an attempt towards the improvement of the method of approximating, in the extraction of the roots of...

An Attempt towards the Improvement of the Method of Approximating, in the Extractionof the Roots of Equations in Numbers. By Brook Taylor, Secretary to the Royal SocietyAuthor(s): Brook TaylorSource: Philosophical Transactions (1683-1775), Vol. 30 (1717 - 1719), pp. 610-622Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/103306 .

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t dgO ) peded>greacer things-fkem }is pregnsnt and fagactou s M{itw l;or he sYaS Scarce ô Years of Age sYhen he held tnefe Correfpondenctes svith Mr Crabtriet And at the Xge of ^3. he was killed at Dr/ionSMoor Battle, on 2l} t. 64.4 68htitig for lRing fhJrks J His Father

utas HeXry GaJ2otfne E; of SX/etnx, betwcen Leeds and hYafce-lfil

\7, Xttz Sttean#; tomtrds tte Strouement of tSe S- thod of attroxintZating) :tz tlse Extrd^}son of the toots of qdttOzas ie N7ternDers. ty Brook Tay- in3r) SvcretdRy to the toyal Society

'St Phvt'lt'Rrg. N0o tt0 T)rt Malley} novrSecretaryof 1 the Roval 30cicry. has pub't&zsd a s7ery compcJldious and u4 cful ?4ethod of extra;>ttug rlle Roots-of aSeded Fguarions of ttle common Fortus in Aunzberst Thls Mernod preceeds by aSming tlle Roor defred nearly tr--un to cyne 0t tvo Places t£] Decimals (wllich is done by a GSson¢Ctrical Conll:rudinn} or by lome other co.n vencat Btay) a>>;d corret&ng the A5unaprion by com- paring vlle DifEetenê butw7cen the true Root and C}le

a-tEumeds by meatss of a new Equatton wllofe Roor is

that DiSrence, and wllich lle {hWws tlow to fiorm. ffom

the -Equation propoled by SulzSitutioll of the 57alue of thn Etoot loughr9 partiyz in known and partly in unF

*

kn0\^tl X w ermse Tn ceing thiS he makes afe <>f a Table of ProduEs

(which he calls SPAG#I#aX 27Xglti;aw) by wllicll 11e comX a,zutesTllP CocEeients in abe new Equatiots for finding tlle DiSerence mentioned. This Tables I obServeds tas fortned in tlle fame lblanner from the Equatiotl

pro-

t dgO ) peded>greacer things-fkem }is pregnsnt and fagactou s M{itw l;or he sYaS Scarce ô Years of Age sYhen he held tnefe Correfpondenctes svith Mr Crabtriet And at the Xge of ^3. he was killed at Dr/ionSMoor Battle, on 2l} t. 64.4 68htitig for lRing fhJrks J His Father

utas HeXry GaJ2otfne E; of SX/etnx, betwcen Leeds and hYafce-lfil

\7, Xttz Sttean#; tomtrds tte Strouement of tSe S- thod of attroxintZating) :tz tlse Extrd^}son of the toots of qdttOzas ie N7ternDers. ty Brook Tay- in3r) SvcretdRy to the toyal Society

'St Phvt'lt'Rrg. N0o tt0 T)rt Malley} novrSecretaryof 1 the Roval 30cicry. has pub't&zsd a s7ery compcJldious and u4 cful ?4ethod of extra;>ttug rlle Roots-of aSeded Fguarions of ttle common Fortus in Aunzberst Thls Mernod preceeds by aSming tlle Roor defred nearly tr--un to cyne 0t tvo Places t£] Decimals (wllich is done by a GSson¢Ctrical Conll:rudinn} or by lome other co.n vencat Btay) a>>;d corret&ng the A5unaprion by com- paring vlle DifEetenê butw7cen the true Root and C}le

a-tEumeds by meatss of a new Equatton wllofe Roor is

that DiSrence, and wllich lle {hWws tlow to fiorm. ffom

the -Equation propoled by SulzSitutioll of the 57alue of thn Etoot loughr9 partiyz in known and partly in unF

*

kn0\^tl X w ermse Tn ceing thiS he makes afe <>f a Table of ProduEs

(which he calls SPAG#I#aX 27Xglti;aw) by wllicll 11e comX a,zutesTllP CocEeients in abe new Equatiots for finding tlle DiSerence mentioned. This Tables I obServeds tas fortned in tlle fame lblanner from the Equatiotl

pro-



( 6'11 3 propos'dj as the,Fluxions,are, taking the Root bught br rhe on!y tElowing Q#antity} its Flalxion br Unicy and aficr everyv )perarson- dividlng thC PrQDuE XCeS 1SveW by the Numbers T, t*t 34, 4.} &o Hence I fvon fbund that thi$ Method mlgl<t eafily and natvarally be- drawn fkom Grf 2. Prot 7* of my Methoda herez¢nto rhsn, and tiat it WtAS capable af a irthos degree of Cb nerality, it bei¢g Appltcabl-e, nor ol<;y to Equstions of the common Form (viz fuoll as cotlf0 otX Te¢j wherb in tlle Powers of the Root fought arv poRtiv a\d InteX gral, without any Radical Sign) bllc aiX tO ,31t SXpReS

f3ons in generalt wherein- any thlng istpropoiede as given whicll by any knoxvtl lletllod migL; be c*-mputedS tS sivc tersd,, tle Root svere con{idertd as gtsten : ich a*SE ;Rtc all Radical Expreltons of lZialonlials. Tritlonllalvt, or or any oth.r Nomiall svlich mala be cotn)uted ey the Roor giv^n, ar leal by logarithMs, svhatever l3e t1<e lndex of ri-e Power of [hkE No-mtat; as lsKesviCe Expreflsons of Logaritllms, of Arcsles ty slle Sines or tfan{nts) o* Areas of Curves by thv abf;;S or any other FiuentsS or itoocs of Flussonal Equa£ions &v.

For tlle ialce of this great GerttMalltyfi it may- not be in3proper to Ihew lzow this htethod is Sderivecl fiom rhc- forefaid C¢O24tea X t-herefore t and % ing two flosrxng Q!lantXCics (tv}lO2> ReEtion to Onv anottler Tney be ex- preR;+lsy anv--f;qmton rharSocv^r) by ths Gro#vrys while z by i|o*St}g unstmfy b;onacst £4-Vi X N^?i

z , *

-become x +-w->* , v*} _. P z.s l ¢<, *. *. w

or x 1 XV 'l- ,X v -i- V , *¢- 5, * Eor Z putting te

X X s, L + Y r

Hence if y be the Roor of any Expre(Elon formed of eXand -Rknosrn (Qlantsticst and tuppoSed cqal to nothingv



( 612 ) and z Xot a part of , and xsibe formed of z and the known Qiuantities, in the fame manner as the Expreffilon nlade equal to nothing is formed of y; and ler y be equal to z + v: the diSerence v will be found by4Extradting

tlleRootoftllisexprelSonx ,xvX_xv + xv X t.t 1.X.3

F Sc-o. + Foritl tliuis Cafe z bei-ngbecome z + s-,

x, wllich is now become x + xv4-- ; +,

muR become equal tO llOtlling.

TIleRootsintheli:quations 1- +_ + x [ I .> 1 .z.3

-3 c G-* O, iS 0 be found tapon rlle SUPPQfiiOn of its

besng very fimall witb refpeEt to z, (as it muR be, if z be. taken tolerably exad) by hicll-means the Terms

Xv _F -*' _t tr. may benegled:ed, upon ac- X ** .3 I * t* 3 *4 count of their fmallnels vvith reEped to the other Tcrms9

' 2

ie as to leave tlle Equation x + _ + x 0^ fOr * x t.

finding the firft approxtmation of v. By extrad;ting tlle Root of this Equation, we have

x

¢} -s/ ..--. - -. That is, X x x

x2 2X X X ,>2

Firll, P ... _-. ... -., if x+;cfa- c>.

x Jc x X

.; . Z

x X Jc X

Thirdly



( 613 ) _ X

X z , , x v 3. -_2 X _., if w_Xs+,a.o*- G X X X

_ r

JC X X X J * X V 4. __vz-->, lf -xXt+"--- -,&;;° X X NC X

This approximation gives s exadt tO Ewice as Tnany

places as tbere are true Figures in z and therefore tre- bAes tlle number of true Figurcs in the ExprefElon of y by z -f vs whicl may be taken for a nexv Value of z, for computing a fecond s} feeking otiler Values of nc, Dc, 9¢ etc. Tho) xvllen z is tolerably exad (whicil it may b: efieetn'd when it coIltains two or three or more true Figures in tlle Value of , according tO the Number o Figures the lloor is propofed to be compured tog) t11e Calculation may be reSor'd nvithout fo much trouble

only byrakingV xz 4_ X _ zx t3 _ Jc X 2, 3 X ,z

* ,

i:. ln0cad of- v- >-. taking every timke fior X

its Ntalue laft computed. . c ss

From tlle fame Equation >c -1- X v ->--> - X --

3 ; q ;* o, may be gatl:erd alfo a ratiQnal Form, viz,

v-- - For negleEing ta: Terms , 650r. X nc I z -)

X -- .

2, Ic o itX

te have v _ -which ss rlearly -4. Tl1erew

4

X -t -v _ X

fore in thw Divlfor nllead of v wrltlug-* wee have

more

B b b b b



wben X + pc s -1--dt6. = Q

X V: .

wllen _ X -F X t 4--dS, O

*. z

, whenv_ xri- 2 F^-°

- X lt * 9 . - rb

r JC J¢ -

C

C

-*- @

{ 'S

:1C F

C

3 .^-.

t. sx

_

z ,oc

_-

4 ..

.. nCX

:X _L- 2 X

wllen _ x-x t ',--Ct'rv * O

This hrmg14 will alfe triplicate tlle rAunlber o^ tru Figures in z. And the C^alculation mar be repatedg af

ter every C)peration, taking fior a Divifor X 2-- v ;

,. n ..

x 2 _ _ tc. in0ead of nc +-..

.z.3 jeXe3*4 2 nc

I:)rM filallee has fully explain'd the manner of uEing bcth tllefe FortntBla's in LEqua£ons *>f the common Fom wllerefore I Ihall be the {horter in cxplaining two or three Examples of anotller fort

Ex 1. Lct it ba propofed to und the Root of tns E qtiOnt 8 Iiv -'.-y- - 16C -ntbiSCA''ts srOR

7 WRitiNg 9 and iX O vritxng x xrc have zz -,- 1vz

( 614 ) _ X

mor-e exaEly s -_ .. , thar is X X

X_-



(615 ) + Z-I6 X. Whence by taklng the Fluxions,

we have x-ws/z xz x + x1v2-' -F x, and X

S , _

w2^ x 8-4vx tt x zz g- vl2z-2* For finding tEse firA Figures of the Root , for v X take7, and wew

have t1le Equation yz + Il-i _> y _ I6-0t A7hiCtibeing

expanded gtves 76 - 3 74 - X 7t -1- ^ 5'-xis-o

3y this Equatton I lind tllat for tlle Srk fiuppofition we may take z w. TIlerefore ln order to xEnd zr let us now make s/ X = 7s (wllicb is nearer tllan befbre*

and we have x zz Jr Ies 4- X _ T6 at + x1S^_x4

-5-v4--)48; w zo,,66; X _+,7X. V7lzence

by the fecond rational Forms _^ 4 48 IOS66+4s7L X 4XAS

X x lo, 66 -o} 38 , wrhich muR be too bigf becalSe 5* < 2 ws and vllerefbre will require a larger Value of y to exllault th: Equation, than u7here v 2 iS exadt For tllt fiCCQlld UpF poSition tllerefore let us take z= -, 3 alwd makeP a ;= z 414wX 36, and by llelp of rhe Logarithms we {hall

have H -1- 1 12 ̂ ^ ^ I 3, 47294 svhetlce X = o ̂ ^* o6; X }, g34zg} and x= y y X gA X o EicacA by r zd.

I A 9 3 4zi Q454I X zrational hr7SxlJ v --v - o _t. w _ y) x84Xg 5) I84I9

I4t98 = os ov5X6> x^7hiW1z gives y x -t- v X

w33xst6, wllicltl is true to fix Pl2ces. 15 you de{;re ic more exad than EO tlle ^tent of rlle Tables of tJogarxthms, taklngz-ws3XHt6 for tllE next XppOa fition, the Caiculation nnuR be repeated by computing or fw Z + I 1/2 tO a EuMcfenc number of Plac->*s 9 whicl1 muR bc donc by tll BinomiaX Serses, or by making a LogaZ

13bbbb X riths



t 616 ) titO ON purpofe, true eo as many places as are necefX ary,

DC4 11 For atlotller Example, Iet it be required torlSnd the Number whoSe Logarithm is Q, 2#9, fuppofjng we llad no otller Table of togarithms but Mr. Shxrps of 200 LoZ garitllms to a great many places. This amounts to the refolving tlxls ;quation Z y = o, 29 or 17 _ of w9-o.

Hencetllereforewehavex_Ix-O?g, nca (a

being the Moiglgs belonging to the Table we uSe, viz.

Ov 434t9448I9s Sc.) x _ a .X_ > a . =6a

&a. Tn tllis Caloe becauSe X llas a negative Sign, chatlging tlle Signs of all the CoefficiencsF tlle Canon for s- wsll be found in tllesfourth Cafe, which in the irrational Form

* * z *; ::

X X X X X X wS

glves s- s/ . z -ll_ _ _ xJ _ ................. **,

X X X ^< X t 3 42- z 21Z --o,s8 tt3 tt

aC.=Z 2.Z + - -XZ.+ --- z a 3Z 4

+ S eF In this CaCc to avoid olien dividing by z, ir

will be moN conl7enicnt to compute , ich is goc

frOmtlliSEqUatiOn Z -I-vI+tIZ °^i8q

2, t3 z t4 z vS

3--> , &r. The- neareR Logarithm, in 3Z 4z4 izs the Tables propofed, to the propofed Logaritl; m o) 29 is , zSoo346x 4, its Number be:ng X3 95 Therefore

for the firft fiuppoEltion takng z xn 95, we llalre X ( Iz _ 0n %9 0} ;9O0346XIX _ 03 29 ) _

aX oooo



r di7 ) z Iz - o, 58 o,*esoc69za*S

o>Oooo3461-r4, and =. ^ = 2 ^ * .4 , 434X 944;S s S

*1tOi>f.

0y 0O01+i39T 3 93o and I -'r = >i->^ X 9

3g g p Wizence f8r thv firS ap:roxin<*:on we t;; '?'9

ep - ,_

- t -v } °°Q; S949t 39- o+} oco7y<Sz<

and v = _ Gs o>Oty4Cs3t, and y-x + v ;3 949844s9968 WhiCIl is £rue tO cleven places3 azd

may ea:fily be correded by the- Terms 3 ; &^* wlli !z I

eare to t1a; Readers cu-riofity. Be:ng upon the Suhjed?c of Approxilmations) ; nzay-

not be amifso tO {eC down hemotto Appr-oxinwations l ilAYt fbrmerly hit upons + TIle Otte iS a Series of Terms bL exprelilng the Root of any QBdratxck liquationi and the otller ss a partxcular Method of Aporoxinarig in ne inventlon of Logarithnls whsch lzas no ocraSo ja sor any of the tSranfecudental Me-tllodsX atld- js expeditiotIs ea nough for tmaking the Tables wzitlzo-ut nzucll troublew

A generJl Sedes Jfor exprgng fAc Root of an- aSSK EqadaiXd

Any Qadratick Equatlon being reducRd to this Form wt_mqx + ;> the OOE X will t expreR by this Series of Terms

x --

Sqz *S ; w r b X _- 3

tD x 2. a. \#hicll rnuR te tlus interpretesl

* Tlle Capical Letters A B. C) o*v Rand for ttlw s7hole Terms wlcll tllelr Sigry px-Ccdng rhofE whcre

,n



( 6i;8 )

t-n tlacy are fiound, as B A X r .

f w _ X

2 T1ze ILt1e Letters 4, [, rf tre in tla Di¢7ifors9 are eqtu-al to the atilole Divitors of tile Fradiotz in t1zc Tertus

imnnedxately preceding c tuS b v: ze

For an Exanaple of tlzis, ler st lJe required to End

v 2. putitkg-v=2 X 4 x, ute kave .xz -| 2s t

C>3 \v:th being compared xvitI1 the general nr,#la gives

fsf =* _ >, at3d m y - t: therefbre for 7 taking _ > xtxe havW 2, an-d a -= x, wilich Values tubfiiw

* * . -t I I

cl£ed 1zn the Strles glve X - ^ -

z 2 x 6 2x 6x94 i X, Mel t#_ *w - _

= xd x=}4 x X i54 X wS x 34 X sIi4 X t43I7s4

Crt Tslc U*<-ad tons lzere wrote down giving tlle Root tr-Ct tO tWdSsLy tsce. [)}ace^

4 rerr ltthoX of cor?pting Logarithgs. <

rl5XS Mer zod is iutlded upon tllefe Confidcrations* x. Tuat t1ze Sulal of tlle Fogarithms of any two NumZ

tCrs iS t> Loga<;hm of the Produd of thofe t\VO NUnlF

bvRs NJuiViPIied [ogetiler 2@ lrhAt the LOgArithm Of Unite is notlling; and

6:0nfCqUtntly t:llat tI1C nearer anY Nl;lMber jS to Unite {I1C nearer W7il1 itS LQgarithm be to OJ 3{jlE7} Tl aX t11C PrO d u2 bY 394 uitipliCatDIon Of C{s\TO Nu mbers, NS 1lereof one s bigger afkAd t}] otller le(i than Unite is nearer to 1JAZ

nite Lllan tat 05 the tsro Num.bers svhicll is ot; tlle fame fJe of tJIaite uttb its lElf .<or Example tlle £WO 5'US1e

brs being ;> atld s4 tt1 i-rod2.S iS 1Ci th<n USfiir> t?ut S&afCt tO it tl.13n & W31iCh iS 2;tO 1i>&S t;aN UX3t+o}

Cpo11 t1z ConSdXatotlss i :fouxld the pre2n; A" ptoX^mat]0n.3.



( 6t9 3 ,uroximation wt1a(:rtz wjli 1ic beR explais<'d by an Ex anaple. tAt it thereXrere be propoid to find tbe Re- lation of the Logartthnzs of 2 and of so. In order

to tllis, I take two Fradicns Iv^O8 and L80, viz, v and r

utllofe Numerators are FowPrs of > and their l:?enomiw narors Poxvers of O; one of teM iitg btggger: ad rile other le& than T. 9 Havi;g Ser thete doxsn in D;vtz mal Fradion¢ in ehe fifEt Column of tlle Tab1^ ane>X-, againR rllem- in the fecond Colufrn I Set A and o fot tlleir togarithnas} cxpreilfig 'oy an Equation tlle Xanner llow they are Compo:nded of tt Logarl;am.s of z aSn'^

Ios for wllich I write I 2 and I Xo. Tlaen Malttp$ylnn the tsro NllmberJ on the firft Colu*nn tog;;;*aCr, 1+ 1ilYe

a ttlxrd SXumber C24s anain4;t xshich t svri*^ C fbr i*E;

Logarichtn, expre^ffia+!g lzZe:NviLe 'ty an Eqv}ation Xn NSa>; manner C ss fbEmed of tne ft"ngorng Log«ritllsns A a-.d b And in the [ame manner <*na Calcuiation is continu.ed 9

Only obCerving tiliS C}rSpt^26t}2't t1lt beire I ht ll:pty the t\VO laft Numbers alrvaWy got xn tl:le Tables t cor^fi der \vilAt Pox^?er Of Qne of ti;em mul:\ be uSed to brinC ehe. Produec ihe ntareft to Un.ite tilat cvn beX Thos 5

f6und, afier sre has7e gom a little stvay in tI^e BbleJ onig by Dtv7iding vlae Xtirces of he Numb<rs fFonz Unce one by £he owher and ;;aking ttae ocient atith :he neaLo

e0, for the fnctex Grtile Power wanted. Thus the txro jatl Numbers ln tiC T*able being o, 8 and 1 tw+) rlleir DiSrences fiom Utllt are 7 X6O and , oa4; tlSeteire th tO3 . > t < * > * -gves oUr+le ndex; svhereior>Mul£Xpiyngthe (3, OL

w rith t+oxver of X . c 2 4 r?y < * S S F 1aave vllv tlEXt > mecr

£ 99o3 > zoS 14s Xj+ \v.}1e Logarithtn iz D 9 C t} tve ;Xa feeksng ehe tnctex ln thos manner bn Dtvifiotl oi- the DiScrences, thz ocient caght generalls7 to bc taRe:

h



( 62e )

%7it11 the icaR: but in thw preSent cafe it happens>to be ti xno(;, becauSe in(lead of the Ditirence between o, 8 and x j we ouglzc Fridtlty to 1zav > taken the diffierenGe between therecgpso<al X s t.5 and tswltch x&oulti hav-egts1erstlleIn dex X O 9 a:zd nat vrozld b too b-iq5 becau^i -the Produd by t5lSt tMCaSs wou!d llave bven b:gger than x s as x ffi oz4 rs FT'Rereas tllls>Apposmaton .rceutrcs cbat the Ntalnlbers bn vue 5rlt Colut2ln 5c alternately g+c<Lt-er and lefis tilan t > as nay ke feen i] tI;ze Tabie.

lvilnwl have-n thss mantler coatAned the Cialcu}a- eXena ti3} I lzalfe get t:Ze NutnS¢rs fmall enough ! [up- eefe the laI} Loga-<tcllm tO 1:BC Ci'-u21.-tO nOtlling. Wili(:ll gtves tne-al Equatiotl? firom x^htch Itaving gOt aN7ay the Lerters by nleans- of the foregoing Equationss I have tlne relation Qf the Loga-ritllms propoSed. In rhis manw

ner if I fsppoWe G oS I ha?e z t 36 l 2 F648 1 IO = 0

MJhird givestIze Logarithm of 2 true in feven Figuress and too big xn the E!gllLt}; -wllacll happens becau.6e tlle Sumber correfponding with G is -bigger than UnIte.

There is another Expedient whicll renders this Cal culation Rill orter. Ii ls foundedv upon thts ConEderaw txon, tt1at xvhcn .t is vcry .ISuall t i;Fx8 is very nearSy

X -', a x Eience if t m x, and # x are tlzC tAy0 JaS

Nllmbers already; gOE iN the Frlt Column of tbe Tab!W

and Elz2lr Posrers X -S xand- r Zla are fuch as wes1.

make the ProduEt X v s x I 'r xlr very tlear to Un-it¢

$ andnmay k found thus: x ->xlm-X -;m , ancl

X _ zlt; - X _ n z, and confequently I -> x, x X '3 - k -1- aw X S -S X w; x, or (negIcding on } Z w) I > znx ra:* Moake tbis equal tO I, atld ute 11a6Ze omS O

e S: II_Z: II -t-x.Vt'hencexlI x-IzI1 2-S =Q. TQ give an Example of the Application of thiss ;CC IS C;4 and o, 99°3SX be the laft Numbers 1n tlli TaltIct tneir Loga:itllms being C and D T11Wn we have



t / O24 1 + X, and o, ggo 3 5^ _ X - z, and coslSequently x = o, ow, atlu X

o,ooe648. M7tlencettle Ratio 4-inthe leali Numbersis2500. So tllat for fuding ibe

Logaritlims propef>6A VJ: may lzave yoo D + zo l C 48 s l o l z-I 460 3 1 X o-ot

whscll givcs 1; =o, 30so307, tvhicEl is tOO tig in the laS Figure; but it IS ncarer the rrulil, Li1an vllat is gc;a from the Logariahm F fuppoSed equal to nothing. So that by tlliS means W& have (avcd foulr Multiplications, whicll were nece{Eary tO Hnd the Nuxll-

ber 9t)89s9 Sef'; correbondent tO F, and wllich mufc llavelJeen had if wc would make [lle LogAri£hm tr tO the talue Sumber of places wstliout this Cogpendiw.

l Z :> o, 28 s:0) 3z PO, 3c)o c c), 301o7 P , 3O1020 <<?,Ovo303 > , 30I03996 <: o, 3 o t oJ9997 :> o; 30 t0.2gigS t

< o > 30 t0299959 :> On 4010z99ii63

< o) 30 102999 i67 > o, 30to2g9gtS6g 9 <: O3 3o to2999 9664

_s In 28C;tOOOCOOOQO 1 A -7 Zz - 2 1 IO - - -- Qv 8osonooooc?ocz 1] t 1 2 g t0 - t - . *

" I, o24oooowooo C - B + -t- Io ls _ g I r o >0 o)99og9O3Xaag D=5-^,-fB=gg l2 28 XQ-w * -

_ ) °043 s 6z77KSUt; E 2 9 > 6-19612-fG l IO- --

O) 9989S9gg6XOi 2 E-L 9=48 i Z z I4611° - - , 00016z824t6 S G 4 F z w ? ts5 1 z _ 64; 1 IQ

ov999936z8tS74 H_6 G+F^ x33Ot 13-4O041 to

, O0G0 3 944t2 f S I 2 1- G = A8748 1 2 86SI I ro _ " -c),ggS97t7zoU3s X _ I -Xg+zoXglz l26SSIxo - I} O00O07 i 6104tt5 L R -t I _z73777 W 2 _ 21 g 96 1 IO --- X 999993zO3itt M= X Lt IR 2s447o 1 2-76S7 1 1 0oto003 64 i 1 X N M + L-?, 2 i I 47 1 2-97879 l ffi o

oj 99g9g9764687 V= 18Nt-AX= 6I070I6 12 1838f95 i I0 ^ W

Com. hrZ3s3x.3 o _ 36+S;: 0nt- z3 y s sgN--* z3o2S8S82sx87 12-6g3XA74oczg71ll0

c)

:> ov s to2999S663987



( 6za) lohtve computed this Table fo far, tlzat tlle teader

way fee in whac- manner this Method Approximates; tllis whole Works as it appears) cofting a little more tlan

. s -

three hours tsme.

.

\+ i(3rofrtetaf.es qdt7 f*rplices SeAionum ConiJ cautn ex nal-$era Focotum cec[zg; cn theore

1 w j ++ + t w_ w * -

v;tategeneralde 7erobas te4! tspetts ; q-ffit-o:tn ope Lex iwEirin^sz Centriipetar{an adt i Focos SeStio*Zn tesldene t;418}^, -77electtates (orporXs irl tllgs relol?wentiN>nX

Q I)eicriptio Orbiawn facillisze dete*mi?ZRztxr. (Per Abr. de Mcivres R. S Soc.

g9tt DE Axis Tran^verCes Ellipfees, AtU Axis alters O & C centrum Sedionis. Sic P pundum quodvis tn circamEcrentia eJ*us; P 2 Tange-ns curvz ad PS oc- currens Axi TranrverSo a1 0t punda S, F Foci 5 C P, > kt fnzidiatnetri Conjugatz; iFr] SemiIatus redum ad diamttrum P C'; F G normalis ad Tangcutenz, cui oc- currat AGS perpendiclllaris ipf--1 P C H, in pundEo (;, ut fiac P G tadius Curvaturz EllipEeos in pun do P : fint etiam s 70 GR, F F perpendiculares in Tangentem P StsdwmifEz: Jungatur S O. EE demittatur in Axem normalis F L. iJis pofitis, Dico quod,

t. Rec2vxgxlz Abb diXa13tits db gtroqae Ellxpfeos FovoX f lfe S P x 12 F gl: eS SJdrJto Semidvametrx C X.

Dessafrtiv

-SSg PCq -; CSq _ 2 Cs X C L fer x3 !7XEJemX PEf-PC<1-t- csq-lrzcS x CL par tXt 11.Elem;

Unde PS f ; P Pq ^ PCq + X US f tan] P s _'1 P e p -E-X c D; ac proptere-a

s s s > r E zt > 2 P S X P g 4 C p f. -%uaro

( 6za) lohtve computed this Table fo far, tlzat tlle teader

way fee in whac- manner this Method Approximates; tllis whole Works as it appears) cofting a little more tlan

. s -

three hours tsme.

.

\+ i(3rofrtetaf.es qdt7 f*rplices SeAionum ConiJ cautn ex nal-$era Focotum cec[zg; cn theore

1 w j ++ + t w_ w * -

v;tategeneralde 7erobas te4! tspetts ; q-ffit-o:tn ope Lex iwEirin^sz Centriipetar{an adt i Focos SeStio*Zn tesldene t;418}^, -77electtates (orporXs irl tllgs relol?wentiN>nX

Q I)eicriptio Orbiawn facillisze dete*mi?ZRztxr. (Per Abr. de Mcivres R. S Soc.

g9tt DE Axis Tran^verCes Ellipfees, AtU Axis alters O & C centrum Sedionis. Sic P pundum quodvis tn circamEcrentia eJ*us; P 2 Tange-ns curvz ad PS oc- currens Axi TranrverSo a1 0t punda S, F Foci 5 C P, > kt fnzidiatnetri Conjugatz; iFr] SemiIatus redum ad diamttrum P C'; F G normalis ad Tangcutenz, cui oc- currat AGS perpendiclllaris ipf--1 P C H, in pundEo (;, ut fiac P G tadius Curvaturz EllipEeos in pun do P : fint etiam s 70 GR, F F perpendiculares in Tangentem P StsdwmifEz: Jungatur S O. EE demittatur in Axem normalis F L. iJis pofitis, Dico quod,

t. Rec2vxgxlz Abb diXa13tits db gtroqae Ellxpfeos FovoX f lfe S P x 12 F gl: eS SJdrJto Semidvametrx C X.

Dessafrtiv

-SSg PCq -; CSq _ 2 Cs X C L fer x3 !7XEJemX PEf-PC<1-t- csq-lrzcS x CL par tXt 11.Elem;

Unde PS f ; P Pq ^ PCq + X US f tan] P s _'1 P e p -E-X c D; ac proptere-a

s s s > r E zt > 2 P S X P g 4 C p f. -%uaro



an attempt towards the improvement of the method of approximating, in the extraction of the roots of...

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