing or decreasing areal rates. An increasing areal rate, for instance, corresponds to a deflecting force that deviates fiom the centre in the direction of motion. The opposite would be true for a decreasing areal rate.

Thus a constant areal rate measures the centripetal direction of a force deflecting a body into a plane orbit. Harper points out that Newton's appeal to these sorts of systematic dependencies allowed him to turn the theoretical question of the direction of the force deflecting a planet into its orbit into one that is answered empirically by measurement.

Newton can then use this established Etct that bodies are being deflected into their by centripetal forces along with theorems about motion under centripetal forces in order to have phenomena measure the inverse square variation of forces. For instance one can use a stable orbit to measure the force to vary inversely with the square of the di~tance.~

As we shall see, Newton appeals to Harmonic Law ratios for a system of orbits and to the stability of any particular orbit as phenomena, which measure inverse-square variation

The mechanical philosophy made a0 o priori commitment to explanation by mechanical contact action. refers to this commitment as illustrating a that overthrew Aristotelian exphtion:

In an earlier period [Aristotelian] explanations in tern of occult qualities had been an integral part of productive scientific work. Nevertheless, the seventeenth century's new co mmitment to mechanico-corpuscular explanation proved immensely fhit11 for a number of sciences, ridding

CoroOug I of Proposition 45 of Book 1 states that the centripetal force fis as the (*)' - 3 power of the distance if n is the numbu of degrees of precession per rtvolutioa For a stable orbit - "If the body aftex each rr~oIutionretums to syne apse, and the apse remains unmoved" (I%fm#&z ~146)-- the vdue of n is 0. Thus, Newton uses a stable orbit to measure the force to vary as the -2pow- of the distance, them of problems that had defied generally accepted solution and suggesting others to replace them9

The paradigm shift from the Aristotelian to the 'kchanico-corpuscular" led to investigations of collisions upon which Newton was able to build his laws of motiodO

Universal Gravitation was criticised for introducing occult qualities because it had failed to provide an explanation by contact action for what appeared to be the action at a distance of gravitation as a universal force of interaction Among the problems Newton's successors had inherited was the problem of reconciling inverse-square variation with corpuscular standards of contact action

Richard WestfslL in his classic paper, 'Newton and the Fudge Factor," has argued:

The role of the Pn'ncipiu m establishing the quantitative paradigm of physical science extended weff beyond its dynamic explication of accepted conclusions. Far more impressive was its success in raising quantitative science to a wholly new level of precision. ... Newton enlarged the definition of science to include those very perturbations by which material phenomena diverge fkorn the ideal patterns that had represented the object of science to an earlier age. The Principia submitted the perturbations themselves to quantitative analysis, and it proposed the exact correlation of theory with material events as the uhimate criterion of scientific truth!1 - The evidence which Newton mounts and upon which he infers an inverse-square gravitational law, the centrepiece of the Principiu, is very striking. In Proposition LXVI of

Book I, Newton explored the principal inequalities that occur in a system of three

9 Kuhn (1970), page 104. '0 Kuhn (1970), page 104-105. 1' Wd(1973), plgt 751-752 mutually attracting bodies.I2 In Book III Newton mars& as evidence the inequalities in the motion of the moon, the effects of the sun and the moon on the earth's oceans, the effect of the moon on the shape of the earth, and the precession of the equinoxes.

During the Enlightenment, Newton's theory of gravitation was empirically challenged in basically three important ways: geodesy, perturbation theoryx3,and the return of Halley's . Smith (Smith, 1997) has recently shown that Huygens' empirical challenge to universal gravitation helped settle a problem to which Newton drew attention in the Principia. There, Newton had argued that the rotation of the earth about its axis should cause it to bulge at the equator and to be flattened at the poles. Newton cites as evidence the pendutum measurements made in 1672 near the equator at Cayenne by Jean Richer (1630-1696). Richer had found that pendulums of the same length swung more slowly at the earth's equator than in Fran~e.'~In Proposition XX, Book EI of the

Principia Newton argued that points on the equator were fiather fiom the centre of the earth than points in and therefore the force of gravitation was weaker so that pendulums would swing more slowly.

M Richer repeated his , made on the island of Cayenne, every for ten together, and compared the lengths of the pendulum which he had there noted in the iron rods with the lengths thereof which he observed m France. This diligence and care seems to have been wanting to the other obse~ers.Ifthis gentleman's observations are to be depended on, the earth is higher under the l2 In the Preface Newton explicitly mentions this point and its rekition to the 'lunar motions (being imperfect)." page 5. '3 The lunar perturbation (the sun-earth-moonsystem) and the Saw-Jupiter perturbation (he Sun-Jupiter- Satum system) were the chief probiems in this ttgard. 14 For a dcmiled disc'ussion of dxse and other ezperiments designed to measure the hpcof the earth see Seymour L. Chapin's The Shape of the Earth,, in Taton and Wilson (1995), pages 22-34. Chapin's vtide highlights the importance of the geodetic Qucscicm and its centrality to the debt^^, cspcdly at tfie Royal Academy of Sciences. equator than at the poles, and that by an excess of about 17 miles; as appeared above by the theoryi5

Newton's position was in sharp contrast to the Cartesian account of gravitation which required a vortex of swirling about the earth? As the vortex of matter swirls around the earth, Cartesians such as Jacques Cassini (1 677-1756) reasoned that one ought to expect that the earth would be flattened (or trirmned) at the equator and elongated at the poles."

Recently, Truesdell, while reflecting on the debate surrounding the shape of the earth, remarked:

For Vokaire, who endorsed mathematical philosophy but did not understand it, this [the shape of the earth] proved Descartes wrong and Newton right about everything. The later philosophes followed his judgement; the British gleemy followed them; and somehow this minor and precarious if not puerile side issue has assumed in the foIklore of science an importance it never for a moment deserved or enjoyed among those who knew what was what in rational mechanic^.'^

George Smith (Smith, 1997) has recently shown the dimensions of the geodetic debate and their relevance to Newton's ideal of empirical success. The ramifications of the problem involved better positional navigation at sea if one knew the geodetic features of the earth.

To that end, expeditions were sent to the equator and to the ~rctic.'' lo 1733 Louis

Godin and Charles-Marie de la Condamine suggested sending teams of academicians to

Peru and to Lapland. Maupertuis and Clairaut went to Lapland in the north and Louis

'5 Pn'm 348-349. '6 See Descutes' PnPn~krufPbzhopby,Part III The Vible ," v6-29. 17 See Chapin (1995), page 26; Hlnirins (1985), page 38. Figuratively spe;rking, Newton's model resembled an onion illmad of Dexuhcs' lcmoil. l8 Truesdell in bis Prtfk to Eulcr (1984), plge rr Godin, with the assistance of Charles-Marie de la Condarnhe, led an expedition to the

Amazon. The polar expedition left in 1736 and returned in 173 7. Godin left for Ecuador in 1735 and returned in 1747.*' The Lapland result prompted the Royal Academy of

Science to re-measure the meridian of in L 739-40. The report, MPridienne vPrzjk!e

(The Meridian Verzped), concluded in fivour of the oblateness of the earth Seven years later, the South American expedition would yield results, which would corroborate these

Wings in support of Newton's theory. Interestingiy, due primarily to Maupertuis' findings, different researchers were already using the oblate shape of the earth in their work prior to the retum of Godin fiom South America in 1747.~~

Another incident, which also involved CIairaut, was the predicted retum of

Hdey's Comet. Edmund Halley (1656-1743) had said that the comet of 1682 would return late in1 758 or early in 1759. Halley's prediction was novel in that were not commonly regarded as being fiequent or periodic visitors to our part of the universe. The suggestion that this comet, Halley's, travelled in a closed elliptical orbit implied that these heavenly objects behaved something like the planets of our system and, consequent@, were governed by the same inverse-square gravitational force of Newton's theory.

Wey's speculation was imprecise because the path of the comet would be strongly affected by the attraction of any large planet near which it passed. Applying Newton's theory, Clairaut and Joseph- Jkr6me Lefknqais de Lalande (1732- 1807) publicly declared on 14& November 1758 that the comet wiU return in mid-April, 1759. Clairaut, Lalande, and with the aide of Nicole Lepautk (who carried out some of the longer calculations)

l9 I amnot aware of any expeditions to the Aatvrtic during this time period- It m;lv be that a north-south symmetry was assumed. Hankins (1985), pages 38-39;Chapin (1 995), pap26-27. used perturbation theories to come up with a more precise perihelion point for this comet.

Clairaut added a margin or error of plus or minus one The comet was determined to have passed its perihelion point on 13~March 1759. The empirical challenge was met and Newton's inverse-square force held the day.

The biggest challenge to Newton and his theory was the 1unar perturbation problem. The inverse-square law offered the natural philosophers of the enlightenment an opportunity to obtain new and more precise quantitative results in celestial .

Through the help of the telescope, motions of heavenly bodies could be measured very precisely. In this respect this new set of data fiom which the phenomena are generalised provided the ultimate challenges to the inverse-square law. d7Alembert, for instance, once remarked:

The ~ewtonian]system of gravitation can be regarded as true only after it has been demonstrated by precise calculations that it agrees exactly with the phenomena of nature; otherwise the Newtonian does not merit any preference over the [Cartesian] theory of vortices by which the movement of the plawts can be very we1 explained, but in a manner which is so incomplete, so loose, that if the phenomena were completely different, they could very often be explained just as well in the same way, and sometimes even better. The [Newtonian] system of gravitation does not permit any illusion of this sort; a simple article or which disproves the calculations will bring down the entire edifice and relegate the Newtonian theory to the class of so many others tbat imagination has created and analysis has destroyed?3

- --

2' Chapin (1995), page 30. 22 Waff (1995), page 79; Hmhs (1985), page 40. " Quoted in Hankins (1985), page 37. Thkom Jean d'Alembert, "Elimtns de phitosophie," in Miihgclr h k#iialm, 8Md &p&hqi%k, 5 vols. (Amstrrdun, 1770), IV, 231. 11

The empirical challenge of the return of Halley's comet was, in d'Alembert's words.

"more tedious than diflicult."" The answer to the theoretical question regarding the shape of the earth was derived, partially, through measurements of the length of a degree by triangulation-measurement s of distances between sites on earth which yield measurements of the length of a degree and offers the same answer as a ''bird's eye view"

Eom space of the shape of the earth? Between these two challenges is nestled a problem in which we find Europe's foremost natural philosophers and mathematicians heavily committed to solving what was considered the most important challenge.

The moon moves much more erratically than the planets. Newton was keenly aware of this The calculations of the moon's motion required solving the problem of three mutually gravitating bodies. As early as 1744, Euler in Berlin, and in

1747 with Clairaut and d'Alembert, a solution to this thee body problem was set as the goal. AU three came to the same conclusion: the observed monthly motion of the lunar apogee differed on average form the predicted value by a factor of two. In 1749 Clairaut proclaimed he had solved the problem without giving his solution. By 1752, it was generally claimed that Newton's theory is right after +the work completed by Clairaut, d'Alembert, and Euler helped vindicate the theory.

These three challenges to universal gravitation shed light on various methodological themes. One may argue that their resolutions merely confirm Newton's

" Chapin (1995) goesinto some detail on this. .%long with pendulum measurements, methods of triangulation were used for measuring arcs by means of tcrrestd okations of angles bemeen established sites. tb HPnltlllS (1985), page 39: "Newton has told John Machin that calcdatiag its motion was the only problem bt ever made his head ache, and he uadastcd wh~It was because the moon is attracted strongly by two bodies-the earth and the sun, pulling at diffirmt angles to one another-whereas the planets arc attracted strongly only by the sun." The "lunar problem," ftom Newton's day to the ptesenc has been cast as the "linchpin of the entire qpncnt for univdgrrrvimtiolzW Wtsdd (1973), page 754. theory and, in fact, this is generally how these historical events have been read. Harper, on the other hand, claims that the developments by Newton's successors on helped re& Newton's ideal of empirical success. Clairaut, for instance, was able to solve the lunar precession problem by completely accounting for the known precession through recalculating the action of the sun on the lunar orbit.27 From Clairaut and Euler to Laplace the Newtonian corrections of Keplerian phenomena %came increasingly precise projectable generalisations that accurately M open ended bodies of increasingly precise data."28 Crucial to these developments were increasingly accurate measurements of relative masses of bodies in the In terms of earth masses, Newton's estimate of the mass of the sun in the 2" edition of the Principia was 32% less than the value accepted m 1976 (the 2* edition was published in 1713). Some 83 years later

Laplace gave estimates that turn out to be within 2 percent of the 1976 value.29

Newton's ideal of empirical success did not immediately become part of the practice of natural philosophy. When one traces the developments of Euler, Clairaut, and others working on Lunar precession and perturbation theory, these developments exhibit a better realisation of Newton's ideal of empirical success. We will explore whether this led to a Newtonian revolution that was more radical than the overthrow of the commitment to explanation by contact action envisioned by Kuh. Harper suggests that this radical revolution was the Wormation of natural philosophy into natural science.

WW(1980); W& (1975). Harper (1997), page 62. Harper (1997), page 62 Central to this transformation was having Newton's ideal of empirical success

become a higher level standard that can override even the most cherished metaphysical

commitments. This suggests that later changes of standard through paradigm shifts, such

as the transition from particle to wave conceptions of Light. can be best understood as

applications of the higher level standard. The entrenchment of this ideal may have

occurred after Newton, by his successors on perturbation theory, and may have led

gradually to the entrenchment of this ideal as a standard in the doing of science.

Newton's early remarks on empirical philosophy will be the focus of the second

chapter. Here we will gain some insight into Newton's model of evidential reasoning.

This nahdly carries us into the third chapter where we wiU explore Newton's more mature thoughts surrounding . That is, first with the rules of reasoning and then with the phenomena listed in Book III of the Principia Newton's vision of natural philosophy will be made clearer. Chapter 3 will specifically deal with the progression of propositions Newton derives through his use of rules and phenomena. These lead to wedgravitation and in the process we will see what Newton counts as a successll theory. Pierre Duhem has marked how philosophers of science will consider the marshallinp of evidence in support of or in refirtation of a theory. Since Duhem himself in

The Aim and Structure of Physical nteory makes use of Newton's for universal gravitation, we will clarifl, in the fourth chapter, how I)uhem is mistaken in his analysis of

Newton and in the process strengthen the position regarding Newton's method.

The fifth and sixth chapters will investigate whether the solution to the major empirical challenge to universal gravitation came about through the entrenchment of

Newton's ideal of empirical success. There are three key figures m this story.. Clairaut, d7AIembert7and Eder. The solution to the lunar theory, first developed by Clairaut, will offer the opportunity to examine whether this so iution exemplifies Newton's ideal.

Specifically, since Eder is the central figure we will examine his reactions to Ckut's solution. Chapter 2

Of Colours, Experimeat 7 From Cambridge University Library Add MS 3975.

7 Taking a Prisme. (whose angle fW was about W into a darke meinto w* ye sun shone only one little round hole k in such manner y' ye rays, being equdly retiacted1 at (n & h) their going in & out of it cast colours rstv on ye opposite wall. The mlours should have beene in a round circle were dl ye rays alike refiaaed but their length to about 7 or eight inches. & _v' centers of ye red & blew, (q & p) being distant about 2 3/4 or 3 inches. The dinance of ye wall trsv hmye Prisme being 260~.

In his methodology of science, Isaac Newton stressed a distinction between propositions wZlicb, he claimed, were established by reasoning f?om and propositions, which were mere or hypotheses. Starting with his on Light and colours, through the fist edition of the Principia, the publication of the wicks, and culminating with the third and final edition of the Principu, we find Newton devising a powerfbl new method for estabkhing theoretical claims as scientific kts. I will be arguing in this chapter that this method was not only 'hew," but an understanding of it will shed light on Newton's ideal of empirical success which will be explored in subsequent chapters. The dominant view of natural philosophy at this time maintainsd a method, which yielded nothing more than accumulating evidence for conjectured hypotheses. The development ofNewton's method can be followed, not only as it is constructed in his paper on light and colours, but also through the debate which it fostered?0

Light and Coloum Paper (1672)

Before delving into this celebrated paper we should note that the main community of natural philosophers at both the Royal Society of London (1660) and the Academie Royale des Sciences (1666) advocated three principal modes of investigation: (i) to experiment in order to search for new phenomena; (ii) to search for mechanical hypotheses in order to explain these phenomena; and (iii) to deduce or mathematically derive the consequences of established principle^.^' This is the backdrop against which we must view Newton's work on light and colours. His first public insistence that there is to be drawn a sharp distinction between empirically established claims and mere hypotheses occurs m a letter Newton sent to Henry Oldenburg, Secretary of the Society and publisher of its Philosophical

Trunsactions. The letter, dated 6 February 1672,32 contained Newton's 'Wew Theory about Light and Colours" and was read to the assembled membership on 8 February. The paper prompted intense controversy. Most notable were the criticisms of Robert Hooke (1635- 1702)' (1 629- 16%)' and Ignace Gaston Pardies (1636-1673). The criticisms offered by both Hooke and Huygens centred on what Newton claimed could be concluded fiorn the experiment he called the experimenturn mcis. That

'0 The initial reactions to Newton's paper will be discussed here. The controversy surrounding his theory dragged ouh not only in but in France and Germany, into the eighteenth century and afier the publication of his @&kr in 1704. According to Gu&c (1981) France, in general, was not won over to the new theory of Iight and colours- Desagulim had to repeat some of Newton's colour experiments successfully ia Gont of Roy a1 Society members in 1714 and 1715 with some Freach academicians in the audience in 1715- Desaguliers performed several repetitions of these experiments in France later on and helped the acceptance of Newton's ideas. The @ticks appeared in two French editions in 1720 and 1722. Hakfoort (2995) draws a similar position with respect to Germany (page 19) showing that Newton's theory gained acceptance slightly before the French accepted it 3' See Hvpu and Smith (1995) and Schaffer (1989). 32 The letter was dated 6 February, 1672 (Old Style) and qprson pp. 92-102 of Nrwbrr'I -, VoL I. Nearton's fvnous ppcr is the somewhat amended version Olbburg pdnted in he 19 Febqissue of he Society's Pbihvjbbl T- (80): 3075-3087. Newton engaged in a "new way of 'd3 is suggested by the controversies to which his investigations gave rise. The challenges forced Newton to explicate not only the experiment, but also to spell out what his Experimental Philosophy was.

Experimenturn Crucis The Table of Contents of the Philosophical Transactions (80) of 19 February, 1672 has among its entries the following:

A letter of Mr. Isaac Newton, Mathematick Professor in the University of Cambridge; containing his New Theory about Light and Colon: Where Light is declared to be not Similar or Homogeneal, but consisting of difform rays, some of which are more rehngi'ble than others: And Colors are afl6rmed to be not QuaIiscations of Light, deriv'd fiom Refractions of Natural Bodies, (as 'tis generally believed;) [emphasis added] but Original and Connate , which in divers rays are divers: Where several Observations and Experiments are alledged to prove the said ~heory."

Now, for the educated reader of the time to see this entry on the Society's records would surely have sparked some interest. The relatively recent development of the telescope gave rise to a number of projects. Important among these was the solution to the problems of spherical and chromatic aberrations in telescopes. Consequently, many were working on perfecting the grinding of mirrors and glasses. The Table of Contents does not mention, however, that Newton's new theory and its corresponding use of experiments makes use of prisms. Risms were common in the last half of the seventeenth century, albeit mostly as decorative pieces m chandeliers and toys.35 Oldenburg provides the reader with some context. This editorial comment is scarcely mentioned m modem literature where it is custom to delve immediately into

33 This expression is bornwed from Wrllivn Harper and George Smith. ~4 Pbibsqhid T~c~RI,(W),I 9 February, I67W75. 35 Schaffcf (1989). Newton's theory.36 Newton's first readers were told that his theory claims that light is not homogenous but, rather, is heterogeneous containing or consisting of difforrn rays, some of which are more refkangible than others. The of informed opinion of the time, as Oldenburg's parenthetical remark asserts, maintained that wbit e light is homogeneous and that the colours we see (through prisms, rainbows, etc.) are derived from refractions through or reflections off of natural bodies. That is, the colours we see in a rainbow are due to the beams of light being modified by the droplets of water suspended in air. Oldenburg notes that in Newton's theory ordiwhite light consists of rays differently rehgiile, and that colours correspond to these rays and so are original properties of light. Kuhn has noted that at least four other natural philosophers, including Descartes and Grimaldi, had discussed in opticd treatises the coIoured iris produced by a

The "phaenome~"were indeed "celebrated." Newton, when he repeated them for his own edification, can have had no reason to anticipate a rdthat he would later descni as "the oddest, ifhot the most considerable detection, which hath hitherto been made in the operations of nature.'J8 In his paper Newton reports his initial investigation of the "celebrated phenomena of co lours" in the following:

It was at first a very pleasing divertisement, to view the vivid and intense colours produced thereby, but after a while applying my self to consider them more circumspectly, I became surprised to see them m an oblong form; which according to received laws of Refkaction, I expected should have been circular.39

" For instance Cohen, in his imroduction to h,zac Nm'rPpn d htlnr on Ndwd Pfihqbby, and Kuhn's "Newton's Optical Papers" which precedes Newton's paper on light and colours in Cohen's edition do not mention d2iS at 2U. Howard Stein, as well, neglects this point in "On and Method in Newton." 37 Kuhn (1958), pqg= 29. 38 Kuha (1958), plg~29-30. P.Tmm (W), 19 February, 16723076. The expectation according to the received view, i.e., Snell's Law of sines, 'O is that the image should be circular at minimum deviation. The rays which traverse a prism at minimum deviation subtend equal angles with the incident and emergent faces of the prism, and small variations in the angle of incidence do not cause appreciable variation in the deviation. This is how Newton later described it in Proposition II, Theorem 11,

Experiment 3 of the @ticks- Newton's description is:

The Axis of the Prism (that is, the Line passing through the middle of the Prism fiom one end of it to the other end parallel to the edge of the Rehcting angle) was in this and the following Experiments perpendicular to the incident Rays. About this Axis I turned the Prism slowly, and saw the rehted Light on the Wall, or coloured Image of the Sun, first to descend, and then to ascend?'

In the paper submitted to the Oldenburg and the Royal Society, there does not appear to be any difficulty in apprehending the position of the prism in the experiment for Hooke was able to replicate the experiment just as Newton had described it. This is how Newton descnis minimum deviation as early as 1672 in his desire to explain the oblong form of the image on the wall:

But because this computation was founded on the hypothesis of the proportionality of the sines of incidence, and ReWion, which though by my own Experience I could not imagine to be so erroneous, as to make that Angle but 3 1' [the apparent diameter of the sun], which in reality was 2 deg. 49'; yet my curiosity caused me again to take my

* The law states that the sine of the angle of incidence is proportional to sine of the angle of rekction @th angles being measured Gmm the normal). Descvtes acdy published the law first We do not know whaether he learned it from Snell, who published his versicm around 1620. Descartes does not refer to Saell in his own writings. See Descartes' Opnir, 'Tkcourse Two: ReGaction" in Tbc Pbr;bopbicrJ Wdngr (Dam, I, translated by John Co- et al. (Cambridge: Cambridge University Press, 1985), page 163- In the Pni@& Newton infixred the sine law for a ''hhppoetical'' body moving under the influence of an attractive Force k Book I, Proposition XCV, Theorem XLM and Proposition XCVI, Theorem L. In the Scholiurn following these two Propositioas, N- &aims These attractions beu a great ~~ to the reflections and refiations of light made in a given ntio of secants, as was diSCoveed by Snd and consequmdy in a given mtio of sines, as was nhilrited by Dc~aatk~''hk@k, page 229. Newton, Cjb&kk, page 27. Prisme. And having placed it at my window, as before, I observed, that by turning it a little about its o*is to and from, so as to vary its obliquity to the light, more then an angle of 4 or 5 degrees, the Colours were not thereby sensibly translated from their place on the wall, [emphasis added] and consequently by that variation of Incidence, the quantiity of Rehction was not sensibly varied. j2

In the Opticks, as well, Newton goes on to descni how he isolated that small range of positions such that the image appeared to be stationary. In this position, he claimed, he fixed the prism and, unless otherwise noted, the prism is understood to be m this position in subsequent experiments.

For in that Posture the Rehctions of the Light at the two Sides of the refracting Angle, that is, at the Entrance of the Rays into the Ptism, and at their going out of it, were equal to one another. ... And in the Posture, as the most convenient, it is to be understood that all the Prisms are placed in the following Experiments, unless where some other Posture is descni"

If white light refracted according to the law of sines in this position of minimum deviation then the image on the screen was expected to be circular. Newton's diagram (below) shows the image to be oblong. The law of sines would have similar rays of sunlight subtending an angle Awith each other to emerge fiom the prism inclined at the same angle.

The length of the spectrum, as we will see, is explained in that wbite light consists of rays of different rehgiiility.

42 P&h~-0P6i(;Q/Tmnrdbm (80),19 Febcuvp, 1672: 3077. 43 Newton, Opticks, page 28. The hct that the image was oblong places this experiment in bewith the first of three modes of investigation outlined earlier. That is, the experiment revealed a new phenomenon-the oblong form of the image. It is this phenomenon that needs to be explained. Newton's version of the experiment of separating a light beam was, therefore, different fiom the experiments that other natural philosophers had performed. In these previous experiments, when white light had been passed through a prism the image of the refhcted beam had been observed on a screen placed close to the pk" This proximity of the screen to the prism did not allow for the rays of the beam to Mciently disperse as

Newton's experiment had shown (with the distance between the prism and the wall being approximately twenty-two feet).4s Newton claims (in the passage quoted above) that he was surprised in seeing the oblong form of the image on the screen According to the law of refkction, as it was understood in Newton's time, the spectrum ought to be circular rather than oblong ifthe prism is placed in a minimum deviation position That the prism was in a minimum deviation position is evidenced by Newton's comments (above) and in

Kuhn (1958), page 30. See for instmce Descutes' Discourst VIII of 45 Howard Stein commented on this point in his commentary on a replicadon of this experiment at the University of Western Onmrio, Fall 1990. Stein noted htit may be the case that Newton was the &st to see the vividness of the cotours and of the dispdon of the Light beam. This done would have had a great impact on him. Notice that with the screea dose to the prism what would have been obsemed would be a ntsldy &cuIar image on the screen not unlike, in cimhirg, the image produced by a ham, which had not ken passed through a prism, Furtfiem~rt,in passing through the prism the hmwould have "ac~"a md-orangc hinge on one edge and 2 blue-violet on tbt other edge- his reply to Pardies' first letter.46 This reply to Pardies is interesting in that Newton is responding to a purely experimental question whose implications are crucial to his theory of light and colours. Newton's letter is dated 13~April 1672 and published immediately in the PhiZosophical Transactions. I will briefly go through Newton's reply here in order to dispel conioo criticisms that Newton was less than clear (and may have purposely been unclear) in explaining his experimental set-up!'

Pardies claimed that the length of the solar image (its wn-circularity) is not a new phenomenon and that there is nothing to explain except to point out the fict that the oblong image could be produced by the different incidence of the rays fiom opposite parts of the sun's disk. Pardies fiuther claims that a difference of 30' of incidence (the apparent diameter of the sun) will resuh in a difference in the refkction of 2"23'." Newton is quick to respond that Pardies is mistaken in that he is not keeping the prism at minimum deviation:

But the Rev. Father is under a mistake. For he has made the rehctions by the different parts of the prism to be as unequal as possible, whereas in the experiments, and m the calculation fkom them, I employed equal refka~tions.'~

In the paper on light and colours Newton proceeded to consider some other investigations designed to measure features of the phenomenon and to eliminate some alternative causes.

I could scarce think, that the various Thickness of glass, or the termhation with shadow or darkness, could have any Influence on light to produce such an effect; yet I thought it

- a IS. Cohen, ed. I~aarNrrv~on's P* and Ltlnr on N~rl~alPtribmpby wrd Rc&d Dmmm~r-(Cambridge, Mh. Hmard University Press, 1958), page 105- 47 See Kuhn's "Newton's Optid Pap"and Hakfoort (1995), pages 15-22. Pardies' First Let- () was sent to the publisher of the Pfih@kd Tnmim5m (go), 9 April 1672: 4087-4090. The passage duded to above is on page 4089. "Newton's Reply to Pudits' Fist Lctbgwin Coben (1958), page 90- not amiss, first to examine those circms, and so tryed, what would happen by transmitting Light through parts of the glass of divers thicknesses, or through holes in the window of divers bignesses, or by setting the Prisme without so, that the Light might pass through it, and be rehcted before it was terminated by the hole: But I found none of those circumstances material. The fishion of the colours was in all these cases the same.50

This passage is indicative of Newton's approach to experimental philosophy. Notice the care with which he precludes other explanations. These alternatives, producing these results, could not deliver the explanation needed for the phenomenon of producing the oblong rainbow of colours. Since changing the aperture size in the window shut or changing the rektive media did not alter the phenomenon of the oblong shape of the spectrum (indeed Newton claims the "&hion of the colours was in all these cases the same") he could eliminate the that it was the prism of the experimental set-up which "caused" the shape of the image. Among other alternatives we would find theories which purport to explain light as being homogeneous and as being altered when passing through varying media These modification theories of light could not quantitatively account for the elongation of the image Newton found on his meen under the conditions of his experiment. After discussing these and other considerations he introduces his

I will quote this passage at length, as its place in our discussion is pivotal.

The gradual removal of these suspitions, at length led me to the Experimenturn Crucis, which was this: I took two boards, and placed one of them close behind the Prisme at the window, so that the light might pass through a small hole, made in it for the purpose, and Etll on the other board, which I placed at about 12 feet distance, having first made a small hole in it also, for some of that Incident light to pass through. Then I placed another Prisme behind this second board, so that the light, trajected through both the boards, might pass through that also, and be again refi=actedbefore it arrived at the wd, This done, I took the first Prisme in my hand, and turned it to and fio slowly about its Axis, so much as to make several parts of the Image, cast on the second bodsuccessively pass through the hole in it, that I might observe to what places on the wall the second Prisme would rehct them. And I saw by the variation of those places, that the light, tending to that end of the Image, towards which the refraction of the first Prisme was made did in the second Prisme suffer a Refraction considerably greater then the light tending to the other end. Emphasis added.]''

Newton isolated, successively, different portions of the dispersed spectral light. The outcome of this procedure was that several portions were rehcted differently fiom one another: the light at the end of the spectrum that was rehcted least in the production of that spectrum continued to be rehcted less, when segregated, than the light at the end of the spectrum that was refkcted the most in the production of that spectrum continued to be rehcted more, when segregated. This pattern occurred, as well, for the intermediate parts of the spectrum. The reader will note that Newton made no mention of colour in the above description. He concluded:

And so the true cause of the length of that Image was detected to be no other, than that Light consists of Ruys dzJfierently refiangibIe, which, without any respect to a difference in their incidence, were, according to their degrees of refkmgiiility, transm&ed towards divers parts of the waltS2 We should note his cautious and debirate progression in reasoning. AU that he concludes fiom the experimenm cmcis is that light consists of rays differently rekmgiile. After noticing that the image reding fiom Light (ordinary !stmli&t) passing through a single prism was oblong when a circular image was expected, Newton proceeded to isolate parts of the original light by passmg each of them through a second hole-prism set-up. The following diagram is Newton's depiction of the experimenturn mcis in his reply to Pardies' second letter."

From the observations that light which was most rekted when it was separated fkom

sunlight by the first prism was found to be of considerably greater reikgi'biIity than the light least refiacted, Newton concludes that rays of differing refrangi'bilities were already present in sdght.

...that Light is not sirnilst, or homogeneal, but consists of difform Rays, some of which are more rehngiile than others: So that of those, which are alike incident of the same medium, some shall be more rehcted than others, and that not by any virtue of the glass, or other external cause, but fiom a predisposition, which every particular Ray hath to suffer a particular degree of ~efraction."

Newton claims to have successfidly isolated the cause for the diering refiztions at the second prism, their corresponding differences at the first prism, and, of course, the oblong shape of the image seen hmthe original experiment.

The Doctrine of Colour

It has been mentioned above that Newton drew a very cautious concbioa Although the experiment itself resulted in a most qxctamh display of colour, Newton, m his inference, says nothing about colour. He says that the experimenturn crucis warrants us to conclude that Light consists of rays differently refrangible. Not until the second part of his paper did Newton turn to the theory of colours. The doctrine of colour is presented in 13 propositions @p. 3081-3084). Concerning the origin of colours, Newton first presents the doctrine itself and only then gives us some brief indications of a few of the many experiments on which the doctrine rests. I will quote briefly fiom a sample of these

13 propositions, as these passages will be helpful m informing our discussion of the controversy they generated with Hooke and Huygens. Wah respect to the question of whether wlours belong to the rays of tight or are dependent on something eke, Newton claims in Proposition 1:

As the Rays of Light differ in degrees of Refkngibilayy so they also differ in their disposition to exhit this or that particular wlour. Colours are not QuaI~j?caofionsof Light, derived fiom Rektions, or Reflections of natural Bodies (as 'tis generally believed,) but Original and connate properties, which in divers Rays are divers." Here, Newton posits a qualitative relationship between refhgiibility and colour. From this Newton proceeds to claim a strict relation between refrangiiility and colour (Proposition

To the same degree of Refi=angiiility ever belongs the same colour, and to the same colour ever belong the same degree of Re£bngiiWy. ... And this Adogy 'twixt colours, and refhngibility, is very precise and-strict; the Rays always either exactly agreeing both, or proportionally disagreeing in both"

It is on this proposition that Newton was able m his Optical Lectures and in the @tick to treat a theory of light as a mathematical theorys7 In Proposition 3, Newton asserted the immutability of colour and refhngibdity-a claim that became a special focus of the subsequent controversy surrounding Newton's

The of colour, and degree of Rehgibility proper to any particular sort of Rays is not mutable by Rehction, nor by Reflection fkom natural bodies- nor by any other cause, that I could yet obser~e.'~ In propositions 5 and 6, Newton distinguishes between two types of colours: name% those which are simple and those which are compound. To the former he lists just those colours which obtained fiom separating white light, that is,

Red, Yellow, Green, Blew, and a Violet-puqde, together with Orange, Indico, and an indefinite variety of Intermediate gradationss9 After noting that sub-groups of these primary colours may be composed to produce another colour (e.g., yellow and blue produce green), Newton notes in proposition 7 that the most interesting composition was that which produced whiteness.

But the most surprising and wonderrl composition was that of Whiteness. There is no one sort of Rays which alone can exhibit this. 'Tis ever compounded, and to its composition are requisite all the aforesaid primary Colours, mixed in a due proportion I have often with Admiration beheld, that all the Colours of the Prisne being made to converge, adthereby to be again mixed as they were in the light before it

was Incident - upon the Risme, reproduced light, entirely and perfectly white, and not at all sensibly differing fiom a direct Light of the Sun, unless when the glasses, I used, were not dciently clear, for then they would a little incline it to their colour?

P- P- Trmrr- (W),19 February, 16723081. 59 Phhqhkd Tw& (w),19 February, 16723082. * Pbihpbl Tmaffionr (go), 19 February, 16723083. Although Newton did not provide in the paper all the experiments for the full range of the thirteen propositions, he does claim that these propositions were conclusively demonstrated by experiments. It is worth digressing from the historical progression here. Notice that prior to being rehcted for the first time, sunlight entered through the tiny hole in the shutter and was ostensibly white. That is, the doctrine of colour stipulates that white light (and presumably this includes ordinary sunlight) consists of rays differently rehigible corresponding to the colours of the bow-white light is not homogeneous but heterogeneous. Shapiro (1980, 1995) has argued that Newton was unable to show that sunlight is compounded of different colours.

That is, white Light, particularly sunlight, is a mixture of rays of every color. Newton recognized that one could not directly prove experimentally tbat colors are innate to sunlight, and most of his experiments in support of this principle depend on a simiLarity argument: By various, often ingenious, means he composes white fiom a mixture of the innumerable spectral colors and shows that in all its properties it is similar to direct sunligk6'

Shapiro's aim was to show an underlying but unsuccessll theme m Newton's writings on the subject: that a mapping of colours to refhgibilities would allow, in principle, a mathematical construction of a theory of colour. A mea~u~ementof the angle of incidence and the angle of refhction, for instance, would dow an extension of mathematical procedures through an equivalence of colour with refimgiibility. Shapiro implicitly seems to agree with Newton's basic conclusion that sunlight is a heterogeneous composition of rays that are differently rebngi'ble. Referring to Newton's inaugural Lectiones opticae or

61 ShapitO (1995 (1984)), page 195. Optical Lectures after being appointed Lucasian Professor of Mathematics at Cambridge,

Shapiro says:

Newton begins the Oprical Lectures, as he does the "New Theory," with a demonstration of the central idea of his theory, that sunIight consists of rays of unequal refkmgi'bility. .. . Newton's arrangement of passing a narrow circular beam of light through a prism of minimum deviation was an original, and by no means obvious, one. He deduced that in this situation if all rays were equally refhgible, as was then universally held, then the image should be nearly circular. He then measured the spectrum and found that it was greatly elongated, about five times longer than broad. This was the key-measurement and calculation He had to eliminate other possible causes for the elongation, of course, but once he had established that there could be no other cause than that the Sun's direct light consists of rays of unequal rehgiiility, he had, or so it seemed, a mathematical measure of color: the degree of refiangibility:2 According to Shapiro, this is only one half of what Newton sought. Newton, he claims, succeeded in developing "a theory of unequal rehngibility that could account for the spatial distribution or separation of colors but not for their sensible appearance-tht is, their color.'d3 The innateness of colours to Nnlight follows fiom the claim that the composition of sunlight is of rays of different refkxqgibilities only if it is also maintained that there is a strict association of colours to refrangibility, which is the substance of Proposition 2 above. Newton never proved the latter claim, according to Shapiro7s position, because

it was impossible experimentdly, to prove color immutability for the Sun's immediate Light, and thus that the wloa are beto it. The problem is that before refkction sunlight appears white, afterwards

62 Shapko (1995 (1984)), page 192 63 Shapiro (1995 (1984)), page 201- it displays all the spectral colors; and if the two are compared, the colors do appear to have changed? I take Shapiro to be accusing Newton of having ~ciouslyreasoned his conclusion regarding colours. According to Shapiro, Newton reasons to the colourific properties of the rays even though the experiment only supports differential rehgibilities Stein (1990) has responded to Shapiro's argument in a convincing way. Stein maintains that implicit in Shapiro's position is the notion that refkangibility is a whose observation is not expected but which is detected in the behaviour of light, whereas colour is a property that ought to be directly observable even though it is not observed in sunlight. This distinction misses, according to Stein, Newton's way of thinking: that is, the colours present in sunlight are dispositional properties of the various parts of light to produce sensations of the various spectral colours. Likewise, rehgibilities present in sunlight are dispositional properties of the various parts of light to produce rehtions of a particular degree? To put it as Stem does, neither the colourific character nor the refiangi'bility is "directly" o b~ervable.'*~ This means that Shapiro 's characterisation of Newton's post hoc conclusion would have to be applied to the claim about refhgibility as it does to the claim about colours as depicted in Propositions 2,3, and 7 above. Insofar as Shapiro is willing to grant Newton the conclusion that dghtis composed of rays differently refkngiile (and upon which a mathematical enterprise can be constructed) then he must be willing to grant Newton the conclusion regarding the strict association of refkgiibilities with CO~OU~S.~~

ShapirO (1395 (1984)), page 202. See also Shapiro (1980), page 214-215: 'The problem of establishing the innateness and immumbility of color is altogaether different from that of refrangibility: hrsq Newton had no mathematical law to describe color chulgcs; and second, the color of the sun's incident Iight appears totally different before the ktrefraction and ever ah, once it bas been resolved into colors. As Newton himself was Ultimately to recognize, it is empirically impossible to prove what may be called the strong principle of color immutability for the colors of the sun's light at the 6rst refraction, since the colors are not perceptible before the first rehctioa and so may not be compared with the colors after that refraction to see if they have changed." 65 Stein (1990), page 25- 66 Stein (1990), page 25. 67 Newton uses a simih strategy in the (as we wiIi see) for his argument for univd gravitation where he invokes Rde I11 which reads: "Ihe qualitities of bodies, which admit neither of indon nor of rtmission of degrees, and wttich arc found to belong to all Meswithin the reach of our csperiemtnts, am to Hooke and Huygens: Challenges to Newton a. The Hooke and Newton Debate over Light

Two of the most distinguished of Newton's critics were Hooke and Huygens.

Their criticisms and Newton's reply to these criticisms can inform our understanding of Newton's method- As the curator of experiments for the Royal Society, Hooke was charged with the task of repeating the experiments reported to the society while checking their accuracy.68 A week after Newton submitted the letter to Oldenburg containing his theory of light and colours Hooke submined his critical discus~ioa~~The tone of Hooke's criticism is evident in the opening lines:

I have perused the discourse of Mr. Newton about coIours and rehctions, and I was not a Little pleased with the niceness and curiosity of his observations. But, tho' I wholly agree with him as to the truth of those he hath alleged, as having, by many hundreds of trials, found them so; yet as to his hypothesis of solving the phaenomena of colours thereby, I confess, I cannot see yet any undeniable argument to convince me of the certainty thereof7'

Although Hooke is able to attest to the veracity of the experiments for which he commends Newton, he does take issue with the explanation Newton offers. Hooke's contention in this opening passage is that he is not so much opposed to Newton's

be esteemed the universal qualities of all bodies whatsoever-" Here we are getting agreeing measurements of the angles. The same ordering occurs at the second prism as at the 6rst 68 For a discussion of experiments and the process of repetition see Simon Schaffer- (1989) "GkWorks: Newton's Prisms and the Use of Experimen~"in Tbc Uses OfEq!mknfiShdk in tbe NddSrimar. Gooding, Pinch, and Schaffer eds. (Cambridge: Cambridge University Press)- 69 Thomas Birch, The H.&g of tbc RgdSodct~ofLdm (London: h Millu, 1757), voL 3, pp. 10-15. Robert Hooke's mply is printed in IB. Cohen, ed- IkxNnvtorr'z P+'l card b#rrs OR NdP'' and nbrcd rbaamxr. (Cambridge, MA: Hvvvd University Press, 1958). Page rcfetenccs will be made to Birch's edition as found in Cohen's book 70 gcHOOke'~Critique of Newton's Theory," page 10. explanation as he is opposed to Newton's claim to have established it beyond a reasonable doubt. Hooke's criticism of Newton is essentially a methodological criticism

For all the experiments and observations I have hitherto made. nay. and even those very experiments, which he aledgeth, do seem to me to prove, that white is nothing but a pulse or motion, propagated through an homogeneous, uniform, and transparent medium: and that colour is nothing but the disturbance of that light, by the communication of that pulse to other transparent mediums, that is, by the rehction thereof: that whiteness and blackness are nothing but the plenty or scarcity of the undisturbed rays of light: and that the two colours (than the which there are not more uncompounded in nature) are nothing but the effects of a compounded pulse, or disturbed propagation of motion caused by rehction But how certain soever I think myself of my hypothesis (which I did not take up without first trying some hundreds of experiments) yet I should be very glad to meet with one experimenturn crucis fiom Mr. Newton, that should divorce me fkom it. But it is not that, which he so calls, wiU do the turn; for the same phaenomenon will be solved by my hypothesis, as well as by his, without any manner of difliculty or straining: nay, I will undertake to shew another hypothesis, differing fiom both his and mine, that shall do the same thing."

Near the end of his critical discussion he remarks that although Newton's Hypothesis does "save the phenomena" it isn't the only one to do so.

Nor would I: be understood to have said all this against his theory, as it is an hypothesis; for I do most readily agree with them in every part thereoc and esteem it very subtil and ingenious, and capable of solving aU the pbaenomena of wlours: but I cannot think it to be the only hypothesis, nor so

- 7' "Hooke's Critique of Newton's Thcorp," page 11- certain as mathematical demonstrations. [emphasis addedlR

To which hypothesis was Hooke referring? All that Newton has concluded, as we have seen, is that Light consists of rays differently rehgible. Hooke seems to have something else in mind:

But grant his first proposition, that light is a body, and that as many colours as degrees thereof as there may be, so many sorts of bodies there be, all which compounded together would make white; ... granting these, I say, I do suppose there will be no great difficulty to demonstrate all the rest of his curious theory: though yet, methinks' all the coloured bodies in the world compounded together should not make a white body, and I should be glad to see an experiment of that kind done on the other side.73

Hooke ends his discussion by noting that even Newton would agree that his (Hooke's) hypothesis would save the phenomena as well as Newton's. In this sense, Hooke argued, Newton's paper did not conclusively show this hypothesis to be true. We see fiom this passage that Hooke thought that Newton was concluding that light is a body. This was not Newton's conchsion, as we have already shown. It would appear that Hooke is misreading Newton

It is not surprising, then, to hdNewton responding to Hooke's criticism by showing that Hooke actually misead him Newton's reply appeared some nine months after the initial publication of his theory of light and colours in Philosophical Trurtsacti~ns~It is a lengthy response, approximately twenty pages. What interests us, though, is the second part of this reply, the so-called 'Theorique part" wherein Newton examines Hooke's "Considerations on my neories. And those consist in armiing an

Hpothesis to me, which is not mine, m Asserting an Hpthesiss,which, as to the principal

72 "Hooke's Critique of Newton's Thory," page 14. 73 "Hooke's Critique ofNewton's Theory," page 14. parts, is not against me; in Granting the greatest part of my discourse if explicated by that Hypothesis; and in Denying some things, the truth of which would have appear'd by an experimental examinati~n.""~In the paper on light and colours Newton does in fact suggest that although the claim of the corporeity of light is possible, he merely offered this as a conjecture. Indeed, he was carell to say that "it can no longer be disputed, whether there be colours in the d& nor whether they be the qualities of the objects we see, no nor perhaps, whether Light be a Body." [emphasis added]" In any case it was punling to Newton why Hooke would bother with a discussion of hypotheses. He claims that it matters not whether light is corporeal or pulse-like, as one can just as easily conceive of it as consisting of rays differently rehgible. Although he conjectured that light is a body, he claimed in his reply that a discussion of this hypothesis is beside the issue.

But whatever be the advantages or disadvantages of this Hypothesis, I hope I may be excused from taking it up, since I do wt think it needful to explicate my Doctrine by any Hypothesis at al~'~ Newton's reply to Hooke served to deflect Hooke's objections by showing that the latter's criticisms were misdirected. AU that he concluded £?om the experimenturn cnrcis is that light consists of rays differently rehgiible and that this is all that is warranted fiom an examination of the phenomena. b. The Euygens and Newton Debate over Light

In March 1672 Oldenburg sent to the leading figure of the fledgling Acadernie Royale des Sciences, Christiaan Huygens, the issue of the Philosophical Transactions in which Newton's paper appeared with the explicit request of receiving a commentary from

Huygens. Huygens' initial reply centred primady on what interested him about Newton's

74 Pbib~apWT~~ (88), 18 November, 16725086. 75 Pbib+kdTm& (80), 19 February, 16723085. 76 PbirEvop6ikal T- (88), 18 Novemkr, 16725089. discoveries, namely the invention ofthe reflecting telescope. Wth respect to the theory of colours Huygens merely claimed that it "strikes me as most ingenious; but we must see if

it is compatible with all the experiments."" There was another attempt to draw Huygens into the byat the end of June, 1672. Huygens' concern still focused on the reflecting telescope and said very Little by way of critique of Newton's theory. Due to Oldenburg's persistence Huygens finally sent a statement about Newton's optical discoveries. By July of the following (1673) the Philosophicui Transactions published a letter from

Huygens dong with Newton's discussion of this letter. Whereas Hooke's dispute with Newton is essentially methodological, Huygens's is perhaps better characterised as Hoo ke portrayed Newton as not having avoided hypotheses and as not having conclusively proven his hypothesis of light having certain innate properties. Huygens did not question the of Newton's experiments, nor did he accuse Newton of positing a corpuscular theory instead of an undulatory or wave theory of Light. Huygens, instead, suggested that Newton had Wed to show US the nature of light.79 Huygens was evidently looking for a mechanical hypothesis for he says:

I have seen, how Mr. Newton endeavours to maintain his new Theory concerning Colours. Me thinks, that the most important Objection, which is made against him by way of Quaere, is that, Whether there be more than the two sorts of Colours. For my part, I believe, that an Hypothesis, that should explain mechanicalEy and by the nature of motion the Colors Yellow and Blew, would be suEcient for all the rest, .. .. Neither do I see, why Mr. Niwton doth not content himself with the two Colors, Yellow and Blew; for it will be much more easy to find an Hpthesis by Motion, that may

Quoted in Guerlac (1981), page 81. The o+ French reads: 'Tour ce qui at de sa nouvde Theorie dcs couleurs, eIIe me paroit fort ingenieuse, rmis iI hudra veoir si elle est compatible avec toutes les expkriences." Guerlac's translation of this last part is to render C+Z~IYRCT as expedmena and not experience. I think Huygens acdy meant experience by this term so that his point is to see whether Newton's theory is compldble with experieenct. 1 base &is m what we will see later to be his tuain concern with Newton's theoy. 78 Harper and Smith (1995). 79 It should be pointed out that both Huppand Hook do apethat hypotheses arc not beside the point explicate these two differences, than for so many diversities as there are other Co Iors. And till he hath found this Hypothesis, he hath not taught us. what it is wherein consists the nature and difference of Colours, but only this accident (which ceaainly is very considerable,) of their dzflerent ~efian~ibiiity?

Unlike Hooke, Huygens did not see Newton as having claimed to demonstrate any mechanical hypothesis. Huygens, as is evident fiom the above passage, claimed that

Newton's doctrine did not make the phenomenon mechanically intelligible. Huygens' claim goes Mer: not only did Newton fail in teaching us the nature of Light mechanically, he made the physical understanding of the phenomenon of light more

Hcult than it need be by postulating a greater variety of lights. Thus, Huygens concludes that Newton's claim is hard to reconcile with any acceptable account of the nature of light.

As we have already seen in the discussion with Hooke, Newton has, in fact, set aside the nature of light and of colours. This is the substance of Huygens' complaint.

Huygens claimed that until Newton has shown how the diversity of colours could be explained by ia physique me'chanique8' (what he calls the hypothesis by motion) Newton will not have taught us the nature of light. To this Newton replies:

I never intended to shew, wherein consists the Nature and Difference of colors, but only to shew, that de factto they are Original and Immutable qualities of the Rays which exhibit them; and to leave it to others to explicate by Mechanical

80 Pb&~('~biclJTr-.m (96), 21 July, l673:6O86. Hakfoort (1995), page 17. The impression one gets bmNewton's critics, especially Hooke, is that an assumption in Newton's theo y regvding the immutability of rehngibilitg conesponding to parti& rays are distinguished by qualitative propaties such as colour. For the mechanists, colour is a secondary Wtyof things and, thus, is not fbndama~tal. For a nmcch;mist &kc Huygtns this meant &at Newton's hoq of 0b1ours was not a physical model in dchcolours were rtduccd to the basic mechacktic catcgorie~.'' Hypotheses the Nature and Difference of those qualities: which I take to be no dficuit matter."

Newton and Experimental Philosophy

Newton's replies to both Hooke and Huygens are indicative of the role hypotheses play in experimental philosophy. Notice that in Newton's reply to Huygens he claims that his intention was to show de facto that colours are original and immutable qualities of the rays that exhibit them. Finding hypothetical explanations is totally beside the point. That is, his project is the task needed to be done prior to providing mechanical hypotheses (which, he ch,ought not to be difficult to find). What is informative is how Newton considers variant mechanical explanations. In his reply to Pardies' second letters3 he shows that Huygens's demand for a mechanical explanation depicting the nature of light and difference of colours will not be met without some way of selecting the right expIanation fiom competitor explanations. Newton points out that Pardies' position was that differently rehgible rays need not explain the length of the coloured image. Newton answered this by showing that there are a number of alternative mechanical hypotheses that could explain the properties implied by the experiments." He reiterates that Hooke's hypothesis stipulates that the differently refianglbe rays correspond to the mionand expansion of undulations (waves). To

Hooke's hypothesis he adds Grimaldi's hypothesis which can also explain these properties of light by stipulating that Light is a "certain substance put into very rapid motion"85

82 Pbib~~pbic;o/Trmactionr(96), 21 July, 1673:6109. 83 Gueflac (1981), page 81, draws a connection between Huygens and Pardies, a Jesuit professor of mathematics at the Collige de Clennont (later to be CollZge Louis-le-Gnnd). Pardies was not a member of Royd Academy of Sciences however he was, according to Gu&c, well acquainted with Huygens. 84 In Newton's replies to Pardies he shows an uncharacteristic patience. civility, he repeatedly protested at having his theory or doctrine chvlacriscd as a hypothesis. Paedia' doubts seem to have been easily removed by Newton's careful uphations of his theory, the details of he eqm+t, and his . 85 Pbihmphd T- (85), t 5 July, 16725014. Finally, he considers a defence of Descartes' hypothesis that differently rebgible rays correspond to globules of different sizes and densities.

To which may be added the hypothesis of Descartes, in which a similar diffusion of conatus. or pression of the globules, may be conceived, like as is supposed in accounting for the tails of comets. And the same diffusion or expansion may be devised according to any other hypothesis, in which light is supposed to be a power, action, , or certain substance emitted evey way fmm luminous bodies. [Emphasis added]86 The point Newton is stressing here is that since it is "not ~cultwto mechanically explain the nature and difference of colours, these mechanical explanations really do not answer Huygens' desire that an explanation ought to tell us something about the nature and difference of colour and in what these actually con~ist.~'Wfiout some way of selecting one hypothesis fiom its competitors, Huygens' demand for a more fhr-reachmg conclusion cannot be met. Newton's reply to Pardies' second letter does not merely stop with this. Here we get, in Newton's own words, a wonderfd passage outlining his experimental philosophy.

Primary m this experimental philosophy is the diligent use of experiment to establish properties, after which one may proceed slowly to find hypotheses to explain them.

For the best and safest method of philosophizinig seems to be, first to inquire diligently into the properties of things, and establishing those properties by experiments and then to proceed more slowly to hypotheses for the explanation of them For hypotheses should be subservient only in explaining the properties of things, but not assumed in determining them; [emphasis added]

Pbih~opbiroiTrm- (85), 15July, l672:SO 14. 8' Hakfoort (1995), pages 15-22, offers an interesting description of the events surrounding the exchange over he light and colours paper and its reception not only in England and Frvlct but also in Gerrmny- Hakfoort's account, liLc o&ers, lacks in ik analysis of Nm's me&odological commitment Furthermore, Hakfoort maintains @Ige 17) that Newton was not clar on the details of the crpimcmt and even less dear about the fact that tbc prism was at rrJnimum deviation Newton, we arc nnin- here, was dar on both counts. unless so fix as they may fbmish experiments. For if the possibility of hypothesis is to be the test of the truth and reality of things, 1 see not how certainty can be obtained in any science; since numerous hypotheses may be deawhich shall seem to overcome new dZEcuhies. Hence it has been here thought necessary to lay aside all hypotheses, as foreign to the purpose, that the force of the objection should be abstractly coosidered, and receive a more fid and general answer?' I don't read Newton, here, as proposing that we should abandon hypotheses outright. He

is "bming" the extent to which hypotheses are to be used in natural philosophy. That k, the priority in natural philosophy must be experimental determinations. Hypotheses do have a role in determining properties as we1 They are not to be the arbiter of truth but, rather, suggestive of fivther experiments the results of which may aid in the determination of these properties. The inference to properties from experiment in the experimenturn crucis illustrates an ideal of empirical success which is developed m the Principio. It is this form of evidential reasoning which Harper claims helps explain the transition &om natural philosophy to empirical science in the seventeenth and eighteenth centuriesg9 We recali that in the experimentum mcis Newton claimed that the light

tending to that end of the Image, towards which the refraction of the first Prisme was made did in the second Prisme suffer a Refraction considerably greater than the light tending to the other end. Emphasis added.lgO

The light at the end of the spectrum that was rebted the least continued to be refkcted the least after segregation. Newton, here, is using agreeing measurements of the degree of rehgibihy of the various portions of the unsegregated light with the degree rehgiiility of the segregated portions of light. It is this agreement that allowed him

conchde from the qerintentmn -is that light consists of rays differently rehngiile.

8s Pbi.T~z#~s&(85), 15 J+, 1672-5014. See Harper and Smith (1995)- 9Q P.7'-adour (80), 19 February, 16723078 I have tried to present here a historical incident early in Newton's career that clearly foreshadows his mature thought. We have, in this incident, a philosophical discourse played out in a journal - perhaps the first of its kind.

Newton was the first to advance through this new medium bournal publication] an experimentally based proposal for the radical reform of a scientific theory, and his proposal was the first to arouse international discussion and debate within the columns of a scientific journal9'

Kuhn elaborates on this passage to note that it is through such exchanges, his letters and his unpublished manuscripts, that we discover Newton's creative genius. This genius is, for Kuhn, entwined with Newton's "fear of exposure and the correlated compulsion to be invariably and entirely immune to criticism" which "show throughout the controversial

Kuhn rhetorically asks: "Is Newton honest m rejecting the corpuscular hypothesis that Hooke armis to him? Or, to take a later and fir clearer example, is not Newton convicted of an inationally motivated lie in his reply to Hyugens' remarks about the co~sitionof the color whiteT93 "Non and "yes", respectively, are the answers Kuhn expects although I think our exposition above clearly shows nothing of the sort. Kuhn thinks that Newton's creative genius lay m his (creative) play of retreating fiom metaphysical hypotheses? We have seen that Newton was not avoiding hypotheses for the sake of avoiding them, for he does clahn that they do have some use. He clearly Limits their use in what can and what cannot be claimed &om experiments. This methodology is creative, not because it is sophiscry as Kuhn wouki have us believe, but, rather, because it sets a "playing field" on which theories compete &lye

9' Kuhn (1958), page 28. 9Kuhn (1 958), page 39. 93Kuhn (1958), page 40 9*For detaiIed rtsponses to Kuhn's cfdmges see Stein (1989), aodc 36, pages 17-19. Sttin cMIcnges Khb historical adysis md, conseguendy, refutes he philosophid import Kuhn gIeams hm the controversy surrounding tfrc paper on light and cotours. In his conclusion to his paper Kuhn claims the following:

Much of modem science inherits &om Newton the admirable pragmatic aim, never completely realized, of eliminating fiom the final reports of scientific discovery all reference to the more speculative hypotheses that played a role in the process of discovery. The desirability of this Newtonian mode of presenting theories is well illustrated by the subsequent history of Newton's own hypotheses. ... But Newton's remarks about the role of hypotheses in science were dictated by personal idosyncrasy as often as by philosophical acumen; repeatedly he renounces hypotheses simply to avoid debate.9s

We have seen how Newton does not renounce hypotheses merely to avoid debate but because they were redy beside the point. Kuhn wants to claim that the next milestone to pass in was an adequate account of a wave theory and that this was delayed because of the metaphysical nature of Newton's corpuscular hypothesis. The retreat fiom hypotheses, then on this account, is self-serving in Newton's case. In his explicit account of scientific reasoning as found in the frincipia Newton does, however, offer a method of measuring the success of a theory. Hypotheses, by this thise, do not undercut what a theory is actually supposed to do. Here is the challenge to Kuhn's positioeto look at what Newton thought he was up to, what his position teUs us about evidential reasoning, and what success his method yields. To that end, let us proceed to an analysis of what Newton's work in the Principiu has to teach us about his mature position on how one ought to proceed m experimental philosophy.

95Kuhn (1958), pp. 4445. Chapter 3

...by the propositions mathematically demonstrated in the former Books, in the third I derive fiom the cefestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces, by other propositions which are also mathematical, I deduce the motions of the planets, the comets, the moon, and the sea, Newton's Preface to the First Edition of the Principia-

The Rules

In the General Scholium at the end of Book III, and added in the second edition of the Principia, Newton summarked his experimental philosophy. I will quote this passage here and come back to it at the end of the chapter. After a discussion of how various phenomena are explained by the "power of gravity," Newton admits that no cause has yet been assigned to this power.

But hitherto I have not been able to discover the cause of those properties of gravity fiom phenomena, and I &me no hypotheses; for whatever is not deduced fkom the phenomena is to be cded a hypothesis; and hypotheses, whether physical or metaphysid, whether of occult qualities or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred &om the phenomena, and afternards rendered general by induction" We ended the last chapter on a simikr theme, showing how Newton's notion of the experimental philosophy transfers the burden of warrant from predisposed metaphysical hypotheses to physical information. We saw with his paper on light and colours that

Newton was able to take the agreeing measurements of the refkugibllities of the rays coming through the first prism with the refbngibilities of the segregated rays going through the second phFrom these agreeing measurements Newton inferred that white light is composed of rays of differing rehgibilities. This form of reasoning is coupled, in the Principia, by Newton's rigorous employment of a mathematical style.97 Here Newton will use the mathematical laws of motion of the first book in conjunction with motion particular, to the Iast part of it. From the phenomena of motion, according to Newton, one can infer forces. The argument for Unived Gravitation is contained m Book III of the Principia. The book begins with a statement of four "Rules of Philosophising" followed by six "phenomemn98 Newton's basic argument consists of seven propositions that depend on the phenomena, the mathematical demonstrations of Book I, and on the Rules of Reasoning. In their final versions, the rules are? RULE I

We are to admit no more causes of natura2 things than such as are both true and stlfticient to explain their appearances.

RULE 11:

Therefore to the same natud effects we must, as far as possible, assign the same causes.

9' 9' Cohen (1982). gT&ephenomena are statemeats of astronomical fegul;Ltities (more on this hter). In the first edition they were called by Newton "hypotheses." The designation "phenomena" was introduced in the second edition and maintained through the third. * Pnh+b, page 398. The qualities of bodies. which admit neither of intension or of remission of degrees, and which are found to belong to all bodies within the reach of our experiments. are to be esteemed the universal qua(ities of all bodies wharsoever.

RULE IV

In experimental philosophy we are to look upon propositions collected by general induction fiom phenomena as accurately or very nearly true. notwithstanding any contrary hypotheses that may be imagined till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions. The expression "Rules of Philosophising" (Replaephilosophandi) appears only in the second and third editions of the Principia. What is curious about the first edition is that in a corresponding place Newton stated a series of nine statements under the heading of "Hypotheses." This is curious in light of his distaste for hypotheses (as we have seen in the debate over Light and colours). Koyr6 has pointed out that these nine hypotheses can be classified into three categories. The first category contains the first two hypotheses that proposed formal and general principles of the philosophy of nature. In the second category is the third hypothesis that aflirmed the unity of matter. Fbdly, to the third category is assigned the remaining six "hypotheses" concerning the structure of the system in which we find the earth, moon, sun, and the remaining primary planets and their satellites. In the second edition, Newton "elevated" the first two hypotheses to

"reep~lae.~lW The third hypothesis was dropped in later editions. Finally, Hypothesis IV became Hypothesis I in later editions. The six phenomena, m their final versions, are based on the five remaining hypotheses of the first edition plus new results which came to light prior to either the second or third editions.

I-e des had various formulations, For a discussion of Newton's refi.t~~mtntsto the rules and a formuhtion of a supposed, but never inchKkd, hfeh rule see Maandre KO* Ndzwh S& (Chicago: University of Chicago Press, 1%5), pp. 261-272. Before discussing the rules and the phenomena, I would like to say briefly

something about Newton's change of terminology. As we saw with the debate over light and colours, Newton expressed his methodological principles, in his various letters to Hooke, Huygens, and Pardies and writings in the PhilosophicaI Transactions, which included his position regarding hypotheses and their role in experimental philosophy. In this philosophy, hypotheses were to have no role except to suggest experiments. The movement, in the revisions to Book III, from hypotheses to a combination of Rules and Phenomena, I think, is very much in line with his earlier cautious remarks. These remarks suggest that the role of hypotheses is to be relegated to that of being suggestive of experiments. Hypotheses are to be taken in the sense iuustrated by the following letter to Roger Cotes: As in geometry the Hypothesis is wt taken in so large a sense as to include Axiomes and Postulates, so in Experimental Philosophy it is not to be taken in so large a sense as to include the fkt Principles or Axiomes which I call the laws of motion. These principles are deduced fiom the Phenomena, and made general by inductions: which is the highest evidence that a Proposition can have in this Philosophy. lo'

Newton is clear about his use of the term "hypothesis." He tells Cotes that what is crucial to the signification of this term is the role of phenomena, a role we have seen already applied in the Iight and colours debate. For Newton then claims in this letter:

And the word Hypothesis is here used by me to sign@ only such a Proposition as is not a Phenomenon but assumed or supposed without any experimental

101 Th ~~fIriax NmVoL 5- Ed H.W.Tumbul13 J-F. Scoq A. Rupest Hail, and Law Tilling (Cambridge: at the University Press, 1959-77) pp. 396-397. Also in, Cohen, I.B. and Richard S. Westfa Ntrvron (New Yo& Norton Critical Editions, 1995). Letter from Newton to Cotes is dated 28 Much, 1713, page 219. 1021bi4p- 120. According to Newton hypotheses are propositions not derived fiom phenomena.

Newton's replies to Hooke and Huygens regarding the Light and colours paper kept re- iterating the point that hypotheses were redly beside the point. The hypotheses Hooke accused Newton of holding were beside the point because, as Newton pointed out, they weren't deduced fiom phenomena On the other hand, the conclusion Newton drew was underwritten by the agreeing measurements of the rays rehcted through the two prisms of the -rimenturn Cmcis-

In the Principia Newton's methodology is expressed in his use of Rules of

Reasoning in Experimental Philosophy. The first two rules are concerned with the sort of conclusions we should be looking for in natural philosophy. Rule I states a principle of parsimony by claiming that we are to admit no more causes than "such as are both true and sdEcient to explain their appearances." In Rule II Newton enjoins us to assign the same causes to the "same natural effects-"'03

RULE I

We are to admit no more causes of natural things than such as are both hue and suf/icient to explain their appearances.

To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes.

RULE II

Therefore to the some natd effects we rnrrst, as for as possible, assign the same causes. As to respiration in a man and in a beast; the descent of stones in Europe and in America; the light of our culinary fire and of the sun; the reflection of light in the earth, and in the planets. We will see that the application of Rule I is more than just an a priori appeal to simplicity. Rule I helps undenvrite inferences fiom motion phenomena to forces by demanding not only sufficiency of explanation but an appeal to truth We will be arguing that Newton's inferences f?om phenomena are backed up by systematic dependencies that make the phenomena into measurements of theoretical parameters. The appeal to truth suggests that Newton is demanding something more other than a metaphysical commitment to simplicity. In this sense, Rule I is not simply an "Occamisation" of sorts, for it also requires some stronger sort of .

That Newton begins Rule II with "therefore" suggests that it follows fiom or is implied by Rule I. Taken together with Rule I, Rule II is telling us to choose explanations with common causes, as far us possible, whenever we are able to find them We will see a clear application of these two rules taken together when Newton, in the moon test, infers that the force maintaining the moon in her orbit is terrestrial gravity. The inference is from agreeing measurements of a single centripetal inverse-square acceleration field directed towards the centre of the earth by two different phenomena: the centripetal acceleration exhiiited by the lunar orbit and the length of a seconds pendulum at the dceof the earth, Rule III, on the other hand, is of a different sort.

RULE rn The qualities of bodies, which admit neither of intension or ofremission of degrees, and which ore found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies wharsoever. Of the four Rules the third is provided with the greatest discussion The discussion begins with an interesting defence of universalization, which is in Line with ideas of simplicity and consonance of nature. Newton claims:

For since the qualities of bodies are ody known to us by experiments, we are to hold universal all such as universally agree with experiments; and such as are not Liable to diminution can never be quite taken away. We are certainly not to relinquish the evidence of experiments for the sake of dreams and vain fictions of our own devising; nor are we to recede fiorn the analogy of Nature, which is wont to be simple, and always consonant with itself.

The discussion here is meant to defend universalization At issue m the third ruIe are questions that are of concern when ow is justified m drawing inferences fiom observation reports to claims about unobservable entities. This type of reasoning, which entitles us to generalise some of our knowledge about things "within the reach of experiments" so as to be able to make claims about tbings which @ecause they are too small or too remote) are outside the reach of experiments, has been called "ttansductio11~~~~Newton will use this

IM PTCj* page 398. '0j"Transdiction"was the term used by Mandelbaum in his 1964 book Pbibsophy, Jake, and Smse Pmrpribn (Baltimore The Johns Hopkins Press). Mandelbaum argued &atRule ITI was heavily laden with ontological and epistemologic;L1import and that it was through this rule dmt Newton was able to resolve a tension in his thought That is, on the one hand Newton demanded that dl scientific laws be "deduced Gom the phenomena," but on the other hand Newton spoke ofien of corpuscles which were not only unobsexved, but in principle unobservable. In his " and 'Analogy of Nature': Newton's Third Rule of Philosophising" (Stwdikr iir /be may and Pbib~ophyofJcima I (1970): 3-58) McGuire called the third rule "transduction" because of its reference to seventeenth ccnauy charact.tion of zhe problem underlying it McGuire chhx Wow transduction as conceived in tht seventeenth cmtq was not simply a type of tmnsatepric infuwce. Induction exemplifies the her, since dl inductme inferences go beyond he evidence Erom which we begin. The distinctme feature of rnnsductive infercnccs, I wish to maintain is &at the conclusion not only goes beyond the evidence we hdyhave, but pa beyond any evidence we might posslily acqukt. Put rule to argue (in Proposition VI, CoroIIary II, Book 111) that there is an equivalence of the ratio of a body's weight toward the earth and its quantity of matter (i.e., inertial mass) for all bodies universally at equal distances fiom the centre of the earth. Newton goes on in the final paragraph of his discussion of Rule LII to indicate how this rule is to be used in his argument for universal gravitation.

Lastly, if it univedy appears, by experiments and astronomical observations, that all bodies about the earth gravitate towards the earth, and that m proportion to the quantity of matter which they severally contain; that the moon likewise, according to the quantity of its matter, gravitates towards the earth; that, on the other hand, our sea gravitates towards the moon; and all the planets one towards another; and the comets in like manner towards the sun; we must, in consequence of this rule, universally allow that all bodies whatsoever are endowed with a principle of mutual gravitation. For the argument from the appearances concludes with more force for the universal gravitation of all bodies than for their impenetnbility ; [emphasis added] of which, among those in the celestial regions, we have no experiments, nor any manner of observation. Not that I affirm gravity to be essential to bodies: by their vis insita I mean nothing but their inertia. This is immutable. Their gravity is diminished as they recede &om the eadM This passage is interesting for a couple of reasons. First, since we do not have experimental reason to establish the impenetrability of heavenly bodies, universal gravitation has greater support &om phenomena But impenetrability was considered to be an essential quality of bodies.'07 Newton claims that even though the argument fiom phenomena supports universal gravitation and not the impenetrability of celestial objects it

ohke, it is not that tnnsducdve inferences go beyond tfie initial evidence, but that the evidence is appropriated to a problem to which it strictly does not apply." '06 Pnwpage 399. '07 See, for instance, Newton's De Gr.dne. Koyr6, in Fmu &d Wmkf to an I$mb Umimr,has shown the sim;larity of Newton's thought on impenetrability to the writings of the Cambridge Platonists (especialIy Henry Moore). In De Gmi&%m Newton argues for impenetrability in response to the Cartuian argument for extension as an esstntid quality of dl bodits. AccordiDg to Koyr6, Moore and Wph Cudworth argued that non-corpod entities can be said to have extension and what is more propcdy essential to corpotezl bodies is kppencbnbiliy. does not follow that gravity is an essential quality of all bodies. This is suggested by the condition of not admitting intensification or remission of degrees specified in the body of Rule III and seemingly violated by gravity which is diminished as bodies "recede &om the earth." We will see an application of this rule in Proposition VI. Corollary I1 that at equal distances fiom the centre of the earth the weights of bodies are proportional to their inertid mass. '08 The last of the des, the fourth, was introduced in the third and find edition of the Principia.

RULE N

In experimental philosophy we are to look upon propositions collected by general induction from phenomena us ucnaalely or very nearly me, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they muy either be made more accurate, or liable to exceptions.

It might be considered to be the most important rule of all. The scope of the rule is to show the confidence one can place in the results fiom induction. Inductions are to be regarded as "accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined" until new phenomena may make them more accurate or liable to exceptions. Rule IV has the briefest comment associated with it in the text. The Motte- Cajori translation reads as follows:

This rule we must follow, that the argument of induction may not be evaded by hypotheses. Newton made it clear that there is an important difference between reasoning f?om the truth of consequences to the truths of premises from which they are drawn and reasoning

IwAs KO# points out the fcsemation hat gravity is dimiaished as we recede bm the earth seems bcside the point once one notes the univaal agreement in the mtio of tfie inverse-square adjusted weight towards the car& with inertid mass. fiom phenomena to forces. That is, the point of Rule N, for Newton, is to protect the results of induction fiom being undercut by hypothetical reasoning. Hypo theses. we recall, are identified as propositions that are not deduced fkom phenomena So it red, then, to identify what Newton meant by "deduced f?om phenomena" and how this relates to induction In a recent paper, Howard Stein has cautioned the modem reader not to apply our association of deduction with strict or mathematical inference to Newton's use of the term Stein notes that, for Newtos deductions fFom the phenomena include inductions.'09 According to Stein, Newton characterises pure& mathematical reasoning as "demonstration" Newton uses deduction in a wider sense to signify any reasoning competent to establish a conclusion as ~arranted."~So in order to understand Newton's deductions fiorn the phenomena we need to examine whether the propositions allegedly "deduced" have been given adequate warrant. These four rules reflect what Newton took to be requirements for natural philosophy. But there are remarkably few explicit references to these rules m the Principia. Newton used the first and second rule, usually together, to just@ the claim that the "same sort of cause" must be responsible for the bebaviour of heavy terrestrial objects, for the moon's orbit around the earth,"' for the six planets and earth orbiting the sun, and for the satellites of Jupiter and orbiting their The third rule is invoked in support of the claim that we should count gravity among the universal properties of bodies.' l3 Newton's illustrating paragraph suggests that the third rule allows us to infer that the inverse-square law is a universal law. Likewise, the fourth deis used

lagHoward Stein, "'From the Phenomena of Motions to tfie Forces of Nature': Hypothesis or Deduction?" PSA 1990, Vol. 2: 209-222, IIOIbid, page 219. %ujbr;z, Proposition IV, Theorem IV, Book 111. n2 Pn&ijk Proposition V, Theorem V, Book 111. Pn'mijhk Coroll.up 11, Proposition W, Theorem VI, Book 111. in conjunction with the tirst two rules, to support the general proposition that whatever causes the moon to retain its orbit will also sectall planets.''J

That Newton makes so little explicit use of these rules has been a source of confusion. Even with the long history of Newtonian Scholarship, historians and philosophers of science have rnistakeniy either misinterpreted the rules or misconstrued their ro Ie. For instance, Gower (1 997) recently claimed:

But what is even more surprising is that now of these des refers to the type of reasoning which is characteristic of Book III of the Principia. The 'Propositions' concerning planets and their satellites that Newton established are derived fkom astronomical Thenomem' but not in accordance with the inductive process mentioned in Rule TV. For the six 'Phenomenaf listed immediately after the rules are already expressed m general terms, so presumably they have already been inferred by induction fiom specific observations. And m any case, the reasoning fiom 'Phenomenaf to 'Propositions' is deductive rather than inductive.

Newton was erplidt in his formulation of the rules and in their applications in his argument for universal gravitation. First, Gower is right in noticing that proposiuons are deduced from the phenomena. What Gower lefi out was that these propositions are, as Newton claimed in the General Scholium, "rendered general by induction." Second, I think Gower incorrectly characterises Book III in the above passage. The argument for universal gravitation, the centrepiece of Book 111, makes explicit use of the rules of reasoning.

Newton's use of agreeing empirical measurements to back up his inferences in Book I11 is what distinguishes Newton's methodology &om the sort of mischaractedzation offered by

Gower.

hk@b, Scholium, Proposition V, Theorem V, Book 111. n5 Gower (1997), page 72. Phenomena

In the second and third editions of the Principia Newton cites six phenomena along with the rules. Newton's proof for universal gravitation rests on an application of the four rules along with these six phenomena. We have already remarked that in the first edition the first two rules and the phenomena were grouped as '%hypotheses."Beginning with the second edition Newton distinguished between the des on the one hand and the phenomena on the other. What are these phenomena? Newton's phenomena are not simply individual observations. They are, rather, generalisations based on observations or data Recall that in Rule N Newton declared that inductions are to be considered

"accurately or very nearly true" until new phenomena "by which they may either be made more accurate, or liable to exceptions." I take it, then, that Newton thought phenomena to be stable as new data are added. That is, phenomena are generalisations taken to hold for open-ended bodies of datatt6

We will now turn to these phenomena The first reads as follows:

Phenomenon I

Thar the circumjoviul planets, by radii drawn to Jupiter's centre, describe areas proportionaI to the times of description; and that their periodic times, the faed stars being at rest, me as the sesquiplicate proportion [ie., qth power] of their distancesfrum its centre-

"6Both Koyrci and I-B. Cohen have documented in && writings the various amendments Newton made from one edition to another of &e An+& Phenomenon I was cowcted wi& he second edition to reflect the better data concerning the satellites of Jupiter. As well, Newton introduced in Phenomenon I1 references to the satellites of Samwhose existence he had not ackaowledged in the first edition. Furthermote, although I will not undertake such a task here, the stability of Newton's data can be measured or assessed by statistical method, Such methods as curve fit* can be employed to verify not only whether Newton's generatisations reasonably fit the data he had but also to compare them with modern estimates Gom ephemeris values. Bill Harper has undertaken part of this formidable task and the reader is urged to consult his wxidngs, especially WL. Harper et al. (1994) ,"Unificaaon and Supporc Harmonic Law Ratios Measure the Mass of the Sun," in [email protected]&y OfS&a rir Up~rab,Sytbclrc Li* VoL 236, pp. 131-146, Brawitz, D. and Westerstahl, D. eds. Kluwer Academic Publishers- Harper has also devoted a full chapter to "Phenomena" in his forthcorning book Phenomenon I asserts that the bvian satellires demibe areas in proportion to the times (i.e., Kepler's Area Law) and also that the periods of these satellites as the square root of the cubed power (i-e., the 312 power) of their distances fiom Jupiter (i-e., Kepler's Harmonic Law.)

The following table illustrates Newton's values for the periodic times of the Jovian satewes.

Moon Periodic Times Periodic Times (as listed by Newton) (days)

For the following calculations is calculated &om the first four observations Listed by Newtoo. We should note, as we& that the Rd, value corresponds to K=58.139, where R is measured m semidiameters of Jupiter and T is measured m days."' It appears that

Newton used the distance calculated by Cassini using the of the satellite method R~ for the third moon to fix the value of K for -. Notice that Newton did not use the T~ average K-value (5 1.627) calculated fiom the 16 observations. Instead he uses the distance observed by Cassini to fix K as this was considered to be the most reliable of the bunch? From this he was able to determine the values of R for each moon hmthe periodic times. Notice, fkther, that m fxing this value of K then the other distances &om observations are, except for Borellik distance for Io, too small.

1W"i'illiam Harper has shown that the periodic times Newton uses in the table when compared to recent Astronomical Aknanacs are fairly acute, In the worst case, the periodic time of the fourth moon as listed by Newton is 242 seconds less than modern values. There are two points to take away hrn this: 1. the method used by seventeenth cennuy astronomers, namely obsemations of of the Jovian satellites, yielded accurate results aad 2. The pedods of these satellites have been stable over time. Harper (1994), Sheynin and Schmeidlu (1995). Pound's results were obtained between the second and third edition using Huygens'

123-foot focal length telescope. Newton placed much confidence in Pound's results. I

have included both Pound's values for the R-value for each moon as well as the modem

values from the 19% Astronomical Almanac. Notice the closer & of Pound's results to the more accurate modern values."g So as new data were made available Newton, we see here at least. incorporated them in his phenomena'20 Phenomenon I will be cited explicitly when Newton deduces inverse-square variation between Jupiter and his satellites

in Proposition I, Theorem I of Book EL1*'

"9 The difference between the calculated values of R Gom the modern values for each satellite in order is: -02605, 63696, -0.5827, -1.0396. For each of rhese four Pound's results are even closer to modern values. The differences are: 0.0622, 0.1074, 0.1743, 0.2914. See Harper's fodcorning chapter on "Phenomena" for a detailed statisticai anaIysis of the fit of Pound's results with modern values. Iz0 Relative to modern values in .\stronomicat Almanacs Pound's results were considerably more accurate. Newton shows a great deal of confidence in Pound's data. Fniraj3ia, page 401-402. 121 For an analysis of tiis phenomenon see Harper and Smith (1995) md Harper (1997). R values

Mooa 1: Io

31ooa S: E-p.

Moom 3: hm

*Moon 4: camto K values

Moon 1: lo

Moon 2: Europ

Maom 3:

-Moon 4: callisto

Avenge K br 16cuacital Stradud Deviation t-vh crror(95%)

Phenomenon I1 is expressed in the following way:

Phenomenon 11

Thot the circumsatu1710Z planets, by radii drown to Satm's centre, describe areas proportional to the times of description; ond thut their periodic times, the Bed stars being at rest, are as the sesqu@Iicate proportion of their distances from its centre. As in the previous phenomenon, Newton is giving the harmonic law for the satellites of

Saturn as Phenomenon II. The Harmonic Law for Saturn's sateilites is iUustrated in the fo ilo wing:

Moon Periodic Times Periodic Times (as listed by Newton) f days)

R~ Newton takes the values for the fourth satellite to ftx - = K. T~ R=8 semidiameters of rings T=15.9453 days

DiffkenW R (observed) in R (calculated) Periodic Times btswmR Moon -diameters of 3 KT2) 113 (calculated) and R (days) Saturrrsw (K=2.0 137) (observed) I: Tethys 1.8878 1.95 2.0806 1.9289 -0.021 2: Dione 2.7371 2.5 2.0856 2.4709 -0.029 3: Rhea 4.5 175 3.5 2.1009 3.4509 -0.049 4: Titan 15.9453 8 2.0 137 8.0 0.000 5: lam 79.325 24 2. I%9 23.3 134 -0.687

95% Average K 2.0956 confidence 0.00 185 rntmfal The calculated distance for the fifth moon, Iapetus, is 23.3 134 semi-diameters of Saturn's rings. Newton lists 23.35 semi-diameters as its distance in the Principiu. The ratio of the standard deviation to the average K is 0*0658 = 0 .O3 1. The fit of the harmonic law to the 2.0956 data is pretty good. Newton also discusses the greatest elongation of the fourth moon,

Titan, using Huygens' 123-foot telescope wah an excellent micrometer. The vdue used to fix K in the table above is 8 semi-diameters of the rings. The greatest elongation of Titan with the 123-foot telescope turned out to be 8.7 semi-diameters. Newton then takes this value to compute the K -value for Titan (2.590) fiom which he was able to re-calculate the other satellite distances. These turn out to be m semi-diameters: Tethy~2.1,

Dione=2.69, Rhea=3.75, Titanz8.7, and 1apetus=25.35.~~ We cannot carry out a. analysis of the fit of harmonic law ratios to observations because the only observation given is the greatest elongation of Titan. Had Newton provided observations of the distances of the other satellites using this 123-foot telescope the reader would have been in a better position to evaluate the fit of the harmonic law to the newer data and to determine whether the new data increase the support for the harmonic law.

We now turn to the statement of the third phenomenon, that the orbits of the five primary planets encompass the sun.

Using the K-vaIue calculated for Titan Gom the observation Lorn the 123-foot telescope Newton computed the fifth satellite's distance to be 2335 semi-diameters. His table for the distance without this correction gave the same value. In our table above, we noted that he calcuiated vaiue ought to have been Listed as 23314 and not 2335, which is the conected value kdon the newer obsemaaons. Phenomenon 111

That the jive primary planets. Mercury. Venus. Mars, Jupiter, and Saturn, with their several orbits. encompass the sun.

That Mercury and Venus revolve about the sun is evident Eom their moon-like appearances. When they shiw out with a fidl fke, they are, in respect of us, beyond or above the sun; when they appear half full, they are about the same height on one side or the other of the sun; and they are sometimes, when directly under, seen like spots traversing the sun's disk. That Mars surrounds the sun, is as plain &om its 111 fhce when near its conjunction with the sun, and fkom the gibbous figure which it shows m its quadratures. And the same thing is demonstrab1e of Jupiter and Satum, fiom their appearing full in aU situations; for the shadows of their satellites that appear sometimes upon their disk make it plain that the Light they shine with is not their own, but borrowed £iom the sun,

We first note that Newton does not include m this list the earth- Phenomenon III does not decide between the Copernican system and the Brahean system. In short, at this point, the system where the five primary planets revolve around the sun and these together revolve around the earth is not explicitly ruled out. Newton basically splits the explanation for this phenomenon into two: 1. The behaviours of Mercury and Venus are considered and 2. The behaviours of the remaining planets, Mars, Jupiter, and Saturn are considered. That Mercury and Venus dispky phases (ie., we sometimes see them 11l, half full crescent, and as spots traversing the sun's disk) indicates that these two plawts revolve around the sm Now, regarding Mars, Newton notes that its Eace is fidl near conjunction and gibbous in the quadratures. Therefore, it encornpasses the sua Jupiter and Saturn always have fdl phases. Furthermore, we see the satellites of both Jupiter and Satum crossing the disks of their respective planets leaving shadows such that we can see both the moon and the shadow. At quadrature the moon may cross into Jupiter before the shadow and vice versa This shows that Jupiter and Saturn are not their own light source and are illuminated by the sun. Phenomenon IV

73at the fuced stars being at rest, the periodic rimes of the fie primary planets, and (whether the sun about the earth, or) of the earth about the sun, are as the 312th power of their mean distances from the sun- This phenomenon, Kepler's harmonic law is formulated for both the Copernican heliocentric model and Brahe's geo-heliocentric model. At this point, Newton is care111y maintaining neutrality between these two world . From the table below we see that Newton compares the planetary distances as calculated through the harmonic law to empirical estimates fiom astronomers with the result that the fit is quite ccl~e.'~~

Newton's reasoning in Phenomenon N is similar to that exhibited in Phenomena I and 11. R~ -- K T~- - Harmonic Law for the planets.

Cf- Curtis Wh(1989), The Newtonian Achievement in ." T(d Rke#er RBod@u Log @ bZ - T(ctays) yeam) (AU) (AU) -IpT wer RIkdiau I(1MPan Msmy 87.m 02408423 03806 0.38585 0.386955 4.618267 -0.4I1101 -0.413581 4.4I2.34 Venls 224.6176 0.6149585 0.724 0.72398 0-72399 4.21 1154 4.140261 -0.140273 4.140267 Earth 365-55 I 1 1 1 0 0 0 0 Mils m.9785 1.8NB1 12 1.5235 1.5235 1.5235 0.2743452 0.1828425 0.1828425 0.1828425 ~~lpita 4332514 11.sim 5.~5 S,ZL~~sms 1.w41a o.nsnw o-n81mo.mm

Note that, overall, the average K for the values for both Kepler and Boulliau is

%,,=I -00085 o,,,=O. 0070

We note, as well, how good the fit is by inspection of the logarithmic plot above. By plotting the logarithm of the period versus the logarithm of the distance, we can examine how well the harmonic law fits the observations. The slope of line produced by this technique is the value of the power in the harmonic law. As it turns out the slope is computed to be 3/2, which indicates a very good fit between the harmonic law and observations. Phenomenon V

Then rhe primary p/anets. by radii drawn fo the earth. describe areas in no wise proporfional to the times; but the meas which rhw describe by radii drawn to the sun are proportiomi to the times of description. This is. of course, the expression of Kepleros area law. The two serious contenders for the centre are the earth and the sun. Notice, thus k, Newton has not used Kepler's ellipse law. Nor does he need to for his argument, for Phenomenon V appeals to a result he obtained early in Book I. That is, a uniform description of areas, ie., a planet that satisfies the area law, is a desideratum for the centre of orbital motion. In the Scholium to Proposition III, Book I of the Principia Newton asks rhetorically the following:

Since the equable description of areas indicates that there is a centre to which tends that force by which the body is most affected, and by which it is drawn back from its rectilinear motion, and retained in its orbit, why may we not be allowed, in the following discourse, to use the equable description of areas as an indication of a centre, abut which all circular motion is performed in fiee spaces?lZ4 Following Newton, then, we look for centres such that the area law holds. Having found this we can identifjr the direction of the force deflecting a body into an orbit. We have noted that Newton was neutral between the Copernican and the Tychonic world systems. 12*

By Phenomenon III the Ptolernaic system was eliminated as a contender. This leaves the Copernican/Kepler and the Tychonic models as possibilities. Phenomenon IV tells us that, viewed fkom the earth, the primary planets progress, retrogress, and

12* Pn-, page 45. '3 By this point in history these were the chief contenders. Newton's chthat he saw "further" because be stood on the shoulders of giants is exempiificd by this. It is worthy to note, as well, that Newton's notation of rhe phenomena is neutral between these two contenders, sometimes remain stationary- That is, as viewed fiom the earth, the primary planets do not behave in a dormway.

On the other hand, viewed fiom the sun the primaty planets "Are always seen direct, and to proceed with a motion nearly uniform that is to say, a little swifter in the perihelion and a Little slower in the aphelion distances, so as to maintain an equality in the description of the areas. " So motion with respect to the sun very clearly approximates the area law. The area law is still maintained even when there is a departure fiom uniform motion on concentric circles because the planets move a little swifter in perihelion than they do at aphelion.

We should note an added proof of the area law that Newton tacks on at the end of his brief discussion of this phenomenon. He states:

This is a noted proposition among astronomers, and particularly demonstrable m Jupiter, fiom the eclipse of his satellites; by the help of these eclipses, as we have said, the heliocentric longitudes of that plaoet, and its distances fiom the sun, are determined.'26

Independently of knowing the shape of the orbits ow can gather the heliocentric longitudes for Jupiter at different intervals. These yield different triangulations for which areas can be calculated. From here it is a simple task to see with respect to the SUII, Jupiter is sweeping out equal areas in equal times. I27

Phenomenon VI

That the moon, by a radius &awn to the earth's centre, describes an area propu~-omZto the time of description.lZ8 This is the area law for the moon. Newton's justification reads as follows:

This we gather fiom the apparent motion of the moon compared with 3s apparent diameter. It is

Pn'n+%, page 405. In The chapter on Phenomena in Harper's forthcoming book explores this task in some detail. Im A* page 405. true that the motion of the moon is a little disturbed by the action of the sun: but in laying down these Phenomena I neglect those small and inconsiderable err0rs. 129 At apogee the moon's apparent diameter is derthan at perigee. Furthermore, it is moving more swiftly in that part of its orbit where it is closest to the earth, i.e.. pigee. The moon's apparent motion at apogee and perigee is approximately what we would expect in order to maintain the area law. Newton concedes that there are certain inequalities m the lunar orbit. These inequalities m the lunar orbit were troublesome for astronomers' attempts to reliable lunar tabled3' We have seen how Newton characterises the Rules and the Phenomena. The latter are not simply data but are generalisations that fit not only the best data available to

Newton but are expected to M new data as they become known. Phenomena include Kepler's harmonic and area laws. Newton moves on to deduce universal gravitation fiom these phenomena and we will explore his deduction I will delay, in what follows, a discussion of various critiques levied against Newton and his method. Most importantly, I will show how our characterisation of Newton's ideal of empirid reasoning adequately responds to hypothetico-deduction, and Duhem's holism. W& this in mind, let us turn briefly to Newton's argument for universal gravitation The Path to Universal Gravitation Newton's deductions fiom phenomena are used ultimate$ to show universal gravitatioa Newton's initinitial volley is to establish two propositions which are very similar: 1. Tbat the moons of Jupiter and of Saturn are kept in their respective orbits by an inverse-square force directed toward Jupiter and Saturn respectively. 2. Likewise, the primary planets are

'Z9 Pn'* page 405. 130The apparent diameter of the moon varies Gom 29 1/2' of arc at apogee to about 33 1/2' of arc at perigee. Densmore (1995), p 282. 13'& we will se later, there was a pratical advantage to having accurate lunar tables. In pardculara they would facilitate aavelling at sea. PhiIosophicaIIy, however, we d argue that one of the empirical challenges to universal gravitation was an adequate luav theory. Clairau~d'Alembeq and Euler solved this some sixty years after the publication of the Pnir+zk kept in their orbits by an inversesquare centripetal force directed toward the sun. Let us examine these in turn Newton reasons that the moons of Jupiter are deflected into their orbits by an inverse-square centripetal force because by Phenomenon I we know that these moons satisfy the area and harmonic laws. Because their orbits satisfil these laws Newton can apply Proposition II or III of Book I and Corollary W,Proposition N of the same book. The first part of the reasoning yields that there is a force directed toward the centre of

Jupiter and, fiom the second part we arrive at the fact that this force varies inversely as the square of the distance. Newton's argument for the second proposition is similar to the preceding argument. In the second proposition Newton turns to the forces acting on the planet except, here, he turns to the harmonic law and area law for the planets, Phenomena N and V respectively. In addition, Newton notes that according to Corollary I, Proposition XLV,

Book I the slightest deviation fkom inverse square centripetal force would manifest itselfin a motion of the heof apsides. We noted in our discussion of the Phenomena that Newton proceeded fiom the harmonic law and area law of Jupiter, Satum, and their respective satellites to the planets, and fidly to the moon Likewise, the progression with his deductions proceeds accordingly. In Proposition III Newton turns to the moon Here, Newton argues that the moon is held in her orbit around the earth by an inverse-square centripetal force in the direction of the earth's centre.13' h this proposition Newton pointed out that given that the lunar apogee precesses 3O3' per revolution (in consequentia), the measure of the centripetal force would not be

'32In "Newton" scholarship the first three propositions of Book XI1 arc ohdealt with quickly. It is with Proposition IV, the famow moon test that historians of science have noted &at Newton's system stands apart Gom previous scientific dinking. Proposition IV is taken to be the crucial Link: in the atgumeat to universal gravitation. As we will see later, it was Proposition 111 of Book 111 htoffwd a major empkical challenge to universal gravitation. It was due to the work of Clairaut, d'-Vembert, and Eder that this particular challenge was resolved- but inversely as power of the distance. Furtherrnore, Newton suggested that we might neglect this precession as being due to the action of the sun on the moon in her orbit around the earth. He fixrther suggested that he wiu show this further on, It turns out, as we wilI show in our discussion of Clairaut, d'Afernbert, and Euler, that this lunar theory posed a serious empirical challenge to universal gravitation This doesn't undermine the task at hand. What is of value here is that Newton is clearly remarking that we can use precession to measure an inverse square relationship. That the moon's orbt does not exemplify this relationship is due to a perturbathe effect. In short, if precession is to measure inverse-square variationt3' one must show that the precession is due to perturbations. Newton suggests that this is the case with the moon but it was not success~yshown until Clairaut's address in 1749.

Reposition N Mer caries the discussion of the moon and confirms the previous proposition's assertion of inverse-square variation in the earth-moon system Newton argues, here, that the moon is continually deflected fkom rectilinear motion by the force of gravity and by this force is retained in her orbit around the earth. Newton has thus inferred that the force holding the moon m her orbit is the same force, which we count as terrestrial heaviness, i-e., gravity. Prior to the publication of the Principia "gravitas" literally meant "terrestrial heaviness. " To identifL the centripetal force that pulls the moon off of tangential motion with gravity, terrestrial heaviness, was a radical departure. It was an admission that what causes heavy objects to E11l to the earth also kept the moon fiom (naturally) following her tangential motion. Historically, the planets had been considered to be of a different sort than terrestrial objects. The "Newtoniann revolution was a revolution that resulted in seeing terrestrial bodies and celestial bodies as of the same kind. Newton infers this "same cause" by appealing to Rules I and 11.

Wna sense, we have non-Kepkrian orbits. This may be why Newton does not include the ellipse law in his list of phenomena. The ellipticity of the orbits is a conclusion the fin@& establishes- Newton computes how far the moon, at 60 earth radii fiom the earth's centre. would fkll in ow minute if it were deprived of all forward motion

And now if we imagine the moon, deprived of all motion to be let go, so as to descend towards the earth with the impulse of all that force by which it is retained in its orb, it will in the space of one minute of time, describe in its fd 15 1/12 Paris feet.IJ4

Assuming, as in Proposition III, that the centripetal force holding the moon m orbit obeys an inverse-square law then we can calculate the centripetal accelerative force operating at the distance of the moon Newton imagines what would happen if the moon were brought down to the shceof the earth. The moon is at a distance of 60 earth radii &om the dscentre. At just above the earth's surfke it would be at one earth radius away eorn the centre. The ratio of the accelerative force at the current orbit to what it is at the surface is as 1 to 60x60 on the assumption of inverse-square variation. Therefore the force at the earth's dacewould be 3600 times greater. On the assumption that gravity is an inversesquare force that extends to the moon, it follows that a heavy object on the earth's surface would fieely fall, in ow minute, 3600 of these 15 1/12 Paris feet. In the increment of ow second this heavy object, then, would fi~elyW 15 111 2 Pmir feet, or 15 feet, 1 inch, and 1 line 4/9, which Newton claims to be more accurate."' These calculations are made on the assumption that the lunar dktance is sixty earth radii away. Newton cites other estimates of the lunar distance. To accommodate these estimates I will derive a general equation, so that the substitution of these various estimates into this equation will permit some analysis. c=Circumference of the &- 123,249,600 Paris feet.

4iusof the earth= 123249600 = 19615783.07 Paris feet. 21t

lU Pnmjhiz, page 408- 5page 408. R=Lunar distance= xr. T=lunar =27d7h43 rn=3 9343 minutes. L=Diarneter of lunar orbit=2xr. C=circderence of lunar orbit=2xR=~L=2xxr=2nrx Since 2xr is the earth's circumference, c, C=cx=l232496OOx. We know the distance travelled by the moon in one orbital period. Therefore we know the distance she will travel in one minute.

C Distance in one minute

Let y= distance travelled in one minute. Therefore,

C*1 123249600~ y=-= = 3 132.6945~Paris feet (this is the distance travelled in orbit in T 39343 one minute). Now suppose the moon was deprived of this tangential motion. We could then calculate how fix it would f2d.l in one minute under the influence of the same centripetal force, which held the moon m her orbit. Newton informs us that this calculation can be done using either Proposition XlUWt or Proposition IV, Cor. IX of Book I. Corollary IX of Proposition N, Book I states:

From the same demonstration it likewise foUows, that the arc which a body, uniformiy revohring m a circle with a given centripetal force, describes in any time, is a mean proportional between the diameter of the circle, and the space which the same body f&Uing by the same given force would describe in the same given time. 13'

According to this corollary7the ratio of the distance the moon would f5ll m ow minute to the distance travelled in orbit in one minute, y7 is equal to the ratio of the distance travelled in orbit in one minute, y, to the diameter of the lunar orbit, D. So, letting ddistance the moon would Fall in one minute,

= 0.250 15x. (Paris feet)

Newton also informs us how to correct for the action of the sun on the moon. We are trying to isolate the action of the earth on the moon and so a correction (Corollary to

Proposition ID, Book m) is needed to ofkt the action of the sun on the moon This

40 ,178,725 correction amounts to - = 1 .OO563. So, more accurately, the distance the 177-29 177.725 40 moon would fkll in one minute is

D=O.2S 1 01 5(l .OO563)x=O -2s1 56x Paris feet, where x refers to the number of earth-radii used in the calcuIation. This is the general equation for the ow-minute fan of the moon at the location of her orbit. But suppose the moon was brought down to an orbit of ow earth-radius. We can now calculate how fkr she will drop in ow second and compare this to the length of a seconds pendulum. Let F=fiorce on the moon in orbit, Horce on the moon at the earth's surfhce. 1 fa- 1 r2

D where d=the distance the moon would fhll at the distance of owearth-radius. Thus, Recall that R=xr and M.25 156x. Further T is measured in minutes and t is measured in seconds. So T=60t. Making these substitutions and solving for d, which is the distance the moon would fd m one second at one earth-radius, we have the general equation:

Note that this general equation incorporates Newton's correction for the action of the sun on the moon Without this correction and depending on the estimate of the lunar distance in earth-radii, the distance the moon would fall in one second is 0.250 15x3

Newton sets 60 as the value for x. The following table Lists the computed distances corresponding to a one second fd at the surface of the earth corresponding to the values listed in the ikst edition, the third edition, and the System of the Wodd. Distance Mean distance of Distance corresponding to a the moon from the corresponding First Edition one second fall at f earth at syzygies one second fall earth-radius (with (in earth radii) earth-radius two bodv correction)

Most Astronomers VendeI in Copernicus Kirc her

Tycho (corrected for parallax)

Mean Sample standard devintiou t-confidence (95%)

Third Edition

Ptolemy and most astronomers 59 Vendel in 60 H~gens 60 Copernicus 60.33 Street 60-4

Tycho (corrected for parallax) 60.5

Mean 60.03 8 Sample standard deviation t-confidence (95%)

System of the World

Ptoiemy, Kepler, Boulliau, Hewelcke, and Riccioli 59 Flamsteed 59.33 60 or 6 1 (choose Tycho (corrected for parallax) 60.5) Vendelin 60 Copernicus 60.33 Kicher 62.5

Mean 60277 Sample standard deviation t-confidence (95%)

Mean of all values Sample standard deviation t-codidence (95%) Huygens had shown that, at the latitude of Paris, a seconds pendulum will be 3 Paris feet, 8 lines 1R inch in length (i-e., 3.06 Paris feet).

And the space which a heavy body dermis by falling in one second of time is to half the length of this pendulum in the duplicate ratio of the circumference of a circle to its diametre (as Mr. Huygens has also shown), and is therefore 15 Paris feet, 1 inch, 1 line 7/9."' Thus d circumference -=( 1/2 diametre I Fioally, d = - 2,where f=3 .O6 Paris feet. 2 d= 15 .O9 Paris feet. The thought experiment result (all values without the two body correction) of a fiee fall of the moon just above the earth's sdkeresulted in a drop of 15.138 f 0.404 Poris feet. Huygens' value of 15.09 Paris feet tslls well within the error bounds. We note, first, that Newton chooses to use the value of 60 earth-radii for the mean lunar distance. This due, as can be seen firom the table above, is near the mean values of the in the Principia and in the System ofthe World, Second, the result of the moon-test was not dependent on Newton choosing 60 as the value corresponding to the number of earth-radii for the lunar distance.13* Third, if we turn our attention to the results obtained when the correction for the sun's action on the moon is made, we notice again that Huygens' value falls well within the bounds. The positive result of the moon-test is not dependent on this correction factor. 13'

n7 Pnnajxh, page 408. '38 The value of 60 ceRainly made the computations easier. Our dysis above indicates that the due Huygens obtained clearly fell in between the axor bounds. '39 Harper's forthcoming Chapter 3 of his book deals with this issue. There he rakes Westfdl (1973) to task regarding the latter's accusation that Newton "hdged" his results to get a dose fit bemeen his moon-test and the length of Huygens' seconds pendulum- Our analysis here is consistent with Harper's point that carrykg out the moon-test, as per Newton's instructions, yields the values Newton claims and that Wesrfall's accusation is not credible. These results agree so well that, by Rules I and 11, "the force by which the moon is retained in her orbit is the very same force which we commonly call gravity."140 Here we have two phenomena that yield agreeing measurements of the same inverse-square force- gravity--toward the earth's centre. The length of a seconds pendulum and the centripetal acceleration of the lunar orbit are two phenomena which measure the same force. The agreement in measured values is another phenomenon, which relates the two phenomena in question. By Rule I if we do not claim that the same force accounts for the centripetal acceleration of the moon and the length of the seconds pendulum at Paris. then we will have to claim that there are two separate causes for these phenomena. The centripetal acceleration of the moon and the length of a seconds pendulum each measure a force resulting in accelerations at a distance of one &-radius (i-e., at the surface of the earth). The moon-test shows that these acceierations are not only equally directed toward the centre of the earth, but that they are equal m value. Now we have this higher order phenomenon of the agreement in measurements of the two phenomena The parsimony invoked by the use of the £ktrule informs us not to infer another cause for this agreement, Notice Newton's use of Rule II, to the same effect assign the same cause. That is, we note that something attracts the moon toward the earth and something attracts heavy objects to the earth By Rule II, that "something" is the same. We note that it is not just a qualitative effect that is generally the same but that it is the same as close as experiments show that they are the same (in this case, Huygens' pendulum experiment). It is Newton's ideal of empirical success which drives his reasoning to show that what we call terrestrial gravity reaches to the moon and, therefore, does not discriminate between terrestrial objects and, thus k,the moon, That is, we have agreeing measurements of the same inversesquare acceleration field £?om the length of the seconds penddum and the moon- test. Newton proceeds to say that

were gravity another force different fiom that, [fiom the centripetal accelerative force on the moon] then bodies descending to the earth with the joint impulse of both forces would fill with a double velocity, and in the space of one second of time would describe 30 1/6 Paris feet; altogether against experience. 14' This indirect argument reinforces the appeal via Rules I and II, to the unification of the measure of the centripetal acceleration of the moon with the measure of the length of a seconds pendulum. Notice that the indirect argument would not count against an alternative hypothesis, say, one which posited a force that maintained the moon in her orbit (an inverse-square force to boot) but which did not on terrestrial bodies. So Newton's appeal to Rules I and 11 help rule out just this sort of alternative. This alternative hypothesis would demand a separate account of cause for each of the two (or whatever number) basic phenomena142Not ody would this alternative hypothesis require a separate cause for each of the phenomena, it would need to accommodate or explain the agreement of the measurements of these phenomena. Furthermore, according to Rule 11 we are to assign to the same effects the same cause. Until we have evidence to the contrary, there is nothing to indicate that the phenomena are of SufEcientIy dfirent kinds to warrant us to claim that they have different causes. Placed in the context of the proof for universal gravitation, Proposition IV carries a new constraint for Newton's theory and for any alternative to the theory. The unification of these phenomena in order to identify the lunar centripetal force with terrestrial gravity forces Newton to constrain systematically the development of the

14' Pn'n(ribkq page 408. 142iis we d see later, Buffon used a similar be to argue against Claitaut's initial lunar theory which 1 k postdated a force which vazies according to + - rr4- theory. Newton transformed the notion of terrestrial gravity, heaviness, to count as varying inversely with the square of the distance fiom the earth's centre. Gravity applies not only to terrestrial objects but to the moon as wen. Atter this proposition Newton is committed to counting any phenomena which measure gravity as also measuring the centripetal force on the moon. Rules I and II impose systematic constraints on theory development such that the measures of the parameters of phenomena which are to be expIained by a theory count as accurate measurements of the theory. A rival hypothesis to the claim that the moon is heId in her orbit by the very same force, which accounts for terrestrial heavioess, would have to account for the equivalence between the centripetal force on the moon and the length of a seconds pendulum Newton raised the stakes considerably for . Before moving on we should also note that Proposition N emphasises the "empirical" aspect of Newton's ideal of empirical success. That is. in answering the theoretical question regarding tk force holding the moon in her orbit and drawing her away &om tangential motion, Newton drew our attention to two phenomena which give us agreeing measurements of the same theoretical parameter.

In the next propositioq Rule II is now used to prove that the satellites of Jupiter and of Satum have heaviness, or gravitate toward their respective centres, Jupiter and Saturn. Furthermore, Proposition V also informs us that the circumsokr planets gravitate toward the sun. Newton has already shown m his list of phenomena that the satellites of Jupiter are drawn off of rectilinear motion in their orbits about Jupiter as their centre. The same holds for the satellites of Saturn. And finally, the planets orbit the sun. Recall that Newton showed this via the harmonic and area laws. The unification of phenomena shown in Proposition N is now used again to chim that

The circUrnpvial planets gravitate towards Jupiter; the ckc~turnaltowards Saturn; the circumsolar towards the sun; and by the forces of their gravity are drawn off &om rectilinear motions, and retained in curvilenear orbits."' Since Jupiter's satellites about Jupiter and the planets about the sun behave with respect to their centres, or primaries, as the moon behaves with respect to the earth then, by Rule II, we ought to assign them the same cause, gravity. Again, though, the application of this rule is quite specific:

especially since it has been demonstrated, that the forces upon which revolutions depend tend to the centres of Jupiter, of Saturn, and of the Sun; and that these forces, in receding fkom Jupiter, fiom Saturn, and fiom the sun, decrease in the same proportion, and according to the same law, as the force of gravity does m receding ftom the earth.lu Newton cites two ways in which their behaviour is like that of our moon. First, Propositions I and II of Book III demonstrated that Jupiter is the centre of force for its satellites, Saturn is the centre of force for its satellites, and the sun is the centre of force for the planets. By Proposition III we know that the centre of force on which the moon depends is the earth. Second, recall that from Proposition I and KI of Book III Newton determined that the respective forces acting on the satellites of Jupiter and Saturn and on the planets about the sun vary as inversely to the square of the distance. This is the same as what was determined for the moon m Proposition III. By Proposition N the force on the moon was shown to be the same as the force which accounts for heaviness on the earth, gravity. We now have an equivalence of the measure of the cenmpetal force on the moon with terrestrial heaviness. By Rule IT, then, we are warranted in concluding that since gravity is the force which retains the moon in her orbit and draws the moon off of tangential motion and since the satellites ofJupiter and Saturn and all the planets are Like

'43 Pn'* page 410. 'U page 410. the moon (i-e., compare Propositions I and I1 to Proposition III and then use Proposition IV)then this force must also be gravity for all these bodies.'45 Newton gives three corollaries to Proposition V. The first corollary explicitly states that "there is, therefore, a power of gravity tending to aU planets." (p.329) But notice that Mercury, Venus, and aIl "moonless" planets do not have satellites affected by centripetal acceleration so that we can compare effects of the same kind. It is not clear how Newton is to apply Rule II here unless we grant him that "doubtless, Venus, Mercury, and the rest, are bodies of the same sort with Jupiter and Sam"(p.329)

Admittedly, we have not obsewed bodies m the vicinity of these "moonless" planets but we may believe that such bodies would be drawn to these planets, by Rule II. The counterfactual claim is in line with the sense of parsimony enshrined in Rule I such that we ought to be inclined to accept the claim that gravity tends to all these planets. There is more to Corollary 1:

And since all attraction (by Law HI) is mutual, Jupiter will therefore gravitate towards all his own satellites, Saturn towards his, the earth towards the moon, and the sun towards all the primary planets.L46

&t is, consider either Jupiter or Satum and their respective moons. Saturn, for instance, orbits the sun by gravity. The satellites of Saturn orbit Saturn by gravity. It stands to reason, then, that an object near one of Saturn's satellites would be attracted to this satellite according to gravity. As it turns out there is such an object, Satum itself Notice, though, that Newton does not appeal to the Rules in his justification but to Law m:

To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon

"SHarper (1989) discusses Newton's application of the Rules of Philosoph;-cirrPas natural kind reasoning. His interpretadon is informed by Whewell's colligation of . Eric .Ahon (1995) in "The Vortex Theory" has shown htthe vortex theorists, notably Leibniz and Huygens, did not dispute inverse-square centripetal acceleration. That is, the explicit use of RuIe I1 in the proof of Proposition V did not pose any problems for tbe vortex theorists. Their task, rather, was to give a vortex account of Newton's results. %ncipzk, page 410. each other are always equai, and directed to contrary parts. 147 Newton is treating gravity as a force of direct interaction. There is mutual attraction between a planet and its satellites. Aiton has pointed out that Huygens

accepted the Newtonian system, though with important reservations. ... Huygens declared that he had nothing against the vis cenhipeta or the gravity of the planets towards the sun, not only because it was established by experience but also because it could be explained by mechanical principles. 14'

What Huygens objected to was not the inversesquare centripetal force characterisation of gravity but to gravity being a force of direct action or muhlal interaction between a planet and its satellites.

I have nothing against Vis Cenlripeta, as Mr. Newton calk it, which causes the planet to weigh (or gravitate) toward the Sun, and the Moon toward the Earth, but here I remain in agreement without dificdty because not only do we know through experience that there is such a manner of attraction or impulse m natureybut also that is explained by the laws of motion, as we have seen in what I wrote above on gravity.'49

Parting company with Newton, Huygems notes:

I say that I agree that the gravity of bodies corresponds to the quantity of their matter, and I have even demonstrated this In the present Discourse. But I have also shown that the gravity can well be imparted to these bodies that we call heavy, by the centdigal force of a matter that does not itself weigh (or gravitate) toward the center of the Earth, because of its very rapid and circular motion, but that tends to move away fiom it.'*'

'4' Pn'mpage 13. l*BAiton (1995), p.7. 149 Huy~nsin Smitb (1997a), page 160. 1% Huygens in Smith (1997a), page 163. According to Huygens, without some such mechanical explanation to back up the account. Newton's appeal to universal gravitation would be occult. A mechanical explanation would show that the proper application of Law III would focus on the interaction between the planet's satellite and the surrounding vortical particles creating a gradient that would deflect the satellite away hmtangential motion.

In the second corollary Newton extends the force of gravity

which tends to any ow planet is reciprody as the square of the distance of place fiom that planet's centre. lS1

The inverse-square variation is shown by the series ofpropositions of Book III which have led us to this point. That is, the inverse-square variation has been demonstrated for

Jupiter and Saturn in Proposition I, for the primary planets m Proposition II, and for the moon in Proposition III. The identification of centripetal inverse-square variation for aU these bodies with gravity takes place m Proposition IV (for the moon) and V (for the rest). Picking up on the theme of mutual interaction in Corollary I, Newton concludes the third corollary to Proposition V with the following:

AU the planets do gravitate towards one another, by Cor. 1 and 2. And hence it is that Jupiter and Saturn, when near their conjunction, by their mutual attractions sensibly disturb each other's motions. So the sun disturbs the motions of the moon; arid both sun and moon disturb our sea, as we shall hereafter exphlS2

The &st part is a generalisation resulting fiorn coroUaries 1 and 2. The Newtonian ideal, as we have already stated, points to aII these systematic dependencies. That is, the action of tbe sun on the moon, and the action of the sun and moon on our oceans are dedin the theory of universal gravitatioa '53

Pn'naiac;z, page 410. 52 fi* page 410. 153See George Smith, "Planetary Perturbations: The Interaction of Jupiter and Saturn," forthcoming in I.B. Cohen's G& to tbe fine, The University of California Press- See also Curtis Wilson (1985) "The Great Newton claims in the Scholium to Proposition V, that gravity will be the term used for the centripetal force that produces and keeps celestial bodies in their orbits. It is worth repeating this schoiium here:

SCHOLIUM

The force which retains the celestial bodies in their orbits has been hitherto called centripetal force; but it being now made plain that it can be no other than a gravitating force, we shall hereafter call it gravity. For the cause of that centripetal force which retains the moon in its orbit will extend itself to all the planets, by Rule 1,2, and 4.Is4

This scholium, added in the third edition, contains the only explicl citation of Rule N m the argument for wedgravitation. Propositions I-V are propositions gathered from the phenomena by induction These propositions, as we have seen, are backed up by the best data available at that time. We know fiom Rule IV that these propositions are to be considered either exactb or very nearly true notwithstanding hypotheses to the contrary. We do this until such time as other phenomena make these propositions either more accurate or liable to exceptions. Newton adds that we do this so that hypotheses do not undercut our inferences. Rule N endorses the inference to gravity between all the planets because it delivers on the explanations of the phenomena being supported by measurements of theoretical parameters- What remains for us is to show how Newton's argument can withstand the challenge of the vortex hypothesis accompanied by the metaphysical commitment to explanations of phenomena by physical contact- To challenge Newton's theory, a rival would have to deliver on all the systematic dependencies to which we have alluded (for exaaple the moontest) and to any new ones not yet mentioned. It turns out there were some serious challenges to Newton's theory. Those that Newton foreshadowed m the hequality of Jupitu and Saturn.- From Kepler to Laplace," Tbe Hiej5r t&e of EkdS&ner, VoL 33:15-290. 1st Pn.w@%~,page 410. third coroUary to Proposition V are Listed here. Chief among these challenges is the effect of the sun on the moon in her orbit about the earth Taking Newton seriously meant following the spirit of Rule N.The natural philosophers following Newton knew perfectly well that Newton's theory accounted for a vast number of phenomena. Furthermore, these phenomena become unified by giving agreeing measurements. To undercut his theory they would need to empbasise that part of Rule IV that until such time as other phenomena make these propositions either more accurate or liable to exceptions. Turning to Proposition VI we find Newton reasoning for a direct proportion of gravitation on a body to the inertial mass of that body. Here is Newton's statement of the proposition:

That all bodies gravitate towards every planet; and that the weights of bodies towards any ow planet, at equal distances fiom the centre of the pkwt, are proportional to the quantities of matter which they severdy contain1s5 Newton offers several for this proposition and, of these, I wish to focus on the oftemcited set of pendulum experiments. In these experiments, pendulums of equal weights of different materials were found to oscillate with equal periods and thus were measured to have equal inertial masses (aU the while accounting for air resistance):

I tried experiments with go14 silver, lead, glass, sand, common salt, wood, water, and wheat. I provided two wooden boxes, round and equal: I t3.M the one with wood, and suspended an equal weight of gold (as exactly as I could) in the centre of oscillation of the other. The boxes, hanging by equal threads of 11 feet, made a couple of pendulums perfectly equal m weight and figure, and equally receiving the resistance of air. And, placing the one by the other, I observed them to play together forwards and backwards, for a long time, with equal vibrations. And therefore the quantity of matter in the gold (by Cor. I and VI, Prop. XXIV, Book 11) was to the quantity of matter in the wood as the action of the motive force (or vis motrix) upon all the gold to the action of the same upon all the wood; that is, as the weight of the one to the weight of the other: and the like happened in the other bodies.'56 Newton's appeal to Corollary 1 of Proposition XXIV is an appeal to some of the major

results of Huygens' work. For two synchronised pendulums A and B, according to this proposition"7 we would have the folIo wing relation:

If tA=tg and, ex hypothesi, w,=w,, then we know that m,~.This is the first corollary.

The equality of periods, a phenomenon, is used to establish the equivalence of gravitational mass and inertial mass. In order to achieve this Newton points out that the experiment itself(m Book III) is accurate in measurement of inertid mass to one part in a thousand. Since WA=WB then

Suppose, now, that according to Newton's cited tolerance we have

and

This difference of 0.0005 is a tolerance of one part in two Thus, in order to determine the equality of inertial masses to one part in a thousand Newton needed to show

%najcr;l, page 411. '5Troposition XXnr, Book I1 reads: "The quantities of matter in pendulous bodies, whose centres of oscillation are equally distant horn the centre of suspension, are in a ratio compounded of the ratio of the weighs and the squared ratio of the times of heoscillation in a vacuum" (p.303) that the periods do not differ by more than one part in two thousand. For illustrative purposes only. let us examine the sort of observations Newton needed to make. Assuming that the centre of oscillation is at the length of the pendulum which Newton cited to be 1 1 feet (=3.3528 metres), let the acceleration due to gravity, g. be 9.82 metres per second

Thus 1.88 sec (roughly) is the time needed to make one swing. To make a back and forth swing we multiply by 2 to get 3.77 seconds. Notice that a tolerance of one part in two thousand would mean that Newton would have had to measure for roughly 53 1 swings, or roughly just over a half-hours worth of measurements. Recall that Newton claimed that he observed the pendulums' synchronisation "for a long time." Although the cited tolerance would be ddZcult to detect in just a few swings, over many such swings it would be detectable. Newton showed sensitivity to this and thus made observations over a long time.159

There are five corollaries to Proposition VI. The kttwo wroUaries extend the argument in Proposition VI, the third corollary argues that all spaces are not equally f~l.l,'~~the fourth argues that provided atoms are identical and that any difference in large bodies can be explained by the diffierent arrangements of their atoms then the existence of a vacuum is plausible, and the fifth corollary lists the differences between gravity and , Let us focus our attention to the first two corollaries. Here is the first:

Cor. I. Hence the weights of bodies do not depend upon their forms and textures; for If the weights could be altered with the forms, they would be greater or less, according to the variety of forms, in equal matter, altogether against experience. ''I

Is9 See Harper (1 991) and (I 993). 160 Pn'n* page 414. 16' Pn'n*, page 413. Newton is claiming that an object's weight cannot be altered simply by changing its shape or texture. Thus melting a piece of wax does not change its weight. In support of this corollary is a Sody of evidence in accord with experience.

The next corollary, the second. explicitly appeals to Rule 111 and in it an inference is drawn that all bodies universally gravitate toward the earth,

Cor. LI. Universally, aJl bodies about the earth gravitate towards the earth; and the weights of all, at equal distances f?om the earth's centre, are as the quantities of matter which they severally contain. This is the quality of all bodies within the reach of our experiments; and therefore (by Rule III) to be aErmed of all bodies whatsoever.

Newton uses the third rule to argue that to all bodies universally at equal diices fiom the centre of the earth there is an equivalence of ratio of weight toward the earth to the quantity of matter (ie., inertial mass). At any given distance fkom the centre of the earth, the proportionality of weight to inertial mass is a "quality of all bodies" which does not lend itself to intensification or to remission and belongs to all bodies within the reach of experiment. If weight were to depend on form (Newton cites , Descartes, and others as holding this point) and could be dtered (increased or decreased) by transformations then Corollary I would be violated. Smce the weights of bodies do not depend on their forms and textures, by Rule lII, at equal distances fiom the earth's centre the weights of bodies are proportional to their ineaial mass.

And now we come upon Proposition W with its corollaries:

That there is a power of gravity tending to all bodies, proportional to the several quantities of matter which they con- Cor.1. Therefore the force of gravity towards any whole planet arises fiom, and is compounded of, the forces of gravity towards an its parts.

162 Pnkzjtkz, page 413. Cor.11. The force of gravity towards the several equal particles of any body is reciprocally as the square of the distance of places fiorn the particles.'63 Proposition VII asserts both that all bodies have a power of gravity in proportion to their masses and that the force of gravity toward several particles of equal mass is proportional to the square of the distance fiom the particles. In his argument, Newton invokes Law III

(to every action corresponds an equal reaction) to identify that all the planets are attracted by and attract all other planets. Wfi Proposition W Newton inferred the equivalence of gravitational mass with inertial mass. Newton extends this result such that the gravitational power of bodies is in proportion to their masses. '"

163 %jUj3i4 page 414-415. "j+or a useW discussion of Newton's reasoning to univesd pvitation see Stein (1991): 209-92. Stein argues that it is open to doubt that Newton showed, in the progression to Proposition VII, that graviy is just the son of force of interaction that warrants the use of Law 111. Stein also points that Newton's deductioas Lorn the phenomena €or universal gravitation do not end with Proposition VII but include the whole of Book 111. The point of our brief discussion here is to show the argument That Propositions I11 and IV figure prominendy 4 form the substance of htcr discussion. Chapter 4

The wandering moon, Riding near her highest noon, Like one that had been led astray Through the heav'n' s wide pathless way John Milton (1 608-1 674)

We have, thus far, explored Newton's method and his model of evidential reasoning.

Salient in Newton's mdeL according to Harper, is the technique of answering theoretical questions empirically by measurement fkom phenomem Newton's Principia can be seen as both the realisation of Kepler's own brand of heliocentric positional astronomy and, at the same time, the use of physical principles that established this heliocentric astronomy as an approximation to account for how the heavenly bodies deviate f?om idealised Keplerian orbits. Systematic deviation fiom Keplerian orbits count as agreeing measurements of the masses involved in perturbing interactions. Universal gravitation immediately implied that

Kepler's laws could not be true. As early as 1684 Newton was acutely aware of this.

Edmund Halley, in August of that year, made his fgmous visit to Newton and asked the latter whether he knew what shape of planetary orbit is implied by an inverse-square attraction We know that eventually the Principia was the reply but prior to this Newton presented a document entitled De Motx There, Newton displays some scepticism regarding a prima facie agreement between universal gravitation and Keplerian motion

First he notices that the sun, as a primary, is not the centre of gravity. This results in the planets deviating fkom Keplerian ellipses and exemplary of this is the motion of the moon:

By reason of [the] deviation of the Sun fkom the center of gravity the centripetal force does not always tend to that immobile center, and hence the planets neither move exactly in ellipses nor revolve twice in the same orbit. So that there are as many orbits of a planet as it has revolutions, as in the motion of the Moon, and the orbit of any one plawt depends on the combined motion of aIl the planets, not to mention the action of all these on each other. 16'

Newton, here, foreshadows a greater than n=2 n-body problem and is sensitive to the difficulties in resolving such problems- He goes on to say:

But to consider simuftaneousfy all these causes of motion and to define these motions by exact laws allowing of convenient calculation exceeds, unless I am mistaken, the force of any mind'66

Not long after this Newton was heavily engaged in such calculations and the result was the Principia, in which deviations fkom idealised Keplerian orbits were taken as evidence for universal gravitation. Newton's successes in his work on deviations served not only to support the law of universal gravitation but also the whole method that

Newton had employed to reach it.

The argument for universal gravitation appeals to agreeing measurements of some theoretical parameters by phenomena such as (among others) the length of a seconds pendulum and the centripetal accelerations of the moon, as exhibited by the moon-test.

Harper has claimed that the transition fiom philosophy of nature to natural science can be understood by noting the acceptance of Newton's model of evidential reasoning in the community of practising natural philosophers.

165 Version 111 of De Mofw in Curds Widson, "From Kepler's Laws, So-called to Universal Gravitarionr Empirical Factors," Arrhiir fw flu &toy .fFYllCi Jcinrm (1970), page 160. ~6 Version 111 of De Mofiv in Curtis Wilson, "From Kepler's Laws, So-called to Universal Gravitation: Empirical Factors," Mi&the kkhy of- Scimar (1 WO), page 160. Harper's claim is similar to the claims made in this century that the revolution of the new science is one which fostered a change in methodology from mechanical explanations to the hypothetico-deductive model which has been attributed to Newton.

According to Hempel (1966) a hypothesis or theory, in the hypothetico-deductive model, is supported when it, along with various other statements, deductively entails a datum.

Thus a theory is supported by its successfid predictions. The hypothetico-deductive model has a number of attractions. First, it allows for the support of hypotheses that appeal to unobservable entities and processes. A hypothesis can be supported by its observable logical consequences. Second, it appeals to our understanding of deductive principles within an inductive hework. Fdy, it seems to reflect scientific practice in that coniirmation of a hypothesis or theory comes hutby finding positive instances of the hypothesis or theory.

Harper has claimed that Newton's ideal of empirical success or model of evidential reasoning did not immediately become a part of the practise of natural philosophy, but that eventually it informed the transition fkom natural philosophy to natural science; Newton's stronger ideal of empirical success informs the transition fiom natural philosophy to natural science. To investigate Harper's claim three points need to be made: 1. The extent to which, on Newton's conception, the solution to the lunar precession problem contributes to the realisation of this ideal; 2. The extent to which solutions developed by

Europe's mathematicians, namely Euler, Clairaut, and d'Alembert, illustrate this ideal and are not merely illustrative of hypothetico-deductivism; and 3. The extent to which these mathematicians were purposely applying a methodology propouuded by Newton and characterised by the phrase "ideal of empirical success." When one traces the developments of Euler, Clairaut, and others on lunar precession and perturbation theory, these developments exhibit a betteri6' realisation of

Newton's ideal of empirical success. What I now propose to do is to explore whether

E der. the key figure in mid-eighteenth century science and mathernat ics, accepted

Newton's ideal. Eder has recently gained a wider audience in the history and not only for his immense contributions to mathematics but also for his work in mechanics. Early m the discussion surrounding the lunar problem Euler was quick to renounce universal gravitation based solely on a small deviation fiom inverse-square variation in the lunar orbit. It would appear Euler's concem with universal gravitation was somewhat motivated by his commitments to an aether theory. Euler never abandoned this commitment although he was quick to reverse his concern with universal gravitation even without reconciling it to the aether hypothesis. By bracketing these metaphysical commitments there is some prelknimqy evidence that Euler exemplifies Newton's method of doing science based on evidential reasoning and lends support for the thesis that the transition fiom natural philosophy to natural science is a revolutionary adoption of

Newton's model of evidential reasoning. Before exploring this, though, it will be usell to contrast Newton's ideal of empirical success to the hypothetico-deductive model.

In defence of the hypothet ico-deductive method, observed:

What made modem science powerfd was the invention of the hypothetico-deductive method, the method that constructs an explanation in the form of a mathematical hypothesis fiom which the observed ktsare deducible.'"

'6' This d be shown in Chapter 6. '68 Reichenbach (1968), page 100. According to Reichenbach Newton epitomises the method of modem science as hyPthetico-deductive. '69 The understanding here is that the method of modem science is the hypo thetico-deductive method. Reichenbach then claims:

Newton himself saw clearly that the truth of his law depended on co~tionthrough a verification of its imp tic at ion^."^

That is, on this account, a hypothesis %s" the red world just in case its (logical) implications match observational data The main idea is that hypotheses are confirmed by verifLing instances of them Wfih respect to Newton, then, this means that UniversaI gravitation is confirmed by verifying that deductions, such as the lunar theory, match observations of the position of the moon In Newton, claims Reichenbach, this method is made abundantly clear

but all the brilliancy of his deductive achievement did not satisfj. him He wanted quantitative observational evidence and tested the implications through observations of the moon, whose monthly revolution constituted an instance of his law of gravitation '" We have seen how Newton conceptualises his approach to natural philosophy as a result of his light and colours paper.'n The short Preface to the first edition of the Principia offers another such statement of the name of his experimental philosophy. There he states:

But I consider philosophy rather than arts and write not concerning manual but natural powers, and consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces, whether attractive or inrpulsive; and therefore I offer this work as the mathematical , for the whole burden of

t69 Reichenbach (1968), page 202. Reichenbach (1968), page 101. Reichenbach (1968), page 101. See Chapter 2. philosophy seems to consist in this-fmm the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena; [emphasis added] and to this end the general propositions in the first and second Books are directed. In the third Book I give an example of this in the explication of the System of the World; for by the propositions mathematically demonstrated in the former Books, in the third I derive &om the celestial phenomena the forces of gravity with which bodies tend to the sun and the several plawts. Then f?om these forces, by other propositions which are also mathematical, I deduce the motions of the planets, the comets, the moon, and the sea 1 wish 1 could derive the rest of the phenomena of Nature by the same kind of reasoning fiom mechanical principles, for I am induced by many reasons to suspect that they may aIl depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards one another, and cohere in regular figures, or are repelled and recede fiom one another. These forces being unknown, philosophers have hitherto attempted the search of Nature m vain; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy.'+J

Notice the far-reaching implications that Newton held to be the hallmark of his system Although he is here taking about his system of universal gravitation, the underlying tone pertains to his method of experimental philosophy. That is, appealing to the mathematical principles of the first two books Newton infers fiom (ceiestial) phenomena the force of gravity tending to the sun, the moon, the planets, comets, and the sea. Further phenomena of nature are deduced fiom these along the method described here: fiom the phenomena of motions to investigate the forces of nature and then to demonstrate other phenomena '" For instance, having established an inverse square centripetally directed force field. Newton can then explain the shape of the earth Newton's Principio exemplifies deductions fkom phenomena where a higher level theoretical claim, i-e., an inversesquare acceleration field towards the sun, is inferred from certain other high level theoretical principles (theorems fiom laws of motion) along with phenomena (e-g., harmonic law, area law, and quiescent aphelia of the planets). Newton shows how on the assumption of the laws of motion such phenomena can measure the values of theoretical parameters. In Book I Newton derived systematic dependencies that showed that the area law measures a centripetally directed deflecting force. Newton then infers fiom the phenomena (that the planets, the moon, the satellites of Jupiter, and of Saturn obey the area law about their respective primaries) that the forces drawing the planets, the moon, the Jovian satellites, and the Huygenian satellites liom rectilinear motion and holding them in their orbits are directed at their respective primaries. Given the laws of motion Newton established that periodic times are as the power n of the distance separating a body fiom its primary is equivalent to the centripetal force varying as the 1-2n power of the di~tance."~Thus fiom the phenomena corresponding to Kepler's harmonic law (ie., that the periodic times vary as the 3/2 power of the distance; see our discussion in Chapter 3 above) Newton deduces that centripetal forces vary as the 3 3 inverse square of the distance (ie., when n = - , I - 2n = I - 2(-) = -2.) Since universal 2 2 gravitation is a function of distance, mass of the p15mqy, and mass of the secondary, a secondary's harmonic law measurements measure the mass of the primary. The mass of the primary can be fixed by the orbital motion of any one secondary and, thus, having agreeing measurements of harmonic law values of various secondaries gives agreeing measurements of the mass of the primary. After Proposition VII, Book III, Newton is

Newton's vgurnent for universal gmvitation comprises the hrst seven propositions of Book III of the Pn'm Stein (1991) cIaims &at the argument for universal gravitation is not complete by this point and that the remainder of Book I11 ought to be viewed as adding to his wentfor universal gravitation. 1'5 Corollary VII, Proposition 4 Book I, Pniujtria able to use the harmonic law ratios to measure masses of the bodies involved, The harmonic values of the planets all count as agreeing measurements of the mass of the sun. Even more importantly for the project at haod Newton established that a measure of the rate of orbital precession could sensitively measure the exponent of the force law.

We will turn to this in great detail in the next chapter, suffice it to note that Newton showed that forward precession means greater than inverse square variation. The moon, as we will explore, precesses forward approximate 35 per revolution- The increment added to the squared power, in this case &, is to be attniuted primarily to the influence of the sun on the earth-moon system Thus quiescent orbits measure inversesquare variation and non-quiescent orbits can be used to measure the deviation f?om inverse- square. IS then, the deviation can be accounted for by the influence of some other body which is itself an inverse-square variation, then precession can also be used to measure inverse-square variation. A measure of precession counts as equivalently measuring the exponent in the force law. The work on deviations or perturbations would eventually provide the most important and forcell evidence for universal gravitation Newton is advocating a new method of empirical reasoni.g+ne, which turns theoretical questions into questions which can be answered empirically by measurements. On this model of empirical reasoning a theory's parameters are measured by the phenomena it purportedly explains. Explanatory success involves, then, cashing m on systematic dependencies that make a phenomenon measure the parameters of a theory. Recall tbat these phenomena are stable genetalisations Wing open-ended bodies of dad6 For instance, as we have seen in our discussion of the argument for universal gravitation, In Newton turned the theoretical question regarding the measure of the force

'76 Bogen and Woodward (1988), page 317: "Indeed, the factots involved in the production of data will ohn be so disparate and numerous, and the details of their interactions so complq that it will not be posslile to coasauct a theory that would dow us to predict their ocmnce or trace in deed how they combine to produce particdar items of data. Phenomcaa, by contrast, are not idiosyncratic to specific experimental contexts- We expect phenomena to have stable, repeatable cbatacteristics which will be detectable by means of a variety of different procedures, which may yield quite different kinds of data." See Chapter 3. holding the moon in her orbit into an empirical question of the measurement of the length of a seconds pendulum Two different phenomena are giving agreeing measurements of the same inverse-square centripetal force fie~d."~Here we have two phenomena that yield agreeing measurements of the same inverse-square force---gravity-toward the earth's centre. The length of a seconds pendulum and the centripetal acceleration of the lunar orbit are two phenomena which measure the same force. The agreement in measured values is another phenomenon, which relates the two phenomena in question Following Harper, we note that Newton's model of empirical reasoning exploits such equivalences in such a way that the stakes are raised for rival theories. That is, this model makes use of these equivalences so that any rival theory purporting to challenge universal gravitation would also have to explain the unification of the centripetal force acting at the place of the lunar orbit and the length of a seconds pendulum at the nvface of the earth.

Stein (1991) has argued that Newton does not stop at Proposition W m his argument for universal gravitation.

La the first place, it is essential to recognize that Proposition W implies a vast range of consequences not implied by the propositions antecedent to it-and, m part, contradictory of the statements of ''Phenomena" on which the initial reasoning of Book III was based. That Newton understood this cannot possibly be called in question: the entire remainder of Book 111 of the Principia is devoted to the derivation of such consequences, and to their confrontation, so fhr as it was possible at the time, with actual phenomena. .. . the remainder of Book II1 can be seen as devoted to the ''proof by phenomena" of the law of gravitation.

That is, the whole of Book III is used to deduce these other phenomena to which Newton referred in the preface to the first edition. One might want to read Newton as a

1'8 See Stein (1970a). 179Stein (1 Wl), p.220. hypothet ico-deductivist on Stein's characterisation since he might be construed as cIaiming something to the effect that universal gravitation predicts all these phenomena that in &ct

The hypothetico-deductivist maintains that explanatory success is contained in matching predictions (which are logically deduced) arising fiom a theory to the observational data arising 60m experiment. Newton's model goes beyond this. First, note that we are not referring simp& to predictions or to this or that bit of observational data but to phenomena which, we red, are generalisations fitting open ended bodies of data This clearly puts Newton m line with what Jim Woodward has recently called a "sophisticated investigator" who makes the following distinction:

Phenomena. ..are relatively stable and general features of the world which are potential objects of explanation and prediction by general theory- .. . Data, by contrast, play the role of evidence for claims about phenomena. As a rough approximation, data are what registers on a measurement or recording device in a form which is accessible to the human perceptual system, and to public inspection. '*O Second, this notion of phenomena informs Newton's ideal in that empirical success is not simply a matter of predictions matching the observational data but, rather, to have agreeing measurements, as exhibited by the moon-test, of the theoretical parameters by the phenomena. As new data are obtained the robustness grows. After Proposition W of Book III, that is after his argument for universal gravitation, Newton begins to assume that it holds in order to generate measurements of the masses of the bodies of the solar system which meet the model he outlined in the Prek to the first edition1*' Harper et. aL (1994) have shown that the harmonic law ratios for the planets give agreeing measurements of the mass of the sun. Harper (1 99 1, 1993) maintains that Newton,

180 Woodward (1989), page 393. 18' Pn'm page 415 and on, immediately after Proposition VII, Book 111, exhibits his model for empirical reasoning by exploiting this proposition (Proposition W) to measure the masses of the sun and the planets with moons by looking at the harmonic law ratios of their respective satellites or,

in the case of the sun, his planets. Wbat would a competitor theory have to do in order to be a legitimate rival to universal gravitation? This reading of Newton suggests that it would wt be sufficient to compare the consequences of the rival theory to those obtained fkom Newton's theory. Newton's model of evidential or empirical reasoning does suggest an inference to the best explanation, however it is an inference tempered by unifying a range of phenomena such that measurements of the theoretical parameters of one count as accurate measurements for another so that theoretical questions are answered by appeal to evidence for theoretical parameters. A rival theory had better be able to compete with the extent to which Universal Gravitation exhiits having agreeing measurements of theoretical parameters. I think it appropriate to always hold what Newton claimed in the General Scholium as indicative of the role he held for evidential reasoning:

]Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed fiom a cause that penetrates to the very centres of the sun and planets, without suffering the least diminution of its force; that operates not according to the quantity of the sufaces of the particles upon which it acts (as mechanical causes used to do), but according to the quantity of the solid matter which they contain, and propagates its virtue on all sides to immense distances, decreasing always as the inverse square of the disbnces. Gravitation towards the sun is made up out of the gravitations towards the several particles of which the body of the sun is composed; and in receding fiom the sun decreases accurately as the inverse square of the distances as iar as the orbit of Saturn, as evidently appears fmm the quiescence of the aphelion of the phnets; any, and even to the remotest aphelion of the comets, if those aphelions are abquiescent. [emphasis added] '"

Quiescent aphelia of the planets measure the inverse-square variation of the centripetal force drawing them to the sun We have seen how Newton unified various phenomena to infer an inverse square centripetal force.lg3 A rival theory would have to deliver on such equivalences and, notably in this scholium, show how the measurement of stable planetary orbits counts as a measure for inverse square variation

Pierre Duhem, m his The Aim and Structure of Physical Theory, criticised Newton's model of evidential reasoning fiom the point of view of hypothetico- deductivism. According to Duhem, Newton inferred the direction of the centripetal force, inverse-square variation of such a force, and that the attractive force would be proportional to mass fiom Kepler's area law, the ellipse law, and the harmonic law respectively. This interesting correspondence between the properties Newton inferred about gravitation and Kepler's prompt Duhem to go on and claim:

The experimental laws established by Kepler and transformed by geometric reasoning yield all the characteristics present in the action exerted by the sun on a planet; by induction Newton generalized the rdobtained; he dowed this result to express the law according to which any portion of matter acts on any other portion whatsoever, and he formulated this great principle: "Any two bodies whatsoever attract each other with a force which is proportional to the product of their masses and inverse ratio to the square of the distance between them" The principle of universal gravitation was found, and it was obtained, without any use having been made of any fictive hypothesis, by the inductive method the plan of which Newton outlined.'84

--- Is2- Is2- page 546- In the moon-test he idenafied the inverse square centripetal force deflecting the moon in her orbit with terrestrial heaviness or gravity. Whhern (1 959, page 191. Upon closer examination, Duhem claims, Newton's inductive method does not yield

universal gravitation fkom Kepler's iaws. That is, universal gravitation cannot be inductively inferred &om Keplerian orbits, for universal gravitation needs corrections to be made to Keplerian orbits because of the problem of perturbation. This leads Duhern to conclude:

The principle of universal gruvity, very far fiom being derivable by generalization and induction fiom the observational laws of Kepler, fonnalij contradicts these taws- If Newton S theory is correct, KepZer 's taws are necessarilyfake.

How then, asks Duhem, can Newton's theory rest on an inadequate set of evidential

Therefore, if certainty of Newton's theory does not emanate fiom the certainty of Kepler's laws, how will this theory prove its validity? It will calculate, with all the high degree of approximation that the constantly perfected methods of involve, the perturbations which at each instant remove every heave* body fiom the orbit assigned to it by Kepler's laws; then it will compare the calculated perturbations observed by means of the most precise iostruments and the most scrupulous methods.'*'

'85Duhern (1954), page 193. 'UMany philosophers of science, incIudIng Popper and Feperabend, have followed Duhern's essential criticism on this point That is, these philosophers of science deny the strength of Newton's inference to universal gravitation by pointing out that it is based on a faulty premise- Forster (1988) and Harper (1989) have shown that this reading of Newton is itself faulty. That Newton was aware of the critique of the type Duhem offers is evidenced by the following denid we find in his "System of the Wodd": "Q&rand BOY^ have, with great care @. 4-04), [this refers to Newton's discussion of Phenomenon IV of Book m] determined the distances of planers from the sun; and hence it is that their tables agree best with the heavens- And in all the planets, in Jupiter and Mars, in Saturn and the earth, as well as in Venus and hfercuy, the cubes of their distances are as the squares of their periodic times; and therefore (by Cor. VI, Prop, IV, Book I) the centripetal circurnsolar force throughout dl the planetary regions decreases as the inverse square of the distances &om the sun. In examining this proportion, we ue to use the mean distances, or the transverse semiaxes of the orbits (by Prop. XV, Book I), and to neglect hose litde ftrctions, whicb, in dehning the orbits, may have arisen Gom the insensible errors of observation, or may be ascribed to other causes which we shall afterwards explain. And thus we shall always 6m-I the said propodon to hold exactfy." [emphasis addedJ Pniw page 559. la7 Duhem (1 954), pages 1934. According to Duhem's hypothetico-deductive account, one way to account for the evidence of the Newtonian theory is to incorporate perturbation methods. That is, through advanced mathematical techniques one can calculate, based on the theory. the expected perturbation and the resulting orbital shape. The degree of precession is then a factor of how well these calcdations match precise observations. Duhem is claiming clearly that the success of the theory involves having the calculated predictions accurately

M precise observations. This is Duhem's hypothetico-deductive account of how perturbation methods validate Newton's theory.

Duhem follows this up with a definitive claim that one can not condemn an isolated hypothesis via experiment, but, rather, that evidential support runs for the entire system of theory.

The only experimental check on a physical theory which is not illogical consists in comparing the entire system of the physical theory with the whole group of experimental laws, and in judging whether the latter is represented by the former in a satisfactory mamer.lS8

This has come to be known in philosophy of science as '%holism" The very next sentence to the passage quoted above along with the remainder of a paragraph is Duhem's argument for stating that evidential support has to be holistic and how this applies to

Newton:

Such a comparison will not only bear on this or that part of the Newtonian principle, but will involve all the principles of dynamics; besides, it will call m the aid of all the propositions of optics, the of , and the theory of , which are necessary to justify the properties of telescopes in their construction, regulation, and correction, and in the elimination of the errors caused by diurnal or annual

Duhem (1 954), page 200. aberration and by atmospheric refkctiom It is no longer a matter of taking. one by one, laws justified by observation and raising each of them by induction and generalization to the rank of a principle; it is a matter of comparing the corollaries of a whole group of hypotheses to a whole group of facts. lg9

In order to demonstrate that the calculated perturbations match the observations. i.e., that what is expected accurately fit the data, appeals to various background assumptions need to be made.'90 That is, a fit between data and predictions does not isolate the hypothesis fkom which the predictions were made but must say something about all the background assumptions needed. In a method of perturbation Newton would have to appeal to the principles of dynamics (the laws of motion) and to theories concerning the telescope. The latter would include optics, statics, and thermodynamics, as these are necessary in the construction of the telescope and in understanding its use.'9' Notice that Duhem concludes this crucial passage by noting that positive instances of a theory do not confirm any part of a theory or the theory itself piecemed For this reason one has to compare a whole group of corollaries or consequences of a whole group of theories to a whole group of kts. Newton needed to appeal to these background assumptions, claims Duhem, in addition to the propositions he inferred &om idealised Keplerian orbits.

Where Newton had gone wrong, on this account, was in the "dual" nature of any law used in . Newton make use of Kepler's laws. Duhem notes that

Kepler's laws are approximations and that Newton made use of them in their symbolic

lag Duhem (1954), page 194. I9O See Harper (1991). form-what I called idealised Keplerian orbits above. That is, Newton translates Kepler's

laws into a symbolic form, which make use of dynamical concepts (e-g.. force. mass).

Now this translation is particular, for it makes use of a particular set of dynamicd laws.

The latter, in Newton's case, were only accepted after Newton made use of then The

laws of dynamics, regarding which we must accord credibility, are wdto just@ a translation while receiving their acceptance based on the use of such a translation. Let me quote the last part of Duhem's polemic because he draws an interesting point connecting appeals to background assumptions with alternative hypotheses. Here is what Duhem claims:

Thus the translation of Kepler's laws into symbolic laws, the only kind usem for a theory, presupposed the prior adherence of the physicist to a whole group of hypotheses. But, in addition, Kepler's Iaws being only approximate Iaws, dynamics permitted giving them an infinity of different symbolic translations. Among these various forms, infinite m number, there is one and ody one which agrees with Newton's principle. The observations of , so felicitously reduced to laws by Kepler, permit the theorist to choose this form, but they do not constrain him to do so, for there is an infinity of others they permit him to choose.'92

What guarantees the Newtonian tramlatio- corrections to Keplerian orbits-is the appropriate use of dynamical background assumptions and universal gravitation. But there were an infinite number of possible background assumptions that could have been

valid use of Kepler's laws and be just as well offas Newton's theory. Now although this

191 Feyetabend makes a similar point in his discussion of Galileo in Agaimt Method There, he claims Galdeo needed not only to sell the authorities on the telescope he needed to convince them of background theories about the telescope which relate, among other things, the image in the telescope with the entity outside. is in principle possible I don't think it does justice to the Newtonian model of evidential

reasoning. Ig3

We shall draw two points fkom this discussion: 1. Hypotheses are not individually

tested by experiment or observations (holism); and 2. Alternative hypotheses can undercut

a theory by adjusting the background assumptions. On both these points Duhem fails to

capture what Newton did in the Principia- Newton made it clear that there is an important

difference between reasoning fiom the truth of consequences to the truths of premises

fiom which they are drawn and reasoning l?om phenomena to their causes. Rule IV

Licenses the latter but not the former. That is, the point of Rule N, for Newton, is to

protect the results of induction fiom being undercut by hypothetical reasoning. If we take

Duhem seriously then we must notice that 1 and 2 above leave us stranded in using evidential support to discriminate background assumptions. Duhem's position, h short, violates the spirit of Rule N. We have been claiming that this rule, central as it is to

Newton's model of evidential reasoning, raises the stakes of what counts as empirical success. It would not simply sutlice for an alternative hypothesis to predict the data.

Hypothetico-deductive success merely yields this predictive power. Duhem' s holirm does

not a110 w us to empirically differentiate between alternative hypotheses. Newton moves beyond these. Consider Newton's use of Keplerian approximations in Proposition III and his extension in Proposition lV of Book EI. Recall that kom the harmonic law and area

law approximations that Newton infers that the moon is deflected fiom tangential motion

192 Duhem (1 9541, page 195. 193 Newton's model of evidential reasoning allows the theorist to discriminate between background assumptions. Under this model measurements of theoretical parameters by a phenomenon count as accurate measurements for another. Newton, for instance, exploited Proposition VII, Book EII to measure the masses of the sun and the planets with moons by looking at the hvmonic law ratios of their respective satellites (or and held in her orbit by an inverse-square centripetal force directed at the earthip4 This inversesquare centripetal force is identified as the same force, which accounts for the length of a seconds pendulum on earth.

In Pro posit ion IV Newton unifies centripetal inverse-square force with terrestrial heaviness or gravity with his reasoning fkom the moon-test. The moon-test unifies the agreement of the length of a seconds pendulum and the centripetal acceleration of the lunar orbit. For an alternative to undercut Newton's claim it would have to deliver the same sort of success as this unification. Now a believable alternative would have to deliver the appropriate corrections or translations of the data which would match

Newton's. A believable alternative would also have to show, through evidential reasoning, that Newton's unification of the centripetal force on the lunar orbit and the length of a seconds pendulum on earth are either mistaken or simply coincidentaL Rule IV, though, provides a model of evidential reasoning on which evidence does allow one to choose between rival theories. Agreeing measurements from a variety of phenomena, for instance, support Newton's arguments for inverse-square variation of the moon about the earth,

Intuitively though there is something appealing in the hypo thetico-deductive account: namely, that we turn to positive instances of hypotheses in codiming these hypotheses. Newton's model seems to appeal to this intuition and goes beyond it.

Important theoretical questions are answered by the phenomena they purportedly explain.

planets for the sun). Such equivalence meets the Duhemian cdtique in being able to discrkninate between background assumptions for a rival would have to deliver results in the same way. 'g4 Theorems I and I1 of Book I indicate &at motions satisfying Kepler's Axea Law relative to an inertial centre measure the centripetal dtection of the force deflecting the body away Gom uniform cangentid motion relative to that centre. Pnjrcjbrit, pages 4043. Harper, it seems to me, has rightIy pointed to a feature of Newton's model that captures the distinction between phenomena and data. Igs What remains then is to see if mid-way through the eighteenth century Europe's leading practitioners of science saw their achievements as hypothetico-deductive or whether they have adopted Newton's higher model ofevidential reasoning.'" More specificaIly, echoing Milton's poem, by 1752 the tide of intellectual opinion maintained that the moon is not led astray but rather, she follows an inverse square variation &Iy precisely. Was the solution to this puzzle considered as part of the model Newton put forward in the Principia as Harper has interpreted it? To this question we now turn our attention.

-

1'5 ,-Uthougb their discussion does not focus on Newton, Bogen and Woodward (1988) and Woodward (1989) offer good discussions of the distinction alluded to here and throughout the dissemtion. '96 We should note that in Stein's "On Metaphysics and Method in Newton" it isn't clear that Newton himself is not hypothetico-deductivist Glymour, TbqmrdEtrihra, has read Newton in a different manner. Glymour claims that Newton's methodology follows a pattern similar to the hotstrap strategy: 'The structure of these demonstrations, I believe, is this: using Newton's three taws, instances of theoretical clairns are deduced Gom the phenomena and from other experimental claims-, Newton's Rules of Reasoning hction as rules of detachment-that is, Newton uses them to move Gom certain kinds of instances of hypotheses to the hypotheses &emselves. These hypotheses are then used as instances of still broader daims, which are then detached, and so on. The total effect is thus a complex and sometimes even beddering mixture of deductive and inductive moves- From instances of a hypothesis Newton proceeds to the hypothesis itself and then, using the hypothesis in new cases, togaether with other laws and empirical relations, he obtains instances of a hther hypothesis, and then, inductively, the further hypothesis itself, and so on." Page 204. According to Glymour, Newton's Rules of Reasoning, as rules of detachment allow one to take specific instances of hypotheses and to generalise *&em. Chapter 5

At lust we Iearn wherefore the silver moon Once seemed to truvel with unequal steps. As ifshe scorned to suit her pace to numbers- TiZI now made clear to no astronomer; From the "Ode to Newton," Sir Edmund Halley

The solar system, as t was known at the beginning of the eighteenth century, contained 17 recognised members: the sun, six planets, ten satellites (one belonging to the earth, four to Jupiter, and 5 to Saturn). Comets were known to have visited on occasion into the region occupied by the solar system and there were reasons to believe that one of them (Halley's Comet) was a regular visitor. But by and large, although there was a great interest in the comets, their action (if any) on the members of the solar system was ignored, a neglect which subsequent investigations has justified.lP7 The number of fixed stars was known to be in the thousands'98 and their places on the celestial sphere determined. They were known to be at very great though unknown distances fkom the solar system, and their influence on it was regarded as insensible. The motions of the 17 members of the solar system were tolerably well known. Until nutation and aberration were properly understood, their actual distances fiom one another bad been roughly estimated, while the proportions between most of the distances were known with considerable ac~lrrac~.'~~Apart from the entirely anomalous ring of

- -

'97 Forbes (1909). lg8 For instmce, we know that Hevelius, tor instance, catalogued 1500 stars, this after his obs-atoq and records were burnt to the ground in 1679 md he an old man. Flamsteed, as Astronomer Royal, left a catalogue of some 2900 stars. See Forbes (1909)- '99 -+berration is the condition that causes a blurring uld loss of clearness in the images produced by lenses or mirrors. Spherical aberration is the failure of a lens or mirror of spherical section to bring parallel rays of light to a single focus; it can be prevented by using a more complex parabolic section. In the case of starlight, on the other hand, aberration is the apparent displacement of a stu's position due to the hnite velociv of Light combined with the velocity of the earth. Diurnal aberration is cawed by the earth's rotation and amounts to approximately O"3; annual aberration, caused by the earth's revolution around the sun, has a due of Saturn most of the bodies of the system were known 60m observation to be nearly

spherical in form Newton, as we know, had shown that these bodies attract one another according to the law of universal gravitation. The astronomers of the eighteenth century inherited fiom Newton the following problem:

Given these 17 bodies, and their positions and motions at any time, to deduce fiom their mutual gravitation by a process of rnalhematical calculation their posirions and motions at any other time.200

approximately 20".47; secular aberration is caused by the motion oc the solar system through space- -\bermtion was discovered in 1728 and not announced until the beginning of 1729 in the Phih~opbid Trmrrrr/liom by the English astronomer J. Bradley. In 1747, Bradley discovered nutation, which is dehned as the sfight wobbhg motion of the earth's axis, superimposed on that which produces the precession of the equinoxes. It has an 18.6-yr period,

One of astronomy's interesting ironies is rhe fircing of planetary distances into some discernible panem- Kepler, we how struggled with the Platonic solids for his pattern. Prior to Herschel's discovery of Uranus in 1781, Bode in 1772 (Bode's law) came up with a curious arbitra.y law of planetary distances. Opposite each planet's name write the figure 4. NOW successively add rhe numbers 0,3,6,12,24,48,96, ... to the 4. This results in planetary distances. Setting the Earth's distance to be 1, we get

Planet Distance from Sun (Ead=l) Computation hf ercurp 0.4 4+0=4 Venus 0.7 4+3=7 Earth I 4+6=10 Mars 1.6 4+12=16 - 4+24=28 Jupiter 52 4+48=52 Saturn 10 4+96=100 (U-us) 19.6 4+192= 196 4+384=388

This fornulation of the project is certainly Laplacian in spirit That astronomers of the day saw thek task in this way is subject to debate. I want to argue that work done in the 1740s through to the 1760s, chiefly by Eulex, is consistent with the formulation of this problem. Consider what Euler had to say in 176 1 in a letter to a young princess. This letter, dated 23* September 1760, concerned the small ineguIarities in the motions of the planets, caused by their rnutud attraction.

Were they [the primary planets] attracted only towards the sun, their motion would be suf&iently regular, and easiiy determined. But the feebler powers of which I have been speaking occasion some slight irregularities in their motion, which astronomers are eager to discover, and which geometricians endeavour to determine on the principles of motion. An important question is here agitated- namely, The powers which act upon a body being known, how to find the motion of that body? Now, upon the principles above laid down, we are acquainted with the powers, to the influence of which every pianet is subjected.z0' Such a calculation would necessarily involve, among other quantities, the masses of the several bodies. Calculations carried out successfidly would also yield the masses of the various bodies involved. In the same way the commonly accepted estimates of the dimensions of the solar system and of the shapes of its members would be fiuther parameters that would be considered in the calculations. Newton adequately solved the problem of the motion of two mutually gravitating spheres. Each body of the solar system could be regarded as moving nearly in an ellipse round some one body, but as slightly disturbed by the action of other bodies. In this sense the problem above (identified as Newton's problem) can be seen to be a special case of the so-called problem of n- bodies:

Given at any time the positions and motions of n- mutually gravitating bodies, to determine their positions and motions at any other time.

201 Letter WU, dated 23d September, 1760, Volume 1, p.209. In the case of the solar system the problem is simplified not only by the consideration that one of the bodies can always be regarded as exercising only a small iduence on the relative motion of the others, but also by the fact that the eccentricities and inclinations of the orbits of the planets and satellites are small quantities. In the case of the system of the sun, earth, and moon the characteristic feature is the great distance of the sun, which (in this case) is the disturbing body, &om the other two bodies. In the case of the sun and two planets, the enormous mass of the sun as compared with the disturbing planet is the important factor. Both problems filU under the problem of three bodies but the cases differ. Consequently two distinct branches of the subject evolved: lunar theory and planetary theory. In the Principiu the problem of two attracting bodies with an inverse square law of force is adequately solved (Propositions I-XW and LW-LXIV of Book I). There,

Newton argued that an inverse 4uafe law is implied from circular, elliptical, parabolic, or hyperbolic orbits - the conic sections. Furthennore, all three of Kepler's laws are in fdl accord with Newton's theory-in fact they are derivable &om the theory.202 In

Propositions LXV and LXVI Newton Iooks at the problem of three bodies but was not satisfied with his solution (and as we have already noted this is the problem over which Euler et d puzzled). It is important to note that Newton had solved the theoretical problem of the motion of two point xnasses under an inverse square law of attraction For more than two point masses only approximations to the motion of the bodies could be found and this line of research led to a large effort by mathematicians to develop methods to attack this three body However, the problem of the act4 motion of the planets and moons in the solar system was highly complicated by other considerations.

'O2 See Forster's (1988) critique of Duhem's probIem. Forster dso exploits this position to argue against the "anti-realist polemics of Cammight" @age 86)- m3~f.Curtis Wilson (1995)- Even if the earth-moon system was considered as a two body problem theoretically solved in the Principio, the orbits would not be simple ellipses. Neither the Earth nor the moon is a perfect sphere and so they do not behave Like point masses. This was to lead to the development of mechanics of rigid bodies.

Newton and the Motion of the Lunar Apse: The Setting of the Problem After listing the Rules of Philosophising and the Phenome~li~,Book Ill of the Principia opens with three similar propositions (as we have seen). These propositions state that the satellites of Jupiter and Saturn, the primary planets, and the moon are attracted toward their respective central bodies by a force which acted mverseiy as the squares of the distances of the places of these various celestial objects fiom the centres of the latter bodies. To establish this inverse-square ratio Newton cited the close approximation of the circumjovial planets around the sun to the harmonic law. He also cites Cor.VI of Proposition TV of Book I, which relates area and harmonic law correspondences to inverse-square variation of a centripetal force. He says:

Cor.VI. If the periodic times are as the 3Rth powers of the radii, and therefore the velocities inversely as the square roots of the radii, the centripetal forces will be inversely as the squares of the radii; and

More importantly, in the next corollary to same proposition Newton relates a relationship of harmonic law ratios other than, but including 3/2, to the power in the force law. He claims:

Cor. W.And univedy, if the periodic time is as any power Rn of the radius R, and therefore the velocity inversely as the power R*' of the radius, the centripetal force wiU be inversely as the power R~*' of the radius; and conversely.z05

*m Pn'mij%q page 46. 205 Ph* page 46. See our discussion of this point in Chapters 3 and 4. That is, for any relationship between the periodic time and the radius. we have the

following: T = an-,where A is a constant. Having established some bamonic law ratio for orbiting bodies we can measure the exponent in the force law governing the centripetal force. These corollaries were stated for a proposition that states: PROPOSITION W.THEOREM IV.

The centripetal forces of bodies, which by equable motions describe dzrerent circles, tend to the centres of the some circles; d are to each other as the squmes of the urcs described in equal times divided respectively by the radii of the

Proposition N, and these corollaries, were written expressly for concentric circular orbits but an extension to con-focal elliptical orbits can be done in a straightforward wayW2O7

Implicit in Newton's citation of Cor.VI and Cor. W, then, are circular orbits concerning which observations of the primary planets did not

Newton also appeals to the precession Theorems and added the following to

Proposition II, Book III:

But this part of the Proposition is with great accuracy, demonstrable fiom the quiescence of the aphelion points; for a very small aberration fiom the reciprocal duplicate proportion would (by Cor. 1, Prop. XLV, Book I) produce a motion of the apsides sensible enough in every single revolution, and in many of them enormously great?

Zo6 Pni.laplr;l, page 45. 20' Harper (forthcoming) "Newton's Argument for Universal Gravitation," in Cohen and Smith, The Cmnbridgc h Ntrvton. M8 Although he does not dte Cor.VIII of this proposition we should point out that Newton had extended Proposition N of Book I to non-ck& orbits. Here is his formulation of Cot. VIII @age 46): The same things hold concerning the times, the velocities, and the forces by which bodies describe the similar parts of any sknilar hgures that have their centres in a sknilar position with those figures; as appears by applying the demonstration of the preceding cases to those. hdthe application is easy, by only substituting the equable description of areas in the place of equable motion, and using the distances of the bodies from the centres instead of the radii-" Pn'nci3r;3, page 406. Newton used this argument in Proposition III, Book 111 to show that an inverse-square centripetal force deflects the moon from following a tangential motion and holds the moon in her orbit. This is evident by a small but measurable forward precession of the moon's apogee amounting to just over 3O forward. We recall that in our discussion of Proposition

[II, Book 111 we noted that such a motion of the Line of apsides was, according to Newto& attniutable to the effect of the sun on the earth-moon system Newton claimed that a non-quiescent line of apsides indicated a deviation fkom inverse-square variation ad, in the case of the moon, the deviation was entirely due to the action of the sun. It remains, then to show how to account for this action,

In Section IX, Book I of the Principia, entitled "The Motion of Bodies in Movable

Orbits; and the Motion of Apsides," Newton eventually demonstrated that if the force exerted by a central body acted in a proportion other than inversesquare, the orbit would in effect rotate.''* In the case of the ellipse this amounted to ciaiming that the apsides would move. (see Newton's drawing below)

210 ''The orbit would in effect rotate" simply means that the line of apsides, which corresponds to the major axis of an elliptical orbi~would rotate. The size and shape of an orbit are specified by (1) the semimajor axis (the mean distance of the smaller body horn the larger body) and (2) the eccentricity (the distance of the larger body Gom the center of the orbit divided by the length of the orbit's semimajor axis). The position of the orbit in space is determined by three factors: (3) the inclination, or dt, of the orbital plane to the reference plane (the ediptic for sun-orbiting bodies; a planet's equator for natural and ardfid satellites); (4) the longitude of the ascending node (measured from the vernal equinox to the point where the smaller body cuts the reference plane moving south to north); and (5) the wentof pericenter (measured &om the ascending node in the direction of motion to the point at which the two bodies are closest). In the case of the Earth- Moon system this is the pen&. These five quantities, plus the time of +center passage, are called orbitat elements. The gravitational attractions of bodies other than the herbody cause perturbations in the smaller body's motions that can make he orbit shift, or precess, in space or cause he sderbodp to wobble slighdy. In Proposition XLV Newton limits himself Yo find the motion of the apsides in orbits approaching very near to circle^."^" That is, Newton is here Limiting himself to an investigation of the motion of apsides m orbits of very small eccentricities. To this end the reader is offered three examples to show how the apsides' motion could be calculated when the centripetal force is assumed to be one of the following:

1, to be constant, or as -- ' A" 2. to be as any power of the altitude, e.g., A"-' or -y and (bAm+ cAn) 3. to be as the (multiple) sum of any m,n powers of the altitude, or as 7 A' where b,c are constants.

What is of importance for our purposes here are the two corollaries Newton developed based on Examples 2 and 3. These corollaries relate centripetal force to the motion of the apsides which they produce. Here is what Newton had to say: Cor. I. Hence if the centripetal force be as any power of the altitude, that power may be found fiom the motion of the apsides; and so contrariwise. That is, if the whole angular motion with which the body returns to the same , be to the angular motion of one revolution, or 360"- as any number as m to another as n, and the altitude calIed A; the force will ?!L3 be as the power Am of the altitude A; the index nn of which power is -- 3. This appears by the mrn second example. ...212

Now, under the assumption that the motion of the apsides of any orbit of small eccentricity arises in the fahion alluded to in Proposition XLV, the first corollary indicates a method for determining the extent of deviation f?om the inverse-square proportion of the centripetal force by the given motion of an orbiting body! Newton illustrates this in concluding Corollary I to Proposition XLV:

Lastly if the body in its progress from the upper apsis to the same upper apsis again, goes over one entire revolution and 3" more, and therefore the apsis in each revolution of the body moves 3" in comequentia; then rn wiIl be to n as 363O to 360° or lm 3 as 121 to 120, and therefore A~-will be equal to --29523 A and therefore the centripetal force will be 29523- 2- 4 reciprocally as A lW1or reciprocally as A 243 very nearly. Therefore the centripetal force decreases in a ratio something greater than the squared ratio 3 [emphasis added]; but approaching 59- times 4 nearer to the squared than the cubed.213 This corollary, then, establishes a correspondence between precession of the line of apsides for a body revolving under the influence of a centripetally directed force and the power of the distance over which the force is varying. Stable elliptical orbits (ie., where there is no precession) imply an inverse-square variation- Forward precession means that the variation of the force is greater than inverse-square and backward precession measures a force varying less than inverse-square. The virtually immobile aphelia of the planets implied, according to the formula of Corollary I of Proposition XLV, a nearly perfect inverse-square centripetal force acting between them respective@with the sun. The apses of the moon, whose motion amounts to roughly 3O per revolution, is obviously more sizeable. The second corollary dowed the measurement of the effect on the motion of the apsides produced by a foreign force.

Cor. II. Hence also if a body, urged by a centripetal force which is reciprocally as the square of the attitude, revolves in an ellipsis whose focus is in the centre of the forces; and a new and foreign force should be added to or subtracted fiom this centripetal force, the motion of the apsides arising f?om that foreign force may (by the third Example) be known; and so on the contrary. As if the force with which the body revolves in the ellipsis be as -- 1 and the foreign force subducted as cA, and AA' A - CA~ therefore the remaining force as ; then (by tbe third Example) A will be equal to 1, m equal to 1, and n equal to 4; and therefore the angle of revolution between the apsides is equal to

W1th respect to this Newton goes on to say:

Suppose that foreign force to be 357.45 parts less than the other force with which the body revolves m 100 the ellipsis; that is, c to be -= A or T being 35745 ' equal to 1; and then 180' -d be

lSOOJE or 180.7623, that is 180°45T44TT. 35345 Therefore the body, parting fkom the upper apsis, will amve at the lower apsis with an angular motion of 180a45'44", and this angular motion being repeated, will return to the upper apsis; and therefore the upper apsis in each revoMon will go forward 1°31'28". The apsis of the moon is about twice as swift. [emphasis added1215

Corollary II of Proposition XLV related the apsidal motion to a mixture of inverse- square centripetal motion and a radial component of a perturbative force (foreign force).

As we said, Newton was able only to account for roughly halfthe observed motion. But later in Proposition III of Book KII Newton addresses the perturbative effect of the Sun on the moon's motion by saying:

The action of the sun, attracting the moon from the earth, is nearly as the moon's distance from the earth; and therefore (by what we have shewed in Cor.2, Prop. XLV, Book I) is to the centripetal force of the moon as 2 to 357.45, or nearly so; that is, as I to 17gr. And if we neglect so 40 inconsiderabie a force of the sun, the remaining force, by which the moon is retained m its orb, will be reciprocally as D~.This will yet more fblly appear &om comparing this force with the force of gravity, as is done in the next ~ro~osition~'~

Newton then adds a corollary to Proposition IlI that will be used later (in Cor.VII,

Proposition XXXW, Book III) in his calculation of the moon's centripetal acceleration

Cor. If we a-nt the mean centripetal force by which the moon is retained m its orb, first in the proportion of 177% to 178%, and then in the

215 Pn'm page 147. 2'6 ~~ page 407, This was added in the second edition. duplicate proportion of the semi-diameter of the earth to the mean distance of the centres of the moon and earth, we shall have the centripetal force of the moon at the surface of the earth; supposing this force, in descending to the earth's surfgce, continually to increase in the reciprocal duplicate proportion of the height."'

It would appear that Newton wished to have his readers believe that the ratio of the action of the sun (i-e.,the mean 'Yoreign" force) to the centripetal force on the moon would be 1 to 178s. kc-the C0r.n of Proposition XLV,Book I formukt for calcdathg the angle of revolution between apsides is equal to 180' . - Doubling this will yield the of the apsides in a complete rev0 lutioa Letting c= we get the mean angular motion to be 363O4'47". The moon's apsis, in one revolution, moves forward just over

3*. So the substitution of this value of c yields a Cor.ll value approximately equal to the observed mean motion of the lunar apogee per revolution Newton nowhere gives any direct explanation as to why he chose to use the ratio of 1 to 357.45 in illustrating Cor.11 or Proposition XLV, Book I. Nor does he suggest that the motion of the moon's apogee might have a cause more complex than that descni in the corollary and in the proof of

Proposition IK2'*

The explanation is to be found in Propositions XXV and XXVI of Book m. In

Proposition XXV Newton undertook to "&Id the forces with which the Sun disturbs the motion of the In Newton's diagram (below), S is the sun, T is the earth, P is the moon, and the moon's orbit is designated by CADB. By extending the line SP, the

--

217 Pn'nw page 407, Wdson (1995), page 2. Whon points out that Newton did have a lunar theory embodies in a manuscript dating Gom 1686 and Grst published by Whiteside in 1974. SL SK' points K and L were determined by making SK=ST and -= -. The point M was SK SP~ defined by the intersection of the extension of the line ST and a line drawn &om L

D parallel to PT. If ST or SK is to represent the accelerated force of gravity of the earth toward the sun, then the accelerative force of gravity of the moon toward the sun will be represented by SL which Newton resolves into two components LM (which is parallel to

PT) and SM. The latter component, SM, acts in a direction parallel to ST and so the actual disturbing force m this direction is TM. Using the result of Corollary XW,

Proposition LXVI, Book I Newton pointed out that the ratio of the mean value of LM to the centripetal force by which the moon is retained in its orbit revolving about the earth at rest at the distance PT eom the earth's centre is equal to the square of the ratio of the periodic time of the moon about the earth to the periodic time of the earth about the sun.

about the Earth

The lan fraction was reduced to the mixed Itaction of 1 to 178$ which is precisely double the ratio of the 'Yoreign" force to the centripetal force used to illustrate Corollary LI. Proposition XLV, Book I. We noted above that the ratio of 178% yields a forward motion of the lunar apse just over 3O per revolution.

We note, though, that in order for Newton to apply property the formula of

Corollary I1 to the case of the moon he needed to choose a value for the "foreign" force, which represented the entire radial component of the sun's disturbing force. That is, the sun's disturbing force had to be broken up into two orthogonal components: one directed along the earth-moon radius vector and the other perpendicular to this.u0 In Proposition

XXVI though, Newton resolved the forces into non-orthogod components. One of the components always acted along the earthmoon radius vector whereas the other component acted in the direction parallel to the straight line connecting the sun and the earth

In Proposition MNI of Book III Newton resolves the solar perturbing force into orthogonal components. The radial component varies depending on where the moon is in her orbit. That is, the quantity and direction of this radial component wiU vary whether the moon is in the syzygies or in the quadratures. I will repeat the diagram here for clarity.

"0 On the assumption that the moon's orbit is nearly circular, we note that the component perpendicular to the earth-moon radius vector would always act in a direction which was approximately tangential to the orbit at the lunar orbital position. SL SK' Redthat the points K and L were determined by making SK=ST and -= - BY SK SP* - cro ss-mult iplicat ion this last equation is equivalent to SL(SP~)= SK~.From the dkqpn~ above it is plain that SL = SP + PL and that SK = SP + PK. Substituting these we have

(SP + PL)(SP') = (SP + PK)' which then becomes:

Sp3+ PL(SP*) = SP~+ 3(Sp2)S?IC + ~SP(PK')+ PK~.

So that

Notice, though, that the ratio of PK to SP is very small and so the last two terms of the expression on the right are negligible. Thus, PL = 3PK.

At quadratures the radial component, LM, will be according to Propostion XXV,

1000 of the centripetal force. Further, this force acts in the same direction as the 178725 centripetal force, toward the earth. At syzygies, though, the radial component will be given by the value of KL. Notice that KL = PL -PKand since

PL = 3PK (approximately) then KL = 2PK. At conjunction and opposition PK = PT.

This means that the radial component, expressed by KL, will be 2PT. However 2000 PT = 'Oo0 and. thus, the radial component will be of the centripetal force and 178725 178725 will be directed away &om the earth.

In Proposition XXVI Newton assumes the lunar orbit is nearly circular. The radial component thus varies sinusoi~.*' The mean values of the ratio of the radial component of the centripetal force will be half way between the two extremes of the quadr;ttures and the syzygies. Thus,

That is, the average radial or foreign force will be directed away fiom the earth and will be

of the centripetal force. The reader will notice that this is precisely the value 3 57.45

Newton uses in Corollary 11 of Proposition XLV, Book I. This value, though, accounts for halfof the motion of the lunar apsis.

Newton, in Propositions III and IV of Book ID relies heavily on the result obtained from Corollaries I and 11 of Prop. XLV of Book I and Rules of Reasoning I and II. These two propositions are key in Newton's argument for universal gravitation.

If there was a weak spot m Newton's theory, the continental philosophers of nature reasoned that it would be in Newton's lunar theory. It is for this reason that we find Euler, Clairaut, and d'Alembert so heavily involved in its correction. In Proposition

LII Newton argued that the moon is maintained in its orbit by a centripetal inverse-square force directed toward the earth

Discussion with WhHarper, December 1998. That the force by which the moon is retained in its orbit tends to the earth; and is inversely as the square of the distance of its place &om the earth's centre.222 In order to show this Newton has to prove two things:

1. That at the centre of the centripetal force urging the moon into an orbit is the earth.

2. That this centripetal force is as the -2 power of the distance of the moon fiom the earth's centre.

The first part is proven by Phenomenon 6-that the moon dermis areas proportional to the times of description around the handPropositions II and III of Book I? Newton moves on to claim the

latter Lime.,2 above] &om the very slow motion of the moon's apogee; which in every single revolution amounting to 3'3' fowards, may be neglected. For (by Cor. I, Prop. XLV, Book I) it appears, that, if the distance of the moon fiom the earth's centre is to the semidiameter of the earth as D to I, the force, from which such a motion will result, is reciprocaliy 4 2- as D 243, ie., inversely as the power of D, whose 4 exponent is 2 -; that is to say, in the proportion 243 of the distance something greater than reciprocally duplicate, but which comes 59 X times nearer to the square than to the cubeu4 The 3O3' fonvard precession is accounted for, via Cor.1 Prop XLV Book I, as measuring a nL centripetal force of -2.0167. Recall in that Corollary we are instructed to use -- 3 in ?n2

m Pn'mpage 406. I will not recall the proof here as it is not substantive for our aim. Since we wish to look forward to Euler, Clairaut, and d'-Alembert's discussion of the lunar theo y then it is Newton's use of Prop. LXLV of Book I that is germane. -2- 4 22d page 406. How much closer is r 243 to inverse square than it is to inverse cube? This index is 4 239 239 4 239 -away from inverse square and -Gom inverse cuk. 59 is equal to the ratio -:- - -. 243 243 243 243 4 calculating the index of the altitude. Since we have a forward precession m=363 "3' and n=360°. Therefore

which is the same result Newton's calculations yield. We recall that in Proposition XLV Newton limits himselfb'tofind the motion of the apsides in orbits approaching very war circles. "US Some of the planetary orbits have appreciable eccentricity. Mercury, for instance, has an of 0.2056. Mercury, one of the primary planets, is obviously included in Proposition II, Book Ill and where he cites Proposition XLV. Vdwi et. aL(1997) show how Newton is able to trust his result fiom Proposition XLV even where the orbital eccentricity is not close to zero, i.e., even where the orbits are not approaching very near In order to show the acceptability of Proposition IlI we have claimed that Newton needed to show that this centripetal force is as the -2 power of the distance of the moon from the earth's centre. In the case of the moon there is a small departure fiom the inverse square variation. Newton fbrther claims in Proposition m:

Ronald Laymon (Laymon, 1983) has chimed that "Newton's descn'puons of the phenomena were typically incompatible with he then accepted observational dam" A stable elliptical orbit, i.e., an elliptical orbit without precession, would correspond exactly to a centxipetal force varying as the inverse square of the distance. Since our moon precesses fornard it would seem btthis example is compatible with Lapon's conjecture. Harper (1989) has conclusivelp shown that Laymon's conjecture is wrong with respect to what Newton explicitly cited as phenomena For example, consider the evidence for the Harmonic Law for Jupiter's moons @e., R3/l?=K) in Phenomenon I (see Chapter 3). In sho~what Newtoa cited as Phenomenon I falls within what we would normally accept as reasonable limits of error. The largest error in the table above is less than a half standard deviation Gom the mean, Likewise, the same can be said for the evidence for the Harmonic Law ratio for the five primary planets as stated in Phenomenon IV (see chapter 3 above). Vallud et al (1997), pges 23-27. The authors point out that "Newton himselfl viewing the systematic dependencies between force Iaw and orbid shape that he had discovered, conduded that the relation betateen force law and orbital figure was one of mutual implication." They go on to say that Newton had found that for orbits differing Gom circles, very special force laws give rise to orbital closure and &at "it must have seemed unlikely that such closure could arise under other laws for isolated values of the eccenmciy-a violation of the systematic order he had discovered." @age 25) But in regard that this motion is owing to the action of the sun (as we shall afternards show), it is here to be neglected. The action of the sun, attracting the moon fkom the earth. is nearly as the moon's distance from the earth; and therefore (by what we have shewed in Cor. 2. Prop. XLV, Book I) is to the centripetal force of the moon as 2 to 357,45, or 29 nearly so; that is, as 1 to 178 -. And if we neglect 40 so inconsiderable a force of the sun, the remaining force, by which the moon is retained in its orb, will be reciprocdy as d."'

At this point we have no reason to think that the sun would have any effect on the moon.

After all this is what universal gravitation would imply and we have not proven that yet.

Nonetheless, we can take Newton to be claiming that the motion of the lunar apsides is due to the influence of the sun and this perturbathe effect ought to be subtracted in order to isolate the centripetal force drawing the moon toward the earth. In Cor. II of Prop.

XLV of Book I, Newton gave the formula for calculating precession &om the centripetal component of a foreign force as a &tion of the main central force. We recall that in this corollary (quoted above) Newton came up with the figure of forward precession of lo

31'28". Doubling this value we get 3O2'56" which is 4 minutes shy of the value Newton cites as the lunar precession in Proposition m, Book El.

L'7 Pn'n- page 407. Chapter 6

Euler, Lunar Theory, and the Vindication of Universal Gravitation

As we claimed earlier of Newton's analysis in the Principia (Prop. XLV, Book I, Props.,

111, IV of Book 111) Newton was able to come up with only half the moon's apsidal motion. ARer the publication of the Principia (all editions) there was some debate as to what Newton had really accomplished. It is with respect to this problem that we find, in the 1740~~three of Europe's foremost mathematicians engaged.

We notice, first, that it took fifty or so years after the publication of the Principia to achieve serious progress in analytical . As early as 1714 the British

Parliament offered to award a handsome prize for a procedure that would give accurate longitudes at sea228 A prize of £20,000, a king's ransom, would be awarded for a procedure that would give longitude to within so,and f 10,000 to within 1d29. Even with such a lucrative incentive, no lwtheory of this was developed within the first half of the eighteenth century. Curtis Wilson has pointed out, rightly, that the lack of lunar theory with the requisite accuracy was due more to ignorance of how the mathematics were to be constructed than to anything elsez3' In fact, the various papers

We know that lunar theory was especially inviting because it offered a possible method for tinding longitude at sea Mariners had to depend on dead reckoning to Gnd thek longitude. The lirst reliable and accurate sea-going chronometers were made by Hdsoa and Berthoud in the 1760s and undl that time it seemed that a mble of the moon's position wouId be the best way to determine longitude. The Paris Academy offered pearly prizes on this and related topics. These alternated pearly between celesdal mechanics and the theory of navigation. Hankins (1970), page 29: "Lunar Theory was also inviting because it offered a possible method for hnding Iongitude at sez Ever since the Age of ExpIoration, mariners had had to depend on dead reckoning to hdtheir longitude. The first accurate sea-going chronometers were made by Harrison and Berthoud during the 1760s and until that time it seemed that a table of the moon's position would be the best way to determine longitude-" ='Hankins (1970), page 30. Sobel and -4ndrewes (1998), page 66- Win(1995), The Problem of Perturbation Analytically Tmted: Euler, Claitau~d'L~embeqn page 89. See dso Bos (1 974) and Youschkevitch (1971). presented to the Paris Academy for the 1748 prize for a theory of Saturn and Jupiter aptly show the and time spent just figuring out how theories of perturbation+ like the

Sun-Jupiter-Saturn system or the Earth-Moon-Sun system, were to be carried out and c~nstructed.~~'

In the 1740s, Euler, Clairaut, and dqAlernbertundertook new analyticd derivations of the Moon's motions f?om the inversesquare law. Initially, all three found a similar anomalous result: like Newton in Proposition XLV of Book I they were able only to recover halfthe observed motion Much of the ensuing debate centred on Clairaut but an account of the debate and the other actors as well will prove to be illuminating.

Euler developed methods of integrating linear differential equations in 1739, fifty or so years after the publication of the first edition of the Principia. Euier drew up lunar tables in 1744 and reflected his study of the gravitational attractions in the earth, moon, and sun system Clairaut and d' Alembert were also studying perturbations of the moon and, in 1747, CIairaut proposed adding a 1/8 term to the gravitational kw to explain the observed motion of the Moon's perihelion. However, by the end of 1748 Clairaut had discovered that a more accurate application of the inverse square law came close to explaining the precession. He published this version in 1752, and two years later, d'Alembert published his calculations going to more tern in his approximation than

Clairaut. This work of the mathematicians of the eighteenth century is of importance for its role in getting Newton's inverse square law of force accepted in Continental Europe.

In a paper entitled "Recherches sur le mouvement des corps celestes en genM

(Research on the Movement of CeIestial Bodies in General) presented to the Berlin

='See Curtis Wdson "The Problem ofPcrturtnaon ,-Inalytically Treated," in Wdson (1995). Academy of Sciences on 8 June, 1747, Euler called into question the accuracy of the

inverse-square law.uz For our present purposes we need to understand what Euler in fact

says about Lunar Theory. Wah respect to the mutual forces among planets Euler expressed some doubts that they followed the inverse-square law exactly.233 His doubts were strengthened by empirical evidence. There were the anomalies in the motions of

Jupiter and Saturn and, more to our point, there was a carell study started in 1744 in which he found a number of disagreements between observation and the results derived eom the inverse-square law. Euler pointed out that up to now other commentators claimed that a perfect agreement existed between Newton's theory and the

Aside fiom the lunar inequalities, Newton's theory accorded fairy well with observations. As observational accuracy increased, perturbations in orbital paths were made easier to observe. For Euler, and anyone sensitive to Newton's confusing comments regarding lunar theory, the motion of the lunar apse was reason enough to call into question universal gravitation and the inverse-square ratio. (Recall that this agreement between observation and Newton's theory was immensely important in encouraging acceptance of the inverse-square law.) The empirid evidence upon which

a2Euler had been in correspondence with Clairaut since 1740 regarding the theoretical determination of the lunar nodes. A summary of the paper he read to the Berlin Academy in October 1744 was entided "Sur le rnouvernent des noeuds de la Lune, et sur la variation de son incliuaison i 1'EdiptiqueW("On the motion of the lunar nodes and the variation of the lunar incliaation). EuIer huther carried out an investigation of the moon's motion in May 1745 as evidenced by his correspondence with a former colleague at the St Perersburg Academy, Jeau-Nicolas Delisle. 233 Euler (1747, "Recherches sur le mouvemen~"pangtaphs 8-10, pp. 4-5. For instance Eder claims in paragraph 8: TOGdonc une raisson assez forte, pourquoi il sera permis de croire que la force, dont les Planetes sont poussies vers le Soleil, ne se rkgle pas parf'aitement sur la &on renversie des quarris des distances: et cette loi sera encore moins certaine, quad il s'agit des forces, dont les Planetes s'atirent musuellernent" ("Here then is a strong reason to maintain &at the force directing the planets towards the sun is not in perfect agreement with the inverse-square lax and this Iaw becomes even less certain when one considers the mutual attraction of the planets,") UI Euler (1747, pp. 4-5. Euler carefully studied and consequently doubted the accuracy of the inverse-square law included observations of the lunar nodes. In paragraph 1 1 of this essay he claims:

Because at first having supposed. that the forces as much f?om the Earth as fiom the Sun, which act on the Moon, are perfectly proportional reciprocally to the squares of the distances, I have always found the motion of the apogee almost two times slower than the observations dictate; and although several small tenns, which I have been obliged to neglect in the calculation, may be able to accelerate the motion of the apogee, I have ascertained after several , that they would be unable by fkr to make up for this lack, and that it is absolutely necessary, that the forces by which the Moon is at present time solicited, are a little different fiom the ones, which I have supposed; because the least ditference in the soliciting forces produces a very considerable one in the motion of the apogee. I have noticed, as well, a small difference in the motion of the line of nodes fiom that expected fiom calculations, which undoubtedly arises form the same source.235

In the next paragraph, 12, Euler describes another unsuccessful test of the theory:

Knowing the weight at the surfke of the earth, I have concluded fiom it the absolute force, which ought to act on the Moon, by supposing that it decreased in doubled ratio of the distances. From this force compared to the periodic time of the Moon, I have deduced the mean distance of the Moon to the Earth, and then its horizontal parallax to this same distance. But this parallax has been

'3j "Car d'abord ayant supposi, que les forces taut de la Terre que du Sole& qui agissent sur la Lune, sont parfairemmt pmportiondes rikiproquement aux quamis des distances, j'ai trouvti toujours Ie mouvement de Pawepresque deux fois plus len~que les observations t rnarquent; et quoique plusieurs petits termes, que j'ai it6 obede negliger dans le dc4puissent acc6lk le mouvement de l'apogCe, j'ai pourtant bien vu aprik plusieurs rechercha, qu'ils ne sauroient de beaucoup pds suppICer i ce deffauq et qu'il faur absolurnent, que les forces, dont la Lune est actudement sollidtie, soient un peu diffkentes de celles, que j'avois supposies; car la moindre difErence dans les forces solliatantes en produit une tres considerable dans le mouvement de I'apoge. J'ai remarqut aussi une petite diffirence enae le rnouvement de la ligne des nocuds, que le cdcul donne, et celui que les observations oat don& i comoia, qui vient sans doute de la mime source." Euler, 6Xecherchessur Ie mouvtment dcs Corps cdestes en ~n~"page 5. The original is in Frtnch and translated here by the author. found a Little too smail, with the result that the Moon is less &ithrn us, than according to the theory, and leaving the force, by which the Moon is impelled toward the Earth, smaller, than I have supposed.236

Euler goes on to describe that his attempt involved separating the component of the total force acting on the moon corresponding to the Earth fiom that corresponding to the Sua The force of the Earth on the Moon ought, then, to result in an action of the Moon in perfect agreement with Kepler's laws (paragraph 13). But his attempt couldn't be accomplished according to cormwdy accepted rules. He, therefore, concludes:

All these reasons joined together appear therefore to prove invincibly, that the centripetal forces which one conceives to apply in the Heavens, do not follow exactly the law established by ~ewton~" Added to this are his researches on the perturbation of Saturn's motion by Jupiter. Euler surmised that there would likely be found fkther discrepancies between the calculated and the observed motions of the other planets of the solar system. We present a discussion of these more general problems here to raise a particular point, viz, that Euler maintained that Newton's inverse-square law could no longer be taken as a given in describing the forces between celestial bodies. For he goes on to say in the fourteenth pmipph:

The theory of Astronomy is therefore still much fiirther removed fkorn the degree of perfecton, to which it has been thought to be already carried. Because if the forces, by which the Sun acts upon

z6 "Comoissant la quantitk de la pesanteur i la surface de la Terre, fen ai conch la force absolue, qui doit agir sur la tune, en supposant qu'elle dCcroit en raison doubliie des distance. De cette force comparie au terns +odique de la Lune, j'ai diduit k distance moyenne de Ia Lune i la Tene, et ensuite sa pvallaxe horizontde i cette mime distance- Mais cette parallaxe s'est trouvie un peut trop petite, de sorte que la Lune est mob doignie de nous, qui suivant Ia thCorie, et parmnt la force, dont la Lune est poussie vers la Texre, plus petite, que j'avois supposi-" Euier, "Recherches sur le mouvement des Corps cdestes en ghhl," page 5. 37 "Toutes ces raisons jointesensemble proissent donc prouver invinciblemen& que Ies forces cenaipetes qu'on consoit dans le Ciel ae suivent pas emtctement la loi Ctablie par Neuton-" Euler, "Recherches sur ie mouvement des Corps c&stes en g&inl," 6. the Planets, and the latter upon each other, were exactly in inverse ratio of the squares of the distances, they would be known, and consequently the perfection of the theory would depend on the solution of this problem- Thal the jbrces. by which a Planet is solicited, being known, the motion of this Phnet is determined. This problem quite difficult as it can be, appertains nevertheless to pure Mechanics, and it can be hoped, that with the assistance of some new discoveries in Analysis, its solution can finally be attained. But as the law itself' of the forces, by which the Planets are solicited, is not yet perfectly known, it is no longer an affair of Analysis alone: and much more than it is necessary in order to work for the perfection of theoretical ~stronorn~.~~~

Supposing the forces on the moon are known we would expect, according to Euler, that its orbital path would be known as well- But we have seen that this is not quite the case with respect to the moon. The task at hand, then, is to test alternative hypotheses against observations in order to see which best describes reality. Immediately after the passage quoted above Euler explicitly states this in what amounts to be a clear expression of a hypothetico-deductive account of how one is to proceed with respect to the three-body problem in gened.

And it seems, that there is no other route for reaching this go4 than to imagine several new hypotheses on the law of forces, and after having applied the calculus to it, to seek, how much each diverges fkom the observations, m order that fkom a large number of errors, the truth can finally be concluded. But it is easily agreed that for this

"La thiorie de L'Astronomie est doc encore beaucoup plus doign&edu de@ de perfection, auquel on pournit penser, qu'elle soit diji portie. Car si les forces dont le Soled agit sur les Planetes, et celles-cy Ies unes sur Ies autres, itoient axactement en raison renversee des quksdes distances, elle seroient connues, et par consiquent la perfection de la thiorie dependroit de la solution de ce probleme: @ hJmas, hnt me Phtc at sokk'e, iltrnlf~yw, on &knnrknnrkek nwmn#n/ & am Phk. Ce probleme tout difficile qu'il puisse itre, appartient n6ananoins i la hliicanique pure, et on pourroit espkr, qu'i hide de quelques nouvelles dicouvertes dam l'~~~se,on sauroit en& pmenir 5 sa solution- Mais comme la loi mime des forces, dont les Phetes sont sollicitees, n'est pas encore pvfaitement connue, ce n'est plus une affaire de l'-.tnalyse seule: et il en laut bien &vantage pout travailler i h perfection de liistronornie thiorique." Euler, YRecherches sur le mouvement des Corps cdestes en &nWwpage 6. purpose, we still lack a quantity of things, as much f?om practical Astronomy as fkom Adysis; and that without the help of several new discoveries &om both sciences, we are unable to flatter ourselves of having made any great progress.n9

In short. for meaninghl progress to be made with the help of applications of mechanics and analysis, the true motions of the planets are to be determined not with a known attraction law, ie., inverse-square, but rather the Jaw is itself to be scrutinised. In another paper completed after "Recherches sur le mouvement des corps dlestes en gkneal" Euler made a similar conclusion. In his "Recherches sur la question des inegalitbs du mouvement de Saturne et de Jupiter," ('aesearch on the Inequalities of the Motions of Saturn and Jupiter") which was his successful entry in the 1748 prize contest of the Paris Academy of Sciences, Euler draws a slightly different conclusion in his discussion of perturbative effects observed m the motions of Jupiter and Saturn.

In 1746 the Paris Academy announced that it was seeking, for the 1748 contest, "a theory of Saturn and of Jupiter by which the hequalities which these plawts appear to cause rnutuaIly, principally near the time of their conjunction, can be explained." We do need to examine it briefly here as it does inform an understanding of the Eulerian Lunar theory.

We find Euler, m the introduction to his essay, reasoning that it was overwhelmingly likely that the academy had m mind Newton's theory of universal gravitation 'khich has been found up to now so admirably well in agreement with all the celestial motions, that whatever will be the inequalities which are found in the motions of the Planets, it can always be boldly claimed that the mutual attraction of the Planets is the cause of them" Whatever doubts Euler was harbouring against inverse-square variations

-- --

39"Et ii sernble rngme, qu'd n'y ait d'auae chemin de parvenir i ce bu~que de s'irnaginer plusieurs nouvelles hypotheses sur la loi des forces, et ap&s y avoir appliquti Ie calcul, de chercher, combien chacune s'icarte des obsemations, aGn que d'un gmd nombre d'erreurs, on puisse enfin conclure la visit& Mais on coaviendra &ment aishent que pour cet effet, il nous manque encore quantitk de choses, tant de I'htronomie pratique que de I'Adyse; et que sans le secours de plusieurs nouveLles dicouvertes de l'une et I'auw science, on ne se puke flatter de faire de grands pro@s." Euler, 'Xecherches sur le mouvement des Corps cilestes en g6n&al," page 6. and the nature of gravity-whether it was attraction or impulsion m some medium--we see Euler here proclaiming the acceptability of universal gravitation by the community of practicing astronomers. In order to solve this problem, claims Euler, one needs to solve a purely mechanical problem.

The inequalities observed by astronomers in the motion of Jupiter and Saturn are readily perceived; to answer satisfactorily the prize questios it is necessary only to determine the motion of three mutully attracting bodies in proportion to their masses and inversely as the squares of their distances and then to put the Sun for one of these bodies and Saturn and Jupiter for the other two.2M

The proposed prize question is reduced, then, for Euler, to a purely mechanical problem.

Euler cautions the reader, though, that although the problem is a problem of mechanics, it was one ofthe most difficult in mechanics to solve. A perfect or exact solution could only be achieved once much funher progress is made in mathematical analysis. Nonetheless, since the Sun's rnass is considerably larger than Saturn's and Jupiter's and since the laner two planets have near-circular orbits, a solution to the problem by approximation is feasible. It is in this fashion that Euler attacked the problem. interestingly at the end of 92

Euler proclaims that this project, Saturn-Jupiter, is more &cult than the Luoar problem

(which to that pint had been judged to be the more difl6cult.)

We now pick up Euler in 97. Here he shows some concern that he has not been able to reconcile theory and observation, ie., that he has not been able to perfectly account for all of Saturn3 irregularities. Nevertheless, he still claimed that the prize ought

Euler, "Recherches sur la question des inkgalit& du mouvement de Satume et de Jupiter": "&]a cause des inigalitk que Ies htxonomes on remarqub dans le mouvement de Saturne et Jupiter, est manifeste; et pour satisfaire 5 las question propode, oa nyaur;l qu'ii diterminer le mouvement de trois corps qui s'attirent mutuellement en &on cornpo~ede cde de leurs masses, et de la raison inverse des quaer6 de leurs to be awarded to him for his work on the Saturn-Jupiter problem, and confirmed by his work in Lunar theory which, showed that a general correction to Newton's theory was in order.24' After having carefidly compared lunar observations with the theoretical predictions, Euler noticed that the Earth-Moon distance is not as large as required by the theory and this in turn implied that the Moon gravitates toward the Earth in a fashion less than inversesquare of the distances. Newton's theory was in need of correction for the above reasons and for the tact that there are certain small irregularities in the Moon's

Mer comparing lunar observations with theory and comparing this work to his work on the motion of Saturn, it seemed that

the Newtonian ratios according to the square of the distances held for small distances and deviated fkom truth with large distaoce~.~~~

Now with respect to Satum, which is itself at an immense distance from the Sun, its irregularities could perhaps be explained by its large distance &om the Sun. Euler even suggests that perhaps Jupiter's action on Saturn might diverge fiom inverse-square, as the distance between them is immense. Valentin Boss (1972) argues that this aspect of

Euler's theory reflects his adherence to an aether theory and makes him anti-~ewtoniau~"

distances, et rnettre ensuite i la place de I'un de ces trois corps le Soleil, et les corps de Sameet Jupiter au lieu des deux autres."§2 page 46. He noted that this correction would require a greater amber of more accurate obsemations and more time to examine them in order to make necessary corrections. 142 It is curious that Euler does not identie the "small irregulzrities" (quetques petiaes k+phritis). We again note that this paper won the prize contest of 1748 even bough the soIution of the prize problem wasn't forthcoming in the paperprr. Eder, 'Xecherches sm la question des in+& du rnouvement de Saturne et de Jupiter": 57. '21 me semble donc que la proportion Newtonienne selon les quatris des distances, n'est vraie qu'i peu pr6s dam le sforces des corps ctlestes, et que peutke elle s'icarte d'autant plus la v&ti que les distances sont grandes." Page 49. 24 Boss (1 WZ), page 215. We will see that although Euler accepted the aether theory, the issue surrounding his alleged anti-Newtoniasm is not resolved by his metaphysical belief in aether.

Two of the judges, Clairaut and d'Alembert, had a particular interest in the contest as they themselves were hard at work on the Lunar Theory and the general three-body problem.

Chitaut and d9A1ernbert

In 1745 Clairaut was working on the problem of the Moon's orbit and on the three-body problem in general. D'Alembert attacked the same problem the following year so that by

1747 both were actively involved in solving it. Even though both were in Paris, they worked in secret giving parts of their theories to the Academy and depositing uncompleted portions with the Secretary of the Academy in the form of sealed plis cachetis

(enve~opes).~~*

Clairaut was undoubtedly pleased in receiving Euler's paper (for by this time Euler had already gained a considerable reputation). Euler's discomfort with the inverse-square law resonated with Clairaut. Instead of looking at the discrepancy between the obsemed lunar distance and its predicted distauce according to the inverse-square law, Clairaut maintained that the more illuminating discrepancy was the one surrounding the motion of the lunar apogee.246

35 D7AIembea, it seems, played in this competition much more enexgetically thaa CIairaut ,\ few times throughout his career, d'rUernbert wouId go so far as publishing incomplete works, mistakes and a4in order to circumvent Clairaut from claim& primacy- (CIaicaut was elected to the Academy in July 1731 at the age of 18; d'Alembert was dected to the Academy in 1741 at the age of 23.) With respect to the Luav Theorp, d'Alembert even had &e Secreq initidk every page of one memoir to guarantee primacy. For a fuller discussion see Hankins (1970). 246 Euler did later communicate to Clakaut that he came to consider the motion of the Moon's apogee to be pivotal in the proof that the forces, which act on the Moon, do not adhere exactly to Newton's law- I would like, now, to offer a brief sketch of the events which involve Clairaut and d0Alembertand surround Euler's essay. Even before reading Eulefs essay, Clairaut (as one of the judges of the prize contest) maintained that his discovery of the motion of the lunar apogee was suBcientIy important to be publicly discwed at the mid-November meeting of the Paris Academy in 1747.247 The greatest possible surprise came at this mid-

November meeting when Clairaut announced, in rather pompous phrases, that the

Newtonian theory of gravity is false! Although Cartesiaoism was beginning to dissipate, even in Paris by this time, to have Clairaut, whose work on the shape of the earth with

Maupertuis had helped fortify Newtonianism on the continent, publicly call into question the inverse-square law surely delighted the Cartesians at the Paris Academy. He added that after carefbl calculations on the motion of apsides of the moon, he had found that the observed motion differed by a factor of two firom the reds predicted by Newton's d'AIembert was by this time in agreement. So we have three of the foremost mathematicians in Europe in agreement regarding the shortcomings of the inverse-square law-that there was a discrepancy between observation and theory for the motion of the apsides. Euler, in fact, wrote Clairaut 30 September, 1747 claiming:

I am able to give several proof5 that the forces which act on the moon do not exactly follow the rule of Newton, and the one you draw fkom the movement of the apogee is the most striking, and I have clearly pointed this out m my lunar theory, ... since the errors cannot be attributed to the observations, I do not doubt that a certain derangement of the forces supposed in the theory is

247 For a carehl reading of these events see Waff, "Clairaut and the motion of the lunar apse," Wilson (1995); Hankins' (1970) bookJean d-k offers a lively and useful discussion of these events. 24 His conuoversial pronouncements at this meeting had been previously deposited in a sealed envelope with the Secretary in September, two months prior to tbis meeting. Weshould note that two months afier Clakaut deposited his results but prior to this meeting, d'rilembert deposited a pbi cucbrti on the same subject and his results agreed with those of Clairaut See HPntinn (1970), Waff (1 995). the cause. This circumstance makes me think that the vortices or some other material cause of these forces ought to be aitered when they are transmitted by some other vorte~."~

It would appear that with Euler's backing, Clairaut would be less hesitant to deny

Newton's theory. This passage is fkther illuminating in that we see Euler, who believed in an aether, returning to vortices in light of a fhdure in Newton's theory.2s0

The academicians who still retained allegiances to Descartes were, obviously, excited that the Newtonians were causing the destruction of their own doctrine.2s'

History offers us rnany twists. Chiraut eventually solved the lunar problem and this placed him f%dy in the Newtonian camp. Another Newtonian, the naturalist Buffon, attacked Clairaut 's previous findings on metaphysical grounds. Clairaut had suggested that Newton's law ought to be modified to contain two terms so that force might vary in the following way:

where k is some constant to be determined. The second term, claimed Clairaut, might account for not only the perturbathe effect but might also account for the phenomena of surhce tension. In his reading at the 20 January, 1748 session of his Reflexions sur la loi de l'attraction (Refectiom on the Law of Gravity), Buffon insisted that the law of

249 M-GI Bigourdan, "Lettres inidids d'Euler i Clairaut Ges des archives de I'Acadimie des sciences," Cornptes-rendus, Co~grhdes sociitks savantes (1930), p29. Quoted in Hvlkins (1970). p.32. Euler in a letter to Clainut later ceded priority to Chut d'AIembert would do the same thing in a letter to Cramer- 51 An interesting sidebar to this and one requiring Merresearch, is the influence d'rYembert exerted on both the Berlia and Paris academies. I would guess, for instance, that a carehl study of the various political allegiances and metaphysical leanings of men like d'Aembcrt dictated vaxious aspects of academic life, For instance, d'Aembert was at the ccnw of a controversy regarding the deckion to turn the Berlin Academy of Science into a Society of '%elles letaes." His association with the literary elite, like Dideroq was later perceived by Clairaut and other "scientifically minded" researchers as a bemy4 of suentism. It would appear gravitation must have only one term, otherwise it would not be a simple function of the

distance and would represent several forces rather than a single force. For Buffon this

suggested that the discrepancy might be due to a magnetic force-that is, gravitation

might not be the only operative force. If gravitation is not the only operative force then it

was conceivable, Buffon argued, that different types of matter may attract according to different laws?2 in the four or five public exchanges between Clairaut and Buffon, it

becomes apparent that Clairaut's Newtonianisrn is quite different fiom Buffon's. Buffon

was defending inverse-square variation largely on metaphysical grounds whereas Clairaut was offering a multi-term modification of inverse-square in order to aid in explaining a wide variety of phenome-both celestial and terrestrial-without the use of any other physical principle besides gravitationz3

In the meanwhile d3Alemberthad begun speculating about Buffon's suggestion and seems to have had a vested interest in the Buffon-Clairaut controversy or debate. dAlembert suggested at one point that it may be worthwhile to investigate a possible correlation between the movements of the moon with the variations of a compass needle, for this would make his conjecture that the force acting on the moon does not simply depend on its distance fiorn the earth but is a fimction of this distance and some other unknown variable.254 d'Alembert points out that this would be a considerable undertaking.

that academic restrucming in the name oE safeguarding a society's existence met opposition much in the same way that restructuring of universities today as well. z2 Hankins (1970). Chapter 3. Waff (1973, p.174. *3 According to Waff (1975) this is bas'% how Ckuttermhated the exchange with Buffon which had contributed nothing to the development of "some sure mans of using the phenomena to know the true laws of Nanue-" 34 Letter of 16 JuneJ748 Gom d'Aembcrt to Gabriel Cramcr- Quoted in Hankins (1970), p.33. It is interesting to note d'Alemkrt's movement with respect to the lunar theory.

In later developments d'Alembert would admit to having misread lunar tables but, even with the necessary corrections, his results accorded well with Euler's. Perhaps d7Alembehat the end of August 1748, best exemplifies the spirit (as Emst Cassirer would have us believe) of the scientific developments:

Although this proves that there is another force besides that of gravitation which acts on the moon, it seems to me tbat the theory of the moon, such as it is, is the most victorious proof of the Newtonian system of

By this point d7Alembert's increased precision in his lunar theory had narrowed the gap between inverse-square and experience. No netheless, he st ill maintained that the motion of the apsides was probably due to an extra force such as magnetism. By December 1748 d' Alembert had completed his book on the theory of the moon. But with unusual restraint he opted to wait for Clairaut to finish in order to see ifhe and Clairaut were in agreement.

Just as he had earlier stood and pompously proclaimed the Mliiility of Newton's theory, Clairaut, on 17 May 1749, publicly declared in the Academy that Newton's law had been correct all along! A mere five days prior to Clairaut's retraction, d'Alembert wrote Cramer that he was embarrassed to have even entertained the notion of overthrowing Newton.

In the next part, I will provide a sketch of Clairaut7s general solution to the problem. Although the solution itself is extremely important it will be m our interest to say something about the developments outliwd here and how they fit into the general

-

255 Quoted in Hankins (1970). thesis that these developments exernplifL what we have been calling the entrenchment of

Newton's ideal of empirical success.

Sketch of Clairaut's ~otution~

Clairaut's retraction came at the end of 1748 and beginning of 1749. It wasn't until 1752, however, that his memoire outlining his solution was printed. I have provided a translation of the two papers comprising these memoires in the appendix. I will refer to the ht memoire, "Concerning the Orbit of the Moon," as Memire (a) and to

"Demonstration of the Fundamental Proposition of My Lunar Theory," as Memoire (b).

The goal is to develop a generd equation for the curve which would be descnkd by a body affected by the action of two forces, X directed toward a centre, T, and Il drawing the body away perpendicularly f?om the centrally directed

He let Mm be that part of the curve described in an selysmall time, dr. Letting

Mm=mn,n is the point where the body would be after another infinitely small time, &, if

* See the appendix [or a more detailed proof. 257 Memoire @) the accelerating force ceased to act at the end of the first instant. The composition of forces n and X gives us mp as the part of the curve descn'bed in the second instant (Mmis the part of the curve described in the £irst instant.) At m the body is under the influence of an accelerating force. In the absence of such a force the body would "fly OR'tangentially toward n. The composition of the radial and transverse forces will result in the body being held in orbit about T by being deflected toward p .

Clairaut makes the assumption that this body moves in an orbital The equations Clairaut derives at the end of this first problem are:

rdh + 2dr&= rIdr2 (1)

rdv2 - ddr = ~&(2), where r is the radius vector Tm, v the longitude, H the sum of the transverse forces, Z the sum of the radial forces, and dx the infinitely small time2sg. Clairaut's solution is to show that if Z, the sum of the radial forces, is composed of an inverse square variation plus some other term, then if the other term approaches zero, the radial force then is an inverse-square variation and any perturbative effect is due to the transverse force. For the earth-moon system this means that Newton was right (as we will see) in Proposition ID,

Book IlI to claim that the small forward precession of the lunar apsides was due to the effect of the sun on the moon and subtracting this effect left the centripetal force drawing the moon toward the earth as proportional to the inverse-square of the distance. Clairaut

-- - ss Curtis Wilson has pointed out that a key difference between Clairaut's solution and later, Euler and d',Uernberr's solution is this assumption of orbital plane. d',Uemk had pointed out that the moon's orbit had double curvature and the more appropriate assurnptioa (the one used by both himself and Euler) would be to project the moon's orbit onto the elliptic. 259 I have kept with Clairaut's notation although it would simplifv matters a bit to hbel time as dt instead of dx. M does this by letting X be composed of an inverse-square variation, -, and some other rr part, . Equation (2) now becomes

By two successive integrations of ( 1), Clairaut amves at a differential of time:

Substituting this expression in (2a) and integrating twice CIairaut deduces

which is the sought after equation, and where f and g are constants of integration and R is a function of r and the perturbing force n.260The left side of (3) along with the first three terms on the right side define r with respect to a fixed elliptical orbit. The second part then carries the perturbative effect and this can be explained through approximation. The other two terms of equation (3),

represent the correction to the orbit due to the perturbative effect of forces and n.

Clairaut cb:

With respect to the second part of this equation

sinvI~dvcosv-cosv~~&sinvwhichexpresses the correction necessary to make to the value

Z6O Inspection of the translation provided in the appendix d conbthis- P 1 - c cos(v - Q)of - when one wants to regard the forces r II and @ , it is evident that it will immediately supply, and without anything neglected, the required correction, when 0 and II would be expressed so that i2 will depend only on v and on constants; and so that it will fiunish a means of knowing this correction by approximation, whatever the values of II and 0 are, provided we initially know the orbit approximately because it will only be necessary in this case to substitute for r in R , its value derived fkom the nature of the supposed orbit-

AU this is by way of showing the hdarnental proposition of his lunar theory. In order that R (a hction of r and the perturbing forces) depend solely on v and on some constants it was necessary to express and n in term of r. To reduce R to a useable expression it would &ce, CIairaut claims, to know the orbit approximately. Now the equation,

will not suffice as it is the equation in polar co-ordinates for a conic section which the

M force, -, acting alone would have the body desmbe. In short, (4) is the equation of a rr fixed ellipse such that the perturbing forces are absent. In just three revolutions the location of the moon at apogee would be out approximately nine degrees.

Clairaut states early on in Memoire (a) that

I will not content myse& as I had in my first memoi?, in supposing that m the value of Q, k r = -- f -ecosmv' nor in the value of T for the time, the quantity

26' Clairaut is here ref* to the Memoire of 1745. k2 2e 3ee (V + -sinrnv + -sin 2mv). 4% 4171 but, rather. I will offer the entire calculation as 1 indicated in the same memoirs (p.352). I will suppose that

And for the equation of the time, or, rather, for the Moon's f?om which it results, the general equation is 2 x = v+ besinmv + ge2sin2rnv+ hasin(--m)v n

The idea here is to test the equation

k 2v 2 2 -= 1-ecosmv + /? cos-y cos(--m) v +S cos(--m)v r n n n (5) 2 2 +Scos(-+m)v +~cQs(--2m)v., n n where n is the ratio of the moon's mean motion to the difference between the moon's and sun's mean motions, and p, y, 6,5 are evaluated m terms of other constants of the theory.

The third through the Iast terms on the right in (5) were delivering the 111 apsidal motion

Notice that the expression

is the expression of a rotating ellipse where k is a parameter of this ellipse and e is its eccentricity, v (as we have already seen) is the angle that the radius vector makes with the line of apsides, and m is a yet to be determined constant allowing for the motion of the apse. In an earlier rnern~ire~~'Clairaut obtained for the radius vector the expression

With a value of rn=O.99%036, Clairaut was able to calculate the motion of the Lunar apse in one sidereal revolution to be 1"30'3 8". That is, 1-m=O.OO4 1964, which means that in one complete revolution the apse will move (1-m) 360°=1 O30'38". This, of course, is only half of the obsened ~notioa*~~Notice, though, that the last three terms of (7) are small in comparison to the other terms that would at first appear to cohthe goodness of equation (4). But with the anomalous result obtained Clairaut undertook a method to obtain more precise values of the variables in (7).

In Memoire (a) we have Clairaut's final equation:

The first level of approximation yielded the following expression:

Notice that the coefficients of (8) do not differ appreciably f?om those in (7). At the time of the public address to the assembly, Clairaut assumed that the value of m, and thus the mean motion of the apogee, would not be much affected by this refined equation of the

'62 Waff (1995), page 45. 263 Waff (1995), page 45. moon's orbit because of the d contributions (due to the smallness of the coefficients) of the new cosmv terms in (6).

Recall that R is a function of r and the transverse perturbing force, as well as the radial perturbing force. It is expressed in terms of v. Letting N be the mass of the sun, 'a' the distance between the earth and the sun. and t is the angular separation between the moon and the sun at any time, then

In order to use (9) in (3), the cos2t and sh2t terms would have to be expressed in terms of v. There would also be required a substitution for r in terms of v (which was the desired goal at the outset). Notice that the last four terms of (7a) contain the constant m in short, the equation for a, equation (9), is required for the equation for r in (7a). Since both (7a) and (9) are in terms of v (and more specifically when substitutions are made, delivering cosmv terms) Clairaut first sets m=l in (7a) and derives an expression for r which can be used in (9). This new expression for R then has cosmv terms whose coefficients wiU be need to re-compute m.

Clairaut moves on to claim-

But is it on an important point on which this new solution differs essentially from the fist: it is the determination of the quantity rn, which gives the motion of the apogee. The term corn in the expression R, and which gives the term of the same kind in the value of r, by which m is determined, is approximately doubled by the addition of the terms 2v 2 Bws--ycos-v+etc to the value 1-ecosmv, with n n which one was satisfied in the first solution, and by this means the motion of the apogee is found to conform well enough to observations without supposing that the moon is drawn toward the earth by any force other than the force which acts inversely as the square of the distance; and there is reason to think that if d the considerations I: have omitted here were to be carried out. the slight difference between theory and observation would be altogether eliminated.

CIairaut thus solved a problem whose solution had eluded Newton. According to Cook, in The Motion of the Moon, the first-order approximation, the term with the coefficient P combines with other terms in the solution to give additional terms in the expression for m. 264 The expression (I-m), after this first approltimatioq was 0.00714, which meant that in one complete revolution the apse will move (I-m) 360°=2034'23". This value is already a 67% recovery of the unaccounted motion of the lunar apse. When Merterms are retained in the expansions, very close agreement is obtained between the calculated and the observed values of the mean motion of the lunar apse? Calculations of m to the sixth power are shown in appendix D dong with polar plots over cycles of fii?y revolutions for the first six terms of the expansion. With his result Clairaut was able to claim that Newton's inversesquare law could be used to account for the deviation of the moon's motion &om an idealised Keplerian orbit.

'M Cook (1988), page 93. 26jCook (1988), page 93. Chapter 7

Newton's Ideal of Empirical Success and Natural Science

The theoretical determination of the mean motion of the lunar apsides played a significant

role in the establishment of the inverse-square formulation of the law describing the force

of gravitation. In the Principia Newton argued &om the harmonic law ratios of the

satellites of Jupiter and Saturn and of the primary planets to an inverse square centripetal

force holding these satellites and the primary planets in orbit. In this argument Newton

made use of Corollary VI and Corollary VII, Proposition N, Book I which relates the

harmonic law to an inverse square centripetal force.

In Proposition XLV, Book I Newton established a mathematical relation between

a centripetal force acting on some revolving body, and the motion of the apsides of the

body's orbit which it would produce. According to the formula, which he developed, a

force varying inversely as the squares of the distances would not produce any motion of the apsides. Quiescent aphelia then can be used to argue for inverse square variation with respect to a revolving body. In other words, the amount of precession measures the exponent in the force law.

The motion of the 1wapsides posed a significant problem. The small but sensible

motion of the lunar apsides, which amounts to roughly three degrees per revolution, means that the exponent in the force law is not exactly 2 but, rather, deviates fiom it slightly. Newton reasoned that the observed motion of the hmar apsides was entirely due to the "action7' of the sun- We have seen that Newton was able to account for half the observed motion of the lunar apsides.266

The most publicly critical of Newton's commentators, Clairaut, was the 6rst to solve the problem surrounding the motion of the lunar apse. CIairaut's achievement regarding the motion of the lunar apsides played an important role in the establishment of the inverse-square formulation of the law of universal gravitation, which depended on an accurate account of Iunar perturbation. Clairaut was able to show that the law of gravitation was capable of accounting not only for the general motions of celestial bodies, but also for a very visible perturbation of these motions.

Euler on the Motion of the Lunar Apse

The problem of the apsidal motion admined an algebraic solution by approximation Newton did not have the analytical took necessary to develop the algorithmic solution required. He, as has been well documented, employed a geometric technique which was wt suited to capture properfy the motion of the lunar apse.267

There was not any significant development with respect to the problem surrounding the motion of the lunar apse until the 1740s when Euler and other continental philosophers applied their analytical techniques to the solution of this problem. Recently, Curtis

has shown that differentiation and integration of trigonometric hctions had not

x6 See Wdson (1995)' Whiteside (1 967), Whiteside (1970). In David Gregory's Tbc Ehmm~rof Atmnony, Pbn'al ond Gcom~~~ilL(1715) London: J. Sicholson, 2 Volumes, Volume 2, page 563, Newton's lunar theory is there presented as solving the motion of the lunar apse. 267 Wilson (1995). Wilson, (1995). been part of the standard procedures in the solution of differential equations prior to 1739 when it was Euler who incorporated them

Eder, we have seen, had grave doubts regarding the exactitude of the inverse- square law. In both "Recherches sur le mouvement des corps cdestes en gMrale" and

"Recherches nu la question des inkgalit& du mouvement de Satume et de Jupiter," with the latter winning the Paris Academy prize of 1748, Euler expressed his reluctance in accepting the inverse-square law. Like Clairaut and d'dernbert, Euler was able to recover only half of the motion of the lunar apse. This discrepamy between theory and observation preoccupied Euler, Clairaut, and d9Alembert until the problem was solved.269

Euler, i believe, offers an interesting case to see whether he adopted a hypothetico- deductivist stance or whetber he accepted what Harper has been calling the entrenchment of Newton's ideal of empirical success.

By 1748 Euler had publicly called into question the inverse-square law on the evidence that it yielded only half of the motion of the lunar apse. What is interesting about

Euler is his metaphysical belief in an aether theory. He maintained that the transmission of all forces had to be carried out in some medium.

Ln an essay, "Reflections on Space and Time," which appeared in The History of the Royal Berlin Academy of Sciences in 1748, Euler suggests that the relationship between physics and metaphysics should be one where knowledge of the former regulates the principles of the latter. That is, instead of having the concepts of physics regulated by metaphysical principles, Euler reversed this relation so that metaphysical ideas are to be

269 The British Parliament prize went posthumously to Tobias Mayer's family. In fact they received only £3000. Euler received €300 for his contribution to Mayer's tables (ie., Mayer used Euler's equations). For his chronometer, H-4 built in 1759, John Harrison was given €10,000. determined by established physical inquiry. This essay is a series of twenty one reflections in which in the first Euier cIaims:

The principles of mechanics have already been established on such a sound basis that one would greatly err if' he wished to encourage any doubt as to their validity. Even if one were not in a position to demonstrate them by the use of the general principles of metaphysics, the excellent agreement of alI the conclusions which one draws fiom them by means of the calculus, with all the movements of bodies both solids and liquids, on the earth, and likewise with the movements of the heavenly bodies, would be sufficient to place the truth of the principles of mechanics, beyond

Euler concludes fiorn this passage that the laws of mechanics are so well con£irmed that one cannot deny their veracity. These laws are Newton's three laws and Euler lists them here as unquestionable kts. From this he claims that it necessarily follows that

Since it is metaphysics which is concerned in investigating the nature and properties of bodies, the knowledge of these truths of mechanics is capable of serving as a guide in these intricate researches (of metaphysics). For ow would be right in rejecting in this science (of metaphysics) d the reasons and all the ideas, however well founded they may otherwise be, which lead to conclusions contrary to these truths (Of mechanics); and one wouid be warranted in not adding any such principles which cannot agree with these same

After the announcement and proof that the Newton's inverse-square law is suEcient for understanding the motion of the lunar apse by Clairaut we find Euler quickly praising this achievement. He wrote to Clairaut in June 1751:

[TJhe more I consider this happy discovery, the more important that it seems to me. .. . For it is very

Eder (1967), page 116. Eder (1967), page 116-117. certain that it is only since this discovery that one can regard the law of attraction reciprocally proportional to the squares of the distances as solidly established; and on this depends the entire theory of astronomy.272

Euler was quick to agree that the law of attraction is established. Among the phenomena

which were discussed in the mid-eighteenth century, (the , the shape of the earth, etc.), perturbation theory and especially lunar theory seemed to restrict the acceptance of

Newton's inversesquare law on the continent. In fiict, the law itself was called into question when the calculated motion of the lunar apse was not brought into agreement with observation. Aside fkom the immense success the inverse-square law had, this disagreement between theory and observation had Europe's leading natural philosophers nearly abandoning the law. In the argument for universal gravitation Newton offered as premises the near correspondence of the satellites of Jupiter, and the primary planets to the harmonic and areal laws. With respect to the moon we are told that a measure of the perturbation will measure the deviation kom inverse-square. Since the lunar orbit is not quiescent, Newton needed a different argument in support of universal gravitation.

Consideration of apsidal motion was seen as crucial for the inverse-square law and for

Newton's ideal as well. That is, the inverse-square law undergoes a test in a such a way that support for it is challenged by the examination of apsidal motion which is to support inverse square variation.

The passage quoted above seems to indicate that Euler, at least prim facie, thought universal gravitation ought to be accepted because it yielded accurate predictions.

That is, Eder's endorsement of Clairaut's achievement can be read as hypothetico-

n2Quoted in Waff (1995), page 46. deductivism. The question is: did Euler see it as much more than this? His claim in the

introduction to the memoir for the 1752 Paris Academy prize (which he won) seems to indicate that although agreement between obsenations and predictions is important, he was aware that the solution to the motion of the lunar line of apsides added a stronger support for universal gravitation He claimed that

because M. Clairaut has made the imponant discovery that the movement of the apogee of the moon is perfectly in accord with the Newtonian hypothesis ..., there no longer remains the least doubt about this proportion .. . . If the calculations that one claims to have drawn fiom this theory are not found to be in good agreement with the observations, one will always be justified in doubting the correctness of the calculations, rather than the truth of the theory.273

This passage seems to indicate that although the support for universal gravitation based on

Clairaut's solution to the motion of the lunar apsides was very strong, it is not clear that

Euler conceived of this support in more than a hypothetico-deductive way.

Even in September 1760, much after the debate surrounding the motion of the lunar apse had ceased, we find Euler still holding on to aether theory. In his fifty-fourth letter to a young German princess, the Princess of Anhalt-Dessau, Eder called the law of universal gravitation an "established" fact which cannot be cccontroverted." He remarks:

It is established, then, by reasom which cannot be controverted, that a universal gravitation pervades all heavenly bodies, by which they are attracted to each other; and that this power is greater in proportion to their prom.

This fkct is incontestable; but it has been made a question, whether we ought to give it the name of irnpuIsion or attraction. The name undoubtedly is a

273 Quoted in Waff (1 995), page 46. matter of indifference, as the effect is the same. The astronomer, accordingly. attentive ody to the effect of this power, gives himself Little trouble to determine whether the heavenly bodies are impelled towards each other, or whether they mutually attract one another; and the person who examines the phenomena only is unconcerned whether the earth attracts bodies, or whether they are impelled towards it by some invisible cause.

But in attempting to dive into the mysteries of nature, it is of importance to know if the heavenly bodies act upon each other by impulsion, or by attraction; If a certain subtile invisible matter impels them towards each other; or if they are endowed with a secret or occult quality by which they are mutually attracted. On this question philosophers are divided. Some are of the opinion, that this phenomenon is analogous to an impulsion; others maintain, with N~on,and the English in general, that it consists of attractiot~"~

So by 1760 it would appear that Euler was resolutely a supporter of universal gravitation and the inverse-square law. Although he claims in this letter that it matters not whether gravitation is properly conceived as an impulse or as an attraction the deciding factor is the bulk of effects observed as a result of gravitation Nonetheless he still finds it necessary to discuss either conception noting, in a fashion reminiscent of Leibniz, that

Newton and the attractionists held that the heavenly bodies act on one another while being endowed with a secret or occult quality. In that Euler, himself, remained an adherent to the aether position, we suxmise that m the letter to the young princess he would side with the irnpulsionist position.

These letters, roughly 243 of them, were written over a period of two years (1760-

1762) to the princess who was 15 years oid when the correspondence began. The Letters can be categorised into three divisions: general science, philosophy, and physical questions. The Letters were quite successfirl having been translated into eight languages

£?om the original French by 1800. By 1840 it ran to over forty editions.275

Euler maintained that gravity had a physical caw, not yet known in detail but certainly arising fiom the action of the fluid matter filling space. In these letters to the young princess, Euler restricted himself to a discussion of the attractionists and the impulsionists. Given his position, by this time, as an impulsionist, it is a curious fict that

Euler does not discuss rivals to Newton's theory although he accepted the role of aether in gra~itatioa'~~While ow may pick one or the other of aether or void, Euler, by 1751, and expressed through these letters of 1760, thought that observations corresponded quite well to the theoretical expectations of Newton's theory. For this reason, the Newtonian theory appears without rival in Euler's letters to the Princess. Planetary vortices (a la

Descartes) do not appear in these letter^.^" Except for Newton's optical explanations found primarily in the @ticks, Eder accepted Newton's system.

The Newtonian System, you will easily believe, made at first a great noise, and with good reason, as no one had hitherto hit upon a discovery so very fortunate, and which diffUsed at once such a clear Light over every branch of science.278

24Letter LIV, 7h September 1760, Volume I, p. 191. "5 BOSS(1972), page 211: Ye\ as far as the Russian public was concerned, the immense authoritg he enjoyed in the Catherinina era was due neither to his renowned work nor to his copious contributions to the A& Erudtmm and other learned journals, buq rather to his littcrr to o Germmr Pn'kzr~- This was a work with literary pretensions and was inspired by the same genre as Fontende's PbarF~dvmdr- z74 In spite of this, some authors, notably Boss (1972), insist that the letters conclusively show Euler as a Cartesian willing to overthtow Newtoniulism. For a Mer discussion than presented here see Ed. ,%ton (1972)- Tbr V&ex ThqojPhfq MD~M, MacDonald: London. &ton points out that by 1740 or so serious publications advocating vortices no longer appeared. Both Youschkevitch (1971) on page 481 of his Euler entry in the Dicrionmy .fS'kj%B&@y and the French biographer DuPasquier (1927) have maintained Euler to be Cartesian in spirit Boss (1972) characterized Euler as pre-Newtonian and as very hostile to Newton. AU of these interpretations do not seem to be well supported by a systematic reading of the Lctkrr as we will see here. '78 Lester LIII, 5h September, 1760. Universal gravitation was established, according to Euler, by the following: the non-spherical shape of the earth (Letters XLVIII-L). a proof of which Newton had given, the movements of the tides being due to attraction and not vortices (Letters XLV-LW(). the orbits of cornets were controlled by attraction and not vortices (Letters XLV-LWC), and that the orbit of the moon is fully explained by universal gravitation (Letter LI).~'~

It is very reasonable to demand, why the moon does not move in a straight line. But the answer is obvious; for as gravity occasions the curved direction of the path pursued by a stone thrown, or a cannon-ball fired 0% there is good ground for maintaining, that gravity acts likewise upon the moon, forcing her towards the earth; and that this gravity occasions also the curved direction of her orbit. The moon, then, has a certain weight-she is, of consequence, forced towards the eaah; but this weight is 3600 times less than t would be at the surfke of the earth. This is not merely a probabIe conjecture, but a truth demonstrated. For this gravity being supposed, we are enabled to determine, on the most established mathematical principles, the path which the moon must pursue; and this is found pertectiy to agree with that in which she actually does move; [emphasis added] and this is a complete demonstration of the truth of the assertio~~*~~

In this passage we find Euler hinting at something more than hypothetico-deduction in his connecting the action of gravity of terrestrial objects with the motion of the moon

However, in the bit that has been emphasised he seems to be advocating that the success of the theory is tied to the mere predictive power it has. In hypothetico-deductive terms,

*9 In Letter LVTII, 19September, 1760, Euler praised the Paris Academy and the chief advances in mechanics "is chieElp to be ascribed to the Academy of Sciences at Paris, which proposes annual prizes to the best proficients in the prosecution of science." (page 201) 280 Letter LI, Zsc Septembec, 1760. the mathematical demonstration of the lunar motion is completely in agreement with the actuai observed motion.

Notice how this passage resonates with Euler's claims in 1747 in the introduction to his memoir on the three body problem in general. In 1753 Euler published his own derivation of the lunar inequalities in his Theoria motus Iunae exhibens omnes eius inaequulitutes. We are told that the aim of this work was to test Clairaut's retraction of

May 1749. That is, Euler was publicly ve-g, for himself, that Clairaut was right in asserting that Newton's inverse-square law sdliced to recover not halfbut the 1I1 motion of the luoar apse. At Euler's suggestion, the St. Petenburg Academy chose for its first prize contest for 1751 the question of whether the motions of the moon281accorded with the inverse-square law of Newton. Euler was one of the judges. As in 1747 where

Clairaut and d' Alembert anticipated Euler entry in the Paris Academy's contest, we now find Euler anxiously waiting for Clairaut's paper on a similar topic.

By late winter or early spring of 175 1 Euler communicated with Clairaut telling the latter that he was in receipt of four essays, one of which is obviously the latter's with the remaining three poor not only in relation to Clairaut's, but m and of themselves. He went on to cIaim:

It is with infinite satkdktion that L have read your piece, which I have waited for with such impatience. It is a magnificent piece of legerdemain, by which you have reduced all the angles entering the calculation to multiples of your angle v, which renders all the terms at once integrable. In my opinion this is the principal merit of your solution, seeing that by this means you arrive immediately at the true motion of the apogee; and I must confess that in this respect your method is k preferable to

28' Pres~~llilblythis meant the motion of helunar apse- the one I have used. However I see clearly that your method cannot give a Merent result of the motion of the apogee than mine; in which I have recently made some change, for having previously reduced d angles to the eccentric anomaly of the Moon, I have now found a way to introduce the true anomaly in its place. Thus while your find equation has its two principal variables the distance of the Moon fiom the Earth and the true longitude, I have directed my analysis to the derivation of an equation between the longitude of the Moon and its true anomaly, which seems to be more suitable for the usage of astronomy.*"

Like Clairaut, Euler satisfied himself that Newton's inversesquare law could account for the motion of the lunar apse. He says in a letter dated 27LhJuly, 1751 to Tobias Mayer:

At least, when I repeated my researches on this subject, I took into account the true anomalies of the Sun and Moon as well as the true distance of the Moon Eom the Sun, which seems to me to be fkr more convenient. My actual intention was to investigate thoroughIy how accurate the theory of Newton concerning the motion of the Moon's apogee agrees with the observations. [emphasis added] For not only I, but others too283who worked on this, have hitherto ahvays found that according to theory the motion of the apogee should be only half as large as it has, in fact, found to be. Now, however, I have found to my great pleasure that this part of Newton's theory is exactly in accordance with the observations: [emphasis added] therefore it is much less to be doubted that the agreement should not be complete: there are,

Translated by Curtis Wdsoa Gom G- Bigourdan, "Lettres inidites d'Euler i ClaLaut" Boss (1972)- pp.153-155, has argued that Euler, even at this point, was a Carte&n and opposed ClaLaut's Newtonkka Boss clakns that Eder opposed Clairaut's nomination to the St Petersburg Academy and that the academy's selection of the topic for the prize competition of 1751 to be the motion of the moon was to show the Cartesians right and the Newtonians wrong. Euler supported the deasion to choose this topic, he even suggested it, not to vindicate , but in order to help himself in his own quest for a lunar theory in accordance with Newton's law of universal pviation That Eder maintained an aether theory ought not to be confused with king an anti-Newtonian as Boss has wrongty done here. -3 Euler was surely referring to both Clairaut and d'Alembert here. He may have also been refening to Bernoulli whose own work on this topic never gained the level of attention that we have seen with Euler, Clairaut and d'Alembert however, still nearIy i~~smmuntabledficulties connected with the calculation for determining accurately all the lunar inequaiities fiom the theory?

The agreement is there, the refinements are to come. Although Eder realized the importance of solving the lunar precession problem for the justScation of Newton's theory. the passage above indicates that he thought of the project in hypothetico-deductive terms. That is, the approximations needed to carry out the calculations of the lunar inequalities were dficult. The image one gets here is that as assumptions are stripped away the parameters involved in the calculation increase. As these increase, the calculations become more and more dfi~ult.~~~

Euler, in a letter to Mayer nearly a half-year later mentions the dficulty in carrying out such calculations. In this letter, Euler notes that the lunar inequalities are complicated by the shape of the earth and whether the Moon has a declination. Even with all these complications, Eder is confident in proclaiming the exactitude of inverse-square. In March of 1752 Euler examines these complications and claims:

Apart fiom those riequalities] thus originating fiom the [non-spherical] shape of the Earth, and still just large enough to be taken into account, different [ones] which seem rather substantial also originate fiom the inclination of the lunar orbit to the . As to these, however, they depend only upon the Moon's mean anomaly p, the Sun's mean anomaly s,

284 Forbes (1971) page 38. 285 Again in the Ieners to the princess Eder relates this position. For instance he claims: "The motion of the moon has accordingly in all ages greatly embarrassed philosophers; and never have they been able to ascertain, for any future given time, the exact place of the moon the heavens. -.. Now in calculating eclipses formerly, there was Gequendy a mistake of an hour or more, in the ecfipse actually taking place an hour eder or later than the cdculation. Whatever pains the ancient astronomers took to determine the moon's motion, they were always very wide of the truth. It was not until the great Nandiscovered he real powers which act upon the moon, that we began to approach nearer and nearer to truth, after having surmounted many obstacles which retarded our progress." Lettcr LX, 23d September, 1760, p211. and the mean distance of the Sun 6om the moon, w.286

Euler pointed out that CIairaut was about to publish in the St. Petersburg transactions his solution which makes use of these parameters. He offers Mayer a table that compares

Clairaut's series for approximating lunar positions to Mayer's reformulation.287 Finally, in order to determine the motion of the line of apsides correctly he claims that no reference need be made of the shape of the orbit but it is only necessary to introduce the term 2wp into the calculation as both he and Chimut have done.

The influence Euler exerted with such a pronouncement is remarkable. By 175 1, the tide of informed opinion was clearly hvouring the agreement of inverse-square with the motion of the lunar apogee. Mayer went on to publish more accurate lunar tables without Euler's help.

The reader will recall that Newton placed a great deal of weight behind his argument for universal gravitation on the proposition that the Moon's motions can be accounted for with an inverse-square variation Propositions III and IV (the Moon Test) of Book 111 are important in this respect. That Newton left lunar theory in a state of codhion allowed for non-Newtonians to attack Universal Gravitation. We have seen that

Euler, Clairaut, and dYAlembertsaw lunar theory as the "jugular" so to speak. Their efforts went from trying to account for the motion of the lunar apogee, along with

Newton's theory, to showing that since the latter was not successfiS the theory was in need of correction. In the process they discovered that the theory actually did yield the

2~ Forbes (1971), page 54 287 The series is 30 terms long of which I offer, here, the £kt few: - 22797"sin p + 80 lsin 2p - 37sin 3p + 608sin s - 2 lsin 2s - 1 13 sin w + 2394sin2w which taken, in to&, yield the true longitudes of the moon motion of the apogee. In the case of Euler, his metaphysical commitment to aether theory did not detract him from accepting inverse-square variation. His letters to a young princess showed him accepting inverse-square while still claiming he was an irnpulsionist.

He had no scientific grounds for such a belief and he was ready to admit so. He thought conceptually that it made more sense than attraction at a distance but it had no bearing on his acceptance of inver~e-s~uare.'~~

In the end theq Clairaut's solution to the lunar precession problem helped buttress universal gravitation Newton, himself, had placed much weight on such a solution to the problem not only because of its role in supporting his argument for universal gravitation but also for its role in his ideal of empirical success. Euler, as a key player in the tide of informed opinion, certainly appreciated the importance of the solution to the lunar precession problem. The empirical success enjoyed by universal gravitation and the role this solution played in its proof allowed him to bracket his a priori commitments to an aether. Nonetheless I do not believe he myrealised this ideal of empirical success. He focused, repeatedly, on the agreement between theoretical predictions and observation in his letters to Chiraut, Mayer, and the young princess. We do see, by the mid-1700s, a greater emphasis placed on empirical success and less emphasis on hypothetical thinking.

I should say, then, that Euler is on a path toward the entrenchment of Newton's ideal of empirical success. This suggests a fbrther project to be carried out: namely, to investigate

288 In "Eder on action-at-a-distance and fuadamental equations in ," Wilson (1992) explores Euler's metaphysical leanings. Particularly usem is the analysis of the correspondence between Euler and Daniel Bernoulli. At every turn Bernoulli would dispute Euler's argumene in favour of aether. Wrlson's argument is consistent with the position taken here, namely that aether, nowithstanding, Euler accepted the inverse-square law. the work of kter 'Wewtonians," such as Amp&e, Laplace, and Fresnel to see whether the idea gets entrenched in their writings. Appendix A

Clairaut. 1 748 Memoires paper.

Concerning the Orbit of the Moon

Without neglecting the squares of the quantities of the same order as the perturbing forces.

by Mr. CIairaut

As in the solution I gave in 1747, I will suppose here that the two orbits are in the same plane, that the Sun's orbit is without Deposited at the eccentricity; and I will have no consideration for the terms which Academy on the would be introduced m the values of the forces and if one 21 n, Jaw, did not neglect the square of the ratio of the distances of the Sun 1749, and read and the Moon to the earth respectively. on the 15 March, 1752. Thus the forces @ and l? win, once again, be

where N is the mass of the Sun, M is the combined masses of the Earth and of the Moon, r is the radius vector of the Moon's orbit, I is the radius of the Sun's orbit, and T is the angular separation between the Sun and the Moon

As in my firsi memoir, I have the following as the general equation of the orbit produced by forces: by supposing, firstly, that v is the angle subtended between the given radius vector r and that which is passed by the Moon at apogee, at the moment when the forces @ and rI started to act; secondty, that

expresses the conic section descrikd by the Moon in the absence of @and rI; thirdly, that is the quantity Or2 fBdv 2 +--Ir3dv M Mdv pM

or more simply

and neglecting the higher powers ofp.

Finally, I take the general equation for time to be:

by neglecting the third powers of p.

This stated, I will not content myself, as I had in my first memoir, in supposing that in the value of Q, nor in the value of T for the time, the quantity

but, rather, I will offer the entire calculation as I indicated in the same memoirs (p.352).

I d suppose that k 2v 2 2 2 - = 1 - ecosmv + p cos-y cos(- - m) v + S cos(- + m)v + < cos(- - 2m)v. r n n n n

And for the equation of the time, or, rather, for the Moon's mean anomaly £kom which it results, the general equation is

Now supposing that

is the ratio (soit le rapport) of the mean motion of the Sun to that of the Moon, from the value of x we will get the following, by neglecting only those terms which would greatly lengthen the calculation for those terms which carry a greater exactitude, 2v 2 2 2 2 sin 2T = Zsin-gsin(- + m)v + 5 sin(- - m)v + asin(- - 2m)v - &sin(- + 2m)v + n n ~l ~l n 4 4 sin mv - r sin(- - m)v + 9sin-v. n 11

4 4 T cos mv - cog- - m)v + 9cos-v - 9, in which. We have

3rrdr P 61 2 2 2 -= -3 e sin mv - kern sin 2mv + -sin - v - 33(- - m)sin(- - m)v k3dv n n n n

with some terms neglected on account of their smallness betitesse).

We have, by means of the values: m3 1. For P or -% 2~dv(a being -, as in the first Memoir) the quantity 2~ M13

by likewise neglecting the other tern, and supposing And p=a+b+c-dte.

3mgdrsin 2T 2. For - we have: 2k3dv

+ [e]acos mv - @]a, previously having: 3r2a 3. For -- cos2T we have: 2k'

- (e)a cos mv + @)a, in which.

9'- 3 9' 9 9 9 3 3 {c) =-ea--p--e6+-eeg--lr+-~4 4 4 4 4 9+-72 9--~r,4

3 9! 9 9 3 {d) =-a6+-eg+-eeZ+-ar--79, 2 4 4 4 2

3 9' 9 9 3 3 {p} =-a9--er+-l;i--~5--~ 5--76. 2 4 4 4 4 4

AU these quantities being determined, the value of

r3a 3r3a ---- cos2T - 3rradrsin2T -2P, that is, of the first factor of the value of Q, 2k3 2k3 2k3dv and which differs fiom the true value of inasmuch as 1-2 P+4PP differs fiom 1, is

The second term of the preceding is incorrectly stated in the muscdpt That is, the mvluscdpt enq is: - Ba cos(: - - mv1 which does not accord with the equadon lor P below. Therefore, in order to have the whole value of R, it is no longer a question of doing anything but to multiply the preceding quantity by 1 -2P+4PP; or having already

one will easily derive fion~,by neglecting those terms which can be neglected,

(2pi + k)cos(i - 2m)v + (ab + ac - 2ep - bd)cos rnv, that which gives

+ 2d'a co(: - 2m)v - 2c'a cos mv, a' = a + 2a(2ap - be - ce), b' = b + 2a(2bp + de - ae), c' = c + 2a(2cp - ae),

Thus, having the two factors of Q, it is no longer a question but to multiply them, such that, by supposing,

C' = Cot + (2c'P - Ae' - ~a')a,

D' = DO'+ (2d'P + Be' + b'E)n,

E' = Eo' + (2e'P - Ab' - a'E3 + Bd' + Db' - AC'- caf)a,

P' = Po' + (a'~+ Bb' + Cc' + Dd' + &')a, we get

2v f2 = -Afacos- - Braco{? - rnjv - Cracos(% + rn)v + D'acos(; - Zm)v - Etacos w + P'a :: n for which substitution in the following general equation llc sin v -=--- cow + -IR sin vdv, will give an equation which will be rendered equal to r P P P thesu~~ion k 2v 2 2 2 - = 1 - ecosmv + flcos-y cog- - m)v + Gcos(- + m)v + Ccos(- - 2m)v, by the r n n n n conditions given by the equations,

Regarding the equation of time, we get it by substituting in the following general equation nd(l- P + PP) , for p and r their respective dues, which I do m the following k2 way. First, I suppose

or (1-ecosmv+~)-~into (1-eco~m)y-~-2~(1-ecosm)v~~+3~~(1-ecosrn)v~, or cos2mv in which

Having made this transformation, I reduce easily

whose solution requires us to detemrine ZP and 2'. r=Ty-26--B--c--B--y.aabbc d 222 2

Regarding PP, its value has already been found, and thus we need not do anything but reiterate it, observing only, in order to bridge the calculation, to make

x = a(2pd + be),

Therefore, substituting these values of ZP,PP,Zzin preceding value of

\ / \ 5. - - 3, 1 =a+~~rr-eae-6ae(-3era--er~a+3ava+2a;lcr+~~~cr;2 2 we will fblly have for the expression sought for time, the product of the constant, the product of the constant Z - i k3 dPM by the quantity 3 20-~ea+2Epa--eeca+l2eav+6eA.a 4 sin mv im +Spa-2ara-fiim -3aa4 2 3 \ I + {'e 2 -eca+-eepa-6e@-3era--ena- 1, sin 2mv 2 2 2 sin 3mv + e -- {I :ern} 3mT

I 1 ZatZaB-3ey+Eba.+36e+Zca+3eeC- 2 I I sin 3 3, 3 ,, -v - -eeda-3eaar+-e&z-3epa+- n 2 2 2eoa+ 2~ - 1 I 3, I 12apa + ava in which the coefficients of sin mv, sin 2mv, sin v. must be equal to be, ge'. ha q a. since in seeking T we have supposed x, or the mean longitude. to be equal to

2 v + be sin mv + ge2 sin 2mv + ha sin - - m)v-qnrin-v. ( n

It will be shown by the substitution in numbers, that the terms in the new expression of time. more so than in the original expression to determine T, are so small that they can be neglected in this determination.

In order to determine the numerical constants' of the terms of the preceding equations, I need only two astronomical quantities: one is the ratio of the revolution of the sun to the mean periodical revolution of the moon and the other the eccentricity of the lunar orbit. The former of these two is determined so simply by observations that the manner in which it is used does not bear any difficulty. To make use of it, it is necessary to do nothing except to equate the number 0.005595 18, that-- is the squared ratio of the i ika mean periodic times of the moon and the sun to the quantity -, which, according to P the preceding calculation, expresses the same ratio.

Regarding the lunar orbital eccentricity, it is much more diflicuh to use because of the manner by which astronomers have considered the motions of the orbit of the moon not being in accordance with the preceding theory, a particular and delicate discussion is necessary in order to discover the numerical value of the quantity denominated e above, which expresses the eccentricity which one can consider to be the base or the directrix of the lunar orbit. In the meantime as we wait for this research, one can, without committing a considerable error, even less so for the terms which follow the first two terms in the orbital equation and in the expression of time, consider e as the mean eccentricity of the moon which, according to Mr. Newton, is equal to 0.05505.

Kaving thus chosen these two quantities, still it is a very long and laborious research for determining by their means the coefficients of the two preceding equations, and the calculations would perhaps be entirely repugnant if one were not to use the method of false position

By a first operation regarding which I, here, give the results while waiting for the completion of the second operation, I find that the equation of the orbit would be

""Pour dhminer en nombres les constantes - ." k 2 2 - = 1- 0.05505 cos mv + 0.007 179cos -v - 0.0 1 1 18 1CON- - m)v

And the value of the time, or, rather, of the mean longitude x, compared with the true longitude v, is

2 2 v + 0.1 1206 sin mv - 0.009 167sin -v - 0.0007 19 sin(- + m)v + 0.00224 1sin 2mv n n 2 2 + 0.022684 sin(- - m)v + O.OWO3 84 sin(- - 3m)v + 0.000055 sin 3mv 11 n 2 2 - 0.00 1388sin(- - 2m)v - 0.0000383 sin(- + 2m)v. n El

It can be seen from these two equations that the solution given m the preceding memoir, although less than perfect because of the quantities which we neglected there and to which we drew attention here, would not diverge much f?om the truth with respect to the coefficients.

But it is on an important point on which this new solution differs essentially fiom the fbt: it is the determioation of the quantity rn which gives the motion of the apogee. The term cosmv in the expression R, and which gives the term of the same kind in the value of r, by which m is determined, is approximately doubled by the addition of the terms

2v 2 B cos -- y cos- v + etc to the value 1- e cos rnv , with which one was satisfied in the n n first solution, and by this means the motion of the apogee is found to codom well enough to obse~ationswithout supposing that the moon is drawn toward the earth' by any force other than the force which acts inversely as the square of the distance; and there is reason to think that if all the considerations 1 have omitted here were to be carried out, the slight merence between theory and observation would be altogether ehhated

What gave rise to the error in my first memoir was the attention given to some v 2 k terms such as B cos- ,y ms(- - m)v in the value of - which introduce products such as n n r llc By,ay, etc. in the equation - = - - -cosv-- Sin ff?vdv- etc. and I believed this rPF P attention to be superfluousus.These terms are in ktof very d consequence in the majority of the terms of this equation, but as m the one assiwed with cosmv, and which

Ckutuses the erpressioa "sans supposer la Lune poussie vers la Ttne .. . ". The Lidtransladon wd be: 'bcrithout supposing that the moon is pushed toward the earth...". 176

3ea would be but , without using the terms regarding which a discussion bas just ZP(= - 1) 3 been given has a numerator -ea which on its own is small enough. it happens that the 2 sum of the terms carrying ay ,czB, etc. which are joined there, nearly double it. Appendix B

Clairaut, 1748 Memo ires

Demonstration ofthe Fundamental Proposition of My Lunar Theory

When I wrote, in the volume of 1745, the theory of the moon, concerning which I just gave a supplement, I suppressed the demonstration of its hdamental proposition, either not to enlarge the 15 March volume in which it was to appear or because the solution which led me to the result of this proposition had not yet achieved a degree of simplicity of which I anticipated it was susceptible. No longer restrained byIl either of these reasons, I have decided to voluntarily offer the demonstration of this proposition, as it has become evident it would give pleasure to the Geometers who have read my theory.

Lemma. Given any straight line, take two hfbhely small and equd parts Mm, mn, and drawn f?om the points 02 m, n to a given point T, the straight Lines TU Tm, Tn, I say that one will have

and

As I have already given this lemma along with its demonshation in 1742, on the occasion of a problem in dynamics, where it was necessary, I will content myself here in referring the reader back to this demonstration. PROBLEM I

We require the equation of the meMmO described by a body thrown with some (any) speed and according to some (any?, whatever?) direction, and supposing that this body is subject lo the action of hue forces, Z which is directed toward the centre T, the other II perpendicular to this direction. Having extended one of the given sides Mm of the cweto n such that mn=Mm, we take, on the straight line nT drawn to the centre of the forces, the small part no to descni the effect of the force Z toward the centre ,and on the perpendicular to nT at o the small pas 03 to describe the effect of the force ll ; by these means mp will be the side of the curve requested, subsequent to the side Mm.

That done we will suppose the radius vector Tm ...... =r

The angle which the radius makes with an axis TB by a given position ...... =v

The idkitely small time used to traverse each side Mm, ma ...... = dx

By the preceding lemma we have

and and as Tp = r + 2r + ddr. and

it is clear that the small straight line no wiU be expressed by

rdv2 - ddr, in the same, the angle oTC by the expression

and consequently the straight line OF wiU be expressed by

As the spaces traversed have for expression the same forces muhiplied by the square of the times we get the equations,

rddv + 2drml= kk2,and

for determining as much the curve Mm, as the time required to traverse it.

PROBLEM II

Supposing that the first ufthe two acceleratingforces, one which urges toward the centre M T, is composed of o part -, inversely proportional to the square of the distance, and rr any other part 0 whatever; we intend,

I. To express the curve Mm by a single equation delivered fiom the element of time.

2. To make this equation in such a way that it is composed of a part in M which one recognised the conic section which the sole force - would rr have described and another part separated. which contains the necessuv corrections for the forces 0and l7.

3 To find the expression ofthe the needed to traverse any part of the curve.

By the preceding problem I have the folIowing two equations

rddv+2drdv= IT&' and

I multiply the terms of the first by r then divide by to get,

whose integral is

f being a constant added in integration.

Next multiplying the terms of this equation by nrdr, it becomes whose integral is

(there is no need here of a constant) f?om which one gets

and consequently

which will give the value of the time once the curve is determined.

Returning now to the equation

give it the form

m order to make those differentials we want to be constant. Choosing dv as constant, and ht dr substituting for -and -, their vahes which are dx dk

one will get which I wite thus

by putting for f 2d~its value of LIr3dv. Nea I transform this new equation into

dv'

f2 dds Letting -= 1- s7 the equation reduces to + -+ Q = 0, which I integrate in the Mr dv foUowing way. I mdtiply it by dv cos v .and it becomes ~~COSV+ sdvcosv + R dvcosv 289 whose integd is

where g is some constant.

Next I multiply this equation by -* , ( which is the same as &tan V) ), and I have COS v-

a5 sdv sinv + + * indvfos= gh COSY (COSV)~(COSV)~ (cosv)' ' whose integral is

s+ sinvIRdvcosv- cosvIRhsinv = gsinvc qcosv, which after replacing s by its value 1- L, becomes Mr

M and expresses the sought after equation of the cwedescribed by the forces -+ and rr n.

The first part

dds cosv - +s&cosv+Rducosv B9 The original had the following dV which is clearly a tppographicd error. --f2 - I- gsinv- qcosv Mr of this equation expresses the conic section which would be described by the single force

and giving it this form

One can easily see that the focus must be at T ,p or ff- ,is the semi-diametre of the M major axis; c or ,/-, the ratio of the eccentricity to the semi-major axis is in the Line TC, determined by letting the angle CTB equal to the angle Q, whose sin is g 290 J&g+qq -

2gOTheoriginal is a bit confusing here. ClaLaur claims: "en& que le position de ce grand axe est dam la Iigne g TC, deteanioie, en faisant l'angle CTB tigal i l'angle Q dont le sinus est 3F ." I subsatuted a + cc) With respect to the second part of this equation sin v I R hr cos v - cos v I t2dv sin v which expresses the correction necessary to make to the value

when one wants to regard the forces II and @ , it is evident that it will immediately supply, and without mything neglected, the required correction, when 0 and ri would be expressed so that R will depend only on v and on constants; and so that it wiU fUrnish a means of knowing this correction by approximation, whatever the values of ll and 4 are, provided we initially know the orbit approximately because it will only be necessary in this case to substitute for r in t2 ,its value derived fiom the nature of the supposed orbit. Appendix C

Detailed Sketch of Clairaut's Proof.

I will refer to the first memoire, "Concerning the Orbit of the Moon," as Memoire

(a) and to "Demonstration of the Fundamental Proposition of My Lunar Theory," as

Memoire (b).

The god is to develop a general equation for the curve which would be described by a body affected by the action of two forces, Z directed toward a centre, T, and l3 drawing the body away perpendicularly from the radius.'9 '

He let Mm be that part of the cwe described in an infinitely small time, & Letting

Mm=rnn, n is the point where the body would be after another Welysmall time, dr, if the accelerating force ceased to act at the end of the first instant. The composition of forces II and Z gives us mp as the part of the curve descnid in the second instant (Mmis the part of the curve descnid in the first instant.) At m the body is under the influence of an accelerating force. In the absence of such a force the body would &'fly off' tangentially

toward n. The composition of the radial and transverse forces wiu result in the body being

held in orbit about T by being deflected toward p .

Clairaut then stipulates the following: r is the radius vector TM. v is the angle the

radius vector makes with axis TB (i-e., the longitude), and dx is the infinitely small time

used to traverse Mm and rnCI?92 Clairaut needs to make the assumption, in its application

to the Moon, that this body moves in an orbital plane.293 Clairaut begins with the

following lemma

Lemma:

Tn = Trn + 2d(TM) + TM[~MT~]'(Lemma a)

By Lernma (a), we have

~n = r +2dr+rdv2

and by Lemma (b), we have

Since Tp = r + 2dr + ddr and mTp = dv + ddv, then

3' 1 have kept with Clairaut's notation although it would simplify matters a bit to have labelled time as dt instead of ds. Clakaut made use of a lemma he derived in an earlier memoire, ,+ccording to Waff (1976) his memoire was "Sur quelques PGcipes qui donnent la Solution d'un grand nombre de Problhes de Qnamique," (translation: "On Some Principles which Offer Solutions to a Great Many Problems in Dynamics"). Mmoim & fMk2by& uh JL.imar, Park, 1742, published 1745, Section XXVI. s3 Curds Wdson has pointed out that a key difference bemeen ClaLaut's solution and later, Euler and d',Uemberr's solution is this assumpaon of orbital plane, D'Alembert had pointed out that the moon's orbit had double curvature and the more appropriate assumption (the one used by both himself and Euler) would be to project the moon's orbit onto tfie elliptic- no = Tn -Tp =r+2dr+rdvZ-r-2dr-ddr = rdv' - ddr

and consequently op = rddv + Zdrdv.

Recd that the spaces traversed equals the product of the forces and the square of the time. Under the influence of II and Z, the body travels to p and in the absence of these

forces the body would be at n. So OK the component due to the action of the transverse force n, and no, the component due to the radial force Z, are respectively:

rddv + 2drhr = lMx2 (1)

rdv2 - ddr = z&* (2).

Equations (1) and (2) could be used to determine not only the curve Mm but also the time needed to traverse it. These equations are precisely the sought after god

Clairaut now moves on to a second problem. After first supposing that Z, which tends toward the centre, is composed of two parts, ;;M (an inverse square centripetal force) and a perturbing force a, Clairaut sets the following as his goals:

1. To express the curve Mm by a single equation delivered from the element of time.

2. To make this equation in such a way that it is composed of a part in which one M recognises the conic section which the sole force - would have described and rr another part separated, which contains the necessary corrections for the forces Z and II . 3. To find the expression of the time needed to traverse any part of the curve. The third goal is achieved first.

Clairaut's solution is to show that XZ, the sum of the radial forces, is composed of an inverse square variation plus some other term and, furthermore, if the other term approaches zero, the radial force then is an inverse-square variation and any perturbative effect is due to the transverse force. For the earth-moon system this means that Newton was right in Prop. 111 Book El to claim that the small fonvard precession of the lunar apsides was due to the effect of the sun on the moon and subtracting this effect left the centripetal force drawing the moon toward the earth as proportionid to the inverse-square of the distance. Clairaut does this by letting Z be composed of an inverse-square

1w variation, rr ,and some other part, 0 . Equation (2) now becomes

In equation (1) Clairaut multiplies each term by 2 to get:

rrddv + 2rdrdv = mdx- dx

Integrate this to get

rrdv -=f+ Jndx dx where f is a constant of integration. Multiplying (3) by 17r& gives

and integrating, The next two Lines were not included in Clairaut's proof but seem to be the way he proceeded. First double both sides of this last equatioa rearrange and add f to both sides to get:

Note that the left side is a quadratic. Taking the square root of both sides we get:

The left side of (4) is the right side of (3). Therefore,

rrdv dx = df'=-

m3dv Let p = I-, f

rrdv dx= fJ1+2p

This is the value of the third stated objective above. Clairaut can now substitute this value into equation (2a).

First note the following:

-=-.-=dr dr dv fdrJl+2p dx dv dx rrdv Recall

(2a) can be re-written as

rdv' M

dv dr Choose dv as constant and substitute values for - and - (i.e., * and **) to get & &

or (using the product and quotient rules on the second term)

which can be re-written as

and

Substituting this, we get -- dv' - + . ------. 1 + --- + --- Mr Mi' Mr3 - M Mdv

ah mdr --- + ------2~ Let L? = M Mdv and substitute to get: 1+2p

f dds Now, let -= 1 - s and substitute to get: s + -+ R = 0. Mr dv

Multiply by dvcosv:

sdvcosv + ddscosv + f2ivcosv = g and integrate

ds -cos v + s sin v + Lkivcosv = g, where g is a constant. dv I

dv Next, we multiply this equation by (or what amounts to hv): (COS v)~

ds + sdvsin v + gdv whose integral is: cos v (cos v)~ (cos v)* (cos v)~

or, multiplying by cosv , f" Now substitute s = 1 - - back in to get Mr

f which is rearranged to solve for -: Mr

M (7) is the equation Clairaut sought for the curve described by the forces -and l7. rr + 0

The left side of (7) dong with the first three terms on the right side defme r with respect to a kedelliptical orbit. This can be rewritten as:

Now, recall the following identity: cos(a - b) = cos a cos b + sin a sin b. Then, This is the equation in polar cosrdinates for a conic section which the force, ;;, M acting alone would have the body describdP4 Clairaut notes that the focus must be at T, the length of the semi-major axis is = p , c or 4- is the ratio of the eccentricity to the length of the semi-major axis, and that the position of this axis is along TC.

The second part then carries the perturbative effect and this can be explained through approximation The other two terms of equation (7),

represent the correction to the orbit (equation (8)) due to the perturbative effect of forces

With respect to the second part of this equation

sin v 1Q dv cos v - cos v 1ndv sin v which expresses the correction necessary to make to the value

P 1 - c cos(v - e)of - when one wants to regard the forces r n and @ , it is evident that it will immediately suppiy? and without anything neglected, the required correction, when 0 and II would be expressed so that Q will depend only on v and on constants; and so that it will fUrnish a means of knowing this correction by approximation, whatever the values of ll and O are, provided we initially kzmw the orbit approximately because it will only be necessary in this case to substitute for r in R , its value derived fiom the nature of the supposed orbit.

All this is by way of showing the fundamental proposition of his lunar theory. In order that R depend solely on v and on some constants it was necessary to express (O and ll in

294 Recall that in equation (2), C, the centrally directed force, was assumed to be composed of an inverse M square variation, -;r, and some other p- @. That component of the centrally directed force which is terms of v. To reduce C2 to a useable expression in (6) it would dce,CIairaut claims, to

P know the orbit approximately. Now equation (8). - = 1 - c cos(v - Q), will not suffice as r M it is the equation in polar co-ordinates for a conic section in which the force, -, acting rr alone would have the body describe. In short, (8) is the equation of a fixed ellipse such that the perturbing forces are absent. In just three revolutions the location of the moon at apogee would be out approximately nine degrees.

Clairaut states earIy on in Memoire (a) that

I will not content myself, as I had in my first memoir29s,in supposing that in the value of Q, k r = 1-ecosmv' nor in the value of T for the time, the quantity

but, rather, I will offer the entire cdculation as I indicated in the same memoirs (p.352). I will suppose that k 2v 2 - = I - ecosmv + pcos--y cos(--m)v r n n 2 2 + ~COS(-+m)v + ccos(-- 2m)v. n n And for the equation of the time, or, rather, for the Moon's mean anomaly fiom which it results, the general equation is

The idea here is to test whether in the equation

inverse square would have the body describe the equation in polar co-ordinates of the conic section above. 295 Clairaut is here ref* to the Memoire of 2745. k 2v 2 2 -= 1-ecosrnv +pcos-ycos(--m)v+6cos(--m)v n n r n (9) 2 2 + 6 cos (- + m)v + 5 cos (- - 2m)v. n n the third through the last terms on the right were delivering the full apsidal motion. Notice that the expression

is the expression of a rotating ellipse where k is a parameter of this ellipse and e is its eccentricity, v (as we have already seen) is the angle that the radius vector makes with the line of apsides, and m is a yet to be determined constant dowing for the motion of the apse.

In an earlier rne~noire~~~Clairaut obtained for the radius vector the expression

With a value of d.9958036, Clairaut was able to calculate the motion of the Lunar apse in one sidereal revolution to be 1"30'38''. That is, 1-m=O.OO4 1 964, which means that in one complete revolution the apse will move (1-m)360°=l O30'38". This, of course, is only half of the observed motion297Notice, though, that the last three terms of (1 1) are small in comparison to the other terms that would at first appear to confirm the goodness of

296 Waff (1995), page 45. 297 Waff (1995), page 45. equation (1 0). But with the anomalous result obtained Clairaut undertook a method to obtain more precise values of the variables in (I 1).

In Memoire (a) we have Clairaut's final equation:

k 2v 3 2 -= 1 -ecosmv tpcos--ycos(=-m)v+6cos(--m)v r n n n (9) 2 2 + 6 cos (- + m)v + C cos (- - 2m)v. n n The htlevel of approximation yielded the following expression:

Notice that the coefficients of (13) do not differ appreciably fiom those in (1 I). At the time of the public address to the assembly, Clairaut assumed that the value of m, and thus the mean motion of the apogee, would not be much affected by this refined equation of the moon's orbit because of the small contriiutions (due to the smallness of the coefficients) of the new cosm terms in (1 1). CIairaut moves on to claim:

But it is on an important point that this new solution differs essentially fiom the first: it is the determination of the quantity rn which gives the motion of the apogee. The term cosmv in the expression a,and which gives the term of the same kind m the value of r, by which m is determined, is approximately doubled by the addition of the terms 2v 2 Bcos--ycos-v+etc. to the value 1-ecosmv, with n n which one was satisfied in the first whtion, and by this means the motion of the apogee is found to conform well enough to observations without supposing that the moon is drawn toward the earth by any force other than the force which acts inversely as the square of the disbnce; and there is reason to think that if aI1 the considerations I have omitted here were to be carried out, the slight difference between theory and observation would be altogether eliminated. Appendix D

The LUNAR orbit. by Pierre J. Boulos

The value of m is fixed by observation. (Clairaut uses 'm'as the coefficient of v; what I calling 'mfis denoted as In' in Clairaut's work.) For Newton and Clairaut et. al. it is the ratio of the sidereal measured in days to the measured in days. (27.32166/365.257)

The constant c which would allow for the motion of the Iunar apse is calculated by approximation. The true value of c according to Cook is 0.99 1427427. So (1- c)=0.0085725730 yielding 3.08612628 degrees of precession per revolution or 3 degrees 5 minutes 10 seconds. The first approximation yields just under half of the observed motion of the line of apsides. That is (1 -c)*36O yields a precession of 1 degree 30 minutes 38 seconds, > ActualPrecession:=3.086 12628;

precess := 1.5 106608

Notice the second approxiamtion (to cl) cubes the already small v&e of m. The assumption Clairaut, and Newton before him, made was that this is negligiile. Already, though, (lcl)*360 yields a precession of 2 degrees 34 minutes 12 seconds, or 83% of the observed precession > precess I :=( I -c l)*360;

PrecessionAccounted := 83.27629743

The next value of c I wish to include (c2 below) yields nearly the fUprecession. ie., ( 1- c2)*360= 2 degrees 55 minutes 42 seconds, or 95% of the observed amount. > > c2:=1-(3/4)*mA2-(225/32)*mA3-31-80*mA4;

PrecessionAccounted := 94.888 706%

The next iteration involves a term, which takes m to the fifth power. Precession per revolution=3 -0376675 degrees or 3 degrees 2 minutes 15 seconds, or 98.4% of the observed amount. > c3:=1-(3/4)*mA2-(225/32)*mA3-31 .80*mA4-129.64*mA5;

PrecessionAccounted := 98.42978622 Now moving along to the next term with a power of 6. Precession per revolution=3.0705659 degrees or 3 degrees 4 minutes 14 seconds. This is very close to the actual value (99.5%). > c4:=1-(3/4)*mA2-(225/32)*mA3-31.80*mA4- 129.64*mAS-521 .75*m%;

Recall that e is the eccentricity of the orbit. Newton fixes this at 0.05505. I have multiplied this by a factor of 10 for graphical purposes only. At the correct value it would take many iterrations to see the rotation of the ellipse (I haven'tfigured out how to change scales or perspective m Maple yet to use the correct value).

I will now plot 3 equations r, rl, r2 each corresponding to the different approximations of c (note: eccentricity is greater for illustrative purposes only), 1 r 1 := ------1 + S505 cos(.9928610787 phi)

Notice how much the next example precesses over the same number of revolutions. Since the difference between cl and c2 is not great, the third and final example precesses only slightly more than the second. 1 J-4:= ------1 + -5505cos(.9914706504 phi)

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