Newton's Discovery of Gravitation

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Newton's Discovery of Gravity

Author(s): I. Bernard Cohen

Source: Scientific American , Vol. 244, No. 3 (March 1981), pp. 166-181

Published by: Scientific American, a division of Nature America, Inc.

Stable URL: https://www.jstor.org/stable/10.2307/24964334

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https://www.jstor.org/stable/10.2307/24964334

PORTRAIT OF ISAAC NEWTON was painted by Godfrey Kneller in 1689, when Newton was 46. Four years earlier Newton had devel-

166

oped the concept o f universal gravitation. Newtou's principal work Philosophiae Naturalis Prillcipia Mathematica was published in 1687.

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Newton's Discovery of Gravity How did he come to develop the concept that marked the beginning of modern science? In essence he did so by repetitively comparing the real world with a simplified mathematical representation of it

The high point of the Scient ific Revolution was Isaac Newton's discovery of the law of universal gravitation: All objects attract each oth er with a force directly proport ional to the prod uct of their masses and inverse ly proportional to the square of their separation. By s u b s uming under a single mathematical law the chief physical phenomena of the observable universe Newton demonstrated that terrestrial physic s and celestial physics are one and the same. In one stroke the concept of universal gravitation revealed the phys ical significance of Johannes Kepler's three laws of planetary motion, solved the thorny problem of the origin of the tides and accounted for G alileo Galilei's c urious and unexplained observation that the descent of a free-falling object is independent of its weight. Newton had achieved Kepler ' s goal of developing a physics b ased on causes.

The momentous discovery of univer sal gravitation, which became the para digm of successful sc ience, was not the result of an isolated fl ash of genius; it was the culminat ion of a series of exer cises in problem solving. It was a prod uct not of induction but of logical de d uctions and transformat ions of exist ing ideas. The discovery of universal gravity brings out what I believe is a fundamental character istic of all great breakthroughs in science from the sim plest innovat ions to the most dramatic revo l ut ions: the creation of something new by the transformation of existing not ions.

N ewton developed the concept of uni versal gravity in the first few months of 1685, when he was 42. Phy sicists have usually made their greatest contribu tions at a much earlier age, but Newton was st ill in what he called "the prime years of my life for invention." The doc uments that have enabled me to date the discovery also make it possible to reconstr uct the process that led to it.

A decisive step on the path to univer sal gravity came in late 1679 and early 1680, when Robert Hooke introduced Newton to a new way of analyz ing mo tion along a c urved trajectory. Hooke

by I. Bernard Cohen

had cleverly seen that the mot ion of an orbiting body has two components, an inertial component and a centripetal, or center-seeking, one. The inert ial com ponent tends to propel the body in a straight line tangent to the curved path, whereas the centr ipetal component con tinuously draws the body away from the inertial straight - line traject ory . In a sta ble orbit such as that of the moon the two component s are matched, so that the moon neither veers away on a tan gential path nor spirals toward the earth.

The concept of a centripetal force re placed the older and misleading not ion of a centrifugal, or center-flee ing, force. Rene Descartes and Christ iaan H uygens had analyzed c urved mot ion in terms of such a centrifugal force. Descar tes, for example, had invest igated the move ment of a ball on the inner surface of a hollow cylinder and the movement of water in a b ucket swung in a circle. The ball and the water seemed to flee the center of the system, and so Desc artes attributed their motion to the infl uence of a centrifugal force. It is now clear there is no such force; a center-flee ing force cannot be traced to the interaction of physical objects. The illusion of a centr ifugal force comes about when a moving object is viewed from a rotat ing frame of reference.

W ith the change in outlook from centrifugal to centripetal force came an apprec iat ion of the fundamen tal role of the central body. The centr if ugal analysis had focused on the revolv ing object, whose "endeavor to recede" from the center seems to be independent of the properties of the central body. The concept of a centripetal force, in contrast, depends fundamentally on the central body, toward which the revolv ing object is impelled or attracted. The inter act ion of the central, attract ing body with the revolving, attracted ob ject can obviously be expected to have a part in any theory of gravitat ion.

Hooke ' s analysis of c urved mot ion may seem to be such an obvious and immed iate consequence of the C arte sian principle of inert ia that N e wton

would not have needed Hooke to in str uct him on the s u bj e ct as late as 1679. Newton had more or less accepted the inertial principle some 20 years earli er. N e verthe less, N e wt on, like Descartes and Huygens, was so mired in the con cept of centrifugal endeavor that the full implicat ions of inertial physics were far from obvious to him.

On N ovember 24, 1679, Hooke wrote to Newton suggesting that they engage in a private "philosophical" correspon dence on scient ific topics of m ut ual in terest. Six years earlier they had clashed p u b licly over N e wton's experiments and theories on the prismatic dispersion of light and on the nat ure of color. Hooke was only one of several invest igators who had rejected N ewton's opt ical theo ries. N ewton was so vexed at having to defend his work that he vowed to aban don "philosophy" (physical sc ience) be cause she was "so litigious a lady" that a man who had anything to do with her would have to spend the rest of his life defending his op inions.

Hooke had since become secretary of the Royal Society of London. In spite of the earlier controversy his letter to New ton was friendly and gracio us. The letter invited Newton to comment on any of Hooke's hypotheses or opinions, partic ularly on the notion of "compounding the celest iall mot ions of the planetts [out] of a direct motion by the tangent & an attractive motion towards the central body." This sentence was apparently N e wton's introd uction to the idea of de composing c urved motion into an iner t ial component and a centripetal one. There is no evidence that he had yet reached Hooke ' s level of understand ing of circular motion. Indeed, N e wton still often spoke of orbital mot ion in terms of centrifugal force.

In his letter Hooke vent ured the sug gestion that the centripet al force draw ing a p lanet toward the sun varies in versely as the square of the separation. At this point Hooke was st uck. He could not see the dynamical consequences of his own deep insight and therefore could not make the leap from int u it ive hunch and g u e sswork to exact science. He

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could go no further beca use he lacked both the mathematical genius of N e w t o n an d a n appreciation of Kepler's law of areas, which figured prominently in N e wton' s subseq uent approach to celes tial d yn a m ics. The law of areas states that the radius vector from the s un to a planet sweeps out e q ual areas in equal times.

On N ov e m ber 28 N e wton wrote to Hooke that be fore reading Hooke's let ter of the 24th he did not "so much as heare (that I remem ber) of your Hy potheses of compoun d ing the celestial motions of the Planets of a direct mo t ion by the tan gent to the c urve" and an "attractive" motion toward the s un . Having a d m i t t e d t h a t Hoo ke' s analysis was new to him, Ne wton immediately changed the subject to a fancy of his own: the effect of the earth's rotation on a fre e - falling object. If a dropped object could pass through the rotating earth, what path would the object take? New ton had incorrectly concluded that it would follow a spiral trajectory.

In Hooke's next letter, dated Decem ber 9, he ca ught N e wton's error and pointed out that the path "would resem ble an Elleipse . " Hooke was eager to get N e w ton going on the problem of plane tary mot ion, and so he suggested that the correct descrip tion of an o bject falling through the earth and his own analysis of planetary motion were both cases of "C ircular motions compounded by a D irect motion and an attractive one to a center . "

On D e c e m ber 1 3 , 1 6 79 , Newton responded guardedly to Hooke's cor rect ion but did not comment on his proposed analysis of circular motion . Hooke did not give up. I n a letter written on J an uary 6 , 1 6 80, he returne d to his thesis about c urved mo tion and repeat ed the q uantitative supposition that the centripetal attract ion is in versely pro portional to the square of the distance. From this supposition Hooke concluded that the veloc ity of the revolving body is inversely proportional to the distance from the center. He then pointed out that his analysis "doth very In te l l igibly and tr uly m ak e out all the Appearances of the Heavens. " N e wton did not reply.

On Jan uary 1 7 Hooke sent a short supplemen tary letter in which he wrote: "It now remaines to know the proprietys of a curve Line (not circu lar nor concen tricall) made by a centrall attractive

power which makes the velocitys of D e s c e n t from t h e tangent L ine o r e q uall straight motion at all D istances in a Du plicate proportion reciprocally taken. " In modern term inology Hooke's prob lem can be paraphrased as follows: I f a central attr active force causes an object to fall away from its inertial path and move in a c urve, what kin d of c urve re sults if the attr active force varies in versely as the square of the distance? He concl u d e d : "I d o u b t not but that b y your exce llent me thod y o u will easily fin d out what that C urve must be, and its propr i etys, and suggest a physicall Reason of this proportion . "

Newton evidently did do almost that. He proved that an ell ipse would satisfy the con d itions outlined b y Hooke. N e v ertheless, he did not comm un icate the result of this proof to Hooke or to any one else until A u g u s t , 1 684, when he was visited by E d m un d Halley, the as tronomer and mathematic ian. Halley came to see N e wton in order to ask "what he thought the C urve would be that would be de scribed by the Plan e t s , supposing t h e force of attrac tion towar d s the Sun to be rec iprocal to the square of their distance from it. " The problem had been m uch discussed b y t h e R o y a l Society. Halley a n d Christo pher Wren were unable to solve it, and Hooke never prod uced a solution, al tho ugh he maintained he had found one .

W h e n N e w ton heard t h e question, he respon d e d immediately: an ellipse. Hal ley asked him how he knew and N e wton replie d : "I have calculated it. " N e wton apparently could not find the calcula tions, but at Halley's urging he wrote them up for the Royal Society in the small tract De Motu (Concerning Motion). In De Motu Ne wton described his wor k o n terre strial a n d celestial dynamics, in c l u d ing his ideas on motion in free space and in a res istive medium. N e wton m ust have fin ished De Motu b y December 1 0, 1 6 84 , because Halley told the Royal Society then that N e wton had rece ntly shown him the c urious treatise.

The exact progression of N e w ton's ideas in the time between his corre spondence with Hooke and his comp le tion of the fir st draft of De Motu is not d o c u m e n t e d . N e vertheless, I am certain it was Hooke's method of analy z ing c urved motion that set N e wton on the right track. Although not all historians would agree with me, I be lieve the ap proach N e w ton takes to terre strial and

NEWTONIAN SYSTEM OF THE WORLD was diagrammed by William Whiston, who suc ceeded Newton as Lucasian Professor at the University of Cambridge. The diagram is from Whiston's broadside "Scheme of the Solar System Epitomis'd," published in 1724. The planets and the satellites of Jupiter and Saturn are shown orbiting the sun under the action of universal gravity. Remarkably, Whiston also included the orbits of comets. Newton had shown that or bits of comets are ellipses or parabolas in which a vector from the sun to the comet sweeps out equal areas in equal times. Below the diagram is Whiston's translation of part of the final General Scholium of the Principia (which is from the second edition, published in 1713). There Newton wrote that "This most Elegant System of the Planets and Comets could not be pro duced but by and under the Contrivance and Dominion of an Intelligent and Powerful Being."

celestial d ynamics in De Mow, which he further developed the following spring in the first book of the Philosophiae Nat uralis Principia Mathematica, repre sents his thin king on plane tary dynamics inspired b y his correspondence with Hooke. In a fe w a utobiographical man uscripts N e wton said the correspon dence either prece d e d or coinc i d e d with his demonstration p ublished first in De Motu an d then in the Principia that an object that has an inertial motion and is s u bj e c t to an inver se-sq uare centripetal forc e moves in an e l l iptical orbit.

I t was this de monstrat ion that brought out the physical significance of Kep ler's law of ellip tical or bits (the law stat ing that each p lane t moves in an e l l ipti cal path w ith the sun at one focus of the e llipse). The modern reader may be sur prised that it was not Kepler but N e wton who revealed the fundamental nature of Kepler's laws of planetary motion. Be fore the p u b l i cation of the Principia. howe ver , these laws (which were even called hypotheses) were not as highly respected as they came to be afterward.

Kepler's law of areas in partic ular had a diminished status in the 1 7 th cen tury. Most astronomical work s d i d not e ven mention it. For example, Thom as Stree te' s Astronomia Carolina, from which N e wton copied Kepler's third law (The c u be of the average d istance of a planet from the s un is proportional to the square of the or bital period), never disc usses the law of areas or hints at its existence. M ost 1 7 th - c e n t ury astrono mers calculated plane tary positions not by the law of areas but by a con struc tion based on a uniformly rotating vec tor emanating from the e m p t y focus of the p l anet' s elliptical orbit [ see top illus tration on page 174). Since astronomers rarely employed the law of areas, it r e q u ir e d extraord inary perceptio n for' N e wton to see its sign ific ance. Ne wton was the one who elevated K e p l e r ' s law of areas to the status it enjoys today.

The very first proposition of the Prin cipia (an d the disc ussion at the b e g i n n i n g of D e Motu) d e v e l o p s t h e d y nami cal significance of the law of areas by proving that the c urved motion de scribed by the law is a conseque nce of centripetal force. The proof, which has three parts, shows how well Newton had learned Hooke' s technique of d e c om posing curved motion into an inertial component and a centripetal one .

In the first part of the proof New ton considers a body moving along a straight line with a constant v e l o c ity. The line is d i v i d e d into e q u a l intervals to indicate that the body moves e qu a l dis tan c e s in e q u a l times. A point P is cho sen at a distance Ii a bove the line of m o tion. T h e tr iangles formed by connect ing P to any of the equal intervals all have the same area because the y have e q ual bases and the same altitude Ir. By this simple analysis N e wton r e v e a l e d

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an unexpected relation between inertial motion and the law of areas.

In the second part of the proof the body moves as be fore initially , but at the end of the second interval it r e c e ives an impulsive force-a blow- toward P. Therefore in the third interval the body no longer moves along the orig inal straight line but rather along another straight l ine closer to P. N e wton again showe d b y geometry that the triangl e f o r m e d b y c o n n e c t i n g P to t h e e n d s of the traj e c tory tr aced in the second inter val has the same area as the triangl e

for m e d by connecting P t o t h e e n d s of the traj e c tory tra c e d in the third interval.

In the third part the body is given a blow toward P at the end of each inter val. As a resul t the body moves in a p olygonal path around P. Again the are a relation holds. In the lim iting case where the interval between bl ows ap proaches zero the body is su bj e c t to a continuous force d irected toward P and the pol y gonal path b e c o m e s a smooth curve or orbit. In this way N e wton proved that a c e n tripe tal force generates a curve according to the law of areas.

LETTER TO NEWTON from Robert Hooke includes Hooke's views on the analysis of mo tion along a curved trajectory. (The letter is dated January 6, 1679, according to a version of the Julian calendrical system in which the year started in March; the modern calendrical sys tem puts the date at January 6, 1680.) In the second sentence Hooke proposes that "the Attrac tion is always in a duplicate proportion to the Distance from the Center ·Reciprocall" (that is, the attraction is inversely proportional to the distance squared). As a result "the Velocity will be in a subduplicate proportion to the Attraction and consequently as Kepler supposes Recipro call to the Distance." Hooke states that this analysis explains "all the Appearances of the Heav ens." He stresses the importance of "finding out the proprietys" of curves because longitudes, which are "of great Concerne to Mankind," can be derived from the moon's curved motion.

170

The second propos ition of the Principia proves the conve r s e : M o tion in a curve desc r i bed b y the law of areas implies a centripetal for c e . With these two propo sitions N e wton demonstrate d that the law of areas is a nece ssary and suffi c ient condition for inert ial motion in a cen tral-force fiel d.

The two propositions are part of a se q uence of demonstrations that begins with the law of areas and e n d s w ith a proof that an ell iptical orbit r e qu ir e s an inverse -sq u are centripe tal for c e . This sequence of demonstrations, presented both in the Principia and in De Motu. marks a profo und discontinuity in the history of the exact sciences. The dem onstrations introdu c e d a radically new celestial dynamics based on new con cepts of force, mome n tu m , mass and in ertia and a wholly novel qu antitative me asure of dynamical for c e . The su b t i tle of K e pl e r ' s Astronomia Nova set the goal of creating a "celestial physics based on c ause s." N e wton achieved this goal, of which K e p l e r had had only a visionary gl imp se. N e ither G a l il e o nor Descartes had conceived of such a celes tial dynam ics. And the N e wtonian for mu lation left e v e n the great physicist Huy g e n s far beh ind.

From the early draft of De Motu that Newton probably wrote in N ovem ber, 1 6 84 , it is c l e ar he had not y e t d e v e l o p e d t h e c o n c e p t o f universal grav itation. The draft d iscu s s e s centripetal force directed toward the focus o f an ellipse and conclu d e s w ith the scholium, "The r e fore the major planets revolve in e l l ip s e s having a focu s in the center o f t h e sun, and radii d r a w n [ f r o m t h e plan ets] to the sun describe areas proportion al to the times, entirely as Kepler su p posed .... "

N e wton ne ither proved this scholium nor continu e d to b elieve it for long, and strictly speaking it is false. A s he soon realized, the planets do not move ac cording to the law of areas in simple Kepler ian e lliptical orbits w ith the sun at a focus. Instead the focus l i e s in the common center of mass b e c ause not only does the sun attract each planet but also each planet attracts the sun (and the planets attract one another). I f N e wton had alre a d y formulated his principle of universal gravitation, he would not have propo s e d the erroneous scholiu m.

N e wton qu ickl y real i z e d he had not proved that the planets move prec isely according to the law of elliptical orb its and the law of are as. He had only found that the laws hold for a o n e - b o d y sys t e m : a single point mass moving w ith an initial component of inertial motion in a c e n tr a l - force field. He recognized that the one- body system corresponds not to the real world but to an artifi c ial situa tion that is easier to investigate mathe matically. The one-body system reduc e s the e arth to a point mass and the sun to an immobile center of force.

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