23 Fchruary 1973. Volunme 1 79. Nutmber 40775

Newton and the Fudge Facti

Richard S. West

The Mathematical WaN

"In the preceding Books," Newtonwrote as he introduced Book III of thePrincipia (1, p. 397) "I have laiddown the principles of philosophy;principles not philosophical but mathe-matical. It remains that. from thesanme principles, I now demonstrate theframe of the System of the World.Upon this subject I had, indeed, com-posed the third Book in a popularmethod, that it might be read by many;but afterwards, considering that suLchas had not sufficiently entered into theprinciples could not easily discern thestrength of the consequences. Ichose to reduce the substanice of thisBook into the form of Propositions (inthe mathematical way), which shouldbe read by those only who had firstmade themselves masters of the prin-ciples established in the precedingBooks." The very title he had chosenfor his work. Plilo.sophiae ,zatuiralisprinicipia ,niatlcautica, had underlinedthe same theme, in evident contrast toDescartes' mere, unmodified Prinicipiaphilosophliac. In the happy phrase ofAlexandre Koyre, the dean of historiansof the scientific revolution, the universeof precision had replaced the worldof more or less. Newton's mathematicalway tLirned out to be the road thatmodern science has followed withsteadily waxing success ever since.The quLantitative thrust that dis-

tinguishes modern science certainly

Thic author is a professor in the Departmentof Historv and Philosophy of Science at IndianaUniversity, Bloomington 47401.

23 FFBRUARY 1973

did not origincate withsearch for nitthenmaticalharmony had lain behisuggested reorderinig oand the influential examlaws of planetary motiokinematics enIsuLred thaprecision in the descripphenomena would betheme of the scientiNevertheless, it is notsay that Newton consolfirmed the quantitativemodern science, if he dit. ThroughouLt the 17Pythagorean tradition.Kepler and Galileo, stwith another equally 1the scientific revolution.tradition, represented b)innumerable others. Vthagorean tradition pumetric description of Fmechanical tradition snatural philosophy oftion by seeking meciThe two traditions secevery effort to bringmony. Vortical theoriesdid not yield Kepler's laexplications of the debodies did not yieldmatics. It was Newtoithe two into harmonythe mechanical philosothe concept of forceAlthough mechanicalstrict persuasion resisteof the occult for apower of Newtonian sciquaantitative resuLlts quic

SCIENCE:

into 1h311 ion and crystallized the en]-during pattern of modern science.The role of the Principia in estab-

lishing the quantitative paradigm ofphysical science extended well beyondits dynanic explication of acceptedconclusions. Far mzore impressive was

or its success in raising quantitative scienceto a wholly new level of precision. The

tfall two classic examples of mathematicalfall science had been those mentioned

above, Kepler's laws of planetarymotioni and Galileo's kinematics. Kep-ler's laws applied to celestial phe-

Newtoni. The nomena, the accepted locus of geo-sinmplicity and metric perfection untroubled by the

ind Copernicus' flux of terrestrial events. If Galileo'sIf the heavens, kiiieneatics brought geometry down topies of Kepler's the mundane realm. it proposed idealln and Galileo's relations that were not completelyLt mathematical realized in their material embodiment.)tion of natural When someone objected to his deriva-a fuLndamental tion of the parabolic trajectory byific revolution. pointing to factors, such as the resist-too much to ance of the air, that disturb the path.

idated and con- Galileo replied (2) that such pertur-e character of battions do not "subnmit to fixed lawsLid not originate aind exact descriptioll. [Hlece, in'th century the order to handle this matter in arepresented by scientific way, it is necessary to cuttood in tension loose from these difficulties; and havingbasic theme of discovered and demonstrated the theo-the mechanical rems, in the case of no resistanice, to

y Descartes and use them and apply them with suchVhere the Py- limitations as cxperienice will teach."irsued the geo- In contrast, Newtoni enlarged the:henomena, the definition of scien1ce to include thosetrove to purge very perturbations by which materialoccuLlt supersti- phenonmena diverge from the idealhanical causes. patterns that had represented thenmed to refuse object of science to an earlier age.them into har- The Prinicipia suLbmitted the pertuLr-of the Liniverse bationis themselves to quLantitative

,ws. Mechanistic analysis, and it proposed the exactscent of heavy correlation of theory with materialGalileo's kine- event as the ultimate criterion ofn who brought scientific truth.by expanding And havinig proposed exact cor-

)phy to include relation ,AS the criterion of truth, itat a distanice. took care to see that exact correlationphilosophers of was presented. whether or not it wasd this eruption properly achieved. Not the least partgeneration, the of the Principia's persuasiveniess wasence in deriving its deliberate pretense to a degree ofkly swept them precision quiite beyond its legitimate

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claim. If the Principia established thequantitative pattern of modern science,it equally suggested a less sublimetruth-that no one can manipulate thefudge factor quite so effectively asthe master mathematician himself.

Acceleration of Gravity

Consider three specific examplesthat rank among the Principia's moreimpressive achievements. The law ofuniversal gravitation rested squarely onthe correlation of the measured ac-celeration of gravity at the surface ofthe earth with the centripetal accelera-tion of the moon. Starting from theprinciple of inertia, Newton had shownthat a system of planets orbiting thesun in accordance with Kepler's threelaws entails a centripetal attractiontoward the sun that varies inverselywith the square of the distance fromthe sun. Equally, a system of satellitesorbiting Jupiter in accordance withKepler's laws entails an inverse squareforce toward Jupiter. Before he couldinvoke the uniformity of nature asstated in his second "Rule of Reason-ing in Philosophy" and apply theancient word gravitas to this force,however, he had to show that theattraction holding the moon in itsorbit is quantitatively identical to thecause of heaviness at the surface ofthe earth. Presented initially in Prop-osition IV, Book III, the correlationof the acceleration of gravity, g, withthe moon's centripetal acceleration wasfurther refined in Proposition XXXVII.Measurements with pendulums had setg, the distance a body falls from rest

in I second near the surface of theearth in the latitude of Paris, at 15Parisian feet (= 15 x 1.068 Englishfeet), 1 inch, 1! lines (1 line = 1

inch). Starting with the measured orbitof the moon, Newton calculated thedistance that the attraction of gravitycauses it to deviate from a rectilinearinertial path during 1 minute. This fig-ure required correction by the amountof the sun's disturbance of the moon'sorbit, measured by the motion of themoon's line of apsides, yielding a cal-culated distance of 14.8538067 feet. Abody removed one-sixtieth as far fromthe earth would fall the same distancein 1 second. Since he had set the dis-tarnceof the moon at 605 times themaximum radius of the earth, Newtoncorrected the calculated fall in 1 secondaccordingly. He took the oblate shapeof the earth into account, computing

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the distance of fall at the mean radius(450 latitude), and adding two-thirdsparts of a line to correct to the smallerradius at the latitude of Paris. Hence,at Paris, bodies would fall 15 Parisianfeet, 1 inch, 43 lines in 1 second. Therotation of the earth gives rise tocentrifugal effects which diminish thefall by 3.267 lines at Paris. Hence, bycalculation from the centripetal accel-eration of the moon, -g should equal15 feet, 1 inch, 1j- lines (3). Beneath2its unsettling combination of decimalscarried to seven places and fractions ofvarying complexity, the calculation pur-ports to find a correlation accurate toa fraction of a line, a degree of preci-sion perhaps somewhat better than 1part in 3000.

Velocity of Sound

Newton's calculation of the velocityof sound falls into a different categoryfrom the correlation of g with themoon's centripetal acceleration. In thelatter case, he demonstrated the corre-lation between two independently mea-sured values. In the former, he startedfrom principles of dynamics and ar-rived at the measured velocity of sound.Because he advocated the corpuscularconception of light, we tend to forgetthat Newton carried out the first suc-cessful analysis of what we now callsimple harmonic motion. In extendingthat analysis successfully to embracethe propagation of sound, a demonstra-tion wholly without precedent whichinaugurated a new branch of theoreticalphysics, he achieved another triumphof quantitative science.The demonstration rested, of course,

on his understanding of the dynamicsof the simple pendulum-that the com-ponent of gravity parallel to the tangentat any point on a cycloid is propor-tional to the displacement of that point,measured along the arc, from the posi-tion of equilibrium. This is the dynamiccondition of simple harmonic motion.By extending the analysis to waves ona surface of water, Newton calculatedthe period of a single wave, whichyielded the time for a wave to advanceone wavelength, and hence the velocityof propagation. Boyle's Law allowedhim to translate the results to compres-sive waves propaga-ted through air. Hedemonstrated that the velocity of pulsesvaries as the square root of the elasticforce divided by the density and, com-paring the density of water to that ofair, arrived at a preliminary velocity of

sound of 979 (English) feet per sec-ond. The figure required two correc-tions. In the calculation he had assumeda medium of pointlike particles, butsince the solid particles, through whichthe pulses pass instantaneously, are offinite dimension in comparison with thespaces between them (a property whichhe called the "crassitude" of the parti-cles), the velocity must be increasedaccordingly. From the measured com-parative densities of air and water(1: 870), Newton calculated that thedimensions of the particles of air oc-cupy between one-ninth and one-tenthof linear distances in air (that is, 98 <870 < 10w). Hence, the diameter of aparticle is to the interval between parti-cles as one to eight or nine (which hethen took simply as nine), and to the979 feet per second we must add979/9, or 109, feet per second. A fur-ther correction must be made for vapor,which does not vibrate with the air andthus causes an increase in the velocityproportional to the square root of theamrount of air that the vapor displaces.That is, if the atmosphere consists often parts of air to one part of vapor,the velocity will be increased by thefactor (11/ 10) %, which is nearly equalto 21/20. Sound moves more swiftlyby this ratio through air containingvapor, that is, through the atmosphereas it is, than it would through "trueair." The two corrections brought thecalculated velocity of sound to 1142feet per second (1, pp. 382-383). Themost recent elaborate measurement ofthe velocity of sound, by William Der-ham, confirming the result whichDerham reported that Halley andFlamsteed had obtained in other mea-surements at the Royal Observatory,was 1142 feet per second (4, 5). Al-though he dispensed with fractions anddecimals in this calculation, Newtoneffectively claimed a precision of about1 part in 1000.

Precession of the Equinoxes

With the precession of the equinoxes,Newton undertook to calculate the ex-act quantity of what he treated as aperturbation of the ideal uniform rota-tion of the earth about an axis fixed indirection. In Proposition LXVI, Book I,he developed a general analysis of thethree body problem, which he thenapplied to several phenomena by ad-justing the parameters. Primarily, itformed the basis of his lunar theory;but by shrinking the orbit of the satel-

SCIEBNCE, VOL. 179

lite to coincide with the circumferenceof the planet it circled, and by replacingthe satellite in its orbit with a ring ofliquid or solid matter, he used the sameanalysis to derive the tides and the pre-cession of the equinoxes. The meanmotion of the lunar nodes, the result ofthe sun's action to change the plane ofthe moon's orbit, is 2001 1'46" per year.The mean motion of the nodes of amoon at the equator of the earth revolv-ing with the period of the earth's rota-tion would be less in the proportion ofthe sidereal day to the lunar month,and a ring of matter equal in diameterto the earth and inclined to the eclipticwould experience such a precession ofits nodes as a result of the sun's attrac-tion. Because the ring must carry notonly its own mass but the mass of theentire earth, the precession is reducedin the proportion of 4,590 to 489,813.Because the matter that occasions pre-cession is not concentrated in a ring,the mean precession must be reducedfurther to two-fifths of the value itwould have for a ring. Moreover, themotion must be reduced yet again inthe proportion of the cosine of 23 0°, theinclination of the plane of the equatorto the ecliptic. Hence, the precessionof the equinoxes arising from the sun'sattraction is equal to 9"7"'20iv annu-ally. A greater precession results fromthe attraction of the moon. From themeasured differences of spring andneap tides, Newton had calculated thatthe attraction of the moon on the earthis 4.4815 times greater than the attrac-tion of the sun. The combined effect ofboth yields a precession of 50"0"' 12v,"the amount of which agrees with thephenomena; for the precession of theequinoxes, by astronomical observa-tions, is about 50" yearly" (1, pp. 489-491). As with the correlation of g withthe moon's centripetal acceleration, hepurported to have reached a precisionof about 1 part in 3000.

Paradigm of Modern Science

So completely has modem physicalscience modeled itself on the Principiathat we can scarcely realize how un-precedented such calculations were. Be-fore Newton, only astronomy, the sci-ence that dealt with the eternal heav-ens, had realized accuracy of a similardegree. Some two decades before thePrincipia Robert Boyle had proposedBoyle's Law, one of the few precedentsto Newton's physics, on the basis ofmeasurements that frequently varied23 FEBRUARY 1973

from theory by 1 part in 100, a lack ofprecision, he said, which was not to beavoided in such "nice" experiments (6).Newton insisted that it could be andmust be avoided. He boldly transportedthe precision of the heavens into mun-dane physics, and beyond their intrinsicinterest the calculations in the Principiawere significant as advertisements ofthe new ideal of physical science thusproposed. The new ideal was intimatelyconnected with the central conceptualinnovation of Newtonian science,"force," or action at a distance, quanti-tatively defined as an element in ra-tional mechanics. The correlation ofthe moon's centripetal acceleration withthe acceleration of gravity, calculationof the velocity of sound, and the deri-vation of the precession of the equi-noxes, together with other examples inthe Principia, offered a compellingdemonstration of the descriptive powerof Newtonian natural philosophy.Or was the compelling demonstration

a cloud of exquisitely powdered fudgefactor blown in the eyes of his scientificopponents? I have chosen the threeexamples above because all of themunderwent significant modifications inthe second edition of the Principia. Ihave cited them as they appear in thethird and final edition, the basis of thestandard English translation in generalcirculaftion today. With the exceptionof insignificant modifications of num-bers in the correlation of g, as a resultof the latest French determination ofthe radius of the earth, the final editionrepeated the second edition. The secondedition, however, introduced majorchanges in the treatment of all threeproblems, changes that signally in-creased the level of apparent precisionby the mere numerical manipulation ofthe same basic body of data (7-10).The second edition of the Principia

was at once an amended version of thefirst edition and a justification of New-tonian science. The battle with the con-tinental mechanical philosophers whorefused to have truck with the occultnotion of action at a distance stillraged. The second edition made itsappearance framed, as it were, by itstwo most important additions, Cotes'"Preface" at the beginning and New-ton's "General Scholium" at the end,both of them devoted to the defenseof Newtonian philosophy, of exactquantitative science as opposed to spec-ulative hypotheses of causal mecha-nisms. By 1713, moreover, Newton'sperpetual neurosis had reached its pas-sionate climax in the crusade to destroy

the arch-villain Leibniz. Only a yearearlier the Royal Society had publishedits Commercium epistolicum (11), acondemnation of Leibniz for plagiaryand a vindication of Newton, whichNewton himself composed privatelyand thrust upon the society's committeeof avowed impartial judges. In New-ton's mind, the two battles merged intoone, undoubtedly gaining emotional in-tensity in the process. Not only didLeibniz try to explain the planetarysystem by means of a vortex andinveigh against the concept of attrac-tion, but he also encouraged others toattack Newton's philosophy. His arro-gance in claiming the calculus was onlya special instance of his arrogant pre-sumption to trim nature to the moldof his philosophical hypotheses. In con-trast, the true philosophy modestly andpatiently followed nature instead ofseeking to compel her (12). The in-creased show of precision in the secondedition was the reverse side of the coinstamped hypotheses non fingo. It playeda central role in the polemic supportingNewtonian science.

Fiddling with Sound

In examining the alterations, let usstart with the velocity of sound sincethe deception in this case was patentenough that no one beyond Newton'smost devoted followers was taken in.Any number of things were wrong withthe demonstration. It calculated a veloc-ity of sound in exact agreement withDerham's figure, whereas Derham him-self had presented the conclusion merelyas the average of a large number ofmeasurements. Newton's assumptionsthat air contains vapor in the quantityof 10 parts to 1 and that vapor doesnot participate in the sound vibrationswere wholly arbitrary, resting on noempirical foundation whatever. And hisuse of the "crassitude" of the air par-ticles to raise the calculated velocity bymore than 10 percent was nothing shortof deliberate fraud. The adjustment in-volved the assumption that particles ofwater are completely solid. In fact,Newton believed that they contain thebarest suggestion of solid matter strungout through a vast preponderance ofvoid: One of the most radical aspectsof Newtonian science was its concep-tion of matter, which approached, with-out quite reaching, the notion thatatoms are point sources of force, putforward by the Jesuit natural philoso-pher R. J. Boscovich in the middle of

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the 18th century. The calculation ofthe velocity of sound set this concep-tion of matter aside in order to adjustthe velocity upward by 109 feet persecond. In all of this, Newton did him-self a gross disservice. As I have sug-gested, his attempted derivation of thevelocity of sound from fundamentaldynamic principles was one of thePrincipia's strokes of genius, an achieve-ment wholly withotut precedent. He ar-rived at a figure, 979 feet per second,which is roughly 20 percent too low.Early in the 19th century, Laplace,building from the foundation Newtonhad laid, demonstrated that the correc-tion derives from the heat generated inthe compressions of sound waves; thecorrection is equal to the square rootof the ratio of the specific heat of airat constant pressure to the specific heatat constant volume. When Newtonwrote the Principia, the concept ofquantity of heat had not been distin-guished clearly from temperature, andthe concept of specific heat did notexist, let alone the subtlety of specificheats at constant volume and constantpressure. Meanwhile, there he stoodface to face with the hostile continentalphilosophers, his nakedness made mani-fest by an uncovered discrepancy of 20percent. The very flagrancy of his ad-justment in this case becomes evidencefor the compulsion behind the pretenseof precision in the other cases.

In the first edition he had approachedthe velocity of sound in quite a differ-ent mood. Impressed perhaps by thesignificance of his own achievement inderiving a velocity of sound that ap-proached its measured value, and cer-tainly not yet assaulted with the hostiletaunt of occult, Newton had been will-ing to leave the demonstration as anapproximation within broad limits. Thebasic features of the derivation wereset in the first edition and did not sub-sequentiy change. The numbers weredifferent, however. Using a ratio ofI :850 for the densities of air andwater, he arrived at a calculated veloc-ity of 968 feet per second. Over againstthis valuie were a wildly varying set ofmeasured velocities. Newton reportedtwo-one by Mersenne equivalent to1474 English feet per second, and oneby Roberval equivalent to 600 feet persecond. In view of the discrepancy, hechose to make a measurement of hisown in an arcade of Trinity College,adjusting a pendulum to swing in timewith the return of an echo that traveleda total of 416 feet. He was satisfied to

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determine that the echo was slower inreturning than the vibration of a pen-dulum 5' inches long and faster thanthat of an 8-inch pendulum, yielding a.velocity between 1085 and 920 feet persecond (8, p. 370). The calculatedvalue fell squarely between them, andthe first edition rested its case modestlyon that loose approximation.

Perhaps there was more concealedin the first edition than meets the eye.A faint aura of deception clings toNewton's report of previous measure-ments and to his own measurement.The structure of his presentation im-plicitly justified both the necessity ofa new measurement and its result, sinceits limiting values fell inside the broaderlimits of Roberval's and Mersenne'smeasurements. There were available anumber of other measurements thatNewton could have known about, how-ever, all of them falling well above hiscalculation of 968 feet per second. Infact, Roberval's figure of 600 was ut-terly out of line with other 17th-cen-tury measurements of the velocity ofsound. The majority arrived at a valuehigher than the one we accept (13).One cannot avoid wondering whetherNewton performed his experiment be-fore or after he made the calculationof 968 feet per second. Already inabout 1694, at any rate, David Gregoryfound him planning to change the sec-tion on the propagation of sound inorder to define its velocity "within nar-rower limits" (14, p. 384). A copy ofthe first edition in which Newton en-tered projected emendations apparentlyrecords a new measurement in thearcade at Trinity. He now stated 'thatthe echo was slower in returning thana pendulum of 5 inches and fasterthan one of 7 inches, corresponding toa velocity between 1109 and 984 feetper second (15, p. 531; 16). Thismoved his measurement closer to thegeneral range of measurements, but thelower limit was now above the cal-culated velocity of 968 feet per second.What could he do? He could fiddle

with the numbers. His calculation restedon the density of air, and the densityof air was a difficult thing to measurein the 17th century. The ratio of thedensities of air and water that he usedin the first edition (1: 850) was arbi-trary. In a sheet of "errata" in the firstedition, he tried both 1: 900 and 1: 950and found that they yielded velocitiesof 996 and 1023 feet per second. Hesettled for the higher figure. While itplaced his calculation within the range

of his new measurement, he may havefelt that it stood too close to the bot-tom limit. It is impossible to unravelevery detail in the chronology of hismanipulation of *the calculation, butthe evidence that he tried various de-vices cannot be mistaken. In additionto the density, he thought of the adjust-ment based on the "crassitude" of theparticles; it appears in the paragraphthat records his new measurement,raising the calculated velocity about100 feet per second (iS, p. 532).

It is reasonable to assume that thecorrections above were made before helearned of the measurement by hisfriend William Derham and of thework of Joseph Sauveur in acoustics,since he crossed out the amended para-graph containing his own remeasure-ment and henceforth accepted Der-ham's value of 1142 feet per second.His mention of Sauveur's work, whichcorrelated frequency with the length oforgan pipes, suggests that it served toconfirm Derham's measure in his mind(17).Now he had a solid figure on which

to exercise his mathematical talents.The annotated and interleaved copiesof the first edition in which he preparedthe corrections for the second editionshow him manipulating the "crassitude"of the particles before he finally chosethe size that allowed him to increasethe velocity by one-ninth. After he liton the further correction due to vapors,he fiddled with its magnitude as well(15, p. 533). Finally he decided to re-turn to the density of the air for thefine adjustment, and setting the wander-ing ratio at 1: 870 he emerged in thesecond edition with a calculated veloc-ity of 1142 feet per second. At onepoint in the calculation, although notin the conclusion where it belonged,he did insert a modest circiter. Thathardly covered the extent of the illu-sion he was attempting to foster.

Doctoring the Correlation

In the case of the correlation of gwith the moon, much more was atstake. The correlation was the linchpinof the entire argument for universalgravitation, and we can understand whyNewton wanted it to appear precise.Nevertheless, in the first edition he hadbeen content to present it within areasonable margin of error. After list-ing available calculations of the moon'smean distance, Newton chose 60 times

SCIENCE, VOL. 179

the earth's radius as a just average. Atsuch a distance, he calculated, themoon would fall 15'2 feet in 1 min-

12

ute; hence, at the surface of the eartha body should fall 15 2 feet in 1 sec-

12

ond, the figure Huygens had estab-lished. While the new edition was beingprepared, Newton sent the editor,Roger Cotes, a scholium to PropositionIV in which he carried out the correla-tion substantially as it finally appeared.Since it assumed material from Propo-sitions XIX and XXXVII, Newton even-

tually broke the scholium up andinserted it in those two propositions,most of it in the latter. In that posi-tion the character of the finely cor-

rected correlation is clear. PropositionXXXVII, on the tides, compares theforce of the sun in attracting the sea

to the force of the moon. A series ofcorollaries then apply the conclusion(and earlier calculations of the com-

parative masses of the sun and planets)to computing the comparative densitiesof the earth and the moon and theircomparative masses. Corollary 7, whichcontains the correlation, is actually a

calculation of the mean distance of themoon from the value of g. In Proposi-tion IV Newton had assumed that themoon orbits the center of the earth. InProposition XXXVII, with the relativemasses of the two known, he correctedthe distance of the moon by referringits orbit to the common center of grav-

ity of the two bodies. If the mean dis-tance of the moon is 60! (in edition

4

three, 60 ) times the maximum radiusof the earth, then its centripetal ac-

celeration will correspond to a valueof Lg, in Paris, of 15 feet, I inch, 5.32lines. When the measurement by thependulum is increased by 3.267 linesfor the centrifugal effect in Paris, ityields a value for 2-g of 15 feet, 1inch, 5.32 lines, slightly different figuresfrom those in the third edition with an

even higher degree of precision (18).What had Newton demonstrated? Hehad shown that he could calculate thedistance of the moon from the value ofg, and he had shown, what should beobvious, that he could start the calcu-lation with a value of g stated withwhatever degree of precision he mightchoose. Even in these terms the corre-

lation is delusory since it employs thecomparative masses of the moon andthe earth drawn from a dubious calcu-lation of the tides that I shall discuss in

a moment. Beyond this, the correlationof g with the moon in PropositionXXXVII demonstrates nothing whatever.23 FEBRUARY 1973

Indeed, Proposition XXXVII servesto indicate that even the modest cor-relation of the first edition is somewhatdeceptive. To be valid, the correlationmust start with two quantities that aremeasured independently, the distanceof the moon and g. Huygens had pro-vided the latter with high precision. Forthe former Newton had a range ofvarying measurements. Most astrono-mers, he said, set it at 59 radii of theearth. Vendelin set it at 60, Coper-nicus at 60-', Kircher at 62}, andTycho at 56'. Tycho's result wasfounded on a false theory of refraction,however, and when his observationswere corrected they gave a distance of61. From this set of numbers Newtonextracted an average of 60 whichyielded in turn a centripetal accelera-tion that correlated with a value of-g of 15 feet, 1 inch. It is difficult tobelieve that the value of 'g did not in-fluence his averaging of the lunar mea-surements. Significantly, in edition twohe omitted Kircher's measure andfound that the correction of Tycho'scame, not to 61, but to 601 times the

2

radius of the earth.When he placed it in Corollary 7 to

Proposition XXXVII, Newton did notindeed explicitly present the correla-tion as anything more than an exactcalculation of the moon's mean dis-tance. When he thought originally ofmaking it a scholium to Proposition IV,however, he seemed to be trying im-plicitly to present it as something else.Cotes, who had caught the spirit of theenterprise rather well, saw as much."In ye Scholium of IVtll Proposition,"he suggested, "I think the length of yePendulum should not be put 3 feetand 8' lines; for the descent wouldthen be 15 feet 1 inch 1 line. I haveconsidered how to make yt Scholiumappear to the best advantage as to yenumbers, & I propose to alter it thus"(19, 20). He went on to select valuesfor the distance corresponding to 1°on the earth's surface and for the lati-tude of Paris that led to a very precisecorrelation. Cotes need not have wor-ried. Newton was also considering howto make the correlation appear to thebest advantage as to the numbers. Ulti-mately he settled on different values forthe elements of the calculation, but bytreating the distance of the moon asterminus ad quem instead of terminusa quo, he reached a correlation quiteas satisfactory as Cotes'. In both cal-culations it was more public relationsthan science.

Manipulating the Precession

Newton's concern that the numbersappear to best advantage in his corre-lation with the moon, however, was asnothing beside his efforts on the pre-cession of the equinoxes. Precession, inhis treatment of it, was a quintessentialdemonstration of the ultimate advan-tage of Newtonian science over thereigning mechanical philosophy, aquantitative derivation of a perturba-tion which qualitative mechanical ex-plications did not even dare to men-tion. The very claim of Newtonianscience on the allegiance of naturalphilosophers depended on quantitativeprecision in such cases. In the case ofprecession, moreover, the correction ofa faulty lemma in edition one imposedthe necessity of an adjustment of morethan 50 percent in the remaining num-bers. Without even pretending that hehad new data, Newton brazenly manip-ulated the old figures on precession sothat he not only covered the apparentdiscrepancy but carried the demonstra-tion to a higher plane of accuracy.

Like his derivation of the velocityof sound, Newton's demonstration ofthe precession was a brilliant insighttempered by limitations that the levelof the 17th-century dynamics imposed.He identified the same cause of pre-cession that celestial dynamics still em-ploys-the attraction of the sun andmoon on the bulge of matter aboutthe equator. He lacked the conceptualtools to treat it adequately, however.The Principia itself established the ba-sic equation of the dynamics of linearmotions relating force to acceleration.Although the concept of moment waspresent in the law of the lever, dynam-ics possessed as yet no equation corre-sponding to Newton's second law thatrelated moment to angular acceleration,and it possessed no concept similar tomoment of inertia that played the rolecorresponding to mass, which was alsoNewton's creation, in the equation oflinear motions. His demonstrationtreated precession on the analogy of thelunar nodes, the motion of which heattempted to derive by extending boththe principles used to derive perturba-tions in the shape of the orbit and thequantities thus found. To understandhow limited the basic concepts relevantto rotational motion were, consider hiscorrection of the precessional motionby the cosine of 231o, the inclinationof the equator to the ecliptic. As itappears in his demonstration, the cor-

755

rection amounts to taking the compo-nent of attraction parallel to the equa-tor, just as he had used the effectivecomponent of force in computing thetides. In this case, however, in whichthe very effect depends on the inclina-tion of the equator to the ecliptic, hisprocedure treated the effect as though itwould be maximized if the inclinationwere zero. Since the result of this cor-rection reduced the effect to 0.91706of the value it would otherwise havehad, it may appear to be an applicationof the fudge factor. In this instance,however, which appeared unchanged inall the editions of the Principia and es-caped comment even by the sharp-eyed Cotes, we have to do more prob-ably with a technical limitation. Cor-respondingly, nothing in his derivationutilized the different inclination of theequator from the inclination of thelunar orbit, although the precession wascalculated directly from the motion ofthe lunar nodes.

Although the alteration involved aconsiderable refinement in his treatmentof the tides, the refinement was neces-sitated by the need to make the num-bers appear to best advantage as aresult of other changes. As I have ex-plained, the demonstration began withthe mean motion of the lunar nodes,and adjusted the value first in the ratioof the two periods of rotation and thenin the ratio of the total mass of theearth to the mass of the ring of matterresponsible for the precession. At alater step, he reduced the motion fur-ther in the proportion of the cosineof 232'0 All of this was a straight-forward matter of ratios that allowedno margin for maneuver. Two majoradjustments that remained supplied theprincipal substance of calculation. Onthe one hand, the comparison with thelunar nodes assumed the matter caus-ing the precession to be concentratedin a ring around the equator, whereasit is in fact spread unevenly over thesurface of an oblate spheroid. A sub-sidiary demonstration (or lemma) es-tablished the ratio of the further reduction thus entailed. On the other hand.the comparison with the lunar nodesinvolved only the attraction of the sun.The attraction of the moon is severaltimes more potent in causing preces-sion, however, and the ratio in this casewas established from the variation be-tween spring and neap tides. The nubof Newton's derivation of the preces-sion lay in the relation of these twofactors, and the problem he faced in

756

the new edition lay in the major re-vision he introduced in the first ofthem, the ratio between the effect ofmatter concentrated in a ring and thatof matter spread over an oblate sphe-roid. Once that revision was made,Newton fell back on the tides, whichbecame in his hands as fluid as thewater composing them.

In the first edition, Lemma I, BooklII, provided the adjustment from aring of matter at the equator to anoblate spheroid. Suffice it to say that thedemonstration is perplexed, and any-one interested in the difficulties that ro-tational motion presented to early dy-namics will find it a revealing casestudy. In attempting to compare thevis et efficacia tota of the two distribu-tions of matter to rotate the earthabout a particular axis, he employedamong other things the moment (or'"efficacy of the forces") of a vectorpassing through that axis. The lemmaarrived at a ratio of 1 : 4. Applied tothe motion derived from the lunarnodes, this ratio yielded an annual pro-cession of 6"122"'2", due to the attrac-tion of the sun. From the tides hecomputed that the moon is 61 times

3

more effective than the sun, so that thetwo together produce a precession of45"24"'1 5i'v. Clearly a further adjust-ment was needed, and Newton foundone in the density of the earth. Theearth is denser at the center; as a con-sequence, it is higher at the equatorthan the computation from centrifugalforce alone reveals, with the result thatthe precession is increased in the ratioof 10: 8' or 6: 5. When the resultingfigure was reduced by the cosine of23"°, introduced in edition one at thispoint, it yielded a precession of 49'-58"', in sufficient agreement with the ob-served value of 50" (8, pp. 467-473).

By 1694 he recognized the fault inLemma I (21). In the second editiontwo lemmas replaced it; they arrivedat a ratio, not of 1: 4, but of 2 : 5.Without any further adjustments, thenew ratio would have left the preces-sion more than 50 percent too large.

Fortunately, the other major correc-tion, the ratio of the attractions of sunand moon as established by the tides,was amenable. This adjustment workedout so well that he abandoned the ad-ditional specious correction from thedensity of the earth despite the explicitstatement in edition two that, becauseof the greater rarity of matter near thesurface of the earth, the diameter atthe equator may be nearly 15 miles

greater than the calculation from cen-trifugal force alone would suggest (22).

In all three editions, Newton reliedon the same body of data about thetides, two sets of observations publishedin the Philosophical Tranisactions in1668 in response to the Royal Society'searly diligence in collecting a completehistory of nature. In light of the con-clusions Newton ultimately drew fromthe observations, it is worthwhile tosavor their precision in their ownwords. Samuel Colepresse measured thetides at Plymouth. He found that "theWater usually riseth about 16 Foot (Isay usuially, because it may vary in thisPart from the lowest Neap to the high-est Annual Spring above 7 or 8 Foot)

(23). If the title to the paper iscorrect, the statement rested on obser-vations made during the year 1667alone. Since the Royal Society beganto solicit such information only in theautumn of 1666, there is no reason tothink the measurements were more ex-tended. The same strictures apply toSamuel Sturmy's observations of thetides in the mouth of the Avon belowBristol (24). The highest spring tidethere "flows in height about 71 fath-

2

oms, or 45. foot; the lowest Neap-tydesflowing in height 25. foot." Sturmy'spaper included a table of the flow of aspring tide measured every quarterhour through its total span of 5 hoursand of the ebb measured every hourthrough its span of 7 hours. Eachcolumn is summed the same, "45. feetcirciter"; the numbers in one add, infact, to 44 feet, 1 inch, and in theother to 45 feet, I0' inches. (Of course,the measuLrement of the total tide wasnot reached by adding the two col-umns.) Although entries in the tableinclude measurements to half inches,he remarked that making them alwaysthat accurate "is neither easie, normaterial, or usefull" (25).

In the first version of Book III, whatis known as "The System of theWorld," Newton concluded from theseobservations that the ratio of attrac-tions is I : 53. The spring tide is pro-

3

duced by the combined attractions ofmoon and sun [L(una) + S(ol)], theneap tide, when the moon is in quadra-tures, by the excess of the moon's at-traction over the sun's (L - S). Hencethe basic equation was (L+S)/(L-S) = 9/5. On this occasion, he didnot even mention Colepresse's obser-vations; the ratio of 9/5 comes fromSturmy alone. Since the moon declines2340 from the plane of the equator at

SCIENCE, VOL. 179

the time of the neap tide, whereas thetwo luminaries act in the same plane atspring tide, the force of the moon mustbe diminished correspondingly. Theequation became (L + S)/ (.841 L-S) = 9/5, or L = 51S, a ratio headjusted minimally to 53 (1, p. 593).In "The System of the World" the ratioplayed no further function; although hehad stated the cause of the precession,he did not attempt to derive it. In edi-tion one he did, and the computedamount of the sun's effect required alarger ratio. Colepresse's measurementswere summoned to the aid of Sturmy's;they produced a ratio of 7i3. Newton

3

averaged the two at 63 and extracted3

the value of precession stated above(8, pp. 464-465).

Mending the Numbers

Then came the new lemmas andthe need for a radical reduction inthe calculated precession. In February1712, Cotes' painful progress throughthe text he was editing arrived at Prop-osition XXXVII, on the tides. Appar-ently the ratio had returned to 53 bythis time, probably by preferringSturmy's measurements to Colepresse'sas the later editions of the Principiadid. After correcting Newton's calcu-lation to 5,3 Cotes noted that this ratiowould change the figures in the corollar-ies to the proposition, especially thecomparative masses of the moon andthe earth. "This alteration," he contin-ued, "will very much disturb YourScholium of ye 4th Proposition [thecorrelation of g with the moon] as itnow stands; neither will it well agreewith Proposition .39t"i [the precession]

(26). In his reply, Newton con-fessed that he had lost his copy of theamended Proposition XXXIX and didnot know how to make the further cor-rection. "If you can mend the num-bers," he added candidly, "so as tomake ye precession of the Equinoxabout 50" or 51", it is sufficient" (27).

Cotes had fully grasped the natureof the game by now; and as long asthey chose to play it, he intended toplay it well. He pointed out to Newtonthat the greater bulge at the equatorexplicitly considered in edition two,so that the diameter through the equa-tor would exceed that through thepoles by nearly 32 miles instead of1 7', raised problems. The passage inthe first edition correcting the precessionby the ratio 6: 5 because of variations

23 FEBRUARY 1973

in density had been dropped, but New-ton now proposed a brief statementthat the increase in precession due tothe added length of the equatorialdiameter was compensated by the de-crease due to the greater rarity. "Youhave very easily dispatch'd the 32 Milesin Prop. XXXIXtl . . .," Cotes com-mented with a muted note perhaps ofirony, perhaps of admiration, perhapsof both (28).He had hardly begun to see the ex-

tent of Newton's dexterity. The basicequation establishing the ratio of S to Lhad been rather crude. Newton beganto refine it. The highest tides do notoccur exactly at syzygies but somewhatafter, when the luminaries are approxi-mately 150 out of conjunction or op-position. Thus, the force of the sunmust be reduced at both its appearancesin the equation. With the new equa-tion, Newton got a ratio of 47, andin a manuscript revision of Proposi-tion XXXIX, he calculated from thatratio to a precession of 51 "58"'40'v."If the force of moon in moving thesea were to the force of the sun as 47-to 1," he continued, "(for the propor-tion of these forces cannot yet be de-fined very accurately from the phe-nomena), an annual precession of theEquinoxes of 50"40"'43"' would re-sult" (29). Later he took the cal-culation a step further. If the ratiowere 42, the precession would be50''14'''45'v (30).

Before the full implications of thisline of thought could be explored,Cotes produced a new problem. Ac-cording to Newton's lunar theory, themoon approaches nearer to the earthin syzygies than in quadratures. "Butthis allowance would increase the num-ber 47 so much as to give some dis-turbance to the XXXIXth Proposition& the Scholium of the IVth as theynow stand, unless," he added discreetly,"You think fit to ballance it some otherway, for there is a latitude in thatXXXVIItil Proposition" (31). WhenNewton seemed to demur, he presentedthe arithmetic. If the new factor wereintroduced without other changes, theratio would increase to 57. In order

7

to save the value 47, the ratio ofspring tide to neap would have to beset at 11/6, but 11/ 6 fell outside themeasurements at Plymouth and Bristol."I shall therefore leave it to Your selfto settle ye whole Proposition as Youshall judge it may best be done" (32).

Over a month passed before Newtonsent Cotes a revision of Proposition

XXXIX, a revision which not only in-corporated the varying distances of themoon but achieved the ratio of 4±

2

as well. He had discovered the secret,which lay in the fact that high tidesoccur after the luminaries are insyzygies and quadratures. The observa-tions were sufficiently vague that hecould adjust the exact angles at willto yield whatever ratio he chose. InFebruary he was reducing the force ofthe sun by 6/7 (corresponding to anangle of 1 51 0). Now he increasedthe reduction nearly to 4/5 (an angleof 172 °) and, beginning to savor thepossibility of a truly impressive demon-stration, he shifted to the decimal0.819152 (the cosine of 350, twice theangular distance between the sun andthe moon at the spring tide). With theratio of 1: 42 in hand, he went on

2

to correlate g with the moon, placingthe correlation now in Corollary 7 ofthe proposition, and with the same ratiohe calculated the precession givenabove. Cotes was not yet satisfied. Hethought that the correction for the dis-tance of the moon was incorrect, buthe hoped equally to retain what hadbeen gained. "I could wish when thewhole is settled that the proportionof 4' to 1 may be retain'J for the sakeof Proposition XXXIX. I think thereis no Proposition in Your Book whichdoes more deserve Your care" (33).Newton was willing enough; he hadfound the key. He not only acceptedCotes' correction, but in compensatingby increasing the angular distanceagain, he pushed the ratio below 42

2

and improved on the calculated pTeces-sion still more. Cotes was delighted. "Iam very glad to see the whole so per-fectly well settled & fairly stated, forwithout regard to the conclusion [sic]I think ye distance of 18' degreesought to be taken & is much betterthan 17! or 154 & the same may be

2 4

said of ye other changes in ye prin-ciples from which the conclusion is in-ferr'd" (34). The equation which hadstarted out as (L+S)/(.841L-S)=9/5 now read (35)

1.017342L + .7986355S - 915.9828616 x .8570328L - .7986355S

L = 4.4824SWith that ratio he carried through thecorrelation of g with the moon andcalculated the precession of the equi-noxes, both with an ostensible precisionof about 1 part in 3000. Some mightconsider it a rather ambitious conclu-sion to draw from the measurements

757

of a retired sea captain who hadsummed up two unequal columns to"45. feet circiter."The emendations to the second edi-

iton cast a special light on the revolu-tion in scientific discourse that Newtonconcluded. "I often say," Lord Kelvinasserted more than a century and ahalf later, "that when you can measurewhat you are speaking about and ex-press it in numbers you know some-thing about it; but when you cannotmeasure it, when you cannot express itin numbers, your knowledge is of ameagre and unsatisfactory kind: it maybe the beginning of knowledge, but youhave scarcely, in your thoughts, ad-vanced to the stage of science(36). What Kelvin had in mind wasnot what Cotes called "making it ap-pear to the best advantage as to thenumbers," but the history of the secondedition suggests that the one may fre-quently be indistinguishable from theother. Undoubtedly the new pattern ofnatural science that Newton more thanany other one man raised to dominancerested on profound epistemological in-sights. Undoubtedly the Principia dis-played them apart from the alterationsintroduced in edition two. Undoubtedlythe displacement of the world of moreor less by the universe of precisionwas a remorseless and irreversible proc-ess. Nevertheless, successful polemicsare the necessary condition of everyintellectual revolution. "I am satisfiedthat these exactnesses . . . are incon-siderable to those who can judge rightlyof Your book": Cotes wrote to Newtonas they polished the final decimal pointsof the correlation, "but y5' generality ofYour Readers must be gratified wt'such trifles, upon which they commonlylay ye greatest stress" (37). Whetheror not he considered them trifles, New-ton comprehended perfectly the natureof the polemic he deployed. Andamong the other factors that provokedLord Kelvin's dictum, and all that itrepresents in modern science, not the

'least was the fudge factor, manipulatedwith unparalleled skill by the unsmilingNewton.

References and Notes

1. I. Newton, Mathematical Principles of NatturalPhilosophy, A. Motte, Transl.; F. Cajori, Ed.(Univ. of California Press, Berkeley, 1934).[This is the standard English version in usetoday; it is based on the third Latin edition(10).]

2. Galileo, Dialogues Concerning Two New Sci-ences, H. Crew and A. de Salvio, Transl.(Macmillan, New York, 1914), pp. 252-253.

3. I. Newton, Principia (1, pp. 407-409; 482-483).The computation of the centrifugal effect usedin Proposition XXXVII is found in Proposi-tion XIX (1, p. 425).

4. W. Derham, "Experimenta & observationesde soni motu, aliisque ad id attinentibus,"Phil. Trans. 26, 2-35 (1708-9).

5. Newton also mentioned the work of JosephSauveur, whose measurement of the lengthsof organ pipes sounding known tones seemedto oonfirm Derham's measurement. See Sau-veur's articles "Sur la determination d'un sonfixe," Hist. Acad. Roy. Sci. (1700), pp. 166-178;"Systeme g6n6rale des intervalles des sons, &son application a tous les systemes & a tousles instrumens de musique," Mem. Acad. Roy.Sci. (1701), pp. 390-482; and "Application dessons harmoniques a la composition des jeuxd'orgues," ibid. (1702), pp. 411-437.

6. T. Birch, Ed., The Works of the HonourableRobert Boyle (A. Millar, London, 1744), vol.1, pp. 101-102.

7. There were three editions of the Principiaduring Newton's life (8-10).

8. I. Newton, Philosophiae naturalis principiamathematica (Royal Society, London, ed. 1,1687).

9. , Philosophiae natiuralis principia mathe-matica, R. Cctes, Ed. (Cambridge, ed. 2, 1713).

10. , Philosophiae naturalis principia mathe-matica, H. Pemberton, Ed. (G. & J. Innys,London, ed. 3, 1726).

11. Commercium epistolicum D. Johannis Collins,et aliorunt de analysi promota (Royal Society,London, 1712).

12. See E. W. Strong ["Newtonian Explicationsof Natural Philosophy," J. Hist. Ideas 18,49-83 (1957)] where the methodological posi-tions of a number of Newtonians are describedin detail.

13. See B. S. Finn ["Laplace and the Speed ofSound," Isis 55, 8 (1964)] for a list of thevarious measurements of the velocity of soundin the 17th century. A paper by a Mr. Walkerof Oxford, "Some Experiments and Observa-tions Concerning Sounds" [Phil. Trans. 20,433-438 (1698)], is interesting for its technique,which was very sinmilar to Newton's, thoughit led him to quite a different result. Hislowest single measurement was 1150 feet persecond, and he averaged 11 measurements to1305.

14. H. W. Turnbull and J. F. Scott, Eds., TheCorrespondence of Isaac Newton (CambridgeUniv. Press, Cambridge, 1961), vol. 3.

15. I. Newton, Philosophiae naturalis principiamathematica, A. Koyr6 and I. B. Cohen, Eds.(Harvard Univ. Press, Cambridge, Mass., ed.3 with variant readings, 1972), vol. 1.

16. Nearly all of the figures have been writtenover and altered-that is, fiddled' with-andit is difficult to know what numbers Newtonmeant at any given stage. See the originalmanuscript, Cambridge University Library,manuscript Adv. b. 39.1, folio 370A.To the extent that edition two employs a

different measure of the velocity of sound.it contradicts my assertion that the changesin it were manipulations of the same basicbody of data. It will become apparent thatthe new data in this case, far from justifying

the higher level of precision, only perplexedthe numerical manipulations that led to it.

17. J. Sauveur, "Applications des sons harmoniques[see (5)]. Planche XIl, inserted after

p. 436, gives a table of frequencies with cor-responding pipe lengths.

18. In Proposition XIX,- edition two, ;g is setequal to l274,- lines (15 feet, I inch, 2,lines), corresponding to a seconds pendulum3 fiet, 8X lines in length. In edition threethe seconds pendulum is quoted at 3 feet8, lines, from which Ig is calculated as 2173,lines. Newton now decided that the buoy-ancy of the air subtracted ' of a line.Hence, he adjusted the valtue of -g back upsubstantially to that of edition two, and forthe rest of Proposition XIX he employedthat figure. In Proposition XXXVII, wherethe slightlv different figures of edition threelowered the value of !g required for a goodcorrelation, he simply forgot the buoyancyof the air, which would have made the cor-relation less precise.

19. Cotes to Newton, 16 February 1711/12 (20,pp. 61-62). It transpired that the fractionI in the length of the pendulum had beena slip of Newton's pen. He had intended .

20. J. Edleston, Ed., Correspondence of Sir IsaacNewton and Professor Cotes (John W. Parker,London, 1850).

21. Newton to Gregory, 14 July 1694 (14, p. 380).22. Proposition XX (9, pp. 386-387).23. S. Colepresse, "Of Some Observations made

by Mr. Samuel Colepresse at and nighPlymouth, An. 1667, by way of Answer tosome of the Quaeries Concerning Tides,"Phil. Trans. 3, 632-633 (1668).

24. See R. Moray, "Patternes of the Tables Pro-posed to be Made for Observing of Tides,"ibid. 1, 311-313 (1666), and H. Oldenburg,"An Account of Several Engagements forObserving of Tydes," ibid., pp. 378-379.Oldenburg's squib mentioned that Colepressehad agreed to observe the tides at Plymouth,and appealed for observations from Bristol,suggesting that Sturmy had not yet been setto work.

25. S. Sturmy, "An Account of Some Observa-tions, made this Present Year by Capt.Samuel Sturmy in Hong-road within fourMiles of Bristol, in Answer to Some of theQuaeries Concerning the Tydes," ibid. 3,814-815 (1668).

26. Cotes to Newton, 7 February 1711/12 (20,p. 58).

27. Newton to Cotes, 12 February 1711/12 (20,p. 61).

28. Cotes to Newton, 23 February 1711/12 (20,p. 68).

29. Cotes to Newton, 23 February 1711/12 (20,p. 65).

30. Newton to Cotes, 19 February 1711/12 (20,p. 67).

31. Cotes to Newton, 23 February 1711/12 (20,p. 73).

32. Cotes to Newton, 28 February 1711/12 (20,p. 76).

33. Cotes to Newton, 15 April 1712 (20, p. 94).34. Cotes to Newton, 26 April 1712 (20, pp. 100-

101).35. Newton to Cotes, 22 April 1712 (20, pp. 95-

96). The calculation that Newton gave in thisletter is substantially that which appeared inthe second and subsequent editions. In hiscustomary careful manner Cotes recalculatedit, altering the numbers slightly. The finalnumbers were his.

36. W. Thomson, Popular Lectujres and Addresses(Macmillan, London, ed. 2, 1891), vol. 1,p. 80.

37. Cotes to Newton, 23 February 1711/12 (20,p. 68).

SCIENCE, VOL. 179758