aequationum cubicarum & biquadraticarum, tum analytica, tum geometrica & mechanica,...

Aequationum Cubicarum &Biquadraticarum, tum Analytica, tum Geometrica &Mechanica,Resolutio Universalis, a J. ColsonAuthor(s): J. ColsonSource: Philosophical Transactions (1683-1775), Vol. 25 (1706 - 1707), pp. 2353-2368Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/102699 .

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t 2g5g )

.4,; . ., s .Rj., ... ....i , . .

-II. uationgm C?wbicaram t Sif,udraticv.>}*Xr,.;,,.* >,S,.}f; XnAlytica) ts4ns Geonaetrica Cr Mechawic.a) tuiGiht*o

Hnstfrfals) a JZ G)1fOSe

§. I.]5 Quationis cubict t X3 " 3 p xt + 3 q x + a r, Uniarerfialis \ _ 3 pX + p3

-3Pq- Radices Tres futlt,

3 - 3

x = p + i r < rt _ q + v r -fra _ qa x_ p-1 - < 3 x ^30 r $ rz-(l3-1;+<-3 4 r < -

I +/-g 3 T- v_w > X= p_ . xit + Mi2 rq 2 Xvr_itt _ qi

Vel ut Calculus Atitllmeticus fatilier ac paratiorevaatX fi pofuetis Blnomii irrationalls r + v r2 ---|- q3 Radicem Gubicam efle m + v nX erunt eiufdem Aquationis Radices tresx =p+2n]* kx =p-m 3n. - -

lgItur data Equatione quavis Cubicas inter e js lluFlrZ que Aiquationis Univerfalis termines 6ngulos inftituezlda ept comparatio9 quo pa6io facillime invenientur ipfz p, qt r; & hiSce cot,nitis inrlotefcent Equationis data Raw lKes omnes. FIs jils vero Solutionis Exempla fint feqllen

tiA in Numeris. s. ADquationis.Cubict X5 _^ 2 2 + @ X + 4 ft Rat

d;s x irtdagand.a. Erit primo jlxta prarEcriptum 3 p = zs IG. N SYe



( :t3S4. )

fsvep_ 2, Secundo 3 q-(3 p^) 4-3, five q _ I3,

Terrio a r ( + p ̂ _ 3 q x p ) x 77 _ 4, fiver = 279'

& rX-q3 27-* Et propterea x =-* v 9t i + z/ 29 ' -27-. Reliquz duz Radices funt impoil-

biles. 3 2

2. -In iquatione x-12 X-4I x +442s erit primO g p = I 2, five p-4 Secundo 3 q _- ( 3 p2 ) 48 _

_ 4I,6veq _ 7-. Tertio2r (p;-3 q xp)36- 3

fivc r-3; Etinde r _ q-- 27. AtBinomii furdi IOO

3 + f- 27 (-r + v r; _ q3) Ralix Cubica per Mc-

thodos ex Arithmeticapetendas extraAa, eR I +

s/- +' (=m+{n,)&proindeRadixx(p 2m =4 v =--)2,veletiamx-(p - m+f-3n =4+ X + ( 4 ) :2 =-97 vel 3. Vel rurfus, ejurdem Binomli a + i_ °°. ELadix alia Cubica (tres enim agnoXcit )

elt 3 t )- I2 :- m + i n, ) & proinde Radis X= (p 2 m-4+ 3-979 &etiamx_ (pm v-3 n-4-2 + :f 4 ): 2 3 rel 2. Vel denuo,

cjLlEdem Binomii 3 il\f _ lOO Radix Cubica tertia eR

s-/-I 5 (-m + M; n, ) R -proinde Radix

_



( t3$S¢ ) x = (p + 2 m --4 X ) 3, atque etiamx-(p _ m + < _ 3 n = 4 I + ( v t 5 -) 7 vel z.

In Equatione X3- I5 xt 84 x + tooX erit

3*

P = 5 * q 3 r _ x3j ; & B1nomtL x3s +

v I8252 Radix Cubica eR 3 t v x2. Igtur Radx

X = 5 + 6 I, &X= -S i 3 +v 36 8 + f 361 impflibiles. 4* tn Equatione X3-34 st 3lo x + IGI2} eri-t

34 _ 2 a6 r _ 5 5 @6; & Bln-omlt <5 3

3 9 a7 27 * f 7°75-Radix Cub1ca eft " + v-. Tgitur Radix

27 ; 3 . , . z

@4 + 3 j- 22, &X_- i-+2 -- I0- 6 3 3* 3 3 -

-T0y lNlpO«lblleS. In A£quatione X3 _ 28 Xt + 6I X Il''''='t 4048 trit 28 967 2 50I0 .............. 2 S0I0-

-

v

3'

3 q-- p ,,r- 27 ;&Binomll- .............. 27-

tr _

l e

_*

p_

= v 38Xg47 .RadiYCubica eR 46 + M- 43

Igtturx-3 + 4 = 23, & x _ 28

1

/ -x6vel_x. 2

2$it w We729 __ __ ( e 4

6* In Eqnatione X3--* 23 + I66 X _ 650^ erit

s W - - +

I _ 499 _ 6 <8 & 3i

P =---, q-- 7 r- 27 ;

9658 sf _ 472Os Etadix CubicacLl tt

27 47 3 Igitur x- > -- * 4 _ I

IS ) k X _ '_ 3 + 3 + f - = 7 + f St lrratonalsg

7* n



7. ial iEquatione 1X3 = 634 sX + 9X73 x + 9 q=,I00996 r603x68oi &.Binomii

47887175043X36 Radix Cubica eft a7

183 + i_ S i. Ggitur x- 2I + 3-6t = 387, k

X = 2t--I83 + ( i 529 ) 23 - - I39 vel x8s* -Nec fccus in c*eris procede-ndum : InveXtigatur aumm Theorema ad modum fequentems Pono Equaliotlis cu- jutdam CubicsE Radicem i-a + b, 84 -cubicemulti plicar}do proveniet z3 = ;( a3 + s a; b + 3 a bL + b3 = ^;) a3 + 3 a b x a + b + b3. Jam loco ipfus a + b valo- em ejus z fubfFituendo, fiet-xs _ 3 a b z -t ia3 + b3 qux t{E Aquatio Cubica ex Radlce x-a + b corlAlzvtEtaX cu terminus Secundus deelE. Ut lzac rero al formam magis ecymmodam magifq; concinnatn revocenter, fumo XEqua tionem z3 = 3 q z + 2 ra qux poRhat ipfius z 3 = 3 a b z + a3 + b-3 vices gerat. Igttur tranfnluratione hu jui in illam, fier primo 3 q _ 3 a bs five q3 _ 33 b3 ; & feJ GntldO 2 r - a3 i 53, filve 2 ra3 = (a6 - a3 b3 _) a6 + {*

Et folaxta hac aquatione quaelratica,erita3 = r + / r; - | q3) 8c hinc b3 ;-tx r . . a3 r r < rL _ q3 Ate it

tlLlr tandem a t < r l r^ - q3 & b-- i r i r2_ 9y

Et propterea in gquatlone Cubica z3 3 q z + 2 r er;it

3 -- -= 3 - Radix_(aTb=)z/r;+arq3 +n/rX vr-(t

At vero hc ltadex revers triplex eAf pro tripliCj va. 3

lore quem Indilere preR & v r +-- f r;_ qg 8r ^ .

; r _ f r; _ 53. CujSvis eni< quantrtatis Radix Co bita ttip]@s erit & ipEusU-Xtatis Itadis Cubica rel

efE



( tg5t ) e{} x, vel ffi 2 + 2 <-3,zrel _ _ I af_ 3

Atqut sd adeo, proptrirea quod lwarum alicujus Cu>As fie 3 .... _ 3 _,

Unitas. [giturfi X xVr F rt r 2__+ q3 aut vr -- v^-r* 3 . * . s <s -

X x r + a r - q - t 1 x v r + i r q 9 K a- dicettn alquamo (quatn fupra ncnz;navimus m + at us ant

E x m t v n) ] Cabt r + nf rt q3 defigner; iifa

I + i 3 x +- * & -- x_ aZ ; 3 2 r t ir;qs 2

01 X -F

3 _ . ^

$ ; r + i r2 _ qW ^ 1. e. 2 X N3, l 8 3 =

-I t 3 x m , V nz alias duas <juZelm u; a _ n w * *

2 w W _ 0. dices defignabunu Slmiiiter & < r- v rs q)

-T + <-3 $ * & ; 2 x r-v rt q3) * ̂ *_

.. . .... _

- 3tr .irt-q3 [i.e..m_f n, -I + --- -t .z. . ^. . i _, _T _ _)

x m _ nf ns -*-7 x m-f 11,] tres Clibicz ta;

dices erunt Apsomes r-ir2 q3* A(}S;v.> t<s w9!.'ta->'^rj'

debite: conneAetide, fier z _ i r *t v r : --< . .

3 . 8 __. . + < r+ r; q38 LiXe. z - ln T;-' n + mzJ 2 *iz; - I + / ; 2

z_ - >tr + r _.qs , -<s^. *

2

- 3-- {- 8 / Xi rirt-q ci.e.z= -- -' + 3 X;\<

r*J _

1 -g *t-4 FC

+ 2 X ni ̂ in - m +at _ 3 n>) s Z;

/ 3 - Z > __.v

t < +> 2 ^ * t -l ^ 2 J x v rif r;_ q3+- 2 7 ty@/r;;v-qX

. .

t4 0 t < a



t ^358 )

_-< 3 v 3 i l e. x _- [ n t Y n -

. .

x u->' n = .- m _ v 3 n,] qux tres crutl; &3u*=vlt

iEqilationis.bubicz Z3 = § qz, + 2 r. Debite alrel8s: >.*Wr nUctun£ul Aadices lftx ad modum praccdentcm? ni jpp*

-qE1X IlC conn<XaX & mAtre rulgarl sn ^ lavlCern (O.fn5<-

dll&X,, iEq!^ ati8i3X<em zS = 3 qz + 2 r reRiiunSt l)*+nl

qut hg z X W p, & iE;quav fiCt E:3 ^ 3 p X -- 2 px

_ ps = S q.y . g p q + 2 r, qux unterfal3s ett) tA

cuJus EVad-;ces elradunt ur ftopra tuerXant exhiblrz ^Xc obirgXr 3]oratil dign sm elt, qtlod X;qutian;s Cill)}cz

tjtCdE:qtlvQ Kadices omruts filat pztiibiles & rcalesvql-lc>ries BitIOtnIi llzcmbtllln irrationale i r -q) imst7ofiibititatenz il] i- conzplcEtitur ) hoc e0, qllotits q eR quantitas aR:iro tnativaX & Stzlaxl cubus t js major tS quadrato ex latPrc ;C

At fi nzembram ifiud v rz - q3 rlt p0«lbilev h0C efi fi q fc guantitas ncgarlvaX auL ctiam fl affirmAativt cubus r nainor clvladrato ex laeere r7 tunc unicatn tantum agnoicit

Equatio Wadicem pofElbilem & t:aletmX reliqutquv dut

trunt impoXsbilesk In hoc Theorealate fi fiat p = Q) ]1OC e@3 fi defit Ruab

-tionis teralinlls fWcundusv talnc deventllm erit ad caliJ.tn Regularum qut clicuntllr Card?i, cujus folueio continetu.r

in pracedentibus. $. 2. Gquationis Biquadratict Univerfalis

Zs-4 p X} + 2 q XZ + 8 r x + 4Sf

' 4 P 4 9

Radices quatuor funt x-p _ a 4 i p2 + q _ at-a s

Ax _ p -ta tip; + q_a2 4 2r> u.ia2efERadix

E;quationis t:ubicx a6 - p2 a4 X - 2 p r t2 +*t2*

Aq S Jat data Aqllatione quavis Biquadratica lrater 0tls

hujufque yE;quationis Univerfalis ,terminos fx3gll1 {>attlZ cnda



( 2359 ) xnda-eR compaztlo? qa3o pacto clti(Elrne xlzvenientur spfx p, q, r, s; & hlfce cognicisX non lateSit valor iptius as ex tRlleoremate fuperiori int7eniendusX & tum demamj Lz noteScent A£quationis datz Radices emnes.

Huic Solutiotli illu(tranda Exemplum 1lnum aut alteruttz fufficiat.

I. A£quationis Biquadratica -X4-8x3 + 83xt s65w

6 fint Radices exenhenQl EDrit primv fXa prXfcriptum 4p = 8) tivep _ v. secuLldo zq 4Az^8

6 = 83, five q-99. Tertio Sr - ( 4;3q) 396

- B I62, five r = 7. Quarro 4s ̂ (q3) --

6, five s _ 6067, Hinc pt + q = 1O7, 2pr z s

7969' rL _ I3689, &proindea6_ I°7a+ 7929 a3

+ 3I6 9. Jam ut Rquatio hwe aliquater.us CaXbtCa

in Radices ejus refolvatur, ad Theorema p\XCd6- tiq r^>^u*

rendum eft, in quo erit p-107, (1 2 53 9) r-- --89;?3- & r; b q3 =- 9 6---5- Arqui 31nos<zii -t -+ < b - 94 75 Radix (:uDi crt - -5-sn -P H<

I6 X 2 ' a

& proprerea az = 67 (,- = , a c.>ia;£z aL _ i_W7 + - + C 3/ 400 ) 20 vf --

2 - 4 4^ perinde eft, Aquationis pramiffia t4C\r;WtA,

\T>-1 \ il }'tl }

_ A_ )

_s } _c 7

a

Cubicx fex Radices funt a = + 3 X

& a + 2' quarum qr,nevis xn(IiScrimlatim proll)O-



( 2 360 ) 6eo noftro faciet fatis Pazta fi in prafenti ca fiat

a = 3 , erit 3uxta Theorema X = t p a + Z IllP t * \ - _ Ar95 t q aX -- = X 3 + 4+

a - 2 2

...z.*_.. . . . ..(i aS) 5 04vel 6} &x-(p - a +

2 )> -t q -. at T = = 2 + 3 + / 4 t 2 -- 9 +

S ' ({ 64) 8 = .)+ I3 Yel 3, qUt funt Ruationis

datx Aadic¢S quatuors

8 ^ 2. n gURtlOne X4 = 20X3 4 52X2 6sg2x + :X3X2} erit p > a = 1767 r - 38+ & s = 3o7*>. Hinc pX wq ZOIx ApR S= 9232s &

r _ I47456; &e indea6 = 20I a4-9232 az + s7456.

fiam in Theoremate pro Cubicis crit p _ 67, q 4z@5,

& r = 65 21-9 ;e eritqlleBlnomii6s 2I9 + f a888930707 2 27

Radix Cubica 77 + ' 47 I,gitur aX 67 1 77 = 14+,

five a - 1 ^; & preitlde x = 5 X 2 +. * * # # s e * t - \ i25 + x76 *+ +64=_7 +C{x2t ) fl+- 4 vc1 I8} & x = 5 ,. 1z + v 25 t I76 I44 o4

_ 7 _ <_* 7} impo{HSles. hujusSueem The3rcm;ttis [aven£iv c{} bujufmodi, Ex

dua^alm Ai!uati-vl;ll <; a<:irir :cr t tin] zX + 23£-s 5 = Q,

& ZE 2i-tZ __.t C - v } iN 1c S Clvtcem [nultiplication4 s

Ai!.lationea c r^fi+*>X^o Bii>adraricam zs-4a + b + c ^{ * . r s x Zt t -aC - swAD X z -s Ucr! Cu} 3 rmlX}tRX lecuT}tASseC2tN

qlmMXo n.Xs e iq*lationi zs-ev > iz - g +; tfo tt3i-

p6ltt%* tsI*le prtyso 4 a; e b + c = e Sve

b _ ¢ 4a; *^ tX vecunto 2ac * 2ab = , l1tC eR :at z>wa 1 8a3 X zac-t five c--- z 2a2,

4a 2 &



S ibde lb = 1R +i < 3 f i t 3es t

sa

io-=-9 9 3ti8t + 4 La + 4i24 --$;a

__ _ ls g3#2_ 1[ Qa + w 4g>

wu3itX viiuti3 <f& HiSu

fstn tt gitx h R h-llXm: *.fisrta ¢Xhli po"*> &C£de-s aWe iZ2oteSutt

-pf;r 19 & c kt Eiqu-ationum Sa + aa; w -- b o &

2 _* 2at. c-0- R^adices furlt z _ a + v an + b.

at-a+fa2+C>&vez---

aT<1e-a2 st)

$e z-a + v > e-aL , qua proin-de count ELadices £i:quationiszs = ezs + ft + gi cognitavidelicet a vei a

OX RUat£0tle ^6 = t eas _ 4 ga2 _ efa ea2 1 4, 3atn at

X:quatio iRa evadat univerfaliss & omnlbus f-uis tcrininis

inEtrudaX fac. z x-p, eritque xs -* op'K3 it 6 pZ xa 4 p3 X + p4 -ex2 =2ee + fs fp X g

stem&X.p_a+ r;,e _a2t t as La1 a,;

i> e _ a + t a Ta+ldem concinnitatis & co^vnpendiw

grati fac c= zn + 2pz & i = 8-r; tv1m -4px3 + 4pt x: =- 2qx2 _ PS-s -F 2p2q j ps + 8.x 8pr btgX x -p _ a + < n2 -r q g a3 a > X i?" -t R s

,,W,_ . . _

f P + 1 R + , & a = p + q W a 4 g T a p

d- p2 q i qS sS 12 + t2* Denigue fac g ̂ 4s " qX

+ 8pr ps vpaqs &z tEarlt uatones prCcun*Cs X4 = 4pxg + 2qx2 -i 8rx + 4S a ati-p2a4 2craL + tt -4P -42q qX + 9 s

Ssilicet omrxia (:v3dUNt llt fupra funt l?ofira

T4 P



( 2:3tSt }

§ -. FIa&enus de Fuatxonum Cubicatum & BiqwL dratwarum Refolutioize Analytica tloniam autem carun- denl ia GEQ"XtriCA per Parabolam Vl8C) tradi foler, & nonnullis in pretio eftX ipXm ¢"te1xx} & quidem uni vcrCalius} non pigeSi: htc exhibere

Data iEquatione qllaYis vel C:ubica te1 Blquadratica, in(lituenda ck comparatio inter terminos ejus > termi noRque rebndentes hujus iEquationis X4- Zs + 4P St + -2q-X + pl}t q@p&Xciletfa6s

v 4tX t + 23 + 83 $2

- 1 -E +ta

ctuatur ipCt pf qi t S t; eatuminterim-un aiiqul utw cunque pro lolbitu aXumpta Tum in Parabla tuavis tataAYBi s-utus^VenexprinciplisV AxisV6; &WAx



( ^343 3

Qo



( tW64 )

e1 pent3:tca1arls ltT; capiatur VS p seri;Xs -ntriora Pa

ratz<sX) & in angllle SVT in@rlbatur ST q1 qux pro 1u&sl Parabolatn fecer ln pun&s binis N & C). }Sifecb tUr tN 111 M) & per M agatur MA AXI parallela & Pamv IO1x cccurrens in A IpfX 0N parallels ducatur AL} ut fit tAL Latus Ae&um Paralzolt ad Diamctrum AM} firque h> c:xdem Untas In AL (urrintue fi opus eft produEa) cap*

*n* t}- AC} tX & a punEt(z C; ducatur GR Axi parallvla)

<k^ t)raholanz Her X-) Bv a que capiatur BS - §, A nov jtExre itzvento pundo R dt)ca4ur RE ipfi VT paraI]Pla

8: tsgl*rt3 l}5> (*j t.It tinxAratn verSs jaceat refDe&s ipfitlS & fi q fit fuantitsXs affirm"Xttta a£ terXs dexttam fi q St neg:3- 1va Atque jdetn de xpEs AS & bR il1telligatut} qut acE <O*rarias itidem partes d-uci de5)}z* fi modo valcres ipS: rum r & s pro-4eant ^;;<^atlv1* D<tliqllwn (:entro E &*Radio EC c t deRrlbatur CIrculus CK*^ qui ParatlJolatn } tOti d&n1 @c3!)it T5titleiS) QLlOt fil;a* fEq6X2tsonis datt Wdt< ten.IeS. Etenilla a punEtis iSis C) K7 @a. cltCatzur CT', ta} Ac tE>fi ST paXstllela & ad te&am GK C fi opus cLE redtl&am3 teralZautZ) eritque lzorsttr <8 jxvSs X {Uu Agua- t:onSsdaft &ad -x 9U}1ta; et @Jic<t ad dextram jece1tis trnt Rauiccsa&rmsat;;a) xt-]a wero ad fi<iftratn fnt po fi X e;unt &32iciNS tlCratI->tA PGB\&L52 C;nta&.83S) fiqLlOd

0t1-Cr1t) 1l*' 1uS,9AUR 0 Xerfed}onis punAis duobu3 ad .+ . . . * i .

X tl sW wes m v< t tz 1 X tSll¢ffi* Inter RL-srotd Cubicas R Biqtladrat^cas ra con.

firu&4as 1lec k A}8tt;[an it<tercedst etitcrtiniilis} q lod in prxn-

t142)2 tb tr}njt)Gtn utt}iR)0S jti} pig 6-l(},*l*it ;lr3ti;n}4 d-

ft¢Cttetu Walper Ete ' gs-st -4 tX-o Ere

. . .

t - i s2 + Xq2 pXe Ig*ur G;ntrt) E a A3dio E S

-;r v B&q + ( E Rq ) STq - VSg 3 dwE tt *o Citrla

CK^vA R3d;CS¢l] 0sA CP IN r}W}I C0nLr2iO.-.'4WA :O NiAilGZ 2 '*

*'aD] *-

Hzeautetn delnon(lran.uX ad mlXtu feauntemv Ma tEAnXibUS }AlN CQnArXAiS, & \2rO4!sV9A C^-J; & O})X>13 (fE) >6 Sewt AM in Ht trt ti O}Yinam Fdlab¢)tw ad lDis

,mF>t2^tn



t XW67 ̂ ) metrum -AH, &t psolsnde CHq-241; x AH-AR, ob AL-I. tt tH--CP + AGy & A$H = GB + Bt & pr<>pterea:cpq + a AG x CP + AGq-GB + BP; {ed o1) naturam *Parabolx erit Mq = GBs unde CPq + 2 Ag X CP--BP. Jam a pllnAoCad ipEam SP demittatur ortna s tE), cluX occurrat etiam ;ipfi- St, ad BP a&se pa- ra.ilclas, inw pun&o [ Propter Smlia Triangula CDP &

TVS, erit DP VS5^1CP 8; CD = W X CP &

-inde CPq + 2AG x CP = BP-DP t- BD = VSsTCP

+ BR IEX fiveCPq + 2AG x CP T CE BR = >1E.- :AOt -tEq <:=>CEq Cfq- CEtl _ CDq

b - | FTg. 2CE) x vr-CEq- srq< - q

_t 2VTtr CP _.:ob-#Tq _ Sti _ \Tq) CEg + CPq

; SgV,q 9q 3ST< d- SNq _- 2St X Cp s1n- CP.

<quae igitur aqualis erit Quadrato tx Latere>CPt -t 2Xig X CP -CP _oRX Arqaie 1l2c Aq.atio ad tertnlX

. L]oS pX q r) -sX t rcsocata ipf^Esm-fi iiluativ propo&a.

Eiinc liquet, <t^d eadeal qltvis EqXatia Biquadratica V*nul; lmras ,er Parabolanz conlirudtionts fortiri poElit, pto l^d*filzto ralore tuantittts f(-tiusX qu5mX ad arbitrisuz aISX- nn:i poffe jrm di:Xlmu5. SHd carus eft fiinplxcillimur faciendc} t\JS p-oX & migrat cotwAruRio} fi rem ipEan:pees Xn stuXtal^em itiam ) in ditla }kadiclltll -reprafieZ}tatrice§ tt& P3 &, int ad Axnn perptIadfculares taLia

tCln fi¢ s4.- -' 4XX3 4,Xt + *sX _ q2z qua fj;¢ t2) _2g S}

t_ I ; tS

X2kvt....Cit.g2rt rZe

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( t366 ) r$ 4* Svd ne Par2b31 defert;?tto Orbnics difi;cils n<

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9ua iieX 3 t^t + Q3 + t9) <

Atgue cs FXecipas 3ilc p1E prilXte mt InXno tntN haSd iSannQM & 9ugEtumvtS 3tSBrtam f3ithr-

tuus -e2e5CIG huXuiedl Ruationum quaramcunqvle- iRadicu: nuilo fere negptto inlreniri pofEnt} & pra ocuo Es ewlAiberi. Ho£ alltem quilibet, fi id Curx fite variis m;odis pro ingenio fuv efficere pote0, & die his 3an<-

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l1I. 2dtionGm fGarz4tlddm {Potej}4tis tertiKj gNXItt4

Jeptirn, ssone, & lsperior2g, dd isafintxm |que per,endo, {tt terminis finitis, ad inf at ffiegxlar%^- tro Cubscis qs4 aocantur Cardanl, !&elolBio 4

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dJ1t(4. (Per Ab. Oe b£oisre, L S. SO

CIt n Numerus quicunque, y quantitas incognita, five J Equationis Radixqutfita, Entqlle aquantitas quxvis 0mneno cognitaw five ut vocant Homogeneam Compara- tionis 0 Atqut horuxn inter fe relatio exprimatur per A; quationem

nn-X un-x nn - 9 . un - x ny + - ny3 + - ._ x _ nys + _

zx-3 aX3 4xS 263

nn-g X nn6 2S nys, Acc. a

x

{ 2 368 ) va twat uatlonis dat h&cd omu3ra16) 9 w

pc ad desterunt 3kadica a5ativtX illz vX ad fafiram :Kadices nqansa. ouLS>fio eit nA icRa a prdentibusX lla!biu tutum tatlont Paniz per Punda 8 tg tn v s tranCesstis Nam pfit*o F :u Par]Z) (Xps &fiantis a vs aS g !X ) nS &t

9ua iieX 3 t^t + Q3 + t9) <

Atgue cs FXecipas 3ilc p1E prilXte mt InXno tntN haSd iSannQM & 9ugEtumvtS 3tSBrtam f3ithr-

tuus -e2e5CIG huXuiedl Ruationum quaramcunqvle- iRadicu: nuilo fere negptto inlreniri pofEnt} & pra ocuo Es ewlAiberi. Ho£ alltem quilibet, fi id Curx fite variis m;odis pro ingenio fuv efficere pote0, & die his 3an<-

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:atls.

l1I. 2dtionGm fGarz4tlddm {Potej}4tis tertiKj gNXItt4

Jeptirn, ssone, & lsperior2g, dd isafintxm |que per,endo, {tt terminis finitis, ad inf at ffiegxlar%^- tro Cubscis qs4 aocantur Cardanl, !&elolBio 4

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dJ1t(4. (Per Ab. Oe b£oisre, L S. SO

CIt n Numerus quicunque, y quantitas incognita, five J Equationis Radixqutfita, Entqlle aquantitas quxvis 0mneno cognitaw five ut vocant Homogeneam Compara- tionis 0 Atqut horuxn inter fe relatio exprimatur per A; quationem

nn-X un-x nn - 9 . un - x ny + - ny3 + - ._ x _ nys + _

zx-3 aX3 4xS 263

nn-g X nn6 2S nys, Acc. a

x



aequationum cubicarum & biquadraticarum, tum analytica, tum geometrica & mechanica,...

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