Historical Development of the Newton-Raphson Method Tjalling Ypma Western Washington University, Tjalling.Ypma@Wwu.Edu

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1995 Historical Development of the Newton-Raphson Method Tjalling Ypma Western Washington University, [email protected]

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Historical Development of the Newton-Raphson Method Author(s): Tjalling J. Ypma Source: SIAM Review, Vol. 37, No. 4 (Dec., 1995), pp. 531-551 Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/2132904 Accessed: 21-10-2015 17:15 UTC

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This content downloaded from 140.160.178.72 on Wed, 21 Oct 2015 17:15:03 UTC All use subject to JSTOR Terms and Conditions SIAM REVIEW ( 1995Society for Industrial and AppliedMathematics Vol.37, No. 4, pp. 531-551,December 1995 003

HISTORICAL DEVELOPMENT OF THE NEWTON-RAPHSON METHOD* TJALLINGJ. YPMAt

Abstract.This expository paper traces the development of theNewton-Raphson method for solving nonlinear algebraicequations through the extant notes, letters, and publications of Isaac Newton,Joseph Raphson, and Thomas Simpson.It is shownhow Newton's formulation differed from the iterative process of Raphson,and thatSimpson was thefirst to givea generalformulation, in termsof fluxionalcalculus, applicable to nonpolynomialequations. Simpson'sextension of the method to systemsof equationsis exhibited.

Key words. nonlinearequations, iteration, Newton-Raphson method, Isaac Newton,Joseph Raphson, Thomas Simpson

AMS subjectclassifications. 01A45, 65H105, 65H10

1. Introduction.The iterativealgorithm

(1. 1) x Xi= (xi)/f-(xi)-f forsolving a nonlinearalgebraic equation f(x) = 0 is generallycalled Newton'smethod. Occasionallyit is referredto as theNewton-Raphson method. The method(1.1), and its extensionto thesolution of systemsof nonlinearequations, forms the basis forthe most frequentlyused techniques for solving nonlinear algebraic equations. In thisexpository paper we tracethe development of themethod (1.1) by exhibitingand analyzingrelevant extracts fromthe preserved notes, letters, and publicationsof Isaac Newton,Joseph Raphson, and ThomasSimpson. Much of thesequence of eventsrecounted here is familiarto historians, and thematerials on whichthis paper is based are fairlyreadily available; hence we make no claimsto originality.Our purposeis simplyto providea comprehensiveaccount of the historicalroots of the ubiquitous process (1.1), assemblinga numberof previouslypublished accountsinto a readilyaccessible whole. In ??2 and3 we showthat methods which may be viewedas replacingthe term f'(xi) in (1.1) bya finitedifference approximation of the form

(1.2) f'(xi) ; ht1[f (xi + hi)-f(xi-], andalso thesecant method in which

(1.3) f'(xi) f(Xi)-f Xi-Xi-l wereprecursors to the method (1.1). Froma modemperspective each of the methods described by(1. 1)-(1.3) arisesnaturally from a linearizationof the equation f (x) = 0. In ?4 we review Newton'soriginal presentation of his method,contrasting this with the current formulation (1.1) andwith Raphson's iterative formulation for polynomial equations discussed in ?7. There is no clearevidence that Newton used anyfluxional calculus in derivinghis method,though we showin ?6 thatin thePrincipia Mathematica Newton applied his technique in an iterative mannerto a nonpolynomialequation. Simpson's general formulation for nonlinear equations interms of the fluxional calculus is presentedin ?8,and we discussthere Simpson's extension ofthe process to systemsof nonlinear equations.

*Receivedby the editors October 6, 1993; acceptedfor publication (in revisedform) January 12, 1995. tDepartmentof Mathematics,Western Washington University, Bellingham, Washington 98225 (t j ypma@ nessie.cc.wwu.edu). 531

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In theabsence of directaccess to sourcematerials we havebased thispaper on a recent translationof the relevant works of Viete [19], the invaluable volumes of Newton's mathematical paperscollected and annotatedby Whiteside [21], Newton's correspondence reproduced byRigaud [14] andTumbull [18], the third edition of Newton's Principia Mathematica in the editionby Cajori [4] withcommentary by Koyre and Cohen [11], and copies of Raphson's text [12] and Simpson'sbook [16]. Forreferences to the work and influences of other mathemati- cianswe haverelied on thenotes in [13], [21] andrelated commentary inthe concise summary ofthis material by Goldstine [6, pp. 64-68]. In ourcomments on thework of Newton we are heavilyindebted to the notes in [21],while the recent papers [17] and [7] providemuch insight intothe contributions ofRaphson and Simpson,respectively. The references,particularly [6] and [21],provide detailed bibliographic references to relevantsource materials. The biographyby Westfall [20] includesincisive discussions of Newton's general math- ematicaldevelopment in additionto hismany significant achievements-scientific and other- wise. The fewverifiable facts concerning the life of JosephRaphson are presentedin [17]. The lifeof ThomasSimpson is describedin [5], whilea severelycritical commentary on his characterand achievements appears in [11]. 2. The methodof Viete. By late 1664,soon afterhis interesthad beendrawn to mathematics,Isaac Newtonwas acquaintedwith the work of the French algebraist Fran9ois Viete (1540-1603) (oftenlatinized to "Vieta"). Viete'swork concerning the numerical solution of nonlinearalgebraic equations, De numerosapotestatum, published in Parisin 1600,sub- sequentlyreappeared in a collectionof Viete'sworks assembled and publishedas Francisci VietaeOpera Mathematicaby Fransvan Schootenin Leidenin 1646. A summaryof that material,incorporating notational simplifications and someadditional material, appeared in variouseditions of WilliamOughtred's Clavis Mathematicaefrom 1647 onward.A recent translationof therelevant work appears in [19], whichincludes a biographyof Viete. New- tonhad access to bothSchooten's collection and thethird Latin edition of Oughtred'sbook, publishedin Oxfordin 1652,and madeextensive notes from them. These notes constitute a firstsign of Newton's interest in thenumerical solution of nonlinearequations. Vieterestricted his attention to monicpolynomial equations. In modernfunctional nota- tionwe maywrite Viete's equations in theform (2.1) p(x) = N in whichthe constant term N appearson theright of theequation. Viete's technique can be regardedas yieldingindividual digits of the solution x*. of (2.1) oneby one as follows.Let the successivesignificant decimal digits of x* be ao, a,, a2,. . ., so thatx* = a010k + a I0k- 1 + a2 10k-2+.. , and letthe ith estimate xi of x* be givenby xi = ao 10k+a, a 10k- + . +ai 10k-i. Assuming xi is given,we have xi+, = xi + ai+l 10k-i+1), and Vi&te's techniqueamounts to computingai+I as theinteger part of N - (2.2) .5 (p(xi + 10k-i-1) + p(X,)) p(X, + lOk-i-1) - p(X,) - 10(k-i-I)n' wheren is thedegree ofthe polynomial. Occasionally the quantity .5 (p(X, + 1ok-i-l)+p(X,)) inthe numerator of (2.2) is replacedby the value of p(Xi) . As notedby Maas [10],the integer aji+ mayin factbe negativeor haveseveral digits, thus permitting the correction of earlier estimatesxi ofx*. Rewriting(2.1) as f(x) = 0 wheref(x) = p(x) - N, theexpression (2.2) can be reformulatedas

-.5(f(X, + 1Ok-i-1) + f(x,))

f(X; + lOk-i-1) - f(xi) - 10(k-i-I)n'

This content downloaded from 140.160.178.72 on Wed, 21 Oct 2015 17:15:03 UTC All use subject to JSTOR Terms and Conditions HISTORY OF THE NEWTON-RAPHSONMETHOD 533 henceViete's method is almostequivalent to

(2.3) xi+1 = xi1-1ioki--)f Lf(xi + f(x()]

This closelyparallels the expression produced by substitutionof (1.2) into(1.1) withhi = 10k-i-I. In thissense themethod of Vieteis a forerunnerof thefinite difference scheme (1. 1)-(1.2), whichis oftenpresented as a modificationof (1.1). The subtra6tonof theterm 10(k-i-1)nin the denominator of (2.2) can be motivatedin the case thatp(x) = Xnby consid- eringthe binomial expansion of (xi + l0k-i-l),; formonic quadratic equations the resulting methodcorresponds exactly to theNewton-Raphson method (1.1), butin practicethis term is negligibleand was oftenomitted [21, I]. An interestingcomparison between this technique andthe Newton-Raphson method for the case p(x) = Xnis givenin [10]. Thistechnique was widelyused until supplanted by the Newton-Raphson method. In a portionof an unpublishednotebook tentatively dated to late 1664,reproduced in [21, I, pp. 63-71], Newtonmade extensive notes on Viete'smethod. Figure 1 is a reproduction from[21] ofNewton's modified transcript of Viete's solution of x3 + 30x = 14356197.The exactsolution x*. = 243 is computed.Here Newtonused the "modifiedcossic" notation of Oughtred,in whichA and E representalgebraic variables, while their nth powers for n = 1,2, 3, 4, 5, . . . arerepresented by adjoining the symbols 1, q, c, qq, qc, . . . respectively to thevariables; thus A5 is representedas Aqc. The line-by-lineanalysis below, based on (2.2) withp(x) = x3 + 30x and N = 14356197,amplifies the discussion of thisextract in [21, I, pp. 66 -67] and [6, p. 67]. Binomialexpansions are used repeatedlyto simplifythe computationaltask. (1) Line 1. x = L, 30 = C2, N = 14356197= P3, hencethe phrases on thenext two lines,a relicof Viete's insistence that all termsin a givenequation must be ofthe same degree. (2) Line 3. "pointing"(the lower dots), a techniquedeveloped by Oughtred[21], is used to obtainthe initial estimate x0 = 200; subsequentlycomputed digits are adjoinedto the initial2. (3) Lines4-6. p(xo) = 8006000evaluated termwise as 2003 + 30(200) . (4) Lines7, 9. N - p(xo) = 6350197. (5) Lines 8, 10-12. evaluationof p(210) - p(200) - 103 = 1260300(cf. (2.2) with k = 2, i = 0, n = 3) usingbinomial expansions: [(200 + 10)3+ 30(200 + 10)] - [(200)3 + 30(200)] -103 - 3(200)210+3(200)102 +30(10) = 12(105)+6(104) +3(102) = 1260300. (6) In an unrecordedcomputation the next digit of x*. is computedas 4, beingthe integer partof (N - .5(p(210) + p(200)))/1260300 = 5719547/1260300= 4.538 .., producing x1 = 240. (7) Lines13-17,19. evaluationofN-p(240) = 524997: N-p(240) = [N-p(200)]- [p(240)-p(200)] = 6350197-[p(240)-p(200)] andthe latter term is evaluatedby binomial expansionas [(200 + 40)3 + 30(200 + 40)] - [(200)3+ 30(200)] = 3(200)240+ 3(200)402+ 403 + 30(40) - 48(105) + 96(104) + 64(103) + 12(102) = 5825200. (8) Lines 18,20-22. p(241) -p(240) - 1 = 173550(cf. (2.2) withk = 2, i = 1,n = 3) computedsimilarly to lines8, 10-12. (9) In anotherunrecorded computation (N - p(240))/173550 = 524997/173550= 3.025. . . has integerpart 3 SO X2 = 243. (10) Line 28. N -p(243) = 0 is computedfrom N - p(240) just as N - p(240) was computedfrom N - p(200) in lines13-19. In thefinal paragraph of his notes on themethod of Viete[21, I, p. 71] Newtonuses the Cartesiannotation of lower-case symbols for algebraic variables and superscripts to represent powers(for example x5); he usedthe latter notation in mostof his subsequentwork. He also brieflyshifts the constant term in the equation to the left of the equality symbol, leaving zero on

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Theanalysis of Cubick Equations. The equationsupposed Lc *+ 30L = 14356197.Lc + CqL=Pc. The square coefficient 3 0 The cube affectedto be 14 356 197 (243 {8 0 =AC Solids to be substracted 6 =Ac

Theiresuffie 8 006 0 Rests 6 350 197 forfinding ye 2d side. The extractionof y seacon.dside Coefficient 30 or superiordivisor. The restof ye cube to be 6 350 197 resolved

The inferiordivisors {1 2 3Aq

Theirsufme 1| 260 |30 48l - =3AqE Sollidsto be subtracted 96 = 3AEq f 64 = Ec 1 20| - =ECq Their sufmne 5 j 825120

[The extractionof ye 3d side] The superiorpart of ye divisor 30 or ye square coefficient The remainderfor finding 524 997 yethird side The inferiorpart of ye I 1 2 18 3Aq thatis 3 x24 x 24 divisor | 72 3A or 3 x24 The suie ofye divisors 173 1550 51814 3AqE Solids to be taken f 6 48 3AEq away 27 Ec 90 Ecq Theire sufine - 524 997

Remaines - 000 000

FIG. 1. Newton'stranscript of Viete's solutionof x3 + 30x - 14356197 0.

theright, thus adopting the now conventional zero-finding formulation for solving nonlinear equations. The generalidea of solvingan equationby improvingan estimateof the solution by the additionof a correctionterm had beenin use in manycultures for millenia prior to thistime [6], [10]. Certainancient Greek and Babylonian methods for extracting roots have this form, as do somemethods of Arabic algebraists from at leastthe time of al-Khayyam (1048-1131). The preciseorigins of Viete'smethod are notclear, although its essence can be foundin the workof the 12th century Arabic mathematician Sharaf al-Din al-T.iisi [13]. It is possiblethat theArabic algebraic tradition of al-Khayyam, al-Tisli, and al-Kashisurvived and was known to 16thcentury algebraists, of whom Viete was themost important.

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3. The secant method.In a collectionof unpublishednotes termed "Newton's Waste Book" ([21,I, pp.489- 491] andthere tentatively dated to early 1665) Newton demonstrates an iterativetechnique that we canidentify as the"secant method" for solving nonlinear equations. In modemnotation, this method for solving an equationf (x) = 0 is (1.1) withf '(xi) replaced by (1.3), thatis

(3.1) Xi+i = xi - f (xi)/[X(Xi - f(Xi) Xi- Xi- An interestinghistorical discussion of this technique, and its relationship to theclosely related methodnow commonly referred to as RegulaFalsi, is givenby Maas [10]. Severaldifferent arguments,all froma modemperspective essentially based on a linearizationof the underlying functionf (x), yieldthis technique. The approachbased on Fig. 2 belowis consistentwith Newton'scomputations. In bothof theinstances shown in Fig. 2 we notethat, by similarity ofthe labelled triangles, a/b = c/d,that is (f (xi) - f (xi1))/(xi - xi -1) F f (xi)/d; thus withd e ?(x - xi), we get(with the appropriate choice of signsin each case) theformula Q3.1) forxi+ I -x,

f Zi-i xi Zi+1 : S

b / f~~~~~~~~~

FIG.2.Thesecantmethodusingmltr

FIG.2. Thesecant method using similar triangles.

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The resolsion of y'affected Equation x3+pxx+qx+r=- 0.r A+ lOx2- 7x=44.

Firsthaving found two or 3 of yefirst figures of ye desiredroote viz 2L2 (wChmay bee doneeither by rationallor Logaridunicaltryalls as MrOughtred hathtought, or Geometricallyby descriptionsof lines, or by an instrument consistingof 4 or 5 or morelines of numbers made to slideby one anotherwd maybe oblongbut better circular) thisknowne pte ofye root I call g, yeother unknownepte I call y[,] thenis g+y=x. Then I prosecuteye Resolution after thismanner (making x+p in x=a. a+q in x=b. b+r in x=c. &c.) x 12=x+p a+q=17 x r -b=10=h. bysupposingx=2. x=2, 24=a b=34 2=x yupsig-

Againesupposing x= 21. x+p=12,2 a+q=19,841 24412 3968j2 244 2 3968 2 26,84=a 43,648=b

r -b=00,352=k. h-k=9,648. That is ye

Jlatterr-b substractedfrom the former r-b thereremainesi 9648 {differencetwixt this & yeformer valor of r-b is I 9 & yedifferencetwixt this & yeformer valor of x is 0,2. Thereforemake 9,648:0,2:: 0,352 :y.

Then isisY0,0704= y= 9,648 =0,00728 &c. the firstfigure of w'- beingadded to yelast valorof x makes2,207 =x. Then WIhthis valor of x prosecutingye operacon as beforetis

X+p= 12,207_ a+ q= 19,94084. r -b=-0,00943388. 85449 7 13958588t7 2 44140 02 39881680 02 24414 2 3988168 2 26,940849=a 44,00943388= b

wchvalor of r-b substractedfrom ye precedentvalor of r-b ye diff:is +0,36143388. Also ye diff[:]twixt this & ye precedentvalor of x is 0,007. ThereforeI make 0,36143388:0,007::-0,00943388:y. That is

= -5,9037416 -0,0001633 &c. Y 36143388 2 figuresof vc (because negatve) I substractfrom ye formner value ofx & there restsx-= ,20684. And so mightye Resolution be prosecuted.

FIG. 3. Newtonuses thesecant method to solvex3 + lox - 7x - 44 = 0.

Wenow give a detailedanalysis of Newton's computations, reproduced in Fig. 3 from[211, to solvex3 + lOX2- 7x = 44 usingthis method. We writep(x) = x3 + 1OX2- 7x, N = 44, and f(x) = p(x)- N.

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Assumegiven (by one of a varietyof means)two initialestimates xo = 2 and xl = 2.2 of the exact solutionx* = 2.2068173.... Firstp(2) = 34 is evaluated(by nested multiplication),producing N -p(2) = 10 = -f(2). Similarly p(2.2) = 43.648 and -f (2.2) = .352 arecomputed. Thus f (2.2) -f (2) = 9.648,hence from the proportionality 9.648/.2 = .352/y , thatis (f(2.2) - f(2))/(2.2 - 2) = -f(2.2)/y, where y is the correc- tionto be addedto thecurrent estimate of x*,we obtainthe exact value y = .0072968 ..., givenby Newtonas .00728 and truncatedto .007, thusproducing x2 = 2.207. Then p(2.207) = 44.00945374 is evaluated (withtruncation of one intermediatedigit, using nested and long multiplication)as 44.00943388,producing -f(2.207) = -.00943388. Hence f(2.207) - f(2.2) = .36143388, so that from .36143388/0.007 = -.00943388/y, that is (f(2.207) - f(2.2))/(2.207 - 2.2) = -f(2.207)/y, we obtain(apparently with an er- rorin transcription)y = (.007)(-.00843388)/(.36143388) = -5.903716/36143.388 = -.0001633 .., truncatedto -.00016. ThusX3 = 2.207 - .00016 = 2.20684. ClearlyNewton had progressedbeyond Vi&te's method in no longerconstructing x* digitwise,but the influence of thatprocess is stillevident in thatNewton truncates y to only itsfirst one ortwo significant digits to obtainthe next correction term. 4. Newton'smethod-first formulation. Newton's tract De analysiper aequationes numeroterminorum infinitas (henceforth De analysifor brevity), probably dating from mid 1669, is notedchiefly for its initialannouncement of the principleof fluxions.The extractfrom that tract quoted below, in thetranslation into English of [21, II, pp. 218-223], is thefirst recorded discussion by Newtonof whatwe mayrecognize as an instanceof the Newton-Raphsonmethod (1.1), althoughthe formulation differs considerably from the now conventionalform, the computations are muchmore tedious than in thecurrent formulation, and themethod is givenonly in thecontext of solvinga polynomialequation. No calculus is used in thepresentation, and referencesto fluxionalderivatives first appear later in that tract,suggesting that Newton regarded this as a purelyalgebraic procedure. In severalother instancesNewton is knownto haveused moretraditional methods and notationsin an effort to makehis ideas moreaccessible to a wideraudience, but there is no clearevidence that at thattime he perceivedthis particular technique as an applicationof the calculus or derivedit usingthe techniques of calculus. The generalrole of calculusin thehistorical development of (1.1) is surveyedin [7]. Newton'stechnique may be describedin modemfunctional notation as follows.Let xo be a givenfirst estimate of thesolution x* of f (x) = 0. Writego(x) = f (x), and suppose go(X) = En=0aix'. Writingeo = x- xO we obtainby binomial expansion about the given xo a newpolynomial equation in thevariable eo:

(4. 1) 0 = go(x.) = go(xo+ eo)= ai(xo + eo)'= a,)xie:i] [ ( = gI(eo). i=o i=o 'J7o J _ Neglectingterms involving higher powers of eo (effectivelylinearizing the explicitly computed polynomialgi) produces

n n n 0 = gl(eo) Eai[x6L + ix'-leo] = [ aix] + eo [ aiixi-i

i=O i=O i=O fromwhich we deducethat

eo = - [Eaix x] /[ ai ixo'] and set xi = xo + co. Formallythis correction can be writtenco = -go(xo)/g'(xo) = -f(xo)/f'(xo). Now repeatthe process, but instead of expandingthe original polynomial

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go aboutxl expandthe polynomial gI obtainedexplicitly in (4.1) aboutthe point co, i.e., co is consideredto be a firstestimate of thesolution eo of thenew equationgI (x) = 0. Thus similarlyobtain 0 = gI (eo) = gI (co + e1) = g2(ei ), wherethe polynomial g2 is explicitly computed.Linearizing again producesas beforea correctionformally equivalent to el1 c =-gI(co)/gj(co), correspondingto c = -go(xo + co)/g'(xo + co) = -(xI)/f'(x ) and x2 = XI + Cl . The processcontinues by expanding g2 aboutcl, and so on. InFig. 4 wereproduce the English translation of [21, II, pp.219-220]of Newton's solution ofg(x) = X3- 2x -5 = 0 (a standardtest problem of the era [6, p. 64]) bythis method. Our analysisof this passage uses thenotation introduced above. (1) Line 1. takex0 = 2; thesuccessive estimates of the solution x* = 2.09455148... are accumulatedhere. (2) Lines2-5. expandgo as go(x*) = go(2 + p) = p3 + 6p2 + lOp - 1 = gi (p) = 0 usingthe binomial expansion; omitting higher order terms leaves 10p - 1 0 hencep .1 andxl = 2.1. (3) Lines 6-10. expandgi(p) = gl(.1 + q) = q3 + 6.3q2 + 11.23q + .061 = g2(q); truncatedto 11.23q + .061 ; 0 thisgives q ;-.00543 ..., whichis roundedto -.0054, henceX2 = 2.0946. (4) Lines 11-14. expandg2(q)= g2(-.0054+r) = 6.3r2+11.16196r+.000541708 = g3(r) (wherethe term q 3 ing2, beingsmall, has beenomitted), which produces after lineariza- tionr -.00004853 ... andhence X3 = 2.09455147after truncation. The processdescribed by Newtonrequires the explicit computation of thesuccessive polynomialsgi, 92, . . ., whichmakes it laborious. Clearly calculus is notused by Newton in his presentation,which is basedentirely on retentionof the lowest order terms in a binomial expansion.Also notethat the final estimate of x* is onlycomputed at the end of the process as x* = x0+ co + cl + ** instead of successive estimates xi beingupdated and used successively. Clearlythis process is significantlydifferent from the iterative technique currently inuse. This extracthints that Newton was alreadyaware of thequadratic convergence of thetechnique, as characterizedby theapproximate doubling of thenumber of correctsignificant digits in successivesteps: observethat the number of digitsretained by Newtonin successivesteps doubles. Newtongave no furtherexplanation of his method, though he usedit in an analyticform a fewpages laterin his tractand, as we shall see in ??5 and 6, it reappearsin a laterletter and in his PrincipiaMathematica. A slightlyrevised version of thepassage quoted above was incorporatedin theopening pages of Newton'smore comprehensive tract De methodis fluxionumet serierum infinitarum written in 1671. The unfinishedmanuscript of De methodis was initiallyintended for publication, but the unprofitable nature of mathematicalpublishing combinedwith Newton's reluctance to publishfollowing the controversysurrounding his "New Theoryabout Light and Colors" suppressedthe workat thetime. De analysiwas notpublished until 1711, in Analysis per quantitatumseries, fluxiones, ac differentias...by WilliamJones, by which time its status was largelyhistorical. De methodiswas notpublished until1736 in translationby JohnColson [21, III, p. 13]. Neverthelessvarious copies of Newton'smanuscripts circulated among leading mathematicians. Newton gave a copyof De analysito Isaac Barrow,who senta copyto JohnCollins, who circulatednews of thework amonghis internationalcorrespondents, some of whomwere privileged to maketheir own copiesof portions of the work. Among the most interesting of theseare the extracts made by Leibnizin 1676during a visitto London, reproduced in [21,II, pp.248-259], which include the passageanalyzed above almost verbatim while omitting much of the material on calculus.The earliestprinted account of Newton's method, including essentially the content of Fig. 4, is in chapter94 ofJohn Wallis' A TreatiseofAlgebra both Historical and Practical, London, 1685.

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EXAMPLES BY THE RESOLUTION OF AFFECTED EQUATIONS. Since the The difficultyhere lies wholly in the resolutiontechnique, I wil firstelucidate the 'uteical methodI use in a numericalequation. affrected Supposey3- 2y-5 = 0 is to be resolved:and let 2 be the numberwhich equations. +2-10000000 differsfrom the root sought by less than {-0-00544853 its tenthpart. Then I set 2+P = y and 2-09455147 substitutethis valueforit in theequation, andin consequencethere arises 2p= y y3 +8+ 12p+t62+p3 he new equation -2y -4-2p p3+6p2+ lop-I = 0 -5 -5 0-- whoseroot p mustbe soughtfor Sum -1+lOp+6p2+p3 it to be added to the

01 +q =p +p3 +0-001 +0-03q+0-3q2+q3 quotient: specficaly + 6p2+ 0-06 + 1-2 + 6-0 (when p3+ 6p2 are + lop + 1 + 10 neglectedon account -1 -1 oftheir smallness) Sum +0-061+11 23q+6-3q2+q3 lOp-I = 0 -0-0054+r = q 6-3q2+0-000183708-0-06804r+6'3r2 or p=0*1 very +11-23q-0-060642 +11-23 nearly true; +0-061 +0-061 and so I write Sum + 0-000541708+ 11-16196r+ 6&3r2 the quo- a~~~~~tenti1 n anld sup- -0-00004853 ose0'1+q=p, and on substitutingthis value forit, as before,there arises in consequence q3+6 3q2+ 11F23q+0'061=0. And,since 11*23q+0*061[= 0] approachesthe truth closely or thereis almost q= -00054 (by'dividing,that is, undl as manyfigures are elicitedas the numberof places by which the first figures of this and ofthe principal quotient are distantone fromthe other),I write-0-0054 in the lowerpart of the quotientsince it is negative.Again, supposing-0-0054+r = q, I substitute thisas before,and in thisway continue the operation as faras I please.But if I desireto continueworking merely to twviceas manyfigures, less one, as are noNwfound in thequotient, in place ofq in thisequation 6-3q2+1123q+0 061[= 0] I substitute-0-0054+r, neglectingits firstterm q3 by reasonof itsinsignifi- cance, and therearises 6 3r2+11 16196r+0-000541708= 0 nearly,or (when rejcte)r- 6-3r2is rejected)r = 0-0005417081696 = 000004853 nearly.This I writein the negativepart of the quotient.Finally, on takingthe negativeportion of the quotentfrom the positive part, I have therequired quotient 2-09455147.

FIG. 4. Newton's methodforsolving x - 2x - 5 = 0.

A methodalgebraically equivalent to Newton'smethod was knownto the 12thcentury algebraistSharaf al-Din al-Tiisli[13], and the 15thcentury Arabic mathematician Al-Kdshi useda formof it in solving xP - N = 0 tofind roots of N. In westernEurope a similarmethod wasused by Henry Briggs in his Trigonometria Britannica, published in 1633,though Newton appearsto havebeen unaware of this [21, II, pp. 221-222].

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But yetI conceivethese roots may be easilierextracted by logarithmes. Supposec as nearlyas youcan guesseequall to ye root z: & ifye aquaton be zn=bz+R make VIbc+R (or ?c xb)=d. Rin xb=e R ?e+ x b-f $If+R x b=g &c. Or if ye acquationbe zn? bz-= R,

makel? Rk=,d.+ ?b +d-e= Re=f &c. Andso shallye last found b-i-c \'~-'d I(DJb -e termee,f, or g &c be ye desiredroot z. For instance if z`=-8z-5: suppose c=1, & ye workwill be this /() 8 x 1-5, orV() 3=1 0373-d. 0)3 =2984-1 !7=e, VC(o)3932561t040894=f etc Thereforeye rootz is 1i040894 So ifz=+120z21=3748000 supposec=1, & ye workwill be 3748000 3748000 1 6363=d =163587-e &c. {2)121 12~1916363 Thereforeye rootz is lI63587

FIG.5. Newton's use offixedpoint iterations.

5. Furthermethods for nonlinear equations. Newtondisplayed a continuedinterest inmethods for the numerical solution of nonlinear equations during the years following 1671. Some of his activitiesare summarizedin [6, pp. 65-66]. Forexample, in a letterto Michael Darydated 6 October1674 [18, I, pp. 319-322] Newtonproposed schemes, which we may writeas

xi+l - (bxi + c)(l/n) and xi+l = ( forsolving the equations x' = bx + c andx ? bxn-I = c, respectively.Equations of this sort arisein the analysis of annuities. We nowrecognize these schemes as examplesof fixed point (or functional)iteration. Similar schemes were previously proposed by JamesGregory and communicatedin lettersto John Collins to solve the equations bnc + X,+I = bnx(8 November 1672)and bnc + x"+1 - bn-1(b + c)x (2 April1674) [18,I, pp. 321-322] and [6, pp. 65-66]. In Fig. 5 we reproducefrom [18] a portionof Newton'sletter to MichaelDary, giving theseiterative schemes and applying them to theequations x30 = 8x -5 andx22 + 120X21= 3748000,respectively, in eachcase withxo = 1. The textis largelyself-explanatory once it is understoodthat the notation V/ 0 denotestaking the indicated root of the subsequent quantity, andthat the symbol L is used insteadof thedecimal point. There are minor errors in thefifth and subsequentsignificant digits in Newton'scomputations. Also ofinterest are two letters by Newton addressed to JohnSmith, dated 24 Julyand 27 August1675. Smithwas preparinga table of square, cube, and quartic roots of all theintegers from1 through10000; in a letterdated 8 May 1675 [18, I, pp. 342-345] Newtonsuggested thathe do thisby computing only the roots of every hundredth integer to a sufficientnumber of digits(10), followedby interpolation toproduce all theother desired quantities to the required accuracy(8 digits).Newton presented in his lettersto Smithseveral schemes, which we may synthesizeas

(5.1) X1 = (1/n)[(n - 1)xo + a/x1n-], n = 2,3,4,

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1. When you have extractedany X by common Arithmetickto 5 Decimal places, you may get the figuresof the other 6 places by Dividing only the Residuum by doublethe Quotient square triplethe q of Quotient F forthe RI cube quadruple the c of Quotient) square square Suppose B. the Quotient or R extractedto 5 Decimal places, and C. the last Residuum, by the Division of wch you are to get the next figureof the Quo-

tient,and D the Divisor (that is 2B or 3BB or 4B'c=D) & B+D shall be the R desired. That is, the same Division, by wch you would finde the 6th decimal figure,if prosecuted,will give you all to the 11th decimal figure. 2. You may seek the R if you will, to 5 Decimal places by the logarithm's, But then you must finde the rest thus. Divide the propounded number once twice by yt 1 prosecutingthe Division alvayes to 11 Decimal places, and thrice to the Quotient add once, & halfe square ye said R twice, & a thirdpart- of the summj Cube R desired. thrice,& a quarter shall be the square square For instance IQ Let A be thenumber, and B. its C R extractedby Logarithmsunto 3 decimal QQ places: A 2) B+ , Q

and 3) 2B+ A, shall be the C root desired

4) 3B+ A, QQ

FIG.6. Newton'smethodfor extracting roots of numbers. forfinding the second, third, and fourthroots of a givenpositive scalar a, intendingthem to be used to computethe required roots of everyhundredth integer to therequired accuracy. Formula(5.1) is (1.1) withi = 0 used to solve f(x) = Xn- a = 0 forarbitrary n. This techniquehad previously been used by the Arabic mathematician al-Kash1 [13], and forms of itappear in earlierwritings. This formula may also be obtainedby applying just the first step of thetechnique of ?4 to thepolynomial equation xn - a = 0. We reproducein Fig. 6, and analyzebelow, an extractfrom [18, I, pp. 348-349] of theletter dated 24 July1675 in which Newtondiscusses this technique for finding roots. Further discussion of this material appears in [21,IV, pp. 663-665]. HereNewton is solvingthe equation Xn - A = 0, forn = 2, 3, 4 startingfrom an initial estimatexo = B andwriting C = A - Bn = a - xn. He partlyreverts to themodified cossic notationdescribed in ?2. In thefirst paragraph he givesthe method in a formcorresponding

This content downloaded from 140.160.178.72 on Wed, 21 Oct 2015 17:15:03 UTC All use subject to JSTOR Terms and Conditions 542 TJALLINGJ. YPMA to xl = x0 + co, where C a-x0x -f(xo) I nBn-1 nx f' (xo) andnotes that one should retain all thecomputed digits for use as thecorrection term rather than truncateafter the first few significant digits. In thenext paragraph the method is reformulated correspondingto (5.1), withthe symbol n) denotingdivision by n. Newtonshows no signof usinghis formulae iteratively, given his instructions toselect a firstestimate of the solution with fivecorrect significant digits to geteleven correct digits after one applicationof the formula, ratherthan indicating that any desired accuracy can be obtainedfrom an appropriateinitial estimateby simplyapplying the formula repeatedly. This passage again shows that Newton was awareof the rough doubling of the number of correct significant digits in one stepof his process,characteristic ofthe quadratic convergence of the process (1.1). Forcompleteness we mentionalso a letterfrom Newton to John Collins, dated 20 August 1672 [18,I, pp. 229-234] in whichhe describesthe use of "Guntersline" (that is, a logarith- micallygraduated ruler) to solve polynomialequations. Since thisingenious method is not iterative,we referthe interested reader to [17] and [21,III, pp. 559-561] forfurther details. 6. Newton'smethod. The firstpublished use by Newtonof themethod (1.1) in an it- erativeform and appliedto a nonpolynomialequation is in thesecond and thirdeditions of hisPhilosophiae Naturalis Principia Mathematica, whose first edition was publishedin Lon- don in 1687. In each successiveedition he describedtechniques for the solution of Kepler's equation

(6.1) x - esin(x) = M.

To understandNewton's geometrical technique and relate it to theconventional analytic form ofthe Newton-Raphson method it is necessaryto have an understandingofthe terms involved in (6.1). Ourdescription below is basedon [15,pp. 72-85]. Theorigins of the problem lie indetermining the position of a planetmoving in an elliptical orbitaround the sun, given the length of time since perihelion passage. With reference to Fig. 7, letthe ellipse (the orbit) ABA'B' centeredat the origin 0 be definedby the canonical equation

(6.2) (y2/a2)+ (z2/b2)= 1. Thenthe ellipse has a semimajoraxis AO oflength a anda focus(the sun) at S = -b = -ae, wheree = b/a is theeccentricity of theellipse. Let ACA'C' be a circlecentered on the originwith radius a circumscribingthe ellipse. Let P (theplanet) be a pointon theellipse whoseCartesian coordinates are to be determined,and let QPR be a lineperpendicular to AO passingthrough P and interceptingthe circumscribed circle and theline AO at thepoints Q and R, respectively.Then the Cartesian coordinates of P are definedby the lengths IPRI and IOR!. Giventhe readily proved fact that IPRI/IQRI = e, so thatwe needfind only IQRI and IORI,it is easyto see thatknowledge of the eccentric anomaly, i.e., the angle x = ZAOQ, is sufficientto locateP: sin(x) = IQRj/IQOI= IQRI/aand cos(x) = IROI/IQOI= IROj/a, thusIPRI = ea sin(x) and JOR!= a cos(x). The eccentricanomaly x is to be computed. Supposethat P representsthe position of a planetin an ellipticorbit about the sun S, at a timet afterpassing through the point A (perihelionpassage) at time0. If theplanet has orbitalperiod T, thensince the radius vector SP turnsthrough an angleof 2w radiansin the courseof one orbit,the mean angular velocity of theplanet is n = 27/ T. Duringthe time t theangle swept out by a radiusvector rotating about S withangular velocity n is themean anomalyM = nt. A cleverargument [15, pp. 83-85] exploitingKepler's laws of planetary

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R S 0 A'

FIG. 7. Locatingthe position of a planetin an ellipticalorbit.

A S R r 0 H B

FIG. 8. Diagramto accompany Newton 's solutionof Kepler's equation. motionreveals that the eccentric anomaly x, themean anomaly M andthe eccentricity of the ellipsee are relatedthrough (6.1), whereboth M andx are in radianmeasure. Thus in (6.1) we areto solve for x, beingthe angle ZAOQ, givenM ande, whereM is computedfrom t as M = 2rt/ T. The historicalorigins of thisproblem are discussedin [21, IV, pp. 668-669]. Newton's interestin Kepler's problem is firstrevealed in a letterto Henry Oldenburg dated 13 June1676 [18, II, pp. 20- 47] in whichhe derivesa seriesexpansion for the quantity IP R I butgives no numericalalgorithm for solving (6.1). We reproducehere the text and accompanying figure (Fig. 8) of Book 1, Proposition31, Scholium,from Cajori's edition of 1934 [4,pp. 113-114]of the third edition of the Principia, publishedin Latin in 1726and translated by Andrew Motte into English in 1729. HereNewton presentshis technique for solving (6.1) numerically. "Butsince the description of thiscurve is difficult,a solution by approxi- mationwill be preferable.First, then, let there be founda certainangle B whichmay be to an angleof 57.29578 degrees,which an arc equal to the

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radiussubtends, as SH, thedistance of thefoci, to AB, thediameter of the ellipse.Secondly, a certainlength L, whichmay be tothe radius in the same ratioinversely. And these being found, the Problem may be solvedby the followinganalysis. By anyconstruction (or evenby conjecture),suppose we knowP theplace of thebody near its true place p. Thenletting fall on theaxis ofthe ellipse the ordinate PR fromthe proportion of thediameters ofthe ellipse, the ordinate RQ ofthe cicumscribed circle AQB willbe given; whichordinate is thesine of the angle AOQ, supposingAO tobe theradius, and also cutsthe ellipse in P. It willbe sufficientifthat angle is foundby a rudecalculus in numbersnear the truth. Suppose we also knowthe angle proportionalto thetime, that is, whichis to fourright angles as thetime in whichthe body described the arc Ap to thetime of one revolutionin the ellipse. Let thisangle be N. Thentake an angle D, whichmay be to the angleB as thesine of theangle AOQ to theradius; and an angleE which maybe to theangle N - AOQ + D as thelength L to thesame lengthL diminishedby thecosine of theangle AOQ, whenthat angle is less thana rightangle, or increasedthereby when greater. In thenext place, takean angle F thatmay be to theangle B as thesine of theangle AOQ + E to theradius, and an angleG, thatmay be to theangle N - AOQ - E + F as thelength L to thesame lengthL diminishedby the cosine of the angle AOQ + E whenthat angle is less thana rightangle, or increasedthereby whengreater. For the third time take an angleH, thatmay be to theangle B as thesine of theangle AOQ + E + G to theradius; and an angleI to theangle N - AOQ - B - G + H, as thelength L is to thesame length L diminishedby the cosine of the angle AOQ + E + G, whenthat angle is less thana rightangle, or increasedthereby when greater. And so we may proceedin infinitum.Lastly, take the angle AOq equal to theangle AOQ + E + G + I +, etc.,and fromits cosine Or andthe ordinate pr, whichis to itssine qr as thelesser axis ofthe ellipse to thegreater, we shallhave p thecorrect place of thebody. Whenthe angle N - AOQ + D happensto be negative,the sign + ofthe angle E mustbe everywherechanged into -, andthe sign - into+ . Andthe same thing is to be understoodof the signs ofthe angles G andI, whenthe angles N - AOQ - E + F,and N - AOQ - E - G + H comeout negative. But the infinite series AOQ + E + G + I + etc.,converges so veryfast, that it will be scarcelyever needful to proceed beyondthe second term E. Andthe calculus is foundedupon this Theorem, thatthe area APS variesas thedifference between the arc AQ and theright linelet fall from the focus S perpendicularlyupon the radius OQ." This passagemay be understoodas follows.Implicitly assume that the horizontal axis has beenscaled so that"the radius" a = 1. Letthe angle ZB equal e radians(the ratio of the angle ZB to 57.29578degrees is setequal to 2b/2a = e, and57.29578 is approximatelyone radian, as Newtonnotes, since the "angle . . . whichan arcequal to theradius subtends" (in degrees on thecircle) is 360a/(2ra) = 360/(2w) = 1 radian).Let L be suchthat L/a = l/e (thus L-1 = e). Let p be theposition of the planet, and P be a firstestimate of p. Note(as above) thatto locatep it sufficesto computethe angle ZAOq, theeccentric anomaly. Let theangle ZAOQ = xo be ourfirst estimate of ZAOq. DetermineZAOq as follows.Let theangle ZN be suchthat ZN/(2w) = t/T (thusZN = 2wrt/T = M, themean anomaly). Let theangle ZD be suchthat ZD/ZB = sin(ZAOQ)/a (thusZD = e sin(xo)),and letthe angle ZE (the correctionc0 to x0 = Z AOQ) be suchthat

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ZE L ZN-ZAOQ+Z/D L-cos(ZAOQ) equivalently + ZD) + e (6.3) co = ZE - L(ZN-xo M-xo sin(xo) L - cos(xo) 1 - e cos(xo) whichproduces xi = xo + co = ZAOQ + ZE. Similarlydetermine, using the same formula, successively ZG = c1, ZI = c2, etc. fromxl = ZAOQ+ ZE and x2 ZAOQ+ ZE + ZG, etc.,respectively, and set ZAOq ZAOQ + ZE + ZG + ZI + ,i.e., x* = xo + c0 + Cl +C2 + - ' " Equation(6.3) is preciselyc0 =-f (xo)/f'(xo)with f (x) = x - e sin(x) - M, andthe subsequentcorrections ci aresimilarly equivalent to applicationsof (1.1). Apparentlythe firstto recognizethis passage as an instanceof theNewton-Raphson methodwas JohnCouch Adams in 1882 [1]. Newton'smethod is usedhere for the first time in an iterativeprocess to solvea nonpolynomialequation. This contrasts with the frequently repeatedclaim [3] thatNewton used his method only for rational integral polynomial equations, withthe extensionto irrationaland transcendentalequations being made firstby Thomas Simpsonas describedin ?8 below.However, given the geometrical obscurity of the argument, itseems unlikely that this passage exerted any influence on thehistorical development of the Newton-Raphsontechnique in general. Thereis againno clearevidence that Newton associated his techniquewith the use of thecalculus. Thereare numerousways to derivethis process that do notrequire the use of calculus;for example, a purelygeometric derivation is givenin thethird of Simpson's Essays [16]. The passagejust quotedfrom the third edition of thePrincipia duplicates the correspondingpassage from the second edition of 1713. Thispassage is a modificationof the correspondingpassage from the first edition of 1687. Thesedifferent versions are discussed in [3], [21, VI, pp. 314-318] and [8, I, pp. 191-196]and suggest a derivationof this method consistentwith the approach previously presented by Newtonin De analysidiscussed in ?4. Following[21, VI, pp. 314 -317], set ;x*= xi + ei, so that(6.1) is rewritten

M = xi + ei - e sin(xi + ei) = xi + ei - e(sin(xi) cos(ei) + cos(xi) sin(ei))

= xi + ei - e(sin(xi)[1 - e2 ... ] + cos(xi)[ei - . .1) fromwhich

M - xi + esin(xi) = ei(I - e[cos(xi) - ei sin(xi).. ..) ei(1 - ecos(xi + lei)), whichleads to theiteration xi+ = xi + ci, where

' M -xi +esin(xi) (6.4) ei ci= 1 - ecos(xi + ci- 1)

The latterform was apparentlyintended by Newton in thecorresponding passage in thefirst editionof thePrincipia, but as pointedout by Fatiode Duillierin an annotationdating from about1690 [21, VI, pp. 315-316] Newton'sform there was flawed.Rather than correct the passageto reflectthe form (6.4), Newtonadopted the simpler alternative of omitting the term 2ci l to arriveat theform (6.3) in thesubsequent editions. It is notclear what role, if any, was playedby calculus in thisrevision.

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PROBLEMA. IX.

Proponatur a a a - b a = c Aequatio secundae Formulae Numeris a a a - 2 a = 5

Theor. x = c + bg - ggg

3gg - b

2 = g 5 = c 2,1 =g 2,1000 4 = bg 2,1 -,0054 -8 = ggg 3gg = 12 +9 9 21 2,0946= g b = -2 - 42 2,0946

10)+1,0(+,1 = x 4,41 = gg 125676 2,1 83784 188514 441 418920 882 4,38734916 gg 3gg = 13,23 -9,261 = ggg 2,0946 b = -2, +9,200 2632409496 +11,23) -, 06100(-,0054 - x 1754939664 3948614244 13,16204748 = 3gg 4,1892 = bg 8774698320 -2, = b 5, = c -9,189741550536 = ggg +11,16204748 9,1892 +9,1892 = bg + c

+11,16205) -,000541550536 (-,000048517 = x

2,0946 -,000048517

2,094551483 = g

FIG. 9. Raphson's methodforsolving x3 - 2x - 5 = 0.

7. Raphson's formulation.In 1690 JosephRaphson (1648-1712 ?) publisheda tract Analysisaequationum universalis [12] in whichhe presenteda newmethod for solving polynomialequations. A secondedition of thistract was publishedas a book in 1697,with an appendixwith references to Newton, but omitting a preface with different references to Newton thathad appeared in the original tract. A copyof Raphson'stract, including a handwrittendedication from the author to John Wallis,corrections to thetext that appear to be in thesame handwriting,and includingthe prefacebut without the appendix described in [7] and [17], is availableon microfilm[12] and is thebasis forour comments. A generaldescription of thebook is givenin [17], while [2] containsa reproductionof thetitle page andRaphson's Problem IX. The latterpassage is reproducedin Fig. 9. HereRaphson considers equations of the form a3 -ba - c = 0 in theunknown a, and indicatesthat, if g is an estimateof the solution x*, a betterestimate can be obtainedas g + x, where

(7.1)(7.1) xX c + bg -g ~~~~~~3g2-b

Formallythis is ofthe form g + x = g -f (g)/f'(g) withf (a) = a3 - ba - c. Raphsonthen appliesthis formula iteratively tothe equation a 3-2a -5 = 0. Startingfrom an initialestimate g = 2, Raphsoncomputes successively the corrections x = 0.1, -0.0054, -0.000048517 and -.0000000014572895859and the corresponding estimates g = 2, 2.1, 2.0946,2.094551483 and2.0945514815427104141 of x*.

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Theequation x3 - 2x-5 = 0 waspreviously discussed in ?4,where we analyzedNewton's techniquefor its solution.The twomethods are mathematicallyequivalent; the distinction betweenRaphson's computations and those of Newtonis thatRaphson uses (7.1) repeatedly, appliedto successivelymore accurate estimates g of thesolution and withoutthe need to generateintermediate polynomials as Newtondid. The discrepanciesbetween the numbers computedby Newton and those computed by Raphson are partly due to the deliberate omission by Newtonof a termin thepolynomial expansion for g3 definedin ?4. Raphsonalso retains all thesignificant digits computed in successiveiterations, while Newton uses onlythe first fewsignificant digits generated by each stepof hismethod. Raphsonpresented more than 30 examplesand formulaein his book. In each case they involvepolynomials, up to degree10. His derivationof theexpressions for the correcting termsx, suchas thatgiven in (7.1), is describedin theopening passages of thetract, and is preciselythat used by Newtondescribed in ?4 above,using binomial expansions, but with the firstcorrection formula derived for a particularequation being used iteratively.Thus Raphsonessentially used (1.1), butas in Newton'spresentation, Raphson proceeded purely algebraicallyrather than using the rules of calculusto forma derivativeterm, and in every case he wroteout the expressions corresponding to f (x) and f'(x) in fullas polynomials. The bookconcludes with a longset of tablesgiving the appropriate correction formula for a varietyof polynomialequations. Despite Raphson's subsequent extensive work concerning fluxions,it is convincinglyargued in [7] thathe neverassociated the calculus with his iterative techniquefor polynomial equations, and he neverextended it to otherclasses of equations. It is neverthelessappropriate to considerRaphson's formulation to be a significantdevelopment ofNewton's method, with the iterative formulation substantially improving the computational convenience. The followingcomments on Raphson'stechnique, recorded in theJournal Book of the RoyalSociety and quoted from [17], are noteworthy. "30 July1690: Mr Halleyrelated that Mr Ralphson[sic] had Inventeda methodof Solvingall sortsof Aquations, and giving their Roots in Infinite Series,which Converge apace, and that he had desiredof himan Equation ofthe fifth power to be proposedto him,to whichhe return'dAnswers true to SevenFigures in muchless timethan it could have been effected by the Knownmethods of Vieta." "17 December1690: Mr Ralphson'sBook was thisday producedby E Halley,wherein he givesa NotableImprovemt of ye methodof Resolution of all sortsof EquationsShewing, how to Extracttheir Roots by a General Rule,which doubles the known figures of the Root known by each Operation, So ytby repeating3 or 4 timeshe findsthem true to Numbersof 8 or 10 places." Thus Raphson'stechnique is comparedto thatof Viete,while Newton's method is not mentionedalthough it had now appeared in Wallis' Algebra. The significanceof the reference tothe solution of a polynomialequation of degree 5 is thatwhile analytic solutions in radicals forall polynomialequations up to degree4 wereknown, no generalformula was knownfor degree5 (indeednone exists); Raphson demonstrated that numerical solutions were neverthe- less attainable.Finally it is remarkablethat the property of quadraticconvergence was again notedfrom the outset. The fewverifiable details of thelife of JosephRaphson are discussedin [17]. Contact betweenNewton and Raphsonseems to have been verylimited, although it appearsthat Newtonexploited the circumstances of Raphson'sdeath to attacha self-servingappendix to

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Raphson'slast book, the Historia fluxionum [17]. In thePreface to histract of 1690,Raphson refersto Newton'swork but states that his own method is "notonly, I believe,not of the same origin,but also, certainly, not with the same development" [2], [12]. Referencesto Newtonin theAppendix to his book of 1697apparently refer to Newton's use ofthe binomial expansion, ratherthan his method for solving equations [7], [17]. The twomethods were long regarded byusers as distinct,though in 1798Lagrange [9] observedthat "ces deuxmethodes ne sontau fondque le memepresentee diff6remment" although Raphson's technique was "plussimple que celle de Newton"because "on peutse dispenserde fairecontinuellement de nouvelles transformees."Further historical details, particularly concerning comparisons between the methodsof Newtonand Raphsonand thefailure to recognizethe role of calculusin these methods,appear in [7].

8. Simpson'scontributions. The lifeand some of the significant mathematical achieve- mentsof ThomasSimpson (1710-1761) are describedin his biography[5] and withmuch criticalcommentary in [1 1]. In his Essays ... in ... Mathematicks,published in London in 1740 [16],Simpson describes "A newMethod for the Solution of Equations in Numbers." He makesno referenceto thework of anypredecessors, and in thePreface (p. vii) contrastshis techniquebased on theuse ofcalculus with the algebraic methods then current: "The Sixth[Essay], contains a newMethod for the Solution of all Kindsof AlgebraicalEquations in Numbers;which, as it is moregeneral than any hithertogiven, cannot but be ofconsiderable Use, thoughit perhaps may be objected,that the Method of Fluxions, whereon it is founded,being a more exaltedBranch of the Mathematicks, cannot be so properlyapplied to what belongsto commonAlgebra." Simpson'sinstructions are as follows[16, p. 81]: CASE I Whenonly one Equationis given,and one Quantity(x) to be determined. "Takethe fluxion of the given Equation (be itwhat it will) supposing x, the unknown,to be thevariable Quantity; and havingdivided the whole by x, letthe Quotient be representedby A. Estimatethe value of x prettynear the Truth,substituting the same in theEquation, as also in theValue of A, and letthe Error, or resulting Number inthe fornwe, b dWiskddbytWis nramerical Valueof A, andthe Quotient be subtractedfrom the said former Value of x; andfrom thence will arise a newValue of that Quantity much nearer to the Truththan the former, wherewith proceeding as before,another new Value maybe had,and so another,etc. 'tillwe arriveto anyDegree of Accuracy desired." In additionto applying his technique to a polynomialequation, Simpson gives an example ofthis technique applied to the nonpolynomial equation 1 x + 1- 2x2+ 1 - 3x3-2 - 0 [16,pp. 83-84]:

"Thisin Fluxions will be 2x 2x2x , , and 21 -x v1-2xx 2 13x thereforeA, here,

_ 99 =- 21a=;2x ; whereforeif x be supposed= .5, it will 2 1-x 1 -2 xx 2 /I1-3x 3 become-3.545: And,by substituting 0.5 insteadof x inthe given Equation, theError will be found.204; therefore3204 (equal -.057) subtractedfrom .5,gives .557 for the next Value of x; fromwhence, by proceeding as before, thenext following will be found.5516, etc.."

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Following[7], in theseinstructions fluxions were to be takenas describedby Newton in now-lostletters to Wallis in August 1692 and reported by Wallis in 1693. In modemterms i is essentiallyequivalent to dx/dt;implicit differentiation is used to obtaindy/dt, subsequently dividingthrough by dx/dtas instructedproduces the derivative A = dy/dxof the function. ThusSimpson's instructions closely resemble, and are mathematicallyequivalent to, the use of (1.1). Thisis thefirst formulation of the iterative method for general nonlinear equations, based on theuse of fluxionalcalculus. Simpson'sapplication of fluxionsin thecontext of solvinggeneral nonlinear equations was certainlyhighly innovative. This significantcontri- butionby Simpson received little recognition, except in [3] and[6], until the recent publication of [7]. The formulationofthe method using the now familiar f'(x) calculusnotation of (1.1) was publishedby Lagrange in 1798 [9],though it probably appeared in earlierlectures; it is given in NoteXI of thatbook (Sur lesformules d'approximation pour les racinesdes equations), ratherthan Note V (Sur la MethodedApproximation donnee par Newton).By thistime the familiargeometric motivation for the method was well known.Lagrange makes no reference to Simpson'swork, though Newton and Raphsonare bothmentioned. In Fourier's1831 Analysedes EquationsDeterminees the method is describedas "le m&thodenewtonienne," andno mentionis madeof the contributions ofeither Raphson or Simpson.This attribution in theinfluential book by Fourier is probablya majorsource of the subsequent lack of recognition givento thecontributions ofeither Raphson or Simpson[3], [7], [17]. The previousextract from Simpson's Sixth Essay referredto CASE I. Simpson'sCASE II is equallyremarkable [16, p. 82]: CASE II Whenthere are twoEquations given, and as manyQuantities (x and y) to be determined. "Takethe Fluxions of both the Equations, considering x and y as variable, and in theformer collect all theTerms, affected with x, undertheir proper Signs,and having divided by x, putthe Quotient = A; andlet the remaining Terms,divided by y, be representedby B: In likemanner, having divided the Termsin the latter, affected with x, byx, letthe Quotient be put=a, andthe rest,divided by y, = b. Assumethe Values of x and y prettynear the Truth, andsubstitute in both the Equations, marking the Error in each, and let these Errors,whether positive or negative,be signifiedby R and r respectively: Substitutelikewise in thevalues of A, B, a, b, and let Br-B and AR-ar be convertedinto Numbers, and respectivelyadded to theformer Values of x and y; and therebynew Values of thoseQuantities will be obtained;from whence,by repeating the Operation, the true Values may be approximated ad libitum." Simpsonis heredescribing the technique now generally referred to as "Newton'sMethod" for systemsof nonlinear equations, restricted to thecase oftwo such equations. In modemterms, thisinvolves solving the system of equations F(x) = 0; F : Rn-* WR'by the iterative process xi+1 = xi - di wheredi is thesolution of thesystem of linearequations F'(xi)di = F(x-) in whichF'(xi) is theJacobian matrix of F evaluatedat xi. In theabove passage Simpson describesthe construction of theentries of thematrix F'(x) forthe case n = 2 and then givesthe explicit formula (Cramer's Rule) forthe solution of theresulting system of linear equations.His bookcontains three examples of this method; we quotethe first [16, p. 84]: "Letthere be giventhe Equations y+/2 -x2-10 = Oand x + +x-

12 = 0; tofind x andy. The Fluxionshere being y + . andx ? FX+--

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or , andi + 2 + we have A equal -__ x_

B equal 1 + [sic], a = 1 + 2, andb = Case I. Letx be supposedequal 5, andy equal6; thenwill R equal -.68,r equal -.6, BrbR = A equal- 1.5,B equal2.8, a equal 1.1,b equal9 [sic];therefore Ab-aB= .23, and aRAr= .37, and thenew Values of x and y equal to 5.23, and 6.37 respectively;which are as nearthe Truth as can be exhibitedin three places only,the next Values coming out 5.23263 and 6.36898." Thisextension by Simpsonof the familiar technique for a singleequation to theequally fundamentaltechnique for systems of equations appears to havebeen overlooked in theliter- ature.It is a significantachievement. In passingit seems appropriateto notealso a contributionby Simpsonto theclosely relatedproblem of multivariableunconstrained optimization. In A New Treatiseof Fluxions, publishedin 1737,he giveswhat may be thefirst example of maximizing a function of several variables,obtaining the maximum by essentiallysetting the gradient of the function equal to zero. We quotefrom that text. "Note,When in anyExpression representing the Value of a Maximum,or a Minimumthere are two or more variable Quantities, flowing independent on eachother, the Value of those Quantities may be determined,by making them to flowone by one,whilst the rest are consideredas invariable,according to theMethods used in thisand the following Examples. EXAMPLE XVII Requiredto find three such Valuesof x, y, z as shallmake the given Expres- sion (b3 - x3)(x2z - z3)(xy - yy) thegreatest possible. First considering y as a variable, we have x -2yy = 0, or y =x thereforexy - yy = X. By making z variable,we have x2z - 3z2z = 0, x or z = thereforex2z - z3 = , and substitutingthese Values in the 3 givenExpression it will become (Xx x )x(b3 x3) = (h -X therefore 5b3x4k-8xUx = O, or x = b24b5, thereforey = l bV5 and z =b 253 N.B. The Reasonfor this Process is evident,for unless the Fluxion of the givenExpression, when any of the three Quantities (x, y, z) be madevari- able,be equal to Nothing,the same expression may become greater, with- outvarying the Values of theother two, which are consideredas constant; thereforewhen it is thegreatest possible, each ofthose Fluxions must then becomeequal to Nothing." The idea of settingthe gradient equal to zero,combined with the use of "Newton'sMethod" to solve theresulting system of nonlinearequations, is a keyingredient of manytechniques forsolving unconstrained optimization problems.

9. Conclusions. We havetraced the evolution of the method (1.1) fromthe appearance offorerunners inthe work of the Arabic algebraists and Vi&te to itsformulation inthe modern functionalform by Lagrange, focusing on thecontributions ofIsaac Newton,Joseph Raphson, and ThomasSimpson. In the lightof thishistorical development it would seem thatthe Newton-Raphson-Simpsonmethod is a designationmore nearly representing the factsof historyin reference to this method which "lurks inside millions of modem computer programs, and is printedwith Newton's name attached in so manytextbooks" [17].

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Acknowledgments.I thank Cambridge University Press for permission to reproduce Figs. 1 and3-6. 1am indebtedto the reviewers of early drafts of this paper and the anonymous refereesfor numerous helpful suggestions and references.

REFERENCES

[1] J.C. ADAMS,On Newton'ssolution of Kepler's problem, Month. Not. R. Astro.Soc., 43,2(1882), pp.43-49. [2] N. BICANIC AND K. H. JOHNSON,Who was "-Raphson"? Internat.J. Numer.Meth. Engrg., 14 (1979), pp. 148-153. [3] F. CAJORI,Historical note on theNewton-Raphson method of approximation, Amer. Math. Monthly, 18(1911), pp. 29-32. [4] , Sir Isaac Newton'sMathematical Principles of NaturalPhilosophy and His Systemof the World, Universityof Califomia, Berkeley, 1934. [5] F. M. CLARKE,Thomas Simpson and His limes,Columbia University Press, New York,1929. [6] H. H. GOLDSTINE,A Historyof Numerical Analysis from the 16th through the 19th Century, Springer-Verlag, New York,1977. [7] N. KOLLERSTROM,Thomas Simpson and "Newton'smethod of approximation": an enduringmyth, Brit. J.Hist. Sci., 25 (1992), pp. 347-354. [8] A. KOYREAND I. B. COHEN,Isaac Newton'sPhilosophiae Naturalis Principia, The Third Edition with Variant Readings,Volumes I-II, HarvardUniversity Press, Cambridge, 1972. [9] J.L. LAGRANGE,Traite de la Resolutiondes EquationsNumeriques, Paris, 1798. [10] C. MAAS,Was ist das Falschean derRegula Falsi?, Mitt. Math. Ges. Hamburg,11 H.3 (1985), pp. 311-317. [11] K. PEARSONAND E. S. PEARSON,The History of Statisticsin the 17thand 18thCenturies, Macmillan, New York,1978. [12] J.RAPHSON,Analysisaequationum universalisseu adaequationesalgebraicas resolvendas methodusgeneralis, et expedita,ex nova inflnitarumserierum doctrina deducta ac demonstrata,London, 1690, Microfilm copy: UniversityMicrofilms, Ann Arbor, MI. [13] R. RASHED,The Development of Arabic Mathematics: Between Arithmetic and Algebra,Kluwer, Dordrecht, The Netherlands,1994. [14] S. P. RIGAUD,ED., Correspondenceof Scientific Men ofthe Seventeenth Century, Volumes 1-2, Oxford,1841. Reprint:Georg Olm, Hildesheim, 1965. [15] A. E. Roy,Orbital Motion, 3rd ed., Adam Hilger, Philadelphia, 1988. [16] T. SIMPSON,Essays on SeveralCuriousand Useful Subjects in Speculative and Mix'dMathematicks,Illustrated bya Varietyof Examples, London, 1740. [17] D. J.THOMAS, Joseph Raphson, F.R.S. NotesRec. Roy.Soc. London,44 (1990), pp. 151-167. [18] H. W. TURNBULL,ED., The Correspondenceof Isaac Newton,Volumes I-IV, CambridgeUniversity Press, Cambridge,1959-1977; Volumes V-VII byA. R. Hall and L. Tilling. [19] F. VIETE,The Analytic Art, T. R. Witmer,Transl., Kent State University Press, Kent, OH, 1983. [20] R. S. WESTFALL,Never at Rest: a Biographyof Isaac Newton,Cambridge University Press, Cambridge, 1980. [21] D. T. WHITESIDE,ED., TheMathematical Papers of Isaac Newton,Volumes I-VII, Cambridge UniversityPress, Cambridge,1967-1976.

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