A Discourse concerning the Methods of Approximation in the Extraction of Surd Roots. ByJohn Wallis, S. T. D. and Savilian Professor of Geometry at OxfordAuthor(s): John WallisSource: Philosophical Transactions (1683-1775), Vol. 19 (1695 - 1697), pp. 2-11Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/102270 .

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( t )

I. X Dircoxrfe soncetning the h etbodw of Xp proximJtion in tbe ExtraSioas of S8rd Roots, By John Wallis, S. 7#. D. and hvilxan ProX fe§or of Geornetry at Oxford.

H E fev-eral*N4ethads bf A<pproximationv which hare been mentsoned of latcYearss ffor Exera*ing the Roots of Sinxple or AffeEtes3-Etuations, gives-llle oc-

caflon of faying foniewhat *f that SubieA. Ie is agreed hy a11, (and, I tflink, demonllrated by the

Greek long ago) that if a Nunlher profpoted be notua true Square,: u is in tain for o hope for a pft Q4adrxtick Rooe thereotexplicable by Rational Nunlberss Integers,or-FraAed.- An(3 theretore. in fuch caGes, we muR corltene our felves with Approximations (Comewhat nqar tlle truth) whoue prerend- * ^

lNg tO ACCtlraCy. And So, for theCubick-Root, of whae is not a perfe9c

Ctibe. And the like for Superiour Po;w¢rs. Now tbe- vAncients ( being aware of this ) had their Me-

thods of Approximation in fucll caKes w whereof fiom have bedn deriared down earen to this day. Of which Nve a1^1 fpeak more-anon.

But fince the Meehods of . Decimal Fradtions ( as they are nowwont to becalled) hate come intoPraAice, it hth been nfaal eo pmfecute fiuch lZxtraAions-(beyonal the placew of Unites) in the places of Decinlal P^rts to wvhat,Sccu racy we pleaS; whereby the former Methods of Appro3ches hive been ( not fo much forgotten, as ) neg!eAed

Not that if fuch Approaches by Decimals were alwa)rs the mo fpeedy, or the moR exa&; (for no Man doub.s but that X is a more Simple, and more Intelligible Notation of that Quantity, than 0.l2<s, or TI^>oW * And , not only a more BrieE, but a more Accurate defignation of the Square Root of 's, than o*33);a3, eFc.) BURJ becauCe Fradtions redllced tO the Decimal form, aremore convenient for SubSeqtlentOperaX tsons, when there is occafion for a further ProgreX.

Mr. Newton's Method of Approximatlon for the Extrs8ting Roots, even of AffeEted Egnations, I have given fome Ac-

cotant

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(3 ) CoUht of- in my DgS Algebra ; and fomewhat nzore fullg in the Lattn Eelition : where 1 gire an hccollnt alfo of Mr. Rap* .*Mst Method; which I tleed tsot here repeat becauSe it iS tO

be feen there Since which tinle, - Monfiewr DB L'vgty hath ptlbliliacd his

-Method of Approximation, principally for fingleEquations, or EstraAing the PUOQt of a fingle Powerj

An(l Mo. HaGcl!- hath fince improved this Method, with a further Advantage; elnpecially as to Affcde(l Equations.

But I need not repeat either of them becau(e they are both publifl}t. That of De L7agy un a Treatifi by it tUlf; and that of . Mr. H411er, in the P^I/Dg. mr4^ifd2. INutsb. 2Io} for the Month of Mat) X 694.

Thefie may all, or any of them) be cf goot3 uSe (eafih in their own way ) for making more fpdy Approache¢, and by greater Lcaps, itI many ca(es, than het;)6Mctlzed (pro fecuring the Extradrion in IDecil Parvs j purfued and itupro

ved by A/lr.OgghtreS and Mr.AIart;Wt of c.olJr onYn, and by o- thers abroad; eSpccially as tO S;mp}X Equations, if zve Etlp- pote fuch Extradtions to be paurfised to the full excent.

As for inRance} if we would Extra& the Root of an itn-

perfe* SatSolid, (or a Power of fiveDimenfons) tQ hRz its Root true as far as the hIxth place of DecimalParts. In order to this, wef are ao add (or fiuppote to be added ) fix PunEtations of Cypherss (er3 fix times five Cwsphees., that is Th§rty Cyphers ) b) ond the placo of Uni es in tEse Ntu.lbFr propa{ed. If raow we purfue the wholeoperation to the urmoR of thoSe Thirty places, the Work would be long and redious.

Eut if we make re of Mr. Ougbtred's Expedient, ( for Mul vsplicaton, DiviJion, Extrafticon ^f Rocats, and other iiRe Vperations, ) by raegleding Wo xnuch of tls Ivng Procefs, as is afrerward to be Ctlt off and ttlrown away as utelefs., (whicIJ; I think, is ge;erally pradriSel ) the Work will be nuch ab- bridged, and the Advvntage of the other Methods much l¢Is confiderable.

That is, If (for aCcertaining thJ fixth place of DecimX^tl Parts) we add fix Cyphers (inRea(l of tllirty) sLnd on: nr two nnore (the better to fectlre what fronl the conleqalent places nlay in theOperation betranrn3ittedhither;) na1 p-y+ fiuw tlac Operation ehas far, negledcing the followirg pisibt-

(which)

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( 4 ) ( wich are- not llXely to influence t he Pigpre due to the lGxth plact of Dectmal I3arts of the Root fought:) thts long Pro Ik;will ba mtlch; fkortned.

Arld if we further confider, what Preparatire Operations are to be! made int lome of thofe other Methods, before we come to the prcEcribd Divifion for giving the Root defired; the Advantage (tholjgb confiderable) will not be fogreae a may at firR be -app(ehended; efipecially as to Aies9ced Equa rions, in whish the -Parodical Powers have great CoefficientsO As wil! foon appear in Pradice, if we come to apply them to partscular cates.

;Buts -witheut diSparaging thelE Methods ( which are realljr Sconfidelable, and well worthy encouragement) that which I here intend is tO {hew the truePoundation of theMe tho(ls uSed by the Ancients, ( however fince negle*ed ) and the jut} Improvement of them. Which thoughAnciently fcarce applyed beyond the Quadratick, or perElaps the Cllw biclk Root, (for wiih the Hsgher Powers they did not much trouble thembelves) yet are equally applicable (by due ad uRments ) to the Superiour Powers alfo.

1 {hall begiti with the Square Root: For which the Ancl- ent Method is to this purpofe.

From the propoRed Non-quadrate ( El,ppofe nt ) fubtra<St -(in the utual manner ) the greateR Square in Irtegers therein convained ( SuppoSe Aq.) The remaitlder (fiuppoSe B = 2 AE + Eg ) is to be the Numerator of a Fradti-on. for defigning the near ralue of E (the remaining part of the Ront {oughe A + E _2 N) ) whofe Denominator or I)ivifor is to b: 2 A (r-0ne dotlbleRoot of the fubtradked Sqt1are) or sA+ (tilat double Root increafed by X ) the true value fallil-g be tween theSe tWO; fometime the orjeg fonetinle the orher ;, being nearel} to tlle true value. But (for av.iding of Saco gative Numbers ) the latter is commonly dircEted.

Tllis Method Mon(;tur De L'agny atXirms to be more rtan zoo Years ole3: Anll it is to; for I find it iin L2c P4uiolzz ;( otllerwite called Luca zle iUr:Q) &I' de BsrgQ S4rSi <Vepslshri) 3?rinted at Mesice In tht Year x494 ( if not cYGn toOner thar? fo) for I find tl}Ore have been Severa] E1itions of it. ) And how mt3ch older than to, I Gannor tell: For Ese doth nor deliver it as a new Invention of llis own, but as a recAived PraAice, and derxved frorta the Maors or ArD* from whom

thvy

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( s ) they had theirAkorsfim, or Pr-aftice of Arithlnetick by t1w: Ten Numeral-Figures mw in uSe.

And it;iS cominxd downhitherF :;in.Books.Qf Pradiml Arithmettck in all Languageh wllich teach tive Extradion £s the Square Rooe, and (theziin) thisrMuhod ̂of Approsv mationJ in ca(e of a Non quadrate.

The true ground of the Rule is this: Aq bei-ng ( by CQR firu*ion) the greate(t Integer Square contained in N, 'ssb evident ttlAt E muR be lef$ than I; (otherwtSe not Aq, blur the Square of A + Iv 0 Come greater than So? would be tC

greatefi Integer Square contained in N.) (biow the Ae-

mainder B - 2 AE ->Eq be dierided by 2 A, the ReSult wxll be too great for Es ( the Divifior being tOO iittlO; for it ffiould be 2 A E, to make the Quotient E. ) BLge xf ( to reEtifie this ) we diminiJh the Quorient, by increafing tlle lVivifors adding t to it, it then ecomes too little; bocauSe the Divitvr is now toobig. For ( E being le{s than I ) 2 A+ - is more than z A + E; and thereforc too big.

As for inEance; If thQ Notl-quadrate- prnpofed be 19 = <, the greateft Integer Square therein contained ;5 Aq = 4 ( the Square of A-2:) which being {ubtraded, leaves N&q -S-4 _ I-B-2 A E 4- E q. Wtwich -divided by z A _4, gives : But divided by 2 A I _ 4 q-I = S, gives r.X Tllat tOD great, and this tOO liitle for E. And there fore th.e true RQO? ( A+ E = v N ) is 1e-X than 2 ts = 2.X F) but greater than 2 T _ 2.2 : And this uvas AIlsiently thougtzt an Approacti near enough.

If this Approach be noc now thought n^ar Onoug ebe

fame Procets may be again repeated; and thlt as otst as is thoughE necdary.

Take now for. A, 2 t.- = 2.zx whofe SquarP is 4.84-Agv ( now confidered as an Inceger in the (econd place zor I)Ccs- mal Parts.;) This filbtradEed fronl -.oo, ( orn wllivll xs th, fame, o 84 the excets of this Sguare above the tormcra f£*on Iwhichwas tlzen the renzzinder,) ica*esanewrenl>stkeldi

. .w-6 z

B_o.t6: which e3Xvided by 2 ^$-4.4} gives-4-+ -S-

-o.os636 - too much. Bu6J-di<ided-by z X v X-e - 4.3 *

it gives--- iZ 2 = °'° g 9:S S +s too littic. .tthe trL;e \ a- laIe (betvveen thete two) berng a,2 36 proximen wluotX fq,Uare iS

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( 6 )

It' thiS be not thought nea-r gh; Subfira* this ukre from S.O°°°O°: The Remaitier B_oooo3e4, diiviid by j A^.4?s t -or by 2 A+x-54.47g, gwcs (her way) o sooM8-; which adddd to A-j 236) sikes ^*XJ6o68}

{omewhat toobg; bata.Zg6067+wou14-be mshmor too l-ittle.

Which gives us alae Square Ract °f 5 ad;ued to the fixth place of Decimal iParts, tt threefreps. And by ffie bmw X4eched, if it - be thought ne¢dEu-lX we may prseed filr ther.

1t weie eafie to coinpouml th: Procefs -of two:or more 1Reps into ono a-ntl give ( for the Rule ) thC RefuE of CLompofiltion. Wesich: tould make- it Xem more Intrscate and MyfleriousX to amufe the Reader: Buc I choofe to mako lt as plain as t can (and eherefore keep the Scveral Reps Se- par-are, 3 that - the Reader ( for his {atisfadtion ) may clearly fev th: ;tue gloltd of the ProceSs. 13ut of thi-s tough.

Pro^eed we now to tIse Cabick 4a. hr *c-h (con Xonanc av the QgtadratiPk ) the R¢ls- is- this :-

Fon1 -the Non-Cabick propoid, ( fuppofe DJ, ) fubtr* t}]e geateA CubA in-Integers theretn containi-, (fiuppoG Ac: ) the rema-nder (futopote B 3 AqE--> 3 AEq+Ec,) is tc) be the tdumetator of a Pradtion for defxgning thae value cf Ex (the fenlainlng paFr of the Roa fqht A+ E-Pc N.) To this Nllmerator, Wf (in the:I)enominator or DitiSor) ve Sub joyIl 3 A qJ the R&k will>anainly be too great for iE: becatlle the I)ivifor is too lietle: (For it ffiollld be ; A q + 3 A E E q, to give the true value of E. ) If, for t.ne Divi orv we take ;Aq -> g A + x, it will cereanly be tOO littd, becauSe tbe Divifor is too great. ( For E, by zn- ;tI'U&iQ" iS l>As than Is ) It mufi therefore ( between thele linli*s^) be mors than this lAtte. And tirefom this lAtEr

Re1u1e being ade3ed tO A, will giYd a Rose whoSe Cui may be fuStraAed firoln the Non-£gabick propoSeda n order tO anotll-r fiep.

This ALzploaci} I find in [Ringate's Arithmetickl Publiffieel fn the Y^eal 630, and maPl thcreforo be -at lealt rO 6}d;

;0tW m*suvh; older I cm8ot t;U- t3ut if; for the DiviSor, we take 3 A q + z A) ( or even

}wf, than SO 3 the- Refult may be t grekt j Qr ( iN CAfi B i ,alalJ ) it may bW too litvle, ( and oft is fo. )

Whict

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( 7 ) Whisb mes to pars fro jhence ; beczufe E ( by Con-

§ruEiqn ) is le& tharw x } and tberefore 3 A E ifs tan 3 A ; and/perhaps fo much as that the addRi of -E q will nor reU drets w. An; w it fo hwpens, 3 A q + 3 X, is a bettcr

Divifof ehan g A;q g A I, (or sven fomewhat lefs than

either.- ) But becaufe it doth not always fo happen ( though for the moll part it dah ) the Rule doth rather ditedc ahe ethex; as wSh dotih certaialy.give a Root ler5 than lthe tree value, whote Cube may a.lways.be [ub.trafted Dom the Ntonv Cllbick propofed. The defign being tO have .fuch a Cub as ( being fabtraAed ) m?y Ieave another B, tO be ordered in like manner for a new Approach.

But, for tlle moll partv 3 A q may be {afelyt taken tor the I)ivifot: For, though the Relillt will then be tomewhat too big, yet the excefs may be fo fimall, as to be negled£sd; 0r3 At

lealf, we may the cafily judge wi Ntta*mbsr ( fomewhat le(s than it ) may b: [afely take.a. And isf we chance to take it fomewhat tz bigX ehe Invenience will be balt this, that B for the ngxt Rep will be a N¢gative. 05 which cate we {hall Epeak anon.

Thus, for inlnratlae; if th Non-Clsbe propofed be g-N The greateR Integer Cube therein contained is 8-Ac, ( whofe CLabick Raot is A = 2. ) Which CuSe fubtraAed , leave; g-8_E_ B= 3Aq E g AEq + Ec. This

diviid by 3Atl I2, glVN i1^-o.o83 3 3 +) too big for E.

Buttllefamedividedby 3Aq+ 3A l= I2+6+ r=I9 gives t-9-o.oS26g + toO little. Or lf but by gAqA gA

=I2 + 6 _ I8, lt glVES 1w so-°*°SSSS >) ytt tOo

ittles For the Gulte of A;+ o.o6,= X.o6> is but 8.742-5 which is hortof 9. And fo inuch ffiortof it, that we nzar lafely take 2.o7 as not too big: Or perhaps 2.v8, ( wluich if it chatlce tl;) be tO0 big) it Wlll t].Ot be much too bxg ;- of svlllch cafe we are to {peak atlon ) And, upon tryals c svill b& iound not too big; for the Ctsbe of 2.O8, is but S.9989X2.

If this firflr tlep bc not aear enough: This Cube (ubtraded from g.oooooo, leaves a nevu B_.o.aotoS8> whicb distided by 3 Aq_ sa.9796, g;ves o.oooo84-; which will be foms what eoo bigt but tlOt mach. ( lior E is alow [o frllall) as tll:t 3 AE may 1)e fifely negledsed, and Eq much more 3 So that it' tO 2.08} we add o.oooo84-, the Retallt 2.o80o8o4 wstI i tOQ big, but a.o8oo83 will be more too littlef ( As wil wp-

G 2 p;,r-

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( 8 )

pear if we take tne CabE of ekch.) So that -eitherRof them, ae the Second kep, gites the i true Root within an Unite in

the fixth placeof Decimal hrts But when I fayX -akE tbe CwJc of cacb, ( which r do, that

the thing may be more clearly apprehended) it is not necefl a- ry thar we trouble our [elves with the whole Cube. For, Azc btig already filbtradred,for finding B=3 AqE+3 AEq+Ec, we halre no more to try, f lSt whether 3AqE gAEq+ Ec be greater or - lets -than t ^ AzG6rding as we take -o.Oooo84, or o.sooo8 3 for-E.

Wlzich may conveniently be doflie in :thgs manner: Take 3A->E, antlMultiply thisbyE, (orEbyit) fohavewe fAE+Ef. To this add 3Aq, and Mllltiply the whole by E, ( fo halre W& g AqE + 3 A8qA Ec,) to See whfflher this i greater or leSs than B.

That IS, in the prefent caSe, if we tkke E = o.otoo84,

and add to -this 3 .X 6.z41 thetS is 6.-240084-o3^^ +E;

This multiplied by E_o.onoo84,ss 3AE+Eq=o.ooos24+. 1 o which if we add ; Aq = I2.9792n it 'iS 3 Aq+3AE+Eq

.979724 Which nlultiplied again by E = o.oooo84 S O.oOIQ902 -> 3 t AqE + g AEq + E4iwhich is more than

b = o.ooo88.

But if we take E=oown83,- and proceed as before, we tha11 ha;v^^ 3 A-qE + t4Eq +<iEe _ mooto77-+, which- is ]els tIlan B. - o.tioS8. - ked there6rd ( if we--fiubtra&thSe trotn 23is ) the Remainder, o.ooooil,- will be another*B -for tIe next {-iCPI it we pleales to procced further.

3itittlerto 1 have palrfued the Method moft affeded by the Aacients, in Seeking;s -Squate or-#CuSe ( and the like of other Powers ) always lefs than the jult value, tbat it- might be fuh- xatieil fronl theNumbernpropoSed, leaving B-a PofitiveRe rnalnd&r; thereby avoiding Negative l!Jumbers.

But fince the AI ;thmXt jCk of Negatives is now fo well un- £3Wri{ood, ic may in this ( and other Operations of like Na- ore ) be a{ivritable, to take the next greater ( in cate thae be -X}carer co the crne valee) rather than the rext oleffier Of -wllich-l took n£:>tiX in nly CommerciwmEpi/Folicum, Epi#. 19. J-n. 2. -X6y-7-. n^a coe nlnve intricate than this-is; And w-icta l clwnere Advif>v in- Seeking the Greatef Common Di- <Usor olt two Numbers., in order to the abbridging a Fradrion, tr oth^.-wif.¢.

Accord-

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(9 ) According to this Notion, for the Sqllare ROGt tf 95 i

would fay, it is ( z + ) Somewhat more chan 2 -; and enquire How much more ? But for the Square Root of 83 I wold fiy, it is (-3-) fomewhat leSs than 3; and irlqllire; Hour much leSs? Taking (i£s both caGes) that which xs neaTert (. ahe iuft Value.

Thus, in the Cubick Root beire us; I woul<:l take [tJ; ;'. (in the la0Enquiry) o.oooo84-(where, rOS CEd ntat ilep; we have B--0.0000023) tather than o.ooooU g zJ

( wbere, for the next tlep, we {hould have B -J- o.esoan t X . )

In the lAtttr cafeX we are tO Dvide B--tro^°oool by 3Aq-I2.9SO236-,tofincl(bytheQuotoca.)l<o-waavlWi- iS tO b: added tO o.aooo83. In the other cafe, we a.e a> Divie3e B = + oXoooOoX) by 3 Aq = I 2 .9 80248} to fiiid v8y the Qotient ) what is to bW abated of o.ooooU4. So IazYe wx

0.0000t I

- 802g6 = ° 0o0°°°8S + tO be auded 6 2+o8) * ()£

-0.000002 8 48-0 oooo°° s + to b: ̂ haFed of 6/zToS (r

it nzay fuffice, in ts hers to r)ivide by 12 98->) or cssa by x3._*, without being incutnbreel witha longDivifor) ei- ther of which gives us fior ehe Root . Iougtlta 2.o8008 3 8 f towsme. True ( at the thirii llep 3 tO the EtgE+tb placo of DeZ cimal Parts And if this ba not near enougll, tilC Cvlbe of this, compared with tise Number propo2t1, uvill gisrc us ann- ther B for the next fiea. And io onu7ar1s ab 8.sr as zZc pleafe.

Now, what is faid nf the ;li5 sJ <^afity asnplicable to t-s+ lligher Powers.

I hall omit chat c>f tl1e Bqut(lrae; 1)59tuS iere pArba;8; it may be thought moi} a(+Vs{able, tO Extl-ah trje SguarA Re) of the Number propoSee3; ans3 thvn the Square Itcot cit tha Roor.

But if xve would do it at once) we are EOtN N ( the N+uns ber propoSed ()>ing not a 3Blquadare ) eo SubtraA Aqq (the greatell Biq,uadrare concaiJIed in it ) to find thv Remaintler B_4AcE + 6AqEq + AEc-) Eqq. W[ic:t; Iten.ain(let if we l)ivi(le by aAc, the QSotient wiil cWrtainly bnv too bit tor E,(though pertiap: no: auech:) if -by 4zq-b;Xq*P4AFx) it vPill bertatniy bv too little ( tor rcaf)ns belore mentoned! ) And wc are to uie our ditcre;on in aleing rom^ int

termet3iStt

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--( IO )

termediate Number. hd if we clzance t:ot to hlt otz the neare0, the Inconvenitce will be bxlt this, ehat our Leap will not be -ro great as otherwife it might be. Whtch will be redified by another B at the next llep

For the Sllrfolsde (of fite-D£tnenfions) we area from N (theNumbcrpropo(ed, beingnota pvrfedrSu,folsde) ta Subtra& Aqc (the greaseft Surfolide therein CQucained ) to find tllC Remainder B-S AqqE + IO ^cg + E-O AqEc

AEqq>Eqc. Which ( as bdore) if we Divxd: by yAqq, the ReSult will be fomdwhat tOO big) ( becauS the DiviSot is toolittle:) If by sAqq+IoAcAloAq'sA+r, the Refilt will certainly be lefi than the true E. The juk value of E being Iomewhat -between thefe two, where we are to uSe our diRcretion, what Intermediate Number to take. Which according as it proYeS t00 great or too littlt, i5 tO be sr.eAified at ti next {?ep.

If, to direEt us in tlle choice of fuch internaediate NumS ber, we Ihould Multiply Rules or Precept§ far fucll choice r

the Trouble of obEervlng tham, would be more than vlae Advantage to be gained by r. And, for the nloft pal<q it will be fafe enollgh ( atld leaIt trouble 3 tO Diaride by sAqq which gives a Quotient fomewhat too big: Wllich we Illay either ReEtifie at DiScretion ( bg taking a Number Sonlewhat leSs ) or proceed to anotller B, ( Affirmatiere or NegatilreX as the calb a11 require) and fo onward to what exa*nefs we pleaSe. ( Whch is, tor ltibRance, in a manner coincident with WIr Raphfon's Nlethoel, cren for Affedcd Equations

Tlllls, in tlle preSxn; caCe; If the Nutnber propoSed be N 33,lacnisAqgt,.andE-3-- z I=iAqqE

+ IO AcEq IoAqEs -t S AEqq + Eqc. Which if we

Oivide by sAqq _ T X 16 = So, the Rfult -' =0.0X2y, is Womewhat too big tos E) but not azzech. And if we examine it, by takitlg theSurfoliils of 2.CIsf, or of z-z+, we Jhal-l find a Negative B (ttor the next Itep) but not very cotlficSe rable. Or st we think it confiderable, we may proceed fur- ther to another fiep,- or more than Wo.

The like Method tnay be applied ( and uZith more lidvan- tage );- in the Higher Powers, according as the Compolition of each Power requires.

And the iwame Method may be of ufie (with good Aelvan tageJ in longNtlmbors fif duly applied) cven before we

¢:ons

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( 1! ) come to the place of Unites) for the lime will equally hold there alSo. Which the Reader may eafi}y apprehend withotlt a long Ditcourfe upon its

How far this Method mae be ctincident with Wome of thoSe before mentioned, I de not trouble my Self to enqu-ire; nor whether, or for what cauis, all or any of thofe may bo more eligible. My dPfign being only tO Jhew the erue Na- tural ground, fronz whence Cuch Rales of Approach are ( or might havebeen) derived; and by which (if there be sc-- cafion) tney may- be examined. And if I have done thisX ie ss what I did propoSe.

In Affeded Equations f eXpecially where the Coefiicients are great, and fome Affirmatives, others Negatives,) tho GaSes will be more perplexed. And to Multiply RuleJ fort each CaCe, would (I conceive) increaSe the Trollble, with ns great Advantage. %thish therefore I leave to tlle Prudence of each (as occafim {hall requ-ire J tO take ponze Intermedl- 0 ate, between a grmter and a lelEer. 0r if they pleate t.D accoa2az0date that above m5ntic3ned ( ouX of Commerc.Ee piRol.) e the prelinF cate, which is thewe ap21lcd to a Ca.fd nat lefs inkricate. Or to make uSe of Sotne of the Method;^< delierered by orilers trO this purpofe. Where rlais ( withal > is to be confideren3, That Such Affe&ed Equatieris are capab8: of nlose Roots than on¢, according to the Number Gi' Dis mcnfions to wlaich they arife.

I.t J

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