am tae Pa ex sai poy 8 x EAE oh
+ *)5?
69. But it has been fhewn-in art. 67, that 1 + « + ap is equal to the com-
pound feries
m
1+ —*, Gung) 1D Bvxs) Fex>,—8cc
+ —d, ih OCH pk UE re Gah Se ed, VEC
Oden Dae, Olea”) 10D x3 d*oc&c
Dd, Bex d* tole? dt k&u
Ed‘, ga Kae eC
Fd5, &c.
Therefore the feries 1 + am LE aahe 3 I+ ar x —, = C x ee x
d* d+ 1/4
ae x I ar 4 ates a oa ated oh I xn x
aap? &c, will be equal to the faid compound feries
ifs
I + = x, Clee eel; Ex*; Fes, &c
+ —d, 2Cxd, 3Dx7d, 4Exd, 5F xd, &c
Cdames Dads nk wid ieto HK x3d*, dc
Dd?; “4E#d37 “1oF «*d3,’ &€
Ed‘, od OE Rae hd tg
Fds, &c.
m
70. Now let 1 + x) be fubtracted from the left-hand fide of this laft equa-
tion; and the feries 1 + = x, Cx?, Dx3, Ext, Fx5, &c (which is equal to
E/4
1+ x\*), be fubtracted from the right-hand fide of it. Then, it is evident,
mm
the remainders will be equal to each other; that is, the feries = x 1 + a\=
Vout. Il. aL d
xem
I+,
258 AMD HS CGvR FE LCOINUCIECR B PMS
1/4
‘ wm MD -
8) ies d* nt ds pani
nig ae RS 0 ee xa BX 1+ xe
ds
x a peal ae rE Fs tae , &c, will be equal to the compound feries
Are oC #5203 Dix*d, . 4 Exid, gkixtdy &e¢
C a°,.% Dxd*, (Oe a4 io bee
D d? 64) aged a OP ds oe
Ed‘; 6 Fowdt)) eae
Ka, ec:
And aa if we divide all the terms by 4, we fhall ae the feries.
ox ipa ae Cea: Sep? ae .
x rar cx, = Ud et yale n~ X a So —— tite Cee feries
=, aC x, (2a. “Ere; cc Fax, fee
Cd, 3D«d, 6Ex*d, 10F #3d, &
Dd?, 4B xa?) “joe da eee
Edt) sien kee
Eod* dec.
41. This equation is always true, how fimall foever we may fappofe d to bez
and therefore it will be true alfo when dis = 0. But, when dis = 0, all the
terms on both fides the equation that involve d will be equal to o likewife ; and
mM
confequently the equation will then be as. follows, to wit, i Xi twa x
I . 17/4 °
ra the feries one Cr, 3D, 4E x, 5 Pixt, &c. ) Theretoreyaaawe
wm
multiply all the. terms by the fraction oe we fhall have 1 + x\ x
E x Fi
- the feries 1, —— Cx, 3 = D «7, +4 =~ Bx, a Fix*, &c; and, if we multiply
MH
both fides of this laft equation by 1 -+ x, we fhall have 1 + | = the com-
pound feries
ia Cx » Dx, A" Ex 2= F x4, &e.
1/3
2n ne
———— 498 x3 a + e
ieee nee ee
2
72. Buti + %| 2 is alfo equal to the feries 1 ++ = «, Cx, Dx, Ext,
F «5, &c. Therefore the faid feries 1 + — x, C eos D CL OK is Fes, &c,,
will be equal to the compound feries
6 2
I, nm
Y ‘
THE (BIN -OtI 2 ESD wR Bo RE M; 259
; 2n 3n
j iy a C#, 2] Dentyite Pate 2 Ets &c.
WW 2 m m2
2
+ x, — OF 22D, Ex*, &c;
and confequently (fubtracting 1 from both fides of the equation) we fhall have
the feries = bk, Cx? > Dw? Ext, Fx, écc — the compound feries
Kees Quip be ppm be Ex, 2 Fx, &c,
Mn 7/4 7/4 7m
+n, —— Cx?; 22 Dery 4® Ext) &c:
Wz Wt id
and, laftly (dividing all the terms by «), we fhall have the fimple feries —, CB,
Dx’, Ex3, Fx*, &c, equal to the compound feries
2x a 3 n
Wt im mM
ca 3
2n Bn 4% 7 5
“Tait Shed Wy pnd Ae ta Bx, &c;
by the help of which equation we may difcover both ** which of the figns +
and — are to be prefixed to the feveral co-efficients C, D, E, F, &c (and con-
fequently to the correfponding terms Cx*, Dw?, Ex*, Fxs, &c, in the feries
17/4
1+ — x, Cx*, Dw3, Ext, Fx’, &c, which is equal to 1 + x)*),” and alfo,
<¢ what are the magnitudes of the faid co-efficients C, D, E, F, &c, refpectively.”
This may be done by proceeding in the manner following :
73. In the firft place, fince the fimple feries =, Ce, Dx*, Es, Fi sfc;
is equal to the compound feries
~C, = Dx, Ex, 2" Fx, &c
Ht
2m Qn 4n ;
se Soot 49 tn x*,—— Ex, &c,
and this equation is always true, of how fmall a magnitude foever we fuppofe «
to be taken, it follows that it will alfo be true when «iso. But, when «is =
o, all the terms in the equation that involve x will be equal to o likewife; that
is, all the terms on the left-hand fide of the equation, except the firft term =,
Z
and all the terms on the right-hand fide of the equation, except the two firft
. : 2 .
terms of the two lines of terms, to wit, — C + 1, will be equaltoo. There-
fore — will be = to — C + 1, orto, —C, or to 1 together with oe Cc,
1
either added to it or fubtra¢ted from it, as may be neceflary to produce fuch
equality. Now, becaufe is fuppofed to be greater than m, = will be lefs than
1, and confequently “= C mutt be fubtracted from 1, in order to make it equal
2L2 to
4
260 A’ p T's clo'urrn £2. TOON ER FPN
. * MM
to =. Therefore the fign — mutt be prefixed to =~ C in the equation — =
2
—— C + 1, 0r a= tr =— ; and confequently the fign — mutt alfo be pre-
V7: 7
fixed to the third term C «? of the feries 1 + — wx, Cw?, Dx3, Ext, Fx', &c
PR Ret: 2 b)
4 >
(which is equal to 1 + x\#), from which third term the faid quantity == C
has been derived by the operations of multiplication and divifion in the courfe
of the foregoing proceffes. Therefore the three firft terms of the feries 1 +
m1
m1
. . te . 2
—«x,Cx’, De’, Ext, Fxs, &c, which is equalto1 + «\7, will be p + ad
— Cy.
Oi te BA
And the magnitude of the co-efficient C may likewife be determined by
‘means of the equation = am it — Cosor — = Iie = C. For, by adding
== C to both fides, we fhall have = + = = 1; and, by fubtraéting = from
both fides, we fhall have = C = 1 — = = “—; and (by dividing both fides
by =, or multiplying them into ~~) Sip os = x = = == = x —. There-
; ° m1
fore the three firft terms of the feries 1 + — «, Cx*, De, Batt Rymeeees
mt
° ° et . m id n— mt
which is = 1 + x|7, will be 1 a ieee ie tend 0°, OF TS
2H
a— 2
a Bw. Qu) Baek
74. To determine the fign that is to be prefixed to the fourth term, Dx?, of
. mM . . ; 14
the feries 1 + — &; Cwx?, Dx?, Ext, Fx5, &c, which is equal to 1 + X\—=»
and to find the magnitude of its co-efficient D, we muft proceed as follows :
Since the fign — 1s to be prefixed to the co-efficient C, it mult likewife be
prefixed to all the terms which involve it in the grand, fundamental equation
obtained in art. 72, to wit, the equation between the fimple feries —, Cady Dix
E «3, F x4, &c, and the compound feries
2 n
Cee oie Pie teed Bonds Ex, &c
m m m m
2 uw
- 1,—~ Cx, = De*, S Ex’, &c;
which equation will therefore be as follows, to wit, the fimple feries Pi Cyn
D xv?, Ex?, Fx*+, &c = the compound feries
n n 2
LEC, Et ena, &c
m7 r4 mn 24
pik de TE hear macley Op
+ 1 ae ee Dw, —- Ex, &e.
Let
Tiptg- BIN o:M Pr AeLs HH BO REM, 261
Let 2 Cx be added to the two fides of this equation; and we fhall then have
the fimple feries = + Cx, Dx*, Ex}, Fx*t, &c = the compound feries
: oe 2 Ct Be ye AS bee, SF 3; Bc
WH 7 °// 4 Wn
2n 3” a 4h
4+-I,— — Cx, ae » —— Ex}, &c
+2Cx,
But it has been fhewn in the laft article, that - Ca ees ee = C. Therefore
if we fubtra& — and 1 — —C from the oppofite fides of the iat equation, the
remainders = be equal me each other; that is, the fimple feries C ey Dace
E «3, Fx*, &c, will be. equal to the compound feries
2 ie Be, Hx 5 8c
esis 7% e/4
nie 2" De’, Ex}, &c
2€ m3 and confequently (dividing all the terms by x),
we fhall have the imple feries C, Dx, Ex?, F x3, &c = the compound feries
2“ D, 4" Ex, 22 F x, &c
—-—C, 2 Dx, # Est, &e
ete:
_ And this equation will be true, how fmall foever we may fuppofe « to be:
and therefore it will alfo be true when v is = 0. But, when xv is = 0, all the
terms that involve w will be equal to o likewife, and confequently the equation
willbe C = 23> D— C4 2C. Therefore C + +=C will be == D
+ 2C, and = C will be = 22D +, or C, 3" D; that is, = C will be
equal to C together with 3a D either added to it or fubtra¢ted from it, as may
be neceffary to produce the faid equality, But, becaufe is greater than m,
aoa C will be greater than = C, or than 2 C, and, @ fortior?, greater than C.
Therefore, in order to make C and 4% D be equal to - G3 sm D muft be
‘added to C, and confequently muft haves the fign + prefixed to a Therefore
the fign + muft alfo be prefixed to the fourth term, D3, of the feries 1 +
= x, Cwx?, Dwx3, Ex*, Fx*, &c (which is equal to 1 + 4|*), from which
fourth term the quantity a D has been derived by the operations of multipli-
cation and divifion in the courfe of the foregoing procefles. Therefore the four
firft terms of the faid feries will be 1 + — x— Cx? + Dy, Kos AP
And
262 AOD IS €C OURS E. COMCER NoIWie
And the magnitude of the co-efficient D may likewife be determined by
means of the equation = C=C + a D. For, by fubtraéting C from both
ied we tall are T= #26 EOE HE == Ae
2n2—m
(by dividing both fides by 3 2, or multiplying them into —) D meee x
C, or (becaufeCis= = x“) D2 = x aoe = Qs, or
Therefore the four firft terms of the feries 1 4 — #, ©x?, Dw, Bit, ee,
mM
te a al aret + =" —— x 2" — x? += x* —
2n a= 77 2% —- Mm
3 . iad SES) iene 2 3
x ~ x), O11 4+ — Ax ee Bx* + - A OSs Q, Hike
4s. To determine the fign that is to be prefixed to the fifth term, Ex‘, of
8
the feries 1 + = x, Cx?, Dx3, Ext, F «3, &c, which is equalto1 + a\2,
and to find the magnitude of the co-efficient E, we muft proceed as follows :
It has been fhewn in the laft article, that the ‘fimple feries C, Dix, Erwin
&c is = the compound feries
=D, Ex, 3” Fy, &c
— =, 2 Ds, A” Ex’, &c
+ 2 iG or : (peahsine the fign + to the terms which in-
volve the co-efficient D), that the fimple feries C + Dx, Ex*, Fx}, &c, is =
the compound feries
+d, 2 Ex, 4 Fe, &e
——c+2 Me Dex, = Ex’; &c
+ 2C. And it has been fhewn alfo in the fame article,
that C is = + a D— a C-+2C, Therefore, if we fubtraét C from the
left-hand fide of the laft equation, and + oS D —- = C + 2C from the right-
hand fide of it, the remainders will be equal; that 1 4 the fimple feries 4+- Dx,
Die ab it, ba &c, will be = the compound feries
Ex, 2 Fx, &e
+ — Dx, = ~ Ex, &c; and sontea een (dividing
all the terms by «) the fimple feries ay D, Ex, F x7, &c, will be = the coms
pound feries
4" E, “- Fx, &c
22D, Ex, &e.
And
THE SBSINDOCMOI £2 ES ATE EDO BE M; 263
And this equation is always true, of however fmall a magnitude we fuppofe x to
be taken: and therefore it will alfo be true when xis = o. But, when « is
= o, all the terms in the equation,that involve » will be equal to o likewife,
and confequently the equation will be + D= a E+ 3" D, or + D= +
3" p, 42 E; that is, D will be equal to a D together oy < E, either add-
1/4 72
ed to it or fubtraéted from it, as may be neceflary to produce the faid equality.
But, becaufe w is greater than m, = D will be greater than D, and confequently
_ E mutt be fubtracted from it, in order to make it equalto D. We mutt
therefore prefix the fign — to the quantity i E inthe equation + D= =". E
-. -- D; which equation will therefore be + D= — Ae E +- 2“D, or-+D
—+ 2 D— <2 E. Therefore the ign — mutt alfo be prefixed to the fifth
. m . °
term, E+, ofthe feries 1 + — a, Cx; Dwi; Ext, Fx’, &c, whichis equal
_ ma
| Na 7 + *\#; and confequently the five firft terms of the faid feries will be 1 +
ie Cx aD x3 ie Bieta is ail tr He ie ides taecs
recfureel the magnitude of the co- efficient » may ‘likewife be determined by
means of the equation + D= — + E42 = D, seit a0 ay hatees
E. For, by adding a E to both fides, we hall have D + + is — an D,
and, by fubtra¢ting D from both fides, = | Diem Jn D—-D=+_, x D
— <= & D, and confgieny B= xo" yx DaMEE De
4% 4n
3%—m 2n—m n—m a n— mM 22 —m 3n—m
———— gee = Dormer eS
oe —_— tC—_— x \
4n x 32 iaa x ” Soap opeytapiesty* Bn 4n pfctl
Therefore the five firft terms of the feries t + =i, Cx?, Dx3, Ext, Fx’,
m
&c, which is equal to 1 + x|7, will ber 4+- —% re = x i e? ob “5 -
a—m oni Oy 2) a—m 2n — m 32m ”
2n an #2 n a 2n x 3 4 e*, orl + a Ax
76. To determine the fign that is to be prefixed to Fw, the fixth term of |
1.)
the feries 1 + = wy C v7. Dive? ats Eve, occ, which is equal toa + x,
and to find the Paeaitade of the co-efficient F, we mutt proceed as flows:
In the laft article it was fhewn that the fimple feries + D, Ex, Be Mes as
é&c, was equal to the compound feries
A
264. A'D 1S COfUoR BLE ECO WH RAR LN rENRG
6
a2 BE, 2= Fz, —Gw?, toe, Bec
™m //4 M1 m
Qn An $2 sults ;
eee eee i en Fes; — Gx, &c,
or (prefixing the fign — to the terms that involve the co-efficient E) that the
fimple feries + D — Ex, Fx?, Gx}, &c, was equal to the compound feries
6
7. 72 Ht ™m
BOT) SAR ee PS BE Tr OES es
“+b D — Ex, 2— Fx*, — Gwe, &e.
m
¢
Add 2 E x to both fides of this equation. And we fhall then have the fimpte
feries D + Ex, Fx?, Gx*, &c = the compound feries
6
Bran ie (era ave Ne So lye
Mn Mm 72
6
+ 2 D—* Ex, * Fx, G x3, 8c
MH WW 3
+2En.
But it has been fhewn in the laft article that Dis = — i E + as D, or —
a D — += E. Therefore, if we fubtraét D from the left-hand fide of the
laft equation, and = D— + E from its right-hand fide, the remainders will
be equal; that is, the fimple feries + Ex, Fx, Gx', &c, will be = the com-
pound feries
SP ye os Gee a geese
In oi m2
anes Hae 3” By, sa Gx3, &e’
m1 m ™m oy (7.
+ 2Ew; and confequently (dividing all the terms by.
«) the fimple feries E, F x, G x7, &c will be = the compound feries
2* £2 Gx, Hex, &c
mM m Mm
be Bad: OMe On Get &e
MW 7 we
oe I Oe ;
And this equation is always true, of however {mall a magnitude we fuppofe x to
be taken; and therefore it will alfo be true when wis = 0. But, when w is
= 0, all the terms in the equation that involve x will be equal to o likewife,
and confequently the equation will be E = i F— x E+ 2£E. There-
fore (adding = E to both fides) we fhall have E + *. bees u F + 2&E,
and (fubtraéting E from both fides) we fhall have i” | iim “ F + E, or =”
‘Hoong UA aa F ; that is, = E will be equal to E together with — F either ad-
ded to it or fubtraéted from it, as may be neceflary to produce the faid equality.
But, becaufe w is gréater than m, = E will be much greater than E ; and there-
fore
THE “BION OMIALS PABORE M 265
fore - F muft be added to E in order to make it equal to 4° E. We mut
- wn :
therefore prefix the fign + to the quantity 2. F in the equation ~- E=_E;
7
_ F; and confequently the-faid equation will be a EE + °2°F. There-
fore the fign + muft alfo be prefixed to the fixth term, F x', of the feries that
m2
- [= : : 72
-is equal to 1 + x|*, to wit, the feries 1 + EOyhs, DS, Ett) bse oes
t é 2 2
from which fixth term the quantity 2. F has been derived above by the opera-
tions of multiplication and divifion in the courfe of the foregoing proceffes.
v2
Therefore the firft fix terms of the faid feries will be 1 + — ~ — Cw? + Dw;
a
ee? Ex. Git. '1.
And the magnitude of the co-efficient F may likewife be determined by
means of the equation = E = E + F. For, by fubtracting E from both
4
ox, )
fides, we fhall have ** F = + E—E= i \x Eat x E;: and
| oar See 4 — m _ 44—m _ 4"%—m 3n — m
confequently F will be = Rear at x 5 = x E= x .
2n — 2— Mm mn in 2 — 1 un — Qn — ih 4% — mm
ie str —- __ —, he ee
es 37 ao? bw tide 32 ” 4n a 5a eee
s mM
Therefore the firft fix terms of the feries r + — x, Cw*, Dx3, Ext, Fx’,
m
* . aa In (e4 a — 2 i n2— Mm
&c, which is equal to 1 + w|%, are I + ee ee
72
21 —m mM n— m2 2% —m 3n — m2 mn 2 — mM 2n — m
a wast a x — xX = 1* + — X A X
3% * 2% 3% 4% 2 22 3%
—m n— mM m nt — 72 22 — Mt — m8
ee x’, ort + —Aw — Tine cha siete” (ce
73 3 2% 32
77. Having thus gone through the inveftigations of the values of the four co-
efficients C, D, E, F, at confiderable length, I fhall treat more concifely of the
inveftigation of the following co-efficients G, H, I, K, L, M, N, O, P,Q, R,
S, T, &c, and fhall only obferve concerning them, that the figns + and —,
which are to be prefixed to them, may be determined, and their magnitudes dif-
covered, by means of the following fhort and eafy fimple equations, which may
be eafily derived from the grand, fundamental, equation obtained above in art.
72; to wit, the equations
F=—G, “Ff,
G “4, &
Vor. Il. | 2M =
266 A DISCOURSE CONCERNING
II
S
eh
Saas
&
|
=
at
S
YN
“ee
|e
BS
to
7 a ee 8x
= m K, m I,
Kyi: Salas “Kk,
ae M, Be o
M = — N, —- M,
Noe Biro. ein
ae Pay jain ei
Ones) 221 Sp. Sng
ne OE 5 ’
Pp
Q
R
S
= T, a S, &c,
or, (becaufe F has the fign + prefixed to it, and = F is greater than F, and
confequently = G mutt be fubtracted from a F in order to make it equal to
F),
Fro— ee G + > F,
and (fgr the like reafons),
oe Abe = ee ze
Sf
ad+H=—-— 1+ 44H,
-~-l=y+2#K—— 1,
+K=—--L+ “Kk,
~-Li+~“M--“L,
+M=—-—N+—M,
—-N=4—“0-—N,
+O=>—-—“~P+“0,
Lio pale: -eoeesealgt) tae ph
+ Quine SS oF
nil tres (att iS — es R,
181 17K
&c¢
TCH EY “BS I'N OM 1 ACCS ow SO 2 TEM. 264
&c ; in all which equations the figns to be prefixed to the feveral co-efficients
G, H, I, K, L, M, N, O, P, Q,R, S, T, &c, are alternately — and +. And
the fame thing, it 1s evident, mutt take place i in all the following co-eflicients of
the powers of « in the feries 1 + — eee. es “*, Fixs, &c, which is
-
equal to 1 + X\ # to whatever number of terms the faid feries fhall be conti-
nued. We may therefore conclude that the faid feries will, when z is greater
than m, be 1 +x — Cx? + Dx? — Ext + Fe? — Gre + He? —
Ix? +.K xo — Lx? + Me™ — Na? + Ox? — Px™® + Qa — Rx? +
S#? — Tx" + &c, in which all the terms after the two firft are marked with
the figns — and + alternately, or are to be alternately fubtracted from, and
added to, the {aid two firft terms. ve eae
And, fecondly, the magnitudes of the co-efficients G, H, I, K, L, M, N, O,
P,Q,R, S, T, &c, may be difcovered by refolving the foregoing (hort fimple
equations, to wit,
6n
eG Sp,
—-G=+H-—G,
+H= — = I + me H,
SS ee eee
+K=>-—L+ kK,
—L=+—-"M-—-—L,
+M=—-=~N+-=M,
— Nom. +) 8 ON,
+-Orz—-—“P+“0, .
~-P=i“Qq-—p, —
+Q=—-—-R+ Q,
HRs 128s ol R,
Spee x Bee EB be Sp 8c
For, fince + Fis = — “*G + 4. nae a a S6
ae aaa 4: ee of ma sl odoc
po 4 TE + &e
ae — zie a a a “ ae ve aa 4. 8c
x ae aa Z = + &c
a ae + &c
ee ai oe + plas ABB ELE we
2
Vou. Il. 3 E ged
394 A METHOD OF EXTENDING CARDAN’S FIRST RUBE
B7H e9 2 BDH et? 2 BFHe'3
fay tt ee ee
oe
+—,— + &c
B7xK ett Nees
Sioa anand + = Ges + &c
B2M ¢t3
G ommarumce arte”
B3¢3 3 B*De5 3 BF e? 3 BH e? 3 BK e™ 3B at
an Eos Eaai aN aak Gea ee ae ee
BD7e7 6 BDF e? 6 BpHe!t 6roKe
+ ao + Sree es at eee + —— + &c
D3¢9 3 BF" 6 bFH re
a ee tee ea ee
3 D* Fe" ee ers
+ Sa, ob 3 4+- &c
© wis err jt sort Le
—_-—
This laft ge feries is the cube of the feries ~ — = + = bee rey Ae Es Spe
Ke? Met
SAE eae — + &c ad infinitum.
Therefore 8 s into the cube of the feries ~ + ie FA She —— mh x =<
ae a Lena oe yet &c will be = 8s x the foregoing eke aaa 5 cor if for
the fake ‘of brevity, we denote the eae aii Seis Py the ie capital
letter A, 85 X the cube of the feries = — “+= — + if apepaae = ee ay a
git
be —— = &c wall be =) 8s < the ee fie Aa ac i, < cube of the
OFe
Be ood Met!
sane 27% s xX the feries — + esa ~—e =< ae xs
&c, or the cube of y, will be = 8 5 se the a suet feries A.
0 e3
gil 513
18, If the foregoing compound feries A be multiplied into 8s, the produ&
will be the following compound feries, to wit,
— v 24 oe ue a = 24 —— He 24 pe a tee
Pa 9 IL : 3
as 24 2: e 4s 48 ne ve Nic of ee Re
3,9 2,1t y 13
— in aa 48 == se
z It 2 13
$e + ES + &e
i ae - + &c,
which
St
FOR THE RESOLUTION OF CUBICK EQUATIONS, é&c. 395
which we will denote by the Greek capital letter I. Then will y* be equal the
compound feries II.
19. Since the compound feries A, obtained in art. 16, is equal to the pro-
duct gy, and the compound feries II, obtained in art. 18, is equal toy, it
follows that the fum of the two compound feriefes A and II will be equal to
gy + y3, and confequently to its equal, the abfolute term r; that is, the fe-
riefes A + It will be = 7. Therefore the feries TI will be — 7 — the feries
A. Butz is equal to the firft term, 6 Be, of the feries A. For B is > and
re =, and confequently 6 Be is (= 6 x — x ~ = 2 = yi Pheterore
the feries II is = 6 Be — the feries A; that is, the compound feries IT is equal
to the excefs of the firft term, 6 Be, of the compound feries A above the whole
of the faid feries. But 6 Be — the feries A, or the excefs of the firft term,
6 Be, of the feries A above the whole of the faid feries, will be a compound
feries confifting of all the terms of the compound feries A, except the firft
term 6 Be, with their figns + and — every where changed. Therefore the
compound feries II will be equal to a compound feries confifting of all the
terms of the compound feries A, except the firft term 6 Be, with their figns +.
and — every where changed ; that j is, the compound feries
8 B33 24 87D e5 24. B7F 7 be 24 B7H e9 24 B?K e7% 24 BM et3
“3 a aie + 6 Sate ig Sear t Pern we &c
24 BD7e7 48 BDF e? 48 BDHe™ 48 Bpk e™3
re Taseropl ated Vay orca rir ele
i"
8 p3¢9 2pIl 8
w = HDPE fe ts 24 BFe big teen BFHets + 8c
58 mar ore
24 D? F et! 24 ote"
gt?
Wa DFe"3
a Tee + &c
will be equal to the compound feries
6 Be3 6 Bc eS 6 Bp e? 6 BE ee? 6 Brett 6 BG e!3
aa 54 56 rh re of 510 oF 22 0 &e
6 pe3 6 pepe 6cpde7 6 p29 6 pE e™™ 6 pFe%3 &
Ts g% g*# 56 58 gO an gid 7c Cc
5 qi 9 It 13
1 Oke 6 Bre ‘, Gere _ 6vFe 6EFe 4. Be
s* 5° : 99 10 yt?
6 He? 6 BHe? 6 cu et 6 pels
56 FPHEY: gre T a + &c
6 Ke 6 BK e™ 6 cK e%3
i]
er 38 1 #0 > 4m &c
6 Me™ 6 BM e%3
ay gf? = gi rE &c
6 as
a + &c;
sz
which, for the fake of brevity, we will denote by the Greek capital letter =.
And then we fhall have the compound feries I1 = the compound feries &.
EA es Of
3.96 A METHOD OF EXTENDING CARDAN’S FIRST RULE
Of the figns + and — that are to be prefixed to the feveral
terms of the foregoing compound feries &.
20. In this compound feries = all the terms of the firft horizontal row of
terms are to be added together, and confequently all the terms after the firft
B*¢3 :
, are to be marked with the fign +, to whatever number of terms
s”
6
term,
the faid horizontal row of terms may be continued; and the firft terms of the
fecond, third, fourth, fifth, and other following horizontal rows of terms (to
whatever number of horizontal rows the faid compound feries may be conti-
nued) will be marked with the fign —, and all the following terms of the faid
horizontal rows after the firft terms will be marked with the fign + ; and con-
fequently all the terms of every vertical column of terms in this compound {e-
ties, except the loweft term, will be marked with the fign +, and the faid loweft
term of each vertical column will be marked with the fign —.
This follows neceffarily from art. 15 and 16; becaufe the fizns + and —
that are prefixed to the terms of the compound feries A, fet down in art. 16,
are the fame with the figns of the correfponding terms of the preceeding com-
pound feries T°, fet down in art. 14, from which the compound feries A is de+
rived by only multiplying its terms into 6 5; and the compound feries © is de-
rived from the compound feries A by omitting its firft term 6 Be, and chane-
ing the figns of all its following terms. Confequently the figns to be prefixed
to the terms of the compound feries 2 mutt be contrary to thofe which are to
be prefixed to the correfponding terms of the compound feries T, and there-
fore muft be contrary to thofe which are defcribed in art. 15, or mutt be fuch as
they are defcribed to be in the prefent article.
Of the equality between the co-efficients of the terms of the compound feries 1
and the co-efficients of the correfponding terms of the compound feries x.
° . . 3 .
21. Since ee is = ne and ss is = a + = we may, by leffening the value
of = or of r, without altering that of g, leffen the value of the fraction
La fe
—
rR “, and confequently that of =, as far as we pleafe. Yet in all
a7 4
thefe values of the fraction < it will always be true that the compound feries II,
which involves in its terms the fractions 5, ss e o = =, &c, which will
be equal to the compound feries ©, which involves in its terms the fame fractions.
it therefore follows from this conftant equality between thefe two feriefes in all
poffible
FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 397
x m 2 e3 e e7 e? eit et3
poflible magnitudes (how fmall foever) of the fractions “a PEP res) eS
&c, that the term, or terms, that involve any given powers of ¢ ands in one of
the two feriefes muft be equal to the terms that involve the fame powers of them
in the other feries. And confequently the feveral co-efficients of the terms
which involve any given powers of ¢ and sin one of the two feriefes muft be
equal to the feveral co-efficients of the terms that involve the fame powers of e
and 5 in the other feries. Thus, for example, the co-efficient of the fraction
= in the feries II, to wit, 8 B*, muft be equal to the two co-efficients of the
fame fraction £ in the feries X, to wit, 6 BX — 6D; and in like manner the
co-efficient of the fraction s in the feries TI, to wit, 24 B*D, muft be equal to
the three co-efficients of the fame fraction = in the feries 5, to wit, 6BC +
6 BD —6F; and the two co-efficients of the fraGtion = in the feries II, to wit,
24 B°F + 24 BD*, muft be equal to the four co-efficients of the fame fraction
< in the feries ©, to wit, 6BD + 6CD + 6 BF — 6H; and the fame thing
mutt take place with refpe& to the co-efficients of the following fractions
ae > i &c ad infinitum, or to whatever number of terms the two feriefes
may be continued. —
Examples of the faid equality of the co-efficients of the terms of the
Jaid two compound Jeriefes.
22. Of this equality between the co-efficients of the fame fractions, confifting
of the powers of ¢ and s, in thefe two compound feriefes II and &, it may not
be amifs to give a few inftances by actually computing the values of the faid
co-eflicients ; which may be done in the manner following.
The capital letters B, C, D, E, F, and H, are equal to =
Be TS chat pce
Qn, Br 243 ee
and 324, refpectively. Therefore B? will be = —, and 8 B? will be = —;
19683 27 ¥
and 6 B* will be (= 6 x —jo and 6 D will be (= 6 x 2=2x2)=
9 3 81 27
~“, and confequently 6 B*— 6 D willbe (= = —~— = neuer y= £ si.that
27 : 3 ll 27 27 27
is, 8 B3, the co-efficient of the fraétion = in the compound feries II, and 6 B?
} Ab he
— 6D, the co-efficient of the fame fraction = in the compound feries 2, will,
each of them, be equal to the fame quantity =
4 In
398 A METHOD OF EXTENDING CARDAN’S FIRST RULE
Tn like manner 24 B’D, or the co-efficient of the fraction & in the compound
. ; 8
Teresa ty 1S (> 2a aay x es eae x =) = Fa: and 6 BC + 6BD —
. . Sey e
6 F, or the co-efficient of the fame fraction = in the compound feries 2, will be
found to be equal to the fame quantity. For 6 BC + 6 BD —6Fis=6 x
I I I 5 228 7 t2 10 44 84 30 44
—X—-+6xX=—-xXe—-6xX— (= -+ ep — ee St SK
3 as REPS 729 § 9 8 243 243 243 243
ma) Be Baath 942
(was 243 243 Q. E. D.
And 24 B’F + 24 BD’, or the co-efficient of the fraction 5 in the com-
ae + yy 55S eee
26 Oye Os pi ube ne be 200. Cees 72 0s
ee SET TIRBUTT TU TM TMT e oo and6BD + 6CD + 6BF
. . 7 . .
-— 6H, or the co-efficient of the fame fra¢tion = in the other compound feries
pound feries IT, is (= 24 X 5 x
=, is equal to the fame quantity. For itis = 6 x _ “4 ma + 6x *5 x =
hp 204 cs di 8 374 (= TA ge adh Sa 222X374
=D Sard eaters ie Serer rR Se eH ey 6561
PATO pL AO oy glad 748 __ 810 270 396 748 _. 1476 748 » He
ae 243. | 729 +6561 6561 ‘ 6561 ' 6561 6561 — 6561 6661
6561 Q. E. D.
And the fame equality will be found to take place between the co-efficients of
s ; oF OR sia dette ; : . :
the following fractions = 3° ee &c, ad infinitum, in the compound feries II
and the co-efficients of the fame fractions in the compound feries &, refpec-
tively, to whatever number of terms the faid feriefes may be continued.
The reduction of the compound feries Il to a fimple Series “f “4
aes Re7 8 e9 Tet v e333 , :
rps dice ihe tear ue ear caer sine ESc ad infinitum.
23. In the compound feries TJ, obtained in art. 18, all the terms after the
x are marked with the fign + and added to the faid firft term.
8 B3
firt term, ——
s
Therefore, if we reduce the faid compound feries to a fimple feries, with fingle
letters for the co-efficients of its terms, by denoting the co-efficient, 8 B3, of
8 B3¢3
, by the fingle letter P, and the co-efficient,
24 B*D,
. 3 .
the fraction = in the firft term —;
POR THE RESOLUTION OF CUBICK EQUATIONS, &c, 399
. ec. 24 B7pD eS
- ¢4 B*D, of the fraction role the fecond term, oaks by the fingle letter Q ,
and the compound co-efficient, 24 B*F + 24 BD’, of the fraction & in the
third term, by the fingle letter R, and the compound co-efficients of the fol-
ee eg
lowing fractions rr &c, in the fourth, fifth, and fixth, and other fol-
po yi?
lowing terms of the fame feries, by the fingle letters S, T, V, &c, the fimple
. . . ( 4 3 S 7
feries that will be equal to the faid compound feries II will be —— + -5-
J c
s 6? Tet v e%3 : : Pe
Ag = eat ate &c, in which all the terms after the firft term, =>
are marked with the fign ++, or are added to the faid firft term.
dame? aes Re? s 09 pe vers
The fame fimple Series 2 + 7, +- a + n> +- ao, iia
+ Se ad infinitum will alfo be equal to the compound feries =.
24. But by art. 19, the compound feries IT is equal to the compound feries
=r be lefs, not only than 294 (which is its greateft pofiible magni-
34/3
1g ey ete Wa) Tees q?
tude) but than 2 X ae? or if r be lefs, not only than = but than reser
or
424 A METHOD OF EXTENDING CARDAN’S FIRST RULE
3 3
or a? and ¢ be taken = -, and zz be = r os or ne — ee; the leffer root
I
piss /
of the equation gx — x? = r will be equal to the expreffion 2 z 3 xX the feries
= = =f — - 9 ae ahs sae aa — &c ad infinitum. It is in-
z z & z z ze
deed a very long and complicated eee But I know not how to
make it fhorter without taking from the perfpicuity of the reafonings ufed in it,
which are both various and abftrufe. And ‘ that they fhould be fo” will appear
the lefs furprifing, if we confider that they fupply the place of thofe very obfcure
and intricate operations by which many writers of Algebra endeavour to find the
roots of impoffible quantities, fuch as 81 + — 2700 and 81 — ¥ — 2700.
See upon this fubject Monfieur Clairaut’s Elémens d Algébre, Part V, art. ix,
pages 286, 287, 288, 289, and a paper of Monfieur Nicole in the Memoirs of
the Academy of Sciences at Paris for the year 1738, pages 99 and 100, from
which Monfieur Clairaut has extraéted what he has delivered upon this fubjeét
in the pages of his Algebra juft now cited. And fee alfo Dr. Wallis’s Algebra,
chapter 48, pages 179, 180, of the folio edition at London in 1685, and Pro-
feffor Saunderfon’s Algebra, pages 744, 745, 746, 747, and Mac Laurin’s Al-
gebra, Part I. the fupplement to the 14th chapter, pages 127, 128, 129, 130,
and the Philofophical Tranfactions, No 451.
— —
57- We will now proceed to give an example of the refolution of a cubick
ae of the foregoing or qx—w«3 =r, when s is lefs than ./2 X ii,
or — is lefs than a 7 or , by means of the expreffion 2 32 x the feries
/
Be Dé Fe H e7 K e? Met 0 e338 .
ey Re TE ie eck wit ee which we have fhewn in
the foregoing articles to be equal to its leffer root.
An Example of the refolution of a cubick equation of the
foregoing form, qx—x =r, by means of the expref-
> Be De F ¢ He7
fin 2? =X the feries — — — = ate
zs
K 69 Mert © ¢t3
———- + <3 — &e.
5? git
Let the Cae that is to be refolved by means of this expreffion be 15%
PACte.g sae "ae and ris = 4; and confequently = =is(= = ae and — — is
ee pte oe eS Per
C= +) = 2, and 4 ; is (= 4 oh) 125, and ere orf, isyaee >= >
62.5, and — x is ed = 2\*) = 4, which is lefs than 62.5, or e. Therefore
5 this
1
"
Bika
FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 425
this equation may be refolved by means of the foregoing expreffion 2 ¥?2 x
ve BEe De Fé H e7 K €? M elt 0 e%3
the feries 2 ae Gp alters keer bamiees “oe ea psig —r — &c.
8
Now, fince r is = 125, and — is = 4, we hall have zz (= . as =
125 — 4) = 121, and confequently z (= VY 121) = 11, andy? z (= y? 11)
= 2.223,980,090,569,361, &c, and 2? z(= 2 X 2.223,980,090, 569,361,
&c) = 4.447,960,181,138,722, &c.
And we fhall have ee (= = = 4, and confequently e = 2, and = meypes, Se
aie
Ba Sigh
0.181,818,181,818, and = = oT
io
Therefore —~ will be (= = x — = o 181,818,181,818 x eri oA
0.727,272,727,272
wei) ay 01000,010,5 15,407":
5 s 3 i
And = will. be (= = Xx = = 0.006,010,518,407 X “hi
0,02 4,042,073,6
Ey —O.OOCm Ob, 0045023"
e7 r es e* 4 —
And = will be (= = x = 0.000,198,694,823 xX Tr =
SEY = 0.000,006,568,423 ;
Py ' 7 2 4 a
And | will be (= S xX = = 0.000,006,568,423 X 7 =
Oo. by 6 ,6
as = 0.000,000,21 751 37;
elt . e? em 4 —s
And a will be GS ai ee 0.000,000,217,137 xX eT ee:
: 868,48
ocencconen sa) = 0.000,000,007,178 ;
13 . an ert 2a by 5 4 oa
And <= will be (= = X = = 0,000,000,007,178 xX T=
POP 028712 =
121 >)
Therefore = Will be (= B® x o.1o1,o18,181,318 = S x 0.181,818,181,
818,181,818
818 = —e) = 0,060,606,060,606 ;
0.000,000,000,237-
And 22 will be (= D X 0.006,010,518,407 = = X 0.006,010,518,407
is Sai 25273972035) = 0.000, 371,019,654 3
SI
And ££ will be (= F x 0.000,198,694,823 = => X 0.000,198,694;
C2 esaaeg 289110) = 0.000,005,996,277 5
729
Vou. II. 3 1 And
426 A METHOD OF EXTENDING CARDAN’S FIRST RULE
He? .
And ale be: (="H-X 0:000,006;565,423 = tte X 0.000,006, 568,
=. 0,002,456,590,202, __ ‘
i ge ee cases ) = 0.000,000,1 24,807 5
Kg? is 150
And Pe will be (0K 3<, 0.0GGsn00,2% 77137 a ee X 0.000,000,
’ ?
i.) 01004,669, 5315185, g. .
8 By Se ears ie Gael = 0.000,000,002,928 ;
: M : 1. 7
And — will be (= M xX 0.000,000,007,178 = ae X' 0.000,000,
ie 0.001,058,087,446. __ ; ee
COs ly oie a GoT ee = 0.000,000,000,073 5
oet3 é : 5I 3920
And Te will be (= O X 0.000,000,000,237 = pois oqid X 0-000,000,
? >
000,237 = wait Bake a | = 0.000,000,000,001
9°37 = 387,420,489 FET OS RRA GR has p> ‘
Therefore the feries — ——— + —— ——— 4 = = 4 = Siena
z 23 25 a Ad ou 213
be = 0.060,606,060,606 — 0.000,371,019,654 -+ 0.000,005,996,277, —
0.000,000,124,807 + 0.000,000,002,928 —= 0.000,000,000,073 = 0.000,
000,000,001 — &c = 0.060,612,059,812 — ©.000,371,144,534 = 0.060,
240,915,278; and confequently the expreffion 2 ./3 z x the feries oo
Fes He? rn Ee as ae “< — &c will be = 4.447,960,181,138, &c
X 0.060,240,915,278 = 0.267,949,192,431, &c. Therefore this number
0.267,949,192,431, &c is the lefler root of the propofed equation 15 «— x?
ah 44 Qs. Ee is
25 ‘ zi
The foregoing number 0.267,949,192,431, agrees with the true value of the
lefler root of, the equation 15% — x? = 4 in all its twelve figures, the faid root
being equal to 0.267,949,192,431,122, &c, or 2 — 1.732,050,807,568,877,
&c, or 2 — 3. For, if we fuppofe «to be = 2 — ¥3, we fhall have 15% (=
15 X{2—v3) = 30 — 15 V3, and #*® (= 8 — 3 X4XA3+3X2*X
3—3 V3 = 8— 12 93 + 18 — 373) = 26 — 15 V3, and 154-7 (=
30 — 1573 —[26 — 1573 = 30 — 1573 — 26 + 15 73 = 30— 26)
calf
Another Example of the refolution of a cubick equation by
means of the fame expreffion.
58. Asa fecond example of the foregoing method of refolving cubick equa-
tions, let it be propofed to refolve the equation gov — x? = 98.
Here g, the co-efficient of «, is = go, and the abfolute termris = 098.
Therefore a will be (= = 30, and " will be (= 4 = 301?) = 27,000,
ea ie and
FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 427
3 3 ; ; : 8
and ve or mere will be (= a) = 13,5003 and — will be (= =) = 40,
and * will be (= 40\°) = 2401, which is lefs than 13,500, or ma Therefore
the equation 90“ — «* = 98 ar be refolved by means of the foregoing ex-
De F é§ H e7 K e9 M eT 0 3
3 ERY” yb Te, SRG Se ee DEAR athens $e ah
preffion 2 ,/* z x the feries * paia aren as me =
ane &c. /
q
Now, fince fis = F127 5Q005 and — is = 2401, we fhall have ez (= £ —
vit a |
rr
— = 27,000 — 2401) = 24599, ie Z(= VW 24599) = 156.840,683, and
VE % (= 4/3 156.840,683) = 5.392,865,326,078, and 2/2 z(= 2 X 5.
392,865,326,078) = 10.785,730,652,156. |
_ And we fhall have ee (= a = 2401, and confequently e= 49, and = ae
sete | Sey iA 4OU
156.840,683 ~~ Peg Oe Bee and 5 "" 24599°
e ; ee é 2A0T es
Therefore —= will be (= z Xo FS 0-312,418,940,435 X 24599
750.117,875,984,4 ee.
1$2.017.575,984835) — 0.030,493,836,1713 -
zs : _* 33 e2 (ae : 2401
And — will be (=. 5. X..m =. 0.030,493,836,171 °X hay 0
73-115,600,646,571 :
O06." =~ 1) = 0.002,972,299,713 :
e7 ° es e 2
And om will be (= a XX Fw = 0.002,9725299,713 X Fn a
6,491,6
ee) = = 0.000,290,113,078 ;
0, 7 »
And <'will be (= 5 x, S = 0,000,290,113,078 x 2° =
0.696,561,500,278, __
ere = 0.000,028,316,659
Ir q ; 9 g@ ;
And = willbe: {= =. x ~ = 0.000,028,316,659 x EES =.
ee —o. 000,002, 763,864 ; |
oe he ‘. :
And 5; will be (= S x SF = 0.000,002,763,864 x 2 =
Sieh = 0.000,000,269,768 ;
ae ips tid - 2401 0.000,647,712,968
And = (= az X gq = 0.000;000,269,768 x 24599 24599
= 0.000,000 30265330
Therefore — will pee B X 0.312,418,940,435 = — X 0.312,418,940,
435 = astethqntssy = 0.104,139,646,811 ;
3 1.2 And
—
428 A METHOD OF EXTENDING CARDAN’S FIRST RULE
03 °
And ~- will be (="DD "oid 36,4039;036,171 = = X 0.030,493,836,17%
: »469, 180,8 MN
pore Sot Seeds) ae 0.001,882,335,566 ;
gs 2
And — will be (= F X 0,002,972,299,713 = = X 0.002,972,299,713
_ 0.065, 399,593,586
a ) = 0.000,089,699,031 ;
And 4£ will be (= H x 0.000,290,113,078 = ah X 0.000,290,113,
o78 = ab Op 29' tI") == 0.000,005,512,487
And a ae oe X 0.000,028,316,659 = Seer X 0.000,028, 316,
Ce Pom, oh deae a = 0.000,000,381,948 ;
And aes Wil be 4 =. Vise 0.000,002,763,864 as raya. X 0,000,002,
763,864 = senses ) = 0.000,000,028,393 5
And a will be (= O xX 0.000,000,269,768 = eRe: X 0-000,000,
0.856,491,318,560
387,420,489
And & = will be (= Q_X 0.000,000,026,330 =
269,768 = = 0.000,000,002,2103
70,664,648
10, 460,3 § 3,203
) == 0.000,000,000,177.
X 0.000,
1.860,600,18 1,840
10,460, 3531203
De3 Fe H e7 K e9 Mert 0 e753 qets
Therefore the feries ~ —— Sb Se eee ee
é&c will be = a. eae 646,811 — 0.001,882,335,566 + 0.000,089,699,
O31 — 0.000,005,512,487 + 0.000,000,381,948 — 0.000,000,028,393 +
©.000,000,002,210 — 0.000,000,000,177 + &cC = 0.104,229,730,000 —
0.001,887,876,623 == 0.102,341,853, 3773 and ope ees the expreffion
e e e? ell el3 ae
2 V/?x x the feries 2 — 2S 4 Ff Ae —=r, when the abfolute term 7 is lefs than 42 x iv, Oe is lefs
3 3
than Me or > - 5» by means of the ama 27> z x the feries < — — +
Fes He? KO M e™™ 0 e%3
OG ee rea Ae ters ir aT "4 &e ad infinitum: and eee | fhall
not add to the length of this coils (which is already longer than I could
have wifhed) by applying the faid expreffion to the refolution of any more ex-
amples.
Another expreffion of the value of the leffer root of the cubick
equation qx —«* =r, derived from the foregoing ex~
preffion of tt.
61. But there is another expreffion for the value of the leffer root of the cu-
bick equation gx — x? = r in the cafe here fuppofed, which, as it may be eafily
F, . & ° . ° Be De}
derived from the foregoing expreffion for it, to wit, 2 /3z X the feries — Stile LU —
Fes He? K €9 M ett 0 e78 eaTsREe aS
so See ee ebaabrere ag as &c, ought not, I fete to be
omitted. This expreffion does not confift entirely of an infinite feries (as the
foregoing expreffion does), but partly of a finite algebraick expreffion, and
partly of an infinite feries; and fewer terms of the infinite feries are neceflary
to be computed and added together, in order to obtain the value of ae expretf-
De
fion to any propofed degree of exactnefs, than of the infinite feries = —[£ +
z x3
Fes He? K e? Me™ 0 e%3 qets - : .
er te ae Pigs “aa P&C contained an’ the’ foregoing exe
preffion. It is as follows, to wit, /*(z Aa em V{z—e—4vV7'%z x the infi-
Me™ qets v er9
a ~ . De He
nite feries oar ait aaa ent waite: a &c ; the terms of which feries are
: ° i aaeee .
taken from the feries that is equal to 1 + 2 or the cube-root of the binomi-
; e : . ce pe Ee Fes Ge
al quantity 1 uae —, to wit the feries 1 - ere cconsgaepeh ss EMleoragh va -lherere bx pence oracle
zt 25 ze
H e7 168 K oe Ler? M el N on o 33 P et4 qets R er
z7 28 ea eet zt ik, Wane ott mish ca wit 1S) cyte Keene
430 A METHOD OF EXTENDING CARDAN’S FIRST RULE
8 e>
neni oer ts — &c, by beginning with the fourth term ~~, and taking
Big eT
every fourth term from it. iis expreflion may be derived al the foregoing
D e3- F eS H e7 K e? Me?! oet3
3 —$ ee —__ ee SS ee oe
expreffion 2/722 xX the feries = 3 are, mI a) iE Py
ane = “+ &c in the manner a i
zt
The derivation of the expreffion of the value of the leffer root
of the equation qx —x* == r given in the preceeding ar-
ticle 61, from the former expreffion of it.
62. By the binomial theorem in the cafe of roots we have 4/3 F 4 = Ce
: Be c e De Ee Fes G 6 H é7 re® K 9 Let
fenenapchig a Qockha rah ge tie ca
M ett N el” 0 e3 Pp v4 qe R er s e%7 ig e8 v et9
fant gay glk ce gia hee gh ie gt a
up Be ce pe3 E e+ F eS Ge He7 re Vee
a ENCLEIICS < + &c is (= 0.000,371,
019,654 + 0.000,000,124,807 -- 0.000,000,000,073 + &c) = 0.000,371,
144.
A432 A METHOD OF EXTENDING CARDAN’s FIRST RULE
144,534, &c, and 4 /? 2 xX the faid feries ze + a: ne <> + &¢is (=m
X 2.223,980,090,569,361, &c X 0.000,371,144,534 = §.895,920,362,2775
A444, X 0.000,371,144,534) == 0.003,301,672,217,358, &c.° And z + els
(= 11 +2) = 13, and g—eis (= 11 — 2) = g; and confequently
/3(z + eis(= WV * 13) = 2.3515334,087,721, and V3 {z—eis(= V3 9) =
2,.080,08 3,823,052. Therefore /? (z + e — /3 z—e will be (= 2.3-7,
334,687,721 — 2.080,083,823,052) = 0.271,250,864,669, and 3 [z+ ¢
e3 e ‘
— /3\z--e— 4 Viz X the feries — +— + +oiexc_ will Degtees
0.271,250,864,669 —- 0.003,301,672,217) = 0.267,949,192,452. And
confequently this laft number 0.267,949,192,452 will be equal to the lefler
of the two roots of the equation 15“—w#«3 = 4, Qe Es as
This number 0.267,949,192,452 is exact in the firft ten figures 0.267,949,
192,4, the more exact value of the leffer root of the faid equation being 0.267,
949,192,431,122, &c, or 2 — 1.732,050,807,568,877, &c, or 2 — 473, as
was fhewn above in art. 57.
In the other equation gox— x? = 98 we have feen above that eis = 49,
and 2 is (= / 24599) = 156.840,683, &c, and < 1s? ct
3. be
418,940,435, and —- 1S =-0.001,882,335,566, and — IS = 0.000,005,512,
) = 0.312,
1S
/* % is (= 7% 156.840,683) = §.392,865,326,078. ‘Therefore the memes
Met. ets ,
487, and ar IS = 0.000,000,028,393, and ~ 1S == 0.000,000,000,177, and
e 1S :
ae a +> + = will be (= 0.001,882,335,565 + 0.000,005,512,
487, + 0.000,000,028,393 + 0.000,000,000,177) = 0.001,887,876,623,
He7 M ett qe
and 4/3 2 X the faid feries a + + te will be 4 a ee
865,326,078 X 0.001,887,876,623 = 21.571,461,304,312 X 0.001,887,876,
623) = 0.040,724,25 7,520.
And z +e will be (= 156.840,683 + 49) = 205.840,683; and z—e
will be (= 156.840,683 — 49) = 107.840,683; and confequently Y? (z + e
will be (= 3 205.840,683) = 5.904,417,671,968, and 3 (z—e will be
(= WV? 107.840,683) == 4.759,860,337,980. Therefore “3 (z+e —
/ 3{z—e will be (= 5.904,417,671,968 — 4.759,860,337,980) = 1-144,5 575
333,988; and 73 (ze — J/3(z—e—4 V3 x the feries + ae +
Me™? gets
<< + Sy will be (= 1.144,557,333,988 — 0.040,724,257,520) = 1.103,
833,076,468. Therefore this laft number 1.103,833,076,468 will be equal to
the leffer of the two roots of the equation gox — #3 = 98. Qs Ect
This value of the leffer root of this equation is exact in the fix firft figures
3.103,83, and exceeds the true value of the faid root (which is 1.103,832,
QII,1;)
¥
,
FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 433
911,15) by only the fmall quantity 0.000,000,165,3, which is-fomewhat lefs
than the difference whereby the former value found for this lefler root in art.
. 2 3 5 4
58 by means of the expreffion 2 /? 2 X the feries ee Ee i nd mae rs
K e? Melt 0 et3 qets ,
eer ar — ar &c, to wit, 1.103,831,664,9, falls hort of the faid
true value, that difference being 0.000,001,246,2.
A third expreffion for the value of the leffer root of the
equation qx—x3 =r, derived from the expreffion
obtained for it above in art. 55.
64. We may alfo derive another expreffion for the value of the leffer root of
the equation gx — «* =r in the cafe here fuppofed, from the foregoing ex-
Des Fe He? K e? M ert 0 ¢3
. Be aes) —_—_—
reffio Ps.5 sh Race A (AO, oe 8 Me
p merce’ 2 x the feries sree ee =A os mE zis
Qe
<== + &c, in the manner following. b.
. . é
The quantity /3 (2 +e— /3(z —eis = /3 2X velt “ha Siri BS
e ‘ 2 3 E e+ Fes Ge
J? |1—L = fz x the feries 1 + 2 — SE bg al! AU wig et at
=
23 xt 2s =
He7 Wu K 6? Le M ett N el2 o e%3 Pp el4 qets 8
Pim ste ee et ee Pa ae Pe
. Be ce pe Ee Fes Ge He7 1 K 6
3
/ feries 1 : = =; 3 * “e 3 = =
L eto Melt Netz 0 et3 Pp elt qets : 32 2B€
ets ge 3) en Pe ok ae = V3 ‘ies ——
et a: Si “73 a me &c = /3 z X the feries =
2pe 2Fe5 2He!7 2K e? 2 Met 2 0¢%3 2aQe5
— 3
Dy ag tee ty ts tee yin
e De Fe He K e9 M ett oe ae
feries — + “3 Gather aan < Sphor santos fe Paalirers = + &c. Therefore,
: ite BE De Fes He? Ke Me™ 0 e3
_ — te —e_ _—_—_ = _— — —7_ neal
if we add 2 ? z & the feries = = = = = hs hte
— + &c to both fides, we fhall have WY? |z +e — yi (z—e + 2732 xX
eee
oe Bre Des Fe H e7 K e9 M ett 0 e%3 Qu
eres — 3
the feries — — — A eR eee ee —=7 +&em2y%z
23 zs he et a git zi3
: ° 2 9r° 13 is ce
x the feries — ‘de ~~ + — +. — Eee LA eee LE feries — +
Fe K €? 13 f
yh = a + &c. Therefore, if we fubtratt 3 (x +e — Vile
De Fes He?
* . B
from both fides, we fhall have 2 47? z x the feries — TX Saga hee
Ke M ett mens aes -.* Be Fes Ke? 0 e73
—_— — —— a pe ee aS 3 as i= ee a
=3 eee et tors ae &c = 47? zx the feries 7 ob aaa
e) ee De
+&e + f3(z—e — V3 [z +e. But 2/32 x the feries — — — +
Fes H e7 K @? M ett 0 3 ers
esi or —— — oe — + &c has been fhewn above to be
& 2 & 2
Vou. Il. ak. equal
434 «A METHOD OF EXTENDING CARDAN’S FIRST RULE
equal to the leffer root of the equation gv — x? = r in the cafe here upp
or when oy is or r., Therefore the expreffion 4./32 x the feries = —
+o + << += —— ay &c ad infinitum + ae (z—e — +f? (z “ae or the
5
excefs of the oe 4/32 xX the feries — els a a
and confequently the compound quantity y? — qy (which has been thewn to
be equal to the faid fraction) will alfo be equal to the fraction oe ri rk add
é 275
I
2 2e x ee
therefore = 77" — (XAT 2m x =r. Therefore e + 3)3
muft be equal to the root y of the equation y? — gy = r.
menace q b q Mone Gy,
Q, E. De
Another expreffion for the root of the foregoing equation
ye— qr.
8. By refuming the folution of the foregoing problem we may find another
expreffion for the root y of the aes y? —47y =7, to wit, the expreffion
rr eat ahr
dag Zamnit ers res ame 1 Sunes
4
This expreffion may be found in the following
le
— sJF
or ¢—
manner.
The inveftigation of the faid fecond expreffion for the value
of the root y of the cubick equation yi —qy =r.
g. In art. 5 we fuppofed the line y to be divided into two unequal parts, wv
and x, of which vw was fuppofed to be the greater; and we firft found the value
of the greater part v, and then determined that of the leffer part z by its relation
to v, which is expreffed by the equation 3 vz == q. But we may with the fame
Vou. II. 32M eale
‘
450 A METHOD OF EXTENDING CARDAN’S FIRST RULE ‘ 4
eafe firft determine the value of the leffer part z, and then derive from the faid
value of z the value of the greater part v by means.of the fame equation 3 vz =
g, Which exprefles their relation to each other; whereby we fhould obtain the
fecond expreffion of the value of v + z, or z + v, ory, which was fet forth in
q
; : : : 3 =e
the foregoing article, to wit, /3|—-— yj/— —2 +
2
S|
aie
=a
I
7 q oT q ; :
bd 3 ae ent gine =e — ———~\ Te
or 1/? {e Seen Ge ore—4l 3 t=5 t> Rae This may be done in the
manner following.
: : ; 3 .
Since 3 vz is = g, and confequently v 1s = ro and v3 1s = 2 and it has
3
g PP) + 27,
2723
3 3
Therefore rz? will be = = + 2°, and rz? — z° will be = 7 Therefore, if
been fhewn in art. 5 that ris = v? + 2%, it follows that 7 will be =
we fubtraét both fides of this equation from a (which is fuppofed to be greater
than c, and confequently muft alfo be greater than, its equal, 7z* — 2°) the
remainders will be equal to each other; that is, sa — frzi — z°, or a — rz?
+ 2°, will be equal to oe — So Therefore the fquare-root of the trinomial
quantity ie — 7z> -+- z° will be equal to the {quare-root of = ~ r But the
{quare-root of the trinomial quantity i — rz> + 2° is the difference of the fim-
°° r . . - ia : r
ple quantities > and z*, that is, either — — 23 or z*— —, according as — or
x? is the greater quantity. But, becaufe 7 is equal to v* + 2°, and ¥? is sreatee
than 2, it follows that 2? muft be lefs than one half of v3 + Bs or than one half
of r, or than — =; and confequently the difference. between — and 2? will be — >
— 2°, and not 2° — =. Therefore = — 2° will be the fquare-root of the tri-
nomial quantity a — rz? + 2°, and ge cca will be equal to the fquare-
root of — eer or — — 2? will be = v\7 —_— i Therefore (adding 2? to
both fides) = > will be = 2? 4 ¥ ceo ZL, a (fubtracting 4/ 33 —f from
both fides) 2? will be = —— ¥ (- —f, and confequently ene the
cube-roots of both fides) z will be = vit—yv{z ait Therefore z + v
8 to as HA phere e. OS Tale eS: PL
wihbe = Wt —vE-f+o0e vE-vy--f£445)
af 3
— =
OO _—
:
FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 451
r rr g3
tion y?—¢y =r will be equal to the expreffion W3/— — yj— — © +
7
=, or to the expreffion 4/3 e—s panier 1e The oy |
3 Vd een eae i ¥ gfiens
2 2
I
> 7
ae i).
~
Q. E. bi
A fynthetick demonftration of the truth of the foregoing exprefion.
, r
10. oo again we may demonftrate fynthetically, that this expreffion e — 5] 3.
"ya = > s\3
by fubftituting it for y in the compound quantity y3 — gy, which forms the left-
hand fide of the faid equation. For, if we make this fubftitution, we fhall find
that the value of y* — gy thence arifing will be equal to the abfolute term r.
This may be fhewn in the manner SENS
is equal to the true value of y in the propofed equation y3 — gy = 7,
If y is {uppofed to be equal to pa daa aan 6 <=) we fhall have y3
2
a himcalsacilel < iensoam a g
(=e¢—s +3 x ¢—s53 x : Beaty deena ute fT)
ae 35 9Xe—sF
93
aia qxer VF Magen taixene Me Os
Se ee sl 3: + Sp =s oe pear er + = can: ee and confequently y:
3
I
Sel es,
—W(=e¢—-s+ qx a3 + in: + 27 Xe—s
ei IN Ga Sk a ea Se
= Tie BN atoet seeds 27@ — 278
276% —i54es + 275" + 93
peer mew 2ps
But it has been fhewn in art. 7 that g? is = 27¢*—-275. Therefore the
«| 27e% — 54es + 2757 + (Bors 2 age? — 540s 2757 2762 2759 a She — Sages
fraction er ae nog oo, Th. IMCS
and confequently the compound quantity y3 — gy (which has been fhewn to
be equal to the faid fraction) will alfo be equal to the fraction Sea Sa =
47
3 M 2 $4 Xx a sees
452 A METHOD OF EXTENDING CARDAN’S FIRST RULE
25 x =
ae aS ae lS 2X SS a2 Xe 52 xX = Sr; thats,
27xKe-s ae
ie value of the compound ane y3 — gy arifing from the fubftitution of the
expreffion e— s| 3 han Tye in its terms inftead of y, is equal to the abfo-
lute sae r of the cane equation y? —gy =r, Therefore the faid expreffion
e—A|Z aes TI muft be equal to the value of y in the faid equation.
Qi E. D.
A third expreffion for the root of the foregoing equation
yeor-gy=r.
11. We may alfo, by refuming the folution of the foregoing problem con-
tained in art. 5, find a third expreffion for the root of this equation y? mm is
to wit, the expreflion 3 = + ie iat a Naa bier k aes £ A
I
Ne bck Sirk? GaSe ODS stad Zick tear dl Sap bis expreffion may be found
in the following manner.
The inveftigation of the faid third expreffion for the value
of the root y of the cubick equation y3 —qy =Tr.
12. Since v3 - z3 is = 7, it follows that z3 will be > r—v?. But v3 is
iy
fhewn in art. 5 to be = = + 7 = Piet Therefore r —v? will be = r—
3 3 . .
-— WV 4 —La=-— v/|—— ‘ Confequently z3 (which is equal tor —
2 Aenitete 7 ez 4
v3) willbe = —— V 7. Therefore v will be = 3) “+ vie,
and z will be = vie — v[=—£, and confequently v ++ z, or y, will be
Wipe te aeeae ~—foyhts+ hoe
AT + e@—s Be Q. E. I,
A [y=
FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. | 453
A fynthetick demonftration of the truth of the foregoing exprefion.
13. Here again we may demonttrate fynthetically that this expreffion
I T
@+5\3 + ¢—s\3 is equal to the true value of y in the propofed equation y?
—qy =r, by fubttituting it for y in the compound quantity y* —gy, which
forms the firft, or left-hand, fide of the faid equation. For, if we make this
fubftitution, we fhall find that the value of y3 —- gy thence arifing will be equal
to the abfolute term 7 of the faid equation. This may be fhewn in the following
manner.
ae} ij is peg to oa = ene +. A+—2B> + joao Ss Ha —
‘S 11.G s? 20 17 38 23 5 26 K hag +2 I eMS
F e 21 e7 24 ee 27 J e 30 rs a ~
I 4 1s 3 16 ne 18>
Bn5-Hop+Erpl—#an+8RE- oss ah ts
39 es 2 eX ets 48 ee
infinitum, or (if, for the fake of brevity, we a ‘the feveral at crane fractions
a Ne, Cee Rae @ Dey BES co D 53 E s# F 55
~~ awe — —_— — — — I —_— — os —— ee — ==
rel et tte otek Comte ee &c) to the feries oe € e at 3 et 57 es
G 5° H 57 15° K 59 Abe M st¥ N 5? 0 513 Pp st4 asis R tS
Ce Myelin ee “eo ere eu git ak ers gl es gt
& 577 ake -
<= — ++ &c ad infinitum, And, by the refidual theorem in the cafe of
id 53
Bey fe =; i eatal to the Gia
3
8 s4 II ss 14 Cd 17 20 $8 28 wit 26
12 x0 15 o 18 é 21 7 24 e 27 ae, 39
st? 29 pil 32 gt 35 g33 38 git 4! gis 44 sl&
ero 33 eit 5ay 36 ez 39. Pry ——— 42 el4 45 ik ae 48 Qa
gif ie) st 4 * « i . Bis. oO 5%
a ecvtire S ier &c ad infinitum, or to the feries 1 — — — — —
é e~
ps3 E 54 F 55 G 56 H 57 ts K 59 Ls? M sit N si2 ost3 psi#
e. - en e an Pee, RE gk ee
Q sts wane s 517 To epee. .
—S oe a ae &c ad infinitum, which. confifts of. the very,
fame terms as the feries which. is equal. to »/ f ote -; but with the fign — pre=
fixed to every term after the firft term 1 inftead of every other term.
Now, if thefe two feriefes (which are equal to /? 1 + Land fr — 2) be
c ; : 2cs Est 6 g-
added together their fum will be the feries 2 — —— — ==> — 742 2°
é€ e* e& . ee
ae
2 Ls! 2N 5% 2 pst4 2Rnstt 28 Wey SF sa : . .
Sn ee ae a — =~ ad infinitum, in: which all the
terms, after the firft term, 2, are marked with the fign —,, or fubtra¢ted from
the faid firft term. Therefore /° ot — + 3 f — ~ will be equal to the
faid
FOR RESOLVING THE CUBICK EQUATION &c. 459
fa ferles 2 — 2cs 2E 54 2656 215° a SA 2 Ss ope 2 Pott 2R 56
a oa er Ta ae et art es par nr graw" wiz lz ce elt | ero
27st Spe ; $4 ;
a &c ad infinitum. Therefore \/3{e x Iv: Bs ib Ia vt sels
. : ° 2 cs 2554 265 215° x0
will be = 43 fe 9¢ the faid feries 2 — “= — why Be a Rees
e et Po ° er?
2Nns'% aps 24ais 27 st ; . ; .
ee es — ee ad infinitum = 2 / (ex the feries
Cs? E s+ Gs® 13? hee N 5? P 5%4 R st6 vst! Seer
Ree A ek pt gt Oa
finitum. Therefore the expreffion /3(e +5 + W#fe—-s (which is = ¥2f
x1? [: a = ie vei —-) will be equal to oVile x the feries I a
E 54 G 5° 15° L 5? N 5? p st4 R 5t6 T 58 : J
eg ag eg ee a Ca
Q. E. F.
25. It follows from the foregoing article, that the root y of the cubick equa-
tion y? — gy = 7 (which has been fhewn above to be equal to the expreffion
z
Vife+s + Y3[e—s) will be equal to 27? (e x the infinite feries 1 — =
E s+ Gs® 158 ie N st? p st4 R58 7 st8 pds Sais
Sa PE recency ai &c ad infinitum.
. . . 3 °
26. This feries will always be a converging one; becaufe # — Sy Or ss, is
‘Tr 5 d { 1 f ge SS 5* 56 58 g10
always lefs than —, or ee, and confequently the fractions >, 4, 3 > aes
gi? 4 gi8 g8
git? “eis? 16? G8?
ries), will form a decreafing progreffion, and therefore the terms themfelves,
which are produced by the multiplication of the faid fractions into the numeral
co-efhcients C, E, G, 1, L, N, P, R, T, &c (which likewife are a decreafing
progreffion), will alfo form a decreafing progreffion.
&c (which are the literal parts of the terms of the faid fe-
OS. GA ERT PETE ALE a
nae a — ne = — a + > — a + &c ad-infinitum, which is
equal to y° f + -; we hall find them to be as follows, to wit:
i 35
and B (= = => x =>
and C (== = = XT =ETXPVHp
and D (= 2 C= > x -) = ae
ies tied and
460 A METHOD OF EXTENDING CARDAN’S RULE\
; 8 8 2 wt O18
Decrees ry x §) =a
on, EE a ER SQ EDs SS a ay eee eee oo a
and P(e Ee - X 343 3x5 243 Fx ogee
ag A Wag Stes be 7 lod a
and G es gS g * ag > Mya? = Bree
2 7 17 $y en TX 22) S17 R52 See 374
and H (= 21 Ga 21 x 6561 3x x 6c6r 7 seb sth a 19,683?
+ 225 +20 20 3 Tn nS Bide: te oat 187 X2 _. 5X 187
and 1 (= 24 24 “* 19,683 — 6 as 19,683 — 3x2 19,683 — 519,683)
2835
592049”
523 7 ne agg POS RN oe ase
and 27 27 59,089 145945323"
26 26 21,505 13 21,505 13 § X 4301
=2K ae ~ — H Se xO SK HS
and L ( 30 30 * 1,594,323 15 15594323 3X5 s 13594323
13X 4301) __ _ $5,913
3 X 1,594,323 4,782,969"
42 26 = 29. de $5,913 — 147,407
CG cera By 15782,969) 145348,907”
__ 32 — 32 147,407. _. 8 147,407 151 79,256_
and N (= 36 M= 36 ~~ 14,348,907 149348,907) 129,140,163”
Pasay gat as _15179)256 35 13% 90,712 35 007s
and O (= 39 N= 39 129,140,163 ~ 3X13 129,140,163 — Shea iia
397749920 |
™ 387,420,489” :
85 — 38. ., 35174920 __ 19 39174920 19 39174,920
and P (= 4 42 3875420,489 21 387,420,489 3X7 3875420,489
JAG 7% 4539560 __ _19X 453,560) _ __ 8,617,640 _
Tag 387,420,489 3X 387,420,489" ~ 1,162,261,467?
set PIAL 8,617,540 _ a 8,617,640. _ | ar
and Q (= near ee 45° 3;162,261,407 —~ $x9 1,162,261,467 — 5x9 *
5X1,723,528 _ 41 X 1,723,528 )= 40, 664,648 _
1,162,261,467 ~ 9X 1,162,261,4677 10, 460,35 3)203°
7 L
pup eee ee! 70,644,648 11 70,644,648 ort
and R(=3Q= 10,460,353,203 2 10,460,353,203 3% 4 x
4% 17,666,162 __ 11 X 17,666,162 Me 19453 27;782
10,460,353,203 3X 10,460,362,203’ ~ 31,381,059,609°
2 Rarer 104,327,782, 47 _ 17% 11,431,046
and § (= 51 31,381,059,609 3X17 31,381,059,609
47 X 11,431,046 ae 5372259162
3 X 31,381,059,609 945143,178,827"
° 50 5372259162, __ 35 §375259,162
and T (= &S = ei SE stata
¢ 54 945143,178,827 27 949143178,82)>
13943 19479:050
2) 541,805 ,928,329 - r
Note,
FOR RESOLVING THE CUBICK EQUATION &c. 461
Note, thefe values of the co-efficients B, C, D, E, F,G, H, I, K, L, M, N,
O, P,Q, R,S, and T, are-expreffed in the fmalleit poffible numbers.
28. It follows from art, 25 and'27 that the root y of the equation y? — gy =r
° rn . . . 2 4
is equal to the expreflion 2 4/?(¢ x the infinite feries 1 — — — —-
ge 243.6%
1545° 935 5° §50913.57° 1,179,256 s*? 8,617,640 5%4
6561e° 59,049 4,782,959 e?° 129,140, 163 e 1, 162,261,467 e%4
194,327,782 576 13,431,479,00 538 te as
ELE ou Di nk dat i Mia Se a — &c ad infinitum.
31,381,059,009 e 25541;865,825,3 29 e%
: ‘ css E s4 G 56
29. It has been obferved in art. 26 that the feries 1 — $e — — =
é é
tg us N 5%? p st4 R 516 rst t , eee ,
as a eae &c ad. infinitum will always con-
verge; from whence it follows that the expreffion 2 /? fe x the faid feries 1 —
gal 5S E 54 6 5° 15° 1s s?9 Nn s!? p st4 Fs feat Ch Ree
= wr tyes arta amr er pre Ela gece glean OEE: ——s sas | =a — &c a infinitum
will always truly exhibit the value of the root y of the cubick equation y*? — qy
; 3
=r in the aforefaid firft cafe of it, or when - is greater than a . And, when ss
4 on ’
is confiderably lefs than ee, or m — - is confiderably lefs than re or mi is
3 . . .
very little greater than = , the convergency of the terms of this feries will be fuf-
ficient to make it ufeful. But in other cafes, or when = is much greater than
3 7r . ° 3
“- (as, for example, when — is triple, or quadruple, or quintuple, of i, or of
fome ftill greater magnitude), the terms of this feries will decreafe fo flowly as
to render it very unfit for practice. And, indeed, in the moft favourable cafes,
this expreffion of the value of the root y of the equation y? —¢y =r will be
lefs convenient in practice than the expreflion f'(e+s5 + Y?(e—s, from
which it is derived. But, though its merit in a practical view be but fmall, yet,
as it is the foundation of the method, which is here intended to be explained, of
extending Cardan’s rule to the fecond cafe of the equation y3 — gy = 7, | thall
now proceed to illuftrate the truth of it by applying it to the refolution ofa fingle
numeral equation of the foregoing form y? — gy = r, in the firft cafe of the faid
: rk. :
equation, or when - is greater than 5? in which I have taken care to choofe
fuch numbers for g and 7 as fhall make = be but little greater than f, and
confequently fhall give us only a {mall quantity for the value of the fraction
=, by the continual multiplication of which the terms of the above feries are
generated.
Pa 73
462 A METHOD OF EXTENDING CARDAN’S RULE
An example of the refolution of a cubick equation of the :
foregoing form y? —qy = 1, in the firft cafe of it, (or
. . 3
when r is greater than ud AOE "is greater than 7 )
by means of the exprefion 2 /3(2 x the infinite
A Css E 54 c 56 158 157° N st
SEES A pp Sg age Fog gta
14 16 18
ae &c ad infinitum, or 2 f 3(e
ear eA ; I SS 10 tag
Xx the infinite Series Oh praia Sry okt ey
154. & 935 s _553913 eer
6561 ee 59049 eé 4,782,969 ero
1,179,256 git 8,617,640 58
129,140,163 * ee 1,162,261,467 ee Oe
1943275782 | iad 13,431,479,050 se
31,381,059,609 eX -34541,865,828,329 en
— Sc ad infinitum,
San
30. Let it be required to refolve the equation y? — 3009 = 2108 by means
of this expreffion.
Here g is = 300, and 7 is = 2108. Therefore - is. == to0,8ane VA is
he s ENE Bef pl =
(== v{t = yY 100) = 10, and confequently iyaes (= 100 X10) sium
and uN is (= 2 X 1000) = 2000, which is lefs than 2108, or. Therefore
a Ja A 4
this equation comes under the firft cafe of the general equation y* — gy =7,
and confequently may be refolved either by one of the three expreffions ob-
tained by Cardan’s fecond rule above explained, or by the foregoing tran{cen-
ff > fe x the feri Cf ERS Oe eee
dental expreffion 2 4/? |e X the feries 1 — —) >= => >= Ht Ge eee
12 r ; : : ,
a+ — &c ad infinitum, which was derived from the third, or laft, of them.
ei
; f rr
31. Now, fince r is = 2108, we fhall have —, or ¢, = 1054, and reps:
ee, (= 1054]*) = 1,110,916, And, fince gis = 300, we fhall have < om
3 :
100, and ©, or the cube of > (= 100]*) = 1000,000, and confequently ss,
8)
3
or — —£ (= 1,110,916 — 1000,000) = 110,916. Therefore the frac-
4 27 ?
: ss ; eye TO,OLO wu !
tion — will be Gz a aye,GIE = 0.099,341,932,23
and as will be (= 0.099,8413932,2)) = 0.009,968,411,4 5
5* $
and + will be (= = et 0.009,968,411,4 X 0.099,841,932,2) a
e* ee
0,000,99 5320554 3
and
a
- —-
FOR RESOLVING THE CUBICK EQUATION &c. 463
GEO 56 2
and =; will be (= = x oa = 0.000;995,265,4 X 0.099,841,932,2) =
* ; 0.000,099,369,2
and = will be (= | x = = 0.000,099,369,2 X 0.099,841,932,2) =
0.000,009,92F,2 3
gi? a! 10 2
and = will be (= a x = = 0,000,009,921,2 X 0.099,841,932,2) =
0.000,000,990,5 5.
s* 7
and = — — will be (= ar X J = 0.000,000,990,5 X 0.099,841,932,2) =
: 0.000,000,098,8 ;
L : 14 2
and — will Deve a x > = 0.000,000,098,8 X 0.099,841,932,2) =
0.000,000,009,8 ;
16 z :
and = will be; (<= a x => = 0.000,000,009,8 X 0.099,841,932,2) =
©.000,000,000;0 ;
And confequently -~ will be (= C X 0.099,841,932,2 = = X 0.099,841,
932,2 = o-299 019822) = 0.081,0933,548,0 5
and = will be (= E XX. 0.009,968,411,4 = << X 0.009,968,411,4 =
LO X 0,009,968, 411,4. __ pom Panete, hal De ime zeph rir
4 243 ay 223 itp aan ? >
and = will be (= G X 0.000,995,265,4 = a X 0.000,995,265,4 =
154 X 0.000,99 5126554 — 2:1532270,871,6 (= 0.000,023,360,9 ;
RS Sy ar a ae Caos Aa a . 9 BS Ye a7 2
6561 6561
8 .
and <; will be (= I X 0.000,099,369,2 = ee X 0.000,099,369,2 =
>
935 X 0.000,099,369,2 __ 0.092,910:202,0, __
222 a ——)) = 0.000, 001 ‘.
593049 593049 ; Recipe aah?
Se 55292 HSS 7 teil
an (= L X'0.000,009,921,2 = 47825989 X 0.000,009,921,2 =
55913 X 0.000,009,921,2 0-554,724;055,0
22 ae) 0.000,000,11 5,93:
4,782,909 4,782,969 ) 12 toe »
am 19179925
and ~ 5 ~ will be (= N X 0.000,000,990,5 = Mae Oe X 0.000,0003;990, 5
__ 14179,256 X 0.000,000,990,5 __ 1.158,053,068,0, __ . : .
eRe aT coe PEETRNET HS ) = 0:000;,000,009,0 5.
en a . oie $619,640
and Ara will be (= P X 0.000,000,098,8 = WRC cn X 0.000,009,098,8
__ 8,617,640 X 0.000,000,098,8 __ 0.851,422,832,0. ___ oy
= 1,162,261,467 7 2,162,261,467 ) == 0.000;000,00057 3
— _1949327,782 |
and == e will be (= R X 0,000,000,009,8 = ST x Cet
a 194,327,782 X 0.000,000,009,8 __ 1.904,412,263,6 ) = 0.000,000,000,0.
| 31)381,059;609- “~ 315381,059,609
Therefore
4.64 A METHOD OF EXTENDING CARDAN’S RULE :
cz? Es G 5° as? L st? N st? pist4 R515 eh,
Therefore praia itereiuD ory vinelp: ta us sda Were och, ci lta x &c will be
= 0.011,093,548,0 + 0.000,410,222,6 + 0.000,023,360,9 + 0.000,001,
57354 -- 0.000,000;115,9 -- 0.000,000,009,;0 + 0.000,000,000,7 = 0.000;
= 3 : : c s*
000,000,0 = 0.011,528,830,5 + &c; and confequently the feries 1 — =
# G56 158 x s*S N 57 Pp st4 R.st6 & itt be =
eh GAD RTT gia Son Re Ne
000,0 — 0.011,528,830,5 &c = 0.988,471,169,5 — &c.
Further, fince ¢ is == 1054, we fhall have 3 fe (= (1054) = 10.176,
853,833,7, and confequently 2 3 bhi 2 X 10.176,853,833,7) = 20.353,
ix.s4
Css Est G 56
707,667,4. Therefore'2 3 (e X the feries 1 oe —, aie
laa NW 512 Pp si4 R. s#6 ai ;
> al gerree iat sy mo — &c ad infinitum will be = 20.353,707,667,4
0.988,471,169,5 = 20.119,053,221,6. Therefore the root of the propofed
equation y3 — 300y = 2108 is = 20.119,053,221,6. Q. Bets
32. This value of y is true to nine places of figures. For its true value is
fomewhat greater than 20.119,053,2, as will appear by fubftituting 20.119,
©53,2 inftead of y in the compound quantity y? — 300y. For, if we fuppofe y
to be = 20.119,053,2, we fhall have y* = 404.776,301,664,430,24, and y3
= 8143.715,9475285,920,546,248,768, and gooy(= 300 X 20.119,053,2)
= 6035.715,960,0, and confequently y* — 300y (= 8143.715,947,285,920,
546,248,768 — 6035.715,960,0) = 2107.999,987,285,920,546,248,768,
which is fomewhat lefs than the abfolute term, 2108, of the propofed equation
y? — 3007 = 21083; and confequently 20.119,053,2 muft be fomewhat lefs
than the true value of y in the faid equation. And, if we were to profecute the
value of y fomewhat further by means of Mr. Raphfon’s method of approxima-
tion, by fuppofing y to be = 20.119,053,2 + 2, and fubftituting this quantity
inftead of y in the equation y3 — 300y = 2108, and refolving the new equa-
tion that would refult from fuch fubftitution, as if it was a-fimple equation, (to
wit, by omitting all the terms that involve either the fquare, or cube, of x) we
fhould find z to be equal to the fraétion 2°22 2070-4 53°75 023" or to 0.000, _
914+328,904,9934290,72
000,013. And confequently the firft eleven figures of 20.119,053,2 + 2, or
of the true value of y in the propofed equation y? — 300 y = 2108, would be
20.119;053,2 + 0.000,000,013, OF 20.119,053,213-
33. It appears therefore from the foregoing example, that this expreffion
o/?(e »« the infinite feries 1 — Etro B Shon ig out < ss = —*S
—— ae — “eS — &c does truly exhibit the root y of the cubick equation
“—¢y = rin that cafe of it which falls under Cardan’s fecond rule above ex-
plained, or in which 7 is greater than ive, or a is greater than c.
6 Of
FOR RESOLVING THE CUBICK EQUATION &c. 465
Of the fecond cafe of the cubick equation y>—qy =r, in
which the abfolute term r is lefs than “iv, or “ is
%
9g
lefs than mr
34. We muft now proceed to confider the other cafe of the cubick equation
° : “s Bae 3 ;
y? — gy =7, In which z is lefs than 209 or — is lefs than < . This cafe (as
. 7
we have already feen) cannot be refolved by the aforefaid rule of Cardan, be-
caufe it is impoffible in this cafe to divide the line, or root, y into two fuch parts
vand z, thatthe product, or rectangle, under the faid parts, to wit, the product
v2, thall be equal to%, and confequently that 3 vz fhall be = g, and 3vz.X
v + 2 fhall be = ¢ X v + 2, which is a fundamental ftep in the folution of
Problem 1 given above in-art-5. Andon this account this cafe of the equation
¥? —47y =r has obtained amongtt Algebraifts the name of the irreducible ca/e;
and particularly it is often fo denominated by the French writers of Algebra.
Monfieur Montucla in his Hiffoire des Mathématiques, Tom. 1. page 482, {peaks
of it in thefe words: On doit @ Cardan la remarque de la limitation d'un cas des
équations cubiques, ou il arrive que Vextraftion de la racine quarrée qui entre dans la
formule, weft pas poffible. Cet ce que nous appellons \e cas irréductible ; dout la
difficulté a donné et donne encore la torture aux Analyftes. lt may, however, be re-
folved by means of a certain tranfcendental expreffion (or expreffion containing
an infinite feries of terms) which bears a great refemblance to the foregoing
tranfcendental expreffion which we have fhewn to be equal to the value of y in
the firft cafe of the equation y3 — gy =r, to wit, the expreffion 2 »/: {e x the
=. glans . os E s# G 5° ¥s' Loe ie igs p st4 yi a
infinite feries -1,— — — '—- — —— — — = = SS SS eS
é
— &c, and which was derived from the finite expreffion 4/3 ets + V73fe—s
_ by the help of Sir Ifaac Newton’s binomial and refidual theorems. To affign fuch
a tranfcendental expreffion, and to demontftrate that it will be equal to the root v
of the cubick equation y3 — gy =r in the fecond cafe of it, or when r is lefs
29/9
than
at 3 : Wak eat ;
, or — is lefs than £, with a certain limitation which we fhall men-
tion prefently, is the chief object of the remaining part of this difcourfe.
35. In order to preferve the two cafes of the cubick equation y* —qy = r
29/9
: 34/3”
and in the fecond of which it is fuppofed to be lefs than the faid quantity),
diftinét from each other, it will be convenient to denote the root of it in thefe_
two cafes, and likewife the abfolute term of the equation, by different letters.
I fhall therefore henceforward denote the root of this equation in the firft cafe
/ .
ord. ur by the letter y, and in-
3/3
(in the firft of which the abfolute term 7 is fuppofed to be greater than
of it (or when the abfolute term is greater than
Vot. Il. 3 O the
466 A METHOD OF EXTENDING CARDAN’S RULE
the fecond cafe by the letter x‘, and fhall denote the abfolute term of the hi
tion in the firft cafe by the letter 7 (as in the. foregoing articles) and in the
fecond cafe by the letter ¢; fo that the two cafes of the equation y? — eA pean i
will now be exprefied by the two feparate equations y3 — gy = rand «3 — gx
= f, in the former of which 7 1s greater than = he and in the latter of which
tis lefs than nt, the letter g being fuppofed to denote the fame quantity. in
qs
ay
(as before) by the detters ss, but fhall put the letter g for =, and the letters zz
both equations. And I fhall denote < (as before) by the letter e, and as ase
3 tt
for % Sg? eh = — gg. With this notation the propofition which I fhall now
endeavour to ddmonttrates will be as follows.
PROP. 3+) [A TH E.OR E M,.
36. If in the foregoing tranfcendental expreffion, to wit, 2 4/3 (e x the infi-
f C55 E54 6 56 15° arse Ns? pst R 576 ce
TE eS Pesci aBh of ncthcatn Geen eke ME)
&c ad infinitum (which has been fhewn to be equal to the root » of the cubick
equation y? — gy = 7), we make the following changes, to wit, firft, infertthe
letter g every where inftead of the letter e, and, fecondly, infert the letter x
every where inftead of the letter 5, and, gdly, change the fign —. into the fign
-- in the fecond, and fourth, and Fath: and eighth, and every following even
term of the infinite feries contained in the {aid expreffion, the new tranfcenden-
tal expreffion that will be thereby obtained, to wit, the expreffion’2 /3{9 «
C2z% E24 G2? 128 Lato N 2i2 Pp xt4 eee
the infinite feries 1 +} — — — +4 — Sj + -3 oOo ore toa roe
SS & & & BS s & §
¥8 f 4 Ae . 1
+ — = &c ad infinitum will be equal to the root « of the cubick equation «3
— gx = +t; provided that the abfolute term ¢ of this equation (though it be lefs:
than a4) be greater than 472 X 174, or that oa or gg, (though it be lefs
34/3
g
than £) be greater than — — xX 52 or than 2 aE
Cz Ext
37. This limitation is neceflary to the end that the feries 1 + <7 aca
G x6 12° Est N xt P zit R2t t aire
ree eae ett et — &c (which forms a part of
the expreffion of the value of the root ) may be a converging feries. For if ¢ is
. 3 . 3 «
lefs than 2 X 901, or # is lefs than =e Ene RA quantity = we
33 4 on 4
q
3
hei oteatersthanst-cak Sek OF an pein tet ecaree anal and, 2 ane reater
e 27-54 7 54 a Sorters % ;
54
than
FOR RESOLVING THE CUBICK EQUATION &c. 467
: #3 335 “1
than ri that is, zz will be greater than gg; and confequently the terms of the
4 ; 4 G x6 I pau L xto N giz Pp alt R iS T 18
Na ey a hy ——- — — + —
4 6 8 gl I2 14+ 16 18
. . +. . 5 -
&c will diverge, inftead of converging, and the faid expreffion 2 1/3
8 e 8 6 8 oP Iz &
: : C2 E G2 1% Ds N& Pp zit
x the faid feries 1 -+ — — = + — — — + ert
ES “ gg ie 2° 2° £9 Ys pee
R& T . }
aes —- — &c, will confequently become ufelefs. But, when ¢ is lefs
oS
2 ff. 3
than =i, and yet greater than 4/2 et, Oior—-Is lefs than a and yet
3 . . .
greater than ©, the terms of the aforefaid feries will converge, or decreafe, and
the aforefaid tranfcendental expreffion will be equal to the root x of the equation
ah anne qx — 7. ,
38. Now, in order-to-demonftrate this propofition, it will be neceffary to make
fome further obfervations on the former tranfcendental expreffion 2 4/7? (2 x
’ ; : C55 E54 Gs® 1s? Ls NW st ne Rs
the infinite feries,1.— —— a ee Hi Si By He NR eI ca! Gla Sula
(4
y 518
i &c ad infinitum, which has been fhewn to be equal to the root of the
equationy? —4¢.y =r.
Odfervations on. the-exprefiion 2 »/3 (e x the infinite feries
Css Es* Gs® rs* rise N st re
Se Par te ee eee
ad infinitum, «which is equal to the root of the equa-
tiny? —qy=r.
e? e ert
6 8
“uy 2 Sia Ty ec. % ee Be ae ie a a cc
6 5
ont e x 4 + as Bi oe 4) &e
ton LA _ 24 a a + Later
2,10 if Iz
Pus a is a sche bey
Vt
er 48 = aay
we —— — &c;
which, for the ke of brevity, we will denote by the capital letter A. Then
will y? be = tothe compound feries A.
42. We mult now multiply the expreffion 2+/?{e x the infinite feries 1 —
C35 E 5# G 5° re Es? N 512 he one
ae — & ad infinitum (which is equal to the
ce e* e
root y of the cubick equation y? — gy = 7) into the co-efficient 4.
, 3 3 3
Now, fince ss is = = & a? we fhall have ss + = ——. and = =
= ¢¢—55, an ly g™he= —s = —
Saves fee , and confequently ¢ 27 X €@— 55 27: 4X Utena 7
2
and 9.337% Vilee x {1 Sars 6.3, Kit 2)-5 = (by the refdual
. rom . : : B Ss crm
theorem in the cafe of roots) 3 x e 3 & the infinite feries 1 — <> see
ps9 ES? F37° Gs? H 514 Fi cre 2 nhs
en ae — SAM as So giee ny aaih eee O a at — &c ad infinitum. Therefore
: ‘ ; : Css E s4 G 5° 15°
q x the expreffion 2 /*e x the infinite feries 1 — — — => —~= — 4 —
ee e*. é é
10 2 2
Ls ns? . mie : . . BSS
——. — —— — &c will be equal to 3 x e 3 X the infinite feries 1 — —: —
e é ee
c'st ps° E s§ ere G st? H 514 ae K 518 er
er ast Se ay a er a as ee ad gina x the
Css E s4 G 56 158 Ly?
expreffion 2 /3(e x the infinite feries 1 — — — —- — —
ee e# e és ee
ws?
FOR RESOLVING THE CUBICK EQUATION &c. 471
N st? ‘ : . : . 4 6 8
—— — &c ad infinitum = 6e X the infinite feries 1 —2* — =~ Pe ES
e ee e4 e e®
Fst? Gst2 F 3 y 4 6 8
= sr — &c ad infnitum x the feries1 ~~ SL EL
e e ee e* ra e
E gto N gi2 ‘ * ° *
se a &c ad infinitum. We mutt therefore multiply thefe two feriefes
together; which may be done as follows.
The multiplication of the infinite feries 1 — — — — — .
e J
Dns E 5° F 570 G st A : ie ;
Bete pe Sa EFc into the infinite feries
Css E s+ G 56 z,3° L st? N 512
ce € e€ e € eu
BSS c 54 D 5® ESs® F 510 G 512 &
— ——. =< — — — ere lhe
I ce et e® e er? 12 Cc
Css E st G 5° 15° L st? N 5% &
= aa ra et e® rw heptane 7 ero oe et2 a c
BSS c s* p s° Es® F st G st
ee e* e e er° el2
Css BC 54 ¢75° cp s8 CE s?° CF si?
ee e* e® ee 7% er? ei ~f- &c
E 54 ae BE s® cE s® DEs?° E252 &
oe. 4 2° 28 ero el2 + Cc
6 8 10 12
Gs BGS cGs DGs
@ e ere = ei2 a7 &c
8 10 Iz
Is BIS CIs
ae é ero = elz “Fe &c
L.ste ie
1o 12 oe Gre
e€ é
N gi2
+ &c.
Therefore the product of the multiplication of g into the expreffion 2 4/?(e x
. Css E 54 cue
value of « in the original equation 1 +
TTH EY BIN OM TO ACL” THE Of RTE. Me 339
3333333 &e X 2 = 7.236,736,126,688; by the refolution of which we had
% = 0.087,801, and confequently ~, or 97 + 2%, —= 97.087,801, of which
number the five firft figures, 97.087, are exact. Therefore ore was a
wv
— 10.
feventh approximation to the value of v in the equation 1 + an
167. We then found the arithmetical mean between the two laft values of z,
to wit, 0,086,840, and 0.087,801, which was 0.087,320; and we obferved
that this mean muft be lefs than the truth, becaufe the fecond value of z, to wit,
0.087,801, muft be much nearer to its true value than the firft value of it, to
Wit, 0.086,840. And therefore we conjectured that the true value of z (being
greater than 0.087,320, and, probably not much lefs than 0.087,801) might be
ey nearly equal to 0.087,7, and confequently that the true value of #5 OD
97 + 2, might be very nearly equal to 97.087,7. And thus we obtained
97.0877 for an eighth approximation to the true value of « in the equation
2
1 + ay = 7,
1co0o
168. We then dropped all further confideration of the equation z x
Z— VS 2%— aaa 5 “24 | fa See x 24 i Rian
os x tl + 1000 — a
O. 2S ae ae &c, and ede a me a the exactnefs of the laft
value of « obtained by the foregoing procefles, to wit, 97.0877, by raifing the
binomial quantity 1 ++ i. to the power of which 97.0877 is the index; which
was done by the help of Sir Ifaac Newton’s binomial theorem. And we found
that the faid power of 1 + a WAS = 9.999,979,282,720,950,507,346, which
is very little lefs than 10; and we thence concluded that 97.0877 muft be a
° . > Zz e
very little lefs than the true value of w in the equation 1 + zal ZT
169. We then fuppofed x to be = 97.0877 + y, and confequently
po Zep 087747 to be = 10.
1000
Mon Mad (ayo Tory we 24 seh oe 77 a,
Then, fince 1 +- —— ore is eecat 1 4 — x1 += we had
24 |97-0877 4\y 24 |97.0877
tick 24 XI+ we = 10, and confequently (becaufe 1 ++ zt
has been found to be = 9.999,9793282,720,950,5075340) 9-999,979282,7203
10.009,000,000,000,000,000,000,
6X1 24 )7 = 10, and 1 aay = acai! Le ine
95935073340 XI + = eden 9-9992979s2825720;950,5075340
== 1.000,002,071,73 oft 97,014, &c.
170. Having thus obtained a third equation 1 + =
732,197,014, &c, in which y is much fmaller than z in the former equation,
FAs we
= 1,000,002,071,
340 A MDIIS £O VW RR SIE CONCERN ING
“
we proceeded to expand 1 + - ie into an infinite feries by means of Sir Ifaac
Newton’s binomial theorem, ms pres obtained the equation 1 y x —&
~ 1000
Uma 24 ] ay—3yy +8
2 “ 1000 x 6 “ 7000
— Pewee iol
24 1000
4 24y — Sopy + 35 y3— 1Oy4-f yS y, 24 \5 OT 5 24
120 1000 1000
+ &c = 1.000,002,071,732,197,014, &c, and (by fubtracting t from both
y— 3
fides) the equation y x 24 - ie a Sighs Je = ae x aa
10090 ‘ 1000
Be g —11yy + 6y3 — y4 24 24y — Soy + 3y3— toys $99 a ot
24 1000 120 1000
’
120y—274yy +235 y3 —S85y*t+ 16 y5 —y? 24 aks Pa ipod 2 : as:
720 1000} =: - — 0.000,002,07 73 5 97>
014, &c.
171.° We then proceeded to approximate to the value of y in this equation,
by firft fuppofing the firft term, y x —, alone, of the feries y x
1000
Pp x a4 +- &c (which forms the left-hand fide of this equation), to be
equal to the whole feries, and confequently to the abfolute term 0.000,
002,071,732,197,014, &c; and, fecondly, by fuppofing the two firft terms,
yx = — us x —+ i of the faid feries to be equal to the fame quantity,
and refolving the equations refulting from thefe fuppofitions.
From the firft of thefe fuppofitions we had the fimple equationy x 24 =
0.002,071,732,197,014, &c; by the refolution of which we had y = 0.000,
086,322,1, and confequently « (= 97.0877 +- y) = 97.087,786,322,15 of
which number the firft feven figures 97.087,78 are exact, the more accurate
value of « being (as we have before obferved) 97. 087578753 53,850,001 ,437.
Therefore the number 97.087, clea is a ninth approximation to the
true value of « in the equation 1 + al" = =<60,
1000
And from the fecond of thefe fuppofitions there refulted the quadratick equa-
tion yy + 82.333,333, &c x y = 0.007,193,514,5723; by the refolution of
which we had y = 0.000,087,371, and confequently ¥ (= 97.0877 +7) =
97:087,787,371; of which number the firft nine figures, 97.087,787,3, are
exact.
This number 97.087,787,371, is therefore a tenth approximation to the true
value of x in the equation 1 4- 24)" wall (a
1000
172. We then, in order to obtain the value of y toa ftill greater degree of
; ° . : 2
exactnefs, had recourfe to a different method of refolving the equation y x =
| 4 : Sis
’ : 2
in
FU-Ree BIN OoMIMA. LL > EEO R EM. RAL
2 < 3 3 6y— idigl a. o'
—p2 Batt or 27 ay) poate: = TUyytby3 —y* 24\4 4 eS
1000 1000 24 1000
©.000,002,071,732,197; ue &c; which was grounded on the omiffion of all
the members in each term of the feries that involved either yy, or y*, or y*, or
any other power of y, except the fimple power, or y itfelf: by which means the
; ; . : : ; 24
{aid equation was ee into the peeled ‘sete ee tO. Wiksty ix Toa
y ey 24 |* 24 \3 524\t Ph lis «sph E20 24 |6
2 1000 1000 1060 I 120 1000 7 20 1000
24 y 4 |? y
+ &c = 0.000,002,071,732,197,014, &c, ory X ——~ — — xX at :
ay Spe heyg EE yt JH} myatishe eee me |
* 1000 4 x 1000 a 5 x 1000 6 X 1000 + &c — 0.000,002,07T,
24 I 24 |? I
732,197,014, &c, a uke: ame 2 4 A x
24 l I 24 ork 24 \5 I 24,\° =
1oool 4 os rool | 5 "s a EtrGwr «Too mn &c = 0.000,002,071,
; 24 . ;
732,197,014, &c, or (if we put A for the firft term 1 X — of this feries, and
B for its fecond term -- x A and C, D, E, F, &c for its third, fourth,
fifth, fixth, and other following terms, refpectively), y x the feries 1 x +.
e090
erase libaticp ye tt alg Ye al ooh} 24 Sim sg
2 1000 3 1000 4 : 1000 5 1000 6
ey = Lily ane wal i 2 SARE OPEC SR Ore
b dolore) ot 7 Fx 1060 Fe G 1000 T 9 H x 1000 antes s 1000 + 1k
24 11 Beer
K X ooo a X Fog F KE = 9.000,002 ey ite &c, We
then computed the value of the faid infinite feries 1 os Sh spre oe 2 AX +.
ex — 2 C x ++Dx 3b Te By 2: + &c,. and
1000 4 1000 5 1900 s| 1000
Sand it to be = 0.023,716,527,561,3.16,042,2345 “wehich gave us the fimple
equation y X 0.023,716,527,561,316,042,234 = 0.000,002,071,732,197;014,
&c, by the ‘refolution of which we had y (==_ ——ORORO7 137 TOD OT
0.023,716,527,561, 3 10,04 042,234
0.000,087,353,943,011, and confequently x (= 97.0877 + y) = 97. eye
987,353,943,011 ; of which number the firft eleven figures, 97.087,787,353,
#
are exact, the more accurate value of « in the equation. 1 ++ 84) = 10 being
(as we have before obferved) 97.087,787,353:856,001,437-
This number 97.087,787,353,943,011, is therefore the eleventh cee ae
tion to the true value of x in the equation 1 + ay waht qf
173. And, laftly, to obtain the value of y to a ftill greater degree of exact-
uae we retained in art. 1 58 the-five firftiterms of the infinite feries fet down in
) arte
342 AV FDAT Sib BS 10+ WS ClLOON CME CREN “EAS
art. 157 and 170, that involved the fquare of y, to wit, the five terms x
24 |? 3y 24.\3 , 1 24 |* Ow ., 24 )% | 274y7 24)
24 Nia Pie +2 x 24 Barr gat + 2B x 28) whicls
ee 424°? 24.4% II 24 \* fe)
are equal to yy X the feries ey x <4 _ ~ Pe 4 Gs i x 24) _ — x
24 |$ 274 ty — : 6.728. 260.88
re + se x foo]. — X 0.000,281,236,758,360,883,200. And hence
we obtained the quadratick equation 0.023,716,527,561,316,042,234.X y +
0.000, 26 1,236,758,360,883,200 X yy == 0.000,002,071,732,197,014, &c;
which (being refolved by approximation by fubftituting, inftead of y, in the
quantity 0.000,281,236,758,360,883,200 X yy the value of y before obtained
by the refolution of the fimpie equation, to wit, 0.000,087,353,943,011) gave
us ¥Y = 0.000,087,353,852,524, and confequently « (= 97.0877 + ¥) he
087,787, 353,852,524; of which number the firft thirteen figures, 97.087,
7875353955 are exact.
This number 97.087,787,353:8525524 is therefore the twelfth approximation
24 \%
1000
to the true value of « in the equation 1 + =< STs
174: We then divided 1 by this laft, or twelfth, near value of *, to wit, 97.
087,787,353,852,524, in order to obtain the value of =, or the logarithm of
the ratio of 1 + + to 1 in Briggs’s fyftem; and we found the quotient to be
== 0.010,299,956,639,812. And hence it followed that the logarithm of 2, or
1024 loo, 1024 1000
. . I I
of the ratio of 2 to 1 (being = 7; log. —- = 5 log. = + x log. —
I~ I
I 1024 I RVATS 1024 fee 1.024 hd
10 ©" 1000 a Toh pe teil log. roc0, 1 03 = to OS: a, ee Sages 7
log. 1 + ~4 + 0.3 == X 0.010,299,956,639,812 + 0.3) would be =
. )
0.3 + 0.001,029,995,663,981,2, OF 0.301,029,995,663,981,2 ; which is exact
to 15 places of figures, the more accurate value of that logarithm (according
to Mr. Abraham Sharp’s computations) being 0.301,029,995,663,981,195,
213,738,894,724,493- |
End of the review of the feveral fieps of the foregoing refolution of the
equation 1 + = = 10, and computation of the logarithm of 2.
ST a RT RE a ee Le Ie Wee LIE NOL INCI LIENS
A SCHOLIUM,
175. The foregoing method of computing Briggs’s logarithm of 2 is cer-
tainly fomewhat laborious, but much lefs fo than the methods ufed for the fame
purpofe by Mr. Briggs himfelf, which required many very long stray of the
quare~
Xx wal 4B BEN GO M"h. AL T H'E OR EM. 343
fquare-root. For the difficulty of performing the operations that were neceflary
in thofe methods was fo great, that (according to what Mr. Euclid Speidall in-
forms us (fee above, page 73) he had been told) it was the work of eight perfons
for a whole year to compute the logarithm of 2 by thofe methods exaét to 15 places of
Jigures, or to the degree of exactnefs to which is been obtained in the foregoing
articles. This affertion of Mr. Speidall feems, I confefs, a little ftrange. Yet, as
he publifhed his traét on logarithms (which has been printed above in the for-
mer part of this volume) fo long ago as in the year 1688, it feems probable
that in his youth (perhaps, about the year 1660) he might have converfed with
fome old men who had been acquainted with Mr. Briggs himfelf, who publifhed
his Arithmetica Logarithmica in the year 1624, which was lefs than 40 years be-
fore that time; and this feems the more likely to have been the cafe, as his
father, Mr. John Speidall, was an eminent mathematician, and had very much
cultivated the, at that time, new invention of logarithms; which muft have
given both him and his fon an opportunity of hearing many remarkable parti-
culars relating to them.
We may further obferve that the foregoing method of computing logarithms
by the help of the binomial theorem, and Mr. Briggs’s methods of computing
them by repeated extractions of the {quare-root, are equally founded on the pure
and genuine principles of arithmetick, without any reference to the hyperbola,
or the Jogarithmick curve, or any other geometrical figure, and alfo without any
recourfe to the doétrine of infinitefimals, or of fluxions, or of the limits of ratios,
or in general, of the arithmetick of infinites in any of its modifications; which 1s,
in Dr. Halley’s opinion, the proper way of treating this fubjet, and the way in
which he boafts (though without being fufficiently authorized in his pretenfions),
that he himfelf has treated it in the foregoing difcourfe reprinted above in this
volume in pages 84, 85, 86, 87, 88, 90, and o1. ;
a ee ee
CLO UNC CeLaUlS IO UN
176. I have now completed the inveftigation of the famous binomial the-
orem in all the cafes of fractional powers; which was the propofed fubject of
this difcourfe. This, however, is but a part of that important and moft com-
prehenfive propofition. For it is found to be true likewife in the cafes of nega-
. ° : ° . ° — 7%
tive powers, both integral and fractional, that is,.in the cafes of 1 + | -,
mm I
See ge I es
Hi ite 7 « the knowledge of thefe
and 1 + sor of ye and But, as no g
cafes is not neceffary to the underftanding any of the foregoing methods of
computing logarithms, I fhall not on this occafion enter into any inquiries
concerning them. And therefore I here conclude what I meant to offer to the
reader’s confideration concerning the binomial theorem properly fo called. But,
as the theorem concerning the fractional powers of @ refidual quantity, fuch as
I=,
344. A DISTCOUVRSE Conecrrwimneey &c.
1 — x, is very nearly related to the foregoing theorem concerning the fraétional
powers of the binomial quantity 1 -+- x, infomuch that it is ufually confidered ~
as a branch of it;—and, as the faid refidual theorem, in the firft cafe of it, or
the cafe of the wth root of the refidual quantity 1 —.«, is made ufe of (as welk
as the binomial theorem) in the fAveftigations of fome of the foregoing methods -
of computing logarithms ;—I fhall now proceed to fhew how we may derive
from the theorems above demonftrated concerning the roots, and the powers of
the roots, of the binomial quantity 1 + %, the like theorems concerning the
roots, and the powers of the roots, of the refidual quantity 1—wx. But this
fhall be the fubjec&t of a feparate tract. And therefore I here conclude this
difcourfe concerning the roots, and the powers of the roots, of the binomial
‘quantity 1 + *.
End of the Difcourfe concerning the Binomial Theorem in the cafe of
Frattional Powers.
A DISCOURSE
Fi.
Lae a A
Dee's Cel OVW RISE
Cor N.C Eek Wear NG
Sm ISAAC NEWTON’s RESIDUAL THEOREM,
OR THEOREM FOR RAISING THE POWERS OF THE RESIDUAL
QUANTITY 1~—%, IN THE CASE OF FRACTIONAL POWERS, OR
POWERS OF WHICH THE INDEXES ARE FRACTIONS,
By FRANCIS MASERES, Esa. F.R.S.
CURSITOR BARON OF HIS MAJESTY’S COURT OF EXCHEQUER.
Art. 1. JN the foregoing difcourfe we inveftigated the feriefes that were equal
to the wth root of the binomial quantity 1 + and to the mth
power of its wth root. In the prefent difcourfe we are to inveftigate the feriefes
that are equal to the wth root of the refidual quantity 1 — x, and to the mth
power of its wth root. Now thefe feriefes may be inveitigated by the fame me-
thods which were employed in the foregoing difcourfe to inveftigate the feriefes
I m2
which are equal tor + x\# andi + xc) : but they may likewife be derived
from thofe former feriefes (obtained in the foregoing difcourfe), by a juft and
legitimate train of reafoning, with much lefs trouble than would be neceffary to
the difcovery of them by a new application of all the methods of inveftigation
ufed in the foregoing tract. And therefore I fhall, for brevity’s fake, have re-
courfe to this derivative method of obtaining them, rather than to the methods
employed in the foregoing difcourfe. And, firft, ‘I thall confider the 7th root
of the refidual quantity 1 — *, and endeavour to fhew that it is equal to an in-
finite feries confifting of the very fame terms as the feries which is equal to the
ath root of the binomial quantity 1 + #, but with the fign — prefixed to all
the terms after the firft term 1, inftead of only the third, and fifth, and feventh,
and other following odd terms of the feries, as in the feries which is equal to
ns
/(i +x, ort + xiz.
2. Now it has been fhewn in the E TEOe tract, that, if z be any whole
number whatfoever, the quantity 1 + a, or (/"[t + *, will be equal to the
Ver. Il. infinite
9
346 A DISCOURSE CONCERNING
- : 2 I = ; a ar
infinite feries 1 + Ax — P=! Bet - 2 Ce: ae
” 2n 3% 4%
a > Ex? — oe -~Fx° + &c ad infinitum. We will therefore now proceed
: I
to fhew that the quantity »/*{1 — x, or 1 — x\*, will be equal to the infinite
feries 1 — 2 Ax — P= Be Bay peri (ines 1 Dt — eS ys
n 2% 3% 4” 5a ;
_ “— Fx° — &c ad infinitum, which confifts of the very fame terms as the
I
~ former feries (which is equal to 1 + «\, or "(1 + «,) but with the fign —
prefixed to every term after the firft term, inftead of every otherterm. This
may be fhewn in the manner following.
3. Since the feries 1 + SUN ee [4 Be 4 2D Ce Bee
f a7 3% 4”
I
RE a Ext — &c is equal to1 + x\", or "ft +x, it follows that, if we
raife the faid feries to the wth power, or multiply it 7 — 1 times into itfelf, the
produét of thefe multiplications will be equal to 1 + «; that is, the product of
the feries 1 + Bx — Cux + Dx* — Et + Fx — &c ad infinitum multi-
plied » — 1 times into itfelf (upon the fuppofition that the co-efficient B is =
~Ax -- sChr, OF -, and that the co-eficient C is = — B, and that the
co-efficients D, E, F, &c, are equal, refpectively, to * = Gy tn — ots ets
E, &c) will be equal to 1 +; and confequently the co-efficient of ¥ will be
equal to 1, and the compound quantities which will be the co-efficients of x”, «3,
x*, x, and of all the following powers of x in the terms of the faid product,
will be, each of them, equal to 0, or will confift of feveral members, of which
fome will be marked with the fign +, and others with the fign —, and which will
be of fuch magnitudes that the fum of the terms, or members, which are marked
with the fign — will be equal to the fum of the terms, or members, which are
marked with the fign +. Some of thefe multiplications will be as follow.
I
1+ Bree Cx? + De? —& =1 + xe.
I
1+ Be— Cx? + Dx? — &c = 1+ x2,
I+ Be — Cer? + Dx? — &c
+ Bxe+ Bx? — BC? + &c
— Cr? — BCwr? + &e
+ Dr? + &c
1+ 2Be—2Cx* + 2D»; i 2
+ B*x? —2BCw3 + &c
1+ By
THE RESIDUAL THEOREM. 347
r+ Be— Cx + Dei — &c =1+ 4)".
1+2Byse— 2Cx* + 2D: &ec
+ B*x* — 2BCx: &c
+ Bxe+ 2B*x*? — 2BCx? + &c
+ Bix? — &c
— Cr? — 2BCxe? + &c
+ Dx? + &c
1+3Be— 3Cx* + 3D: &c ra
2 + 3B*x* — 6BC%3 cc =i + «}2.
+ B33 &c
I
1+ Be— Cx? + De? —& =1+ ux)".
1+ 3Be — 3Cx* + Dini &c
+ yb ams 30° DC.x3 &e
+ B?x3 &c
+ Be+ 3B*x* — 3BCx3 + &c
+ 3Bixt — &c
— Cx? — 3BCxi &c
+ Dx? + &c
1+4Bsy — 4Cx*? + 4D»: &c 4
+ 6B*x? — 12BCw; sc =1+x\.
+ 4B?x3 &c
r+ Be~ Cx* + De? —& =I+ xs.
r+4B.e — 4Cx*? + 4D; &c
. + 6B*x? — 12 BCx:; &c
t+ 5Be— 5Cx* + 5 Dx? &c $y:
+10B7*x? — 20 BC x? &$ =I + x42.
+ 10 B3x3 &c
4. By thefe multiplications it appears that the fquare of the feries 1 + Bx —
Cx? + Dx? — Ext + Fx — &c (carried only to the third power of «) is
equal to the compound feries
I1+2Be—2Cxr? +2Dx? &ec
+ B*x? —2BCx: &c;
and that its cube is equal to the compound feries
1+ 3B,e—3Cx* +3Dx3 — &c
+ 3B?*x*—6BCx? &c
+ Bx: &c;
ga ee” and
348 As iD AD BC OW R) Seen CoN CoB R Nowe
and that its fourth power is equal to the compound feries
1+4Bxe —4Cxr? + 4Dx &c
+ 6 B’x* — 12 BC x3 &c
+ 4Bix>? &c;
and that its fifth power is equal to the compound feries
1+ 5Bxe = 5 Ce? + 5 Dx? adc
+ 10 B*x* — 20BC x? &c
+ 10B3x3 &ec.
s. Now, when is = 2, the firft of thefe compound feriefes (which is equal
to the {quare of the feries 1 + Bx — Cw? + Dx? — &c), to wit, the feries
1+2Be-—-2Cx* +2Dx> &c
, B*x? —2BCx? &c,
mutt be equal to 1 + x, and confequently 2 B (the co-efficient of x) muft be
= 1, and B* —2C (the compound co-efficient of **) muft be = 0, and
2D — 2 BC (the compound co-efficient of x?) muft alfo be = 0, and every
following co-efficient of one of the powers of « in the faid feries muft, in like
manner, be = o, if the co-efficients of the terms of the feries 1 + Bx — Cx?
+ Dx? — &c, or 1 + = Ax — = Baz + =" Cy3 — &c, have been.
rightly affigned. For otherwife the faid compound feries cannot be equal to
1 + #, as it-ought to be upon the prefent fuppofition that is = 2, becaufe
upon this fuppofition the feries r + ~ Ax — = Bx? + eee Cae eae Bec
H | I
(which is univerfally equal tor + x\#, or /{t + «), will be = 31 + “2,
or, 47 1 + «, or the fquare-root of 1 +- *, and:confequently the {quaiqiof the
faid feries muft be equal to 1 4 x.
And accordingly we fhall-find that, if 2 be fuppofed to be = 2, the co-effi-
cient 2 B, of the fecond term B.x, -of this compound feries, will-be = 1, and
B? — 2Cand 2D — 2 BC (the compound-co-efficients of the two following
powers of x, to wit, ** andw? in the faid feries) will each of them be equal to
o.*n Mor, Bhs aces we fhall have B(= —X A=+x AS xi}
tT —I I _
go and C (ar Be KX EE eS) gq and D(=> SSC
enh Reiter ET Sh, eee 2— 3 pale ca’ J): te Ey
uently 2B (= 2 X ~) =1,and B? = 2C(=2— a et ee
ee ( x =) ; ae z acta rea 3
= — coet a4 Et Rois fi.
ZT 7) = % and 2D 2BC(=2x = Jah lead, Os — <4)
jaro
6. In like*manner, when # is = 3, the fecond of thefe compound {eriefes
(which is equal to the cube of the feries 1 - Bw = Cx* + Dw? — &c), to
wit, the compound feries
1+ 3Bx
PE WRAESSSLED IU Mod o oR ot fobRn eM, 349
1+ 3Bry~—3Cx* +3D*? &e
+ 3 B**x* — 6 BCw &e
+ Bx? &c
muft be = to 1 + x, and confequently 3B muft be = 1, and 3B? — 3C
muft be = o, and 3 D — 6 BC + B* muft be = 0, or 3 C muft be = 3 B’,
and 6 BC mutt be = B? + 3D.
| —lia-—t —-+ — + a oe Ben
erate Sete x I) = 7, and (= =i Brome
Saf 2 pe: ‘F; a | —ti oe oe hall | eatin
se cae 3 8 sae 1D ¢ Sian re sic ea
pee Di, =P. I
; 5 = 3% a = g73 and confequently 3B (= 3 x *, = 1, and 3B’
—3C(=3 x ©— 3 x >) =o, andgD—-6BC 4+ BI(=3 x S —
6 ale ge, oa a RIN RA IRE COR PAs
Bie Soe chs Ae CUR i ane ihe a ie
7. And, when w is = 4, the third of the foregoing compound feriefes.
(which is equal to the fourth power of the feries 1 + Bx — Cx* + Dx —
Ex«+ + Fx — &c), towit, the compound feries
1+4Bse—4Cx* + 4Dx? &c
+ 6B*x? — 12 BCx? &c
+ 4B3x3 &c,
muft be equal to 1 4+ «; and confequently 4B muft be = 1, and 6B? — 4C
muft be = o, and4D — 12 BC + 4 B? mutt likewife be = o.
And.fo we fhall find thefe quantities to be. For, if z is = 4, we fhall have
B(=— — 2 and C (= 2" B= gee: hikes
f— 2% —1 _ 2X 4—1 3° B41 "3 , F ;
es CG — 3. Eee See b 2 — —_-- >. Oe — aL x = ae x
( 3 3X 4 Gey Mal ae a = sg) and confe
quently 4B (= 4 x Z) = rand 6B* — 4C (=6 x = —4 x 2 =F
—3)=0, and 4D — 12 BC +.4B3 (=4 x + - 125% - x o+4 x
oe ee See
8. And, when is = 5, the laft of the foregoing compound feriefes (which
is equal to the fifth power of the feries 1 + Bx — Cx* + Dw? — Ext + Fx
— &c), to wit, the compound feries
1+ 5Be— 5Cx* + 5Dx3? &e
+. 10 Bw? — 20 BC x? &c
| : + 10 Bx? &c,
muft be equal to 1 -+ x, and confequently 5 B muft be = 1, and 10 B* — 5 C
muft be = o, and 5 D — 20 BC + 10 B? muft likewife be = o,
And
350 AD TSC oO URS BF CON CE R NTN G
And fo we fhall find thefe quantities to be. | For, if is = 5, we fhall have
Bigs A ot eee ap and C (= =“ B= i= x-=
m 5 5 oe 2X5 3 ;
Box Sk oe and D (= = CS SX
x= h ease aby = and confequently 5 B(= 5 x =) = 1,
and 10 BX — 5C(= 10 X= X-—— 5 XH THs and Sa
20BC + 10B' (= 5 x G2 —20X = X 3g ok 19K a ee
10 — 4 #)= 6
€26- FH A25 125
9. Thus we fee in all thefe inftances, that, when the feries 1 + Bw — Cx?
I
+ Dx: —~ Ext + Fx = &c (which is equal to 1 + x|7, or "1 + x)
is raifed to the wth power, or multiplied into itfelf 2 — 1 times, the co-efficient
of « in the compound feries obtained by fuch multiplication is always equal to 1,
and the co-efficients of «* and x? (which are compound quantities, or quanti-
ties confifting of more than 1 term) are, each of them, equal to o, the term, or
terms, which are marked with the fign —, or fubtra¢ted from the other terms,
being equal to the term, or terms, which are marked with the fign +, and from
which they are to be fubtraéted. And the fame thing muft take place in the co-
efficients of all the following terms of the compound feries obtained by fuch mul-
tiplication. For otherwife the faid feries could not be equal to 1 ++. We
may therefore lay it down univerfally, as an undoubted truth, refulting from the
I
nature of powers and roots, ‘ that, if the feries which is equal to 1 + «|", or
—ae F aon if tk
«4/1 + 4, to wit, the feries 1 + - Aw — I Bx? + ary Cxi —
_ — I °
«(eo Det + oe Ex«xs — &c, ortheferies1 + Bx — Cx? + Dx? —
n
4 5 ; : ,
< Ex4¢ + Fxs — &c (in which latter feries the co-efficients B, C, D, E, F,
“¢ &c, are re{pectively equal to the tormer co-efficients — A,-or =, and ral B,
peeiin Wc Oe ry 48-1, &c), be raifed to the mth power, or multiplied into
n a n
cc itfelf n—1 iss the Greene of « in the compound feries produced by fuch
«© multiplication will be = 1, and the co-efficients of «*, 3, «*, «5, and all the
<< following powers of x in the faid feries, will be, each of them, equal too; the
« faid co-efficients being compound quantities confifting of fimple terms, or mem-
«© bers, connected with each other, by the figns + and —, and the fum of the
<< members of each of the faid co-efficients that are marked with the fign —, or
‘¢ fubtraéted from the other members of them, which are marked with the fign +,
«« being always equal to the fum of the faid other members from which they are
« fubtracted.” This fundamental propofition being well underftood, we may de-
rive from it a proof of the propofition afferted above in art. 2, concerning the
quantity
THE RESIDUAL THEOREM. 351
rey; 1
quantity 1 + x|#, or 4/7" 1 — x, or the wth root of the refidual quantity
1 —«, to wit, that it will be equal to the feries 1 —— Ax — [ Bx?
2n—1 nN—-— I nm I
“2 banat QF SE “— Det — o— Ex? — &c, or 1 — Be — Cx? —
a” :
«Dx? — Ext — Fx’ — &c3 which confifts of the very fame terms as the
‘¢ former feries which is equal to the wth root of the binomial quantity 1 + x,
<¢ but with the fign — prefixed to every term after the firft term 1, inftead of
“¢ every other term.”
10. To render the proof of this propofition as eafy as poffible, it will be con-
venient to multiply each of the two feriefes 1-+ Bx — Cx* + Dx} — Ext
+ Fx’ —&c,andi—Bwr — Cx* — Dx} — Ext — Fx — &c, into itfelf,
and then to compare together the f{quares, or produ¢ts thence arifing, Thefe
multiplications will be as follow.
Poesy ae Cx? + Date Extn eRe d) ~ &e
E+ Bw — Cw? +: De?) —: Ext + Fei. — &c
ra Be — Cx? + De? — Ext 4+ Fes —.&e
+ Bre+t+ B*x?— BCx? + BDe+ — BEx? + &c
— BCw3 + Cert — CDi + &e
+ Dx? + BD«+ — CDxr + &c
— Ext — BEx«: + &c
+ Fx + &c
1+: 2Be—2Cx* +2De3 ~—~o2kut + 2F x: &c
+ Bex? —2BCxKw? + 2BDx*+ — 2BEx &c
4+ Crt — 2CDx &c
Poe DB he Cyt DY pele eee LR Eo = bre
1 —- Be—° Ca? — -Dxe — OER* (=~ OF es = &e
t— Be— Cr? — Dei *—" Ext > Fes -— &c
— Bxe-+ Bx? +: BCx? + BDx* + BEw + &c
— Cx> + BCw? + C2x* + CDs + &c
— De? + BDxwr+ +. CDxe: + &c
— Exvt + BEx + &c
— Frys + &c
1—2Be—-2CHe? —2Dx? —2Ewt —2F ew: &c
+ Btw? + 2BCw3? + 2BDx* + 2BEx: &c
+ C*x+ + 2CDx> &e.
11. Now, if we compare this laft compound feries (which is equal to the
fquare of the fimple feries 1 — Bx — Cx* — Dx? — Ext — Fx’ — &c)
with the former compound feries (which is equal to the fquare of the fimple
feries1 + Bx — Cw? + Dx? — Ex* + Fx’ — &c), we hhall find the fol-
lowing obfervations to be true concerning them..
Tir
352 A“ WstctiolutTR siM © f.0R mR NE ete
In the firft place, all the terms of the fecond of thefe compound feriefes are
the very {ame with the correfponding terms of the former of them, but are dif-
ferently connected with each other by the figns + and —. And this, it is evi-
dent, muft be the cafe with all the following terms of both thefe compound’
feriefes (to whatever number of terms thefe feriefes may be continued) as well as
with the few terms here computed. For, fince the terms of the fimple feries
1— Bx — Cx? — Dx? — Ext — Fv &c, are the fame with the terms
of the fimple feries 1 + Bw — Cx* + Dx? — Ex* + Fx — &c, though
differently connected with each other by the figns'+ and —, it follows that the
products of the multiplication of the terms of the feries 1 — Bx —, Cw — ©
Dx3 — Ext — Fx’ — &c, into each other muft be equal to the produéts of
the multiplication of the correfpondent terms of the feries 1 + Bx — Cu? +4
Dei — Ext + Fx* — &c, into each other, that is, the feveral members: of
the {quare of the feries 1 — Bx — Cx? — Dv — Ext — Fai = &e;
muft be equal to the correfponding members of the {quare of the feries 1 44 Buy:
—Cxr + Dxi — Ext + Fes — &c, though differently connected with
each other by the figns + and —. .
And fecondly, we may obferve that the firft term in both thefe compound
feriefes is 1. ,
And, thirdly, we may obferve that the third and fifth terms in the fecond of
thefe products, or compound feriefes, which is equal to the {quare of the fecond ©
fimple feries 1 — Baw — Cx? — Dx? — Ex+ — Fx’ — &c (in which all the
terms after the firft term 1.are marked with the fign —) to wit, the terms — 2C x”
+ Bex? and —2Ex*+ + 2BD«* + C*x*, have the fame figns + and — |
prefixed to their feveral members 2 Cx’, B*x*, 2 Ext, -2 BDx*, and C*x*,
re{pectively, as are prefixed to the fame members of the third and fifth terms
of the former of thefe products, or compound feriefes, which is equal to the
{quare of the fimple feries r + Bx — Cw 4+ Dwi — Ext + Fwi — &c
(in which the fecond and other following terms are marked with the figns +
and — alternately), to wit, the terms — 2Cx%x? + B*x? and — 2Ex* +
2BDx«*t + C*x*. And the fame obfervation is true likewife of the feventh
and ninth, and eleventh, and all the following odd terms of thefe two com-
pound feriefes (to whatever number of terms the faid feriefes may be con-
tinued), to wit, that the figns + and — that are to be prefixed to the feveral
members of any of the faid odd terms in the latter of thefe compound feriefes
will be the fame that are to be prefixed to the fame members of the fame odd
terms in the former of thefe compound feriefes; as will appear from an attentive -
confideration of the two foregoing operations of multiplication fet down in the
foregoing article. 3 |
And, 4thly, we may obferve that the fecond, fourth, and fixth terms of
the fecond of thefe products, or compound feriefes (which is equal to the fquare
of the fecond fimple feries 1 — Bx — Cx? — Dx? — Ex* — Fax’ — &c),
«o wit, the terms — 2 Bx, — 2 Dx? + 2BCw3, and — 2F ses 4+ 2 BEw:
-+- 2CD«x‘, have their feveral members 2Bx, 2Dw3, 2 BCw?, 2 Fx,
2 BE x, and 2CD«:, marked with contrary figns to thofe which are pre-
fixed to the fame members of the fecond, fourth, and fixth, terms of the for-
2 . mer
Tole RESID VU AcL oTeH E-o-R BM. 353
mer of thofe compound feriefes (which is equal to the fquare of the firft fimple
feries1 + Be — Cx? + Dx? — Ext + Fxs — &c), towit, the terms -+
2Bxe, + 2D«? — 2BCxw3, and + 2Fxs — 2BExw' —2CDx5. Andthe
fame obfervation is true likewife of the eighth, and tenth, and twelfth, and other
following even terms of thefe two compound feriefes (to whatever number of
terms they may be continued), to wit, that the figns ++ and — that are to be
prefixed to the feveral members of any of the faid even terms in the latter of
‘thefe compound feriefes will be, refpectively, contrary to thofe which are to be
prefixed to the fame members of the fame even terms in the former of thefe
compound feriefes; as will appear from an attentive confideration of the two
foregoing operations of multiplication fet down in the foregoing article.
12« And if we were to repeat the foregoing multiplications of the two feriefes
1+ Be — Cx? + Dei? — Ext + F xi — &c and 1 — Bx — Cx? — De:
— Ext — Fx’ — &c into themfelves any number of times whatfoever, fo as
to obtain the cubes, andthe fourth powers, and the fifth powers, or any higher
powers of the faid feriefes, the foregoing obfervations would be true of all the
compound feriefes, or of all the powers of the faid two fimple feriefes, which
would be thereby obtained ; to wit, 1ft, That the terms of every power of the
one feries would be the very fame with the correfponding terms of the fame
power of the other feries; and, 2dly, That the firft term of every power of both
feriefes would be 1; and, adly, ‘Chat the figns -+- and —, which would be
prefixed to the feveral members of the third, fifth, feventh, ninth, eleventh,
and other following odd terms of any’ power-of one of thefe feriefes, would be
the very fame that weie to be prefixed to the fame meinbers of the fame odd
terms, refpectively, of the fame power of the other feries; and, gthly, That
the figns + and —, which would be prefixed to the fecond term, and to the
feveral members of the fourth, fixth, eighth, tenth, twelfth, and other follow-
ing even terms of any power of one of thefe two feriefes would be, refpectively,
contrary to the figns that would be prefixed to the fecond term, and to the
fame members of the fame following even terms of the fame power of the
other feries.
13. Of the truth of thefe obfervations in the cafe of higher powers of the two
foregoing fimple feriefes 1 + Bx —Cx* + Dx? —Ew* + Fut — &c and
1— Be —Cxr? — Dx? — Ext — Fx — &c than the fquare, I fhall here
give one example, by raifing the faid two feriefes to the cube, or third power.
We have already feen in‘ art. 10 that the fquare of the feries 1 4+ Bx — Cx’
+ Dx? — Ex* + Fx — &c is the compound feries
1+ 2Be—2Cx*? +2Dx? —2Ex* +2Fx5 — &c
+ Bx? —2BCx? + 2BDx* —2BEx* + &c
+ Crxt —2CDs* + &e.
‘Therefore to find its cube, we muft multiply this compound feries into the
original feries 1 + Bx — Cw? + Dx? —Ewt + Fx' — &c; which may
be done as follows.
Vou. II. 27, 1+ 2B
454 A DISCOURSE CONCERNING
r+ 2Be—2Cr*? +2DKe* —2Exwt + oFxes — &e
+ Bx? —2BCwr? + 2BDx* —oaBExs + &c
é' + C*x* —2CDs 4+ &c
1+ Bee Cx? + De? — Ext + Fre — &c
1+ 2Be—2Cxr? +2Dxe3 —2Ewt +2Fe5 —&e
+ Btx? —2BCxe? + 2BDx* —2BEx® + &c
+ C*x* —2CD«%5 + &e
+ Bw+2B*x? —2BCx? + 2BDx* —2BEx® + &c
+ B?x3 —2B*Cxt* + 2B*Dxs — &c
+ BC*xs —&c-
— Cx? —2BCx? + 2C*x* —2CDxi + &c
— B*Cx* + 2BC*xs —&c
+ De? -2BDse+ —2CDx> + &e
+ B*D«s — &c
me Ext = —2BEx® + &e
+ Fx: — &c
1+ 3Bxe—3Cx* + 3Dx? —3Ex* +3Fx° —k&c
+ 3 Bx? —6BCx? 4+ 6BDx* —6BEx> + &c
+ Bix? + 3C*x* —6CDx> + &c
— 3B*Cx* + 3B°Dxs — &c
+ 3 BC*xs — &c.
Therefore the cube of the feries 1 + Be — Cx? + Det — Ext + Fx®
> &c is the compound feries
1+ 3Be—3Cx? + 3Dxe? —3Ex* +3Fx5 —&e
+ 3 B*x* —6 BCx? + 6BDx* —6BEx® + &c
+ Bix3 +—+ 3C*x* —6CDx> + &c
— 3B°*Cx* 4+ 3 B*Dx — &c
+ 3BC*xs — &e.
And we have already feen in art. 10 that the fquare of the fecond feries 1 —
Ba — Cx? — Dx? — Ex* — Fx — &c is the compound feries’
1—2B.e—2Cx*? —2De? —2Exw+ —2o2Fews — &e
+ B*x? + 2BCw? + 2BDxw* + 2BExs + &c
+ C*x* +2CDs> + &c.
Therefore, to find the cube of the faid fecond feries 1 — Bx — Cw? —
Dx? — Ext — Fx’ — &c, we muft multiply this laft compound feries into
the faid fecond feries itfelf; which may be done as follows.
I—2Byxr
THE RESIDUAL THEOREM 355°
tr—2Buy—oCx*? ~2Dx? —2Ext —2Fxs —&c
+ Bx? + 2BCwi + 2BDx* + 2BExs) + &c
+ C*x+ +2CDxe> + &c
1— Be— Cx? — De? — Ext — Fre — &c
¥—2Bse—2Cx? —2Dxe3 ~2Ex* —22F xs — XC
+ Bex? + 2BCx? + 2BDxe* + 2BEx5 + &c
+ C€C*xt+ + 2CDxs + &c
— Br +2B*x* + 2BCw? + 2BDxe+ + 2BEx5 + &c
— Biys —2B*Cx* —2B* Dx’ — &c
— BC*xs —k&c
— Cr? +2BCx? -2C*x* +2CDx5 4+ &e
— B?Cx+ —2BC*xs — &c
— Dx? + 2BDx+ + 2CDx* + &c
— B*Dxs + &c
— Ext +2BExs + -&c
— xs +. &c
1—3Bx —3Cx*? —3Dxr? —3Ex+ —3Fx &c
+ 3B*x* + 6BCx? + 6BDxt 4+ 6BEx' + &c
— Bix? + 39C*x* + 6CDxes + &
— 3B*Cx*—3B*Dxs &c
. — 3BC*x’ — &c
Therefore the cube of the faid fecond feries 1 — Bx — Cw? — Dx3 ~
Ext — Fx’ — &c is the compound feries
1—3Bxe—3Cr* —3Dx? —3Ex* —3Fx5 —&c
» | + 3B*x* + 6BCw? 4+ 6BDx* + 6BEx® + &c
— Bes + 3C*xrt + 6CD«5 + &c
— 3B°Cx* —3B*Dx' = &c
— 3BC*xs — &c.
14. The four obfervations made in art. 11 and 12 are evidently true of the
two compound feriefes obtained in the foregoing article 13, and which are equal
to the cubes of the two fimple feriefes 1 + Bx — Cx? + Dx? — Ext +
Fx’ — &c and 1 — Be — Cx? — De? — Ext —Fxs — &c. For, in
the 1ft place, the feveral terms of the latter compound feries are exaétly the
fame with the correfponding terms of the former compound feries; and 2dly,
the firft term in both thefe compound feriefes is 1; and, gdly, the figns -- and
— that are prefixed to the feveral members of the third and fifth terms, — 3 C x*
+ 3 Bex? and — 3 Ex* + 6 BDx* + 3 C*x* — 3 B*C x* of the latter com-
pound feries (which is equal to the cube of the feries 1 —- Bx — Cv? — Dx?
— Ext — Fx — &c) are the fame with thofe which-are prefixed to the fame
members of the third and fifth terms, — 3 Cw? + 3 B’x? and —3 Ex«* +
6 BDxt + 3 C2x* — 3 B*C x‘, of the former compound feries, which is equal
to the cube of the feries 1 + Bu — Cx? 4+ Dwi —Eut + Fes — &c;
and, 4thly, the figns +- and — which are prefixed to the fecond term, —
3 Bx, and to the feveral members of the fourth and fixth terms, — 3D? +-
6.BC x3 — Bixs and — 3F xe) + 6BEw! 4+ 6CDx> — 3 B*Dw —
2Z 2 3 BC*x5y
356 Aw D TT SVCTONUTR SIR. “CS CO INF CT ESR WOH C6
3 BC*w’, of the latter compound feries (which is equal to the cube of the fim-
ple feries 1 — Bu — Cx* — Da} — Ext — Fx — &c) are, refpectively,
contrary to thofe which are prefixed to the fecond term, + 3 Bx, and to the
feveral members of the fourth and fixth terms, + 3 Ds} — 6 BC w#3 4+ B?x:
and + 3Fx«* —6BEx*’ —6CDx> + 3 B*D«’ + 3 BC’*x5, of the former
compound feries, which is equal to the cube of the fimple feries 1 + Ba —
Cx? + Dx? —Ext + Fas — &e.
And the fame obfervations will be found to be true.of the feveral members of
all the following terms.of the faid.two compound. feriefes after the fixth terms
(to whatever number of terms the faid feriefes may be continued) and of the
fions 4+-.and — that are to be prefixed tothe faid. members. . And they will be
true alfo of the terms of. the feveral.compound feriefes that are equal to the
fourth powers, and to the fifth powers, and to all higher powers, of the faid two
fimple feriefes 1 + Bx —Cx? 4+ Dx} — Ext + Fx’ — &c and 1 — Bx
— Cx?.— Dx —- Ext — Fes —&c.
15. This relation between the figns that are to be prefixed to the fame*mem-
bers of the. feveral terms of the faid compound feriefes which are equal to the
{quares and the cubes and other higher powers of the two fimple feriefes 1 + Bx
— Cxrte+ Deri — Ext ot) Fai — &c and 1 — Be — Cx? — Dx? —
Ext —Fx’ — &c 1s the confequence of the known rules of Algebraick mul-
tiplication; according to which the product of the multiplication of two.quan-
tities which are both marked with the fign — is to be marked with the fign ++;
as well as the product of the multiplication of two quantities. which are both
marked with the fign +; and the product of the multiplication of two«uan-
tities, of which one is marked with the fign -+, and the other with the fign —,
is to be marked with the fign —. .For it will follow from hence that in the
compound feries which is equal to the fquare, or cube, or fourth power, or
fifth power, or other higher power, of the feries 1 + Bx— Cx? + Dx3 —
Ext + Fxt — &c, all the terms that will involve the odd powers of *, to
wit, #, #3, «5, w7, «9, x'*, &c, will be marked with contrary figns, re+
fpectively, to thofe which are to be prefixed to the fame terms in the com:
pound feries which is equal to the fquare, or cube, or fourth power, or fifth
power, or other correfponding higher. power, of the feries 1 — Ba — Cx? —
Dx? — Ext —Fx' — &c; and all the terms that will involve the even
powers of x, to wit, ~*, x*, *°, x*, «'°, xt7, &c, in the former compound
feries, which is equal to the fquare, or cube, or other higher power, of the
feries 1 + Bx —Cx* + Dy?--Ext+ Fut — &c, will be marked with the
fame figns + and —, refpectively, as are prefixed to the fame terms in the latter
compound feries, which is equal to the fquare, or cube, or other correfponding
higher power, of the feries 1 — Bw — Cx* — Dx? — Ext — Fx — &e.
16. To make this more evident, let the capital letter P be put = Bx +
Dei + Fes + He? + Kxe + Max" + &c, or the fum of all the terms in
the firft feries 1 + Bx —Cx* + Dx? — Ext + Fx' — Gao 4+ He? —
Tx? + Keo —Lx? + Myx" — &c, which involve the odd; powers of x,
and which in this feries have, all of them, the fign + prefixed to them; and
let
e
Ti He E REST DUAL Ty. H EO R EF M. 357:
let the capital letter Q be put = Cx? + Bet + Gut + Le? + Le? + &,
or the fum of all the terms in the fame feries which involve the even powers of
«, and which in this feries, as well as in the fecond feries 1 — Bu — Cx? —
Dw? — Ext — F xe — Gut — He? —I xe? —Keo — Lx? — Mat —
&c, are all marked with the fign —.
Then will ihe feries 1 + Bu — Cx? + Dx? —Ewt + Fxs ~— & be —
1+ P—Q,, and the feries 1 — Be — Cv? — Dx? — Ext — Fu — &
will be = 1 — P —Q;; and confequently the fquare of the former feries will
eat a Pi OP = Lefp2s=Q\} = ST eg. cir x{[P—Q +
P—Q)’) = 1+2P—2Q+ P*— 2 PQ + Q’, and the {quare of the
erences ail) sbe- (=. PO) ee P+Q Sicepy EY 6 ay ops.
P+ Qi+ P + Q’) = 1—2P—2Q4 P?+2PQ+ Q.
17. The terms of this latter quantity (which is equal to the fquare of the lat-
ter feries) are exactly the fame with the terms of the former quantity 1 + 2 P
— 2Q + P*—2PQ+ Q‘, (which is equal to the fquare of the former fe-
ries) but are not connected with each other by the figns + and — in the fame
manner as the terms of the faid former quantity. And the difference between the
terms of thefe two quantities in this refpeé is as follows. The third term 2 O.,
and the fourth term P*, and the fixth, or laft, term Q?, in both the quantities
r+2P—2Q 4 P?’—~2PQ+4+ Qi and 1—2P —2Q+ P* + 2PQ
+ Q’ are marked with the fame figns + and —, being —2Q + P* + Q?in
both quantities; but the fecond term 2 P, and the fifth term 2 PQ , have dif-
ferent figns in the two quantities, being + 2P and — 2 PQ in the former quan-
tity, and — 2P and + 2 PQ in the latter quantity.
18. Now the three terms 2Q, P*, and Q’, which are marked with the fame
figns in both thefe quantities, will be equal to feriefes that will involve only the
even powers of x. '
For, in the firft place, the fecond term 2 Q is = 2 x the feries Cx* + Ext
+1Greoqe le? + Lx? + &c = the feries 2Cw2.4+2Ex* + 2Geo 4+
21xv* + 2 Lx'® + &c, which contains only the even powers of x.
And, in the fecond place, the fixth term Q?’ (being equal to the fquare of the
feries Cx? + Ex*+ + Gwe + Ix? + Lx + &c) will evidently be equal toa
feries that will contain only x*, *°, x°, x"°, and the following even powers of x.
And, laftly, the fourth term P* (being equal to the {quare of the feries B x
+Dxe 4+ Fo + He? + Kx? + Me™ + &c, which contains only the odd
powers of x) will alfo be equal to a feries that will contain only the even powers _
of x; becaufe its terms will be the products of the multiplication of the terms
of the feries Bx + Dx? + Fes + Hx? + Kao + Ma™ + &c (which
involve_only the odd powers of x) into each other: it being evident that the
multiplication of two odd powers of x into each other muft always produce an
even power of it. daig wT
Therefore the three terms 2 Q., P*, and Q’, which are marked with the fame
figns + and — in both the quantities 1 + 2P —2Q-+ P*—2PQ 4+ Qi
and
458 Ae F PY slctoBu’ Ri se “CO WhOSEIR NPN GS
andi —2P—~2Q + P? + 2PQ + Q’, will be equal to feriefes which will
involve only the even powers of «. Q. E. D.
19. And the fecond term 2 P, and the fifth term 2 PQ, of the faid quanti-
ties1 + 2P—2Q+ P?~2PQ+ Qandi —2P—2Q+ P* + 2PQ_
+ Q*, which are marked with contrary figns in the latter of thefe two quanti-
ties to thofe with which they are marked in the former quantity, will be equal to
feriefes which will contain only the odd powers of x. ;
For, in the 1ft place, 2 P is = 2X the feries Bx + Dw? + Fuss + Hx?
+ Kwo + Ma™ + &c = the feries 2Bx + 2D? + 2Fxe3 + 2H? +
2K x9 +- 2 Mx" + &c, which contains only the odd powers of ».
And, 2dly, 2PQ is = Q. x 2P=Q -™ the feries 2Bx + 2Dx? +
2Fxe5 +2Hx7 + 2Kxe° + 2Mx" + &c = the feries Cx? + Ext +
Gee + Ix? + Lx’? + &c (which involves the even powers of x) x the feries
2Be+2Dx3 + 2Fxe' 4+ 2He? + 2Ke? + 2 Mx" + &c (which in-
volves the odd powers of «) = a feries confifting of terms which will involve
only x3, x5, «7, ¥°, x, and the following odd powers of x; becaufe all the
terms of it will be the products of the multiplication of fome of the terms of
the feries Cv? 4+ Ext + Guo + Ix* + Lx? 4+ &c (which involve only
the even powers of x) into fome of the terms of the feries 2 Bx + 2 Dx? +
2Fe5 +2H«? + 2Ke° + 2Mxe" + &c, which involve only the odd
powers of w; it being evident that the product of the multiplication of an even
power of x into an odd power of it muft always be an odd_power of it.
Therefore the two terms 2 P and 2 PQ of the two quantities 1 + 2P —
2Q+ P?—2PQ+ Q* and 1 —2P—2Q+ P* + 2PQ + Q?, which
are marked with contrary figns + and — in thofe two quantities, will be equal
to feriefes which will contain only the odd powers of x. Q. E. D.
20. It appears, therefore, from the two preceeding articles, that, to whatever
number of terms the two feriefes 1 + Bx — Cx? + Dx? — Ext 4 Fxs —
&c and 1— Bx — Cx? — Dx? — Ex* — Fx’ — &c, and the fquares of
thefe two feriefes, may be continued, the terms involving the even powers of x
in the fquares of both feriefes, will be marked with the fame figns + and —,
and the terms involving the odd powers of « in the {quare of the latter feries will
be marked with the contrary figns to thofe with which the fame terms are marked
in the fquare of the former feries.
21. The reafonings ufed in the fix preceeding articles to prove, ‘ that,
«* in the two compound feriefes which are equal to the {quares of the two fimple
“‘ feriefes 1 + Bx — Cx* + Dx? — Ext + Fus — &c andi — Bye —
“Cx? — Dx? — Ext — Fxw' — &c, the feveral terms, and members of
‘© terms, which involve the even powers of x will be marked with the fame
“ fions + and — refpectively, and that the feveral terms, and members of
“* terms, which involve the odd powers of « in the latter of the faid compound fe-
«© riefes will be marked with the contrary figns to thofe which are to be prefixed
“to the fame terms, and members of terms, in the former of the faid com-
«* pound
SME aReFaSulipev A LB ¢T\i> E.0 8 EB M. 359
“* pound feriefes,”” may be extended to the more complicated compound {eriefes
which are equal to the cubes of the faid fimple feriefes 1 + Bu — Cx? + Dx;
— Ext + Fes — &c andi — Be —~ Cx? — De? — Ext — Fw — &e,
and to the fourth powers of the faid feriefes, and to the fifth powers of the faid
feriefes, and to all higher powers of the faid feriefes, fo as to prove, ‘that in
** all thefe compound feriefes the feveral quantities which involve the even
** powers of » will be marked with the fame figns in every two correfponding
“* feriefes, and that in the compound feries which is equal to the cube, or the
“* fourth power, or the fifth power, or any higher power, of the feries 1 — Bu —
“Cx? — Dei — Ext — Fx’ — &c, the quantities which involve the odd
** powers of x will be marked with the contrary figns to thofe which are prefixed
‘* to the fame quantities in the compound feries which is equal to the cube, or
“« the fourth power, or the fifth power, or other correfponding higher power of
“the feries 1 + Be — Cw? + De? — Ext + Fw’ — &c.”? And there-
fore I think, we may now confider the four obfervations fet forth above in art.
11 and 12 as fufficiently eftablithed.
22. Now, if thefe obfervations are admitted to be true, it will follow that,
whenever the co-efficients of the third, fourth, fifth, fixth, and other following
terms of the compound feries which is equal to the fquare, or the cube, or the
fourth power, or the fifth power, or any higher power, of the feries 1 + Bu —
Cx? + Dx? — Ext + Fx’ — &c are all equal to 0, or thofe of the mem-
bers of the faid co-efficients which are marked with the fign —, taken together,
are equal to the other members of the faid co-efficients which are marked with
the fign +, and from which the former members marked with the fign —, are
to be fubtracted, the fame thing will alfo take place in the compound. feries
which is equal to the {quare, or the cube, or the fourth power, or the fifth power,
or the other correfponding higher power, of the feries 1 — Bx — Cx*— Dw:
— Ex*+ — Fx — &c; towit, that the co-efficients of the third, fourth, fifth,
fixth, and other following terms of the faid compound feries will likewife all be
equal to o, or that thofe members of the faid co-efficients which are marked
with the fign —, or are fubtracted from the other members of them which are
marked with the fign +-, will be equal to the faid other members.
For in the faid’correfpondent compound feriefes the members of the feveral
correfponding terms, or terms involving the fame powers of x, will be the very
fame quantities, by the 1{t obfervation in art. 11 and 12. And-in the third,
fifth, feventh, and other following odd terms of the faid compound feriefes
(which will involve the even powers of x) the figns + and —, to be prefixed
to the feveral members of the faid terms, will be the fame in both feriefes; and
in the fecond terms and the feveral members of the fourth, fixth, eighth, and
other following even terms of the faid compound feriefes (which will involve the
odd powers of «) the figns + and —, to be prefixed to the faid fecond terms,
and to the feveral members of the fourth, fixth, eighth, and other following even
terms of the faid compound feriefes, will be, refpectively, contrary in one of
thofe compound feriefes to what they are in the other ; as is evident from the
third and fourth obfervations in art. 11 and 12. Therefore, if in ge
2 ourth,
260 A DISCOURSE, CONCERNING
fourth, fifth, fixth, and other following terms of the compound feries which is
equal to any power of the firft fimple feries 1 + Bx — Cx? + Dx? — Ext
-+ Fws — &c, the fum of the members of each of the faid terms that are
marked with the fign — is equal to the fum of the members of the fame term
which are marked with the fign +, the fame thing will alfo take place in the
compound feries which is equal to the fame power of the fecond fimple feries
1 — Be — Cx? — Dx? — Ext — Fxs — &c, or the fum of the members
which are marked with the fign — in the third, and the fourth, and the fifth, and
the fixth, and every other following term of the faid fecond compound feries,
will be equal to the {um of the members of the fame term which are marked with
the fign +; that is, in other words, when the third, fourth, fifth, fixth, and
other following terms of the compound feries which is equal to any power of the
fimple feries 1 + Bx —Cx? + Dx? — Ext + Fx? — &c, become, all,
equal to o, and the whole feries confifts of only the two firft terms, the third,
fourth, fifth, fixth, and other following terms of the compound feries which is ©
equal to the fame power of the feries 1 — Bw — Cx? — Dx} — Ex* — Fx
— &c will alfo be, all, equal to o, and the whole feries will confift of only the
two firft terms. 0 iE. aD.
23. And hence we may derive a proof of the propofition afferted above, in
1
art. 2, concerning the quantity 1 — |”, or ¥/*{1 —.«, or the ath root of the
refidual quantity 1 — x; to wit, that, if A be = 1, and B be = - A, and C
be = = B, and D be = ore ~C, and E, F, G, H, &c be equal to - -D,
E, ie a “= > G, &c, refpettively, the faid quantity 1 — « =, or
n a : 3 5 2 ne 2 Roe CL: 210
/"{t —«, will be equal to the infinite feries 1 —Ax —= Bx
er Cxi — [a= LD ¥* pS Ext — &c, ort — Bx — Cx? — Dx?
— Ex+ — Fxs — &c, in which all the terms after the firft term 1 are marked
with the ign —, or are fubtracted from the faid firft term.
For, fince the feries 1 + — Ax fms By 4 20 Cys ee
% 2% 3% 42
Dist + 2— Ex: — &c, or 1 + Be — Cx? + De? — Ext +Fes —
&c, has been fhewn in art. 47 of the preceeding difcourfe concerning the Bino-
I
mial Theorem, page 237, to be equal to 1 + x|7, or "(1 + x, it follows
that, if the faid feries were to be raifed to the mth power (” being any whole
number whatfoever) or to be multiplied # — 1 times into itfelf, the compound
feries thence arifing would be equal to t + x, which would be the two firft
terms of it, and confequently the co-efficients of «*, «3, #*, *', &c ad infi-
nitum in the following terms of the faid compound feries would be, each of them,
equal too, or thofe members of the faid co-efficients which would be marked
with
THO RYE SHI Dov AG «HH Eo: R EM. 301
with the fign —, or would be to be fubtracted from the others which would be mark-
ed with the fign +, would be equal to the faid other members, from which they
would be to be fubtracted; of which we have given fome examples above in art. 5,
6, 7; and 8. But, by the foregoing article 22, whenever the co-efficients of the
third, fourth, fifth, fixth, and other following terms of the compound feries which
is equal to any power of the feries 1 + Bx —Cx* + Dw? — Ex* + Fx’ — &c,
are all equal to o, or the fubtratted members of each of the faid co- efficients are
equal, taken together, to the members from which they are fubtrafted, the co-efhi-
cients of the third, fourth, fifth, fixth, and other following terms of the compound
feries, which is equal to the fame power of the feries 1 — Bu — Cx? — Dx —
Ex*+ — Fx* — &c, willalfo be, all of them, equal to 0, or the fubtratted nem-
bers of each of the faid co-efficients will be equal, taken together, to the mem-
bers from which they aré fubtracted. Therefore, if the feries 1 —— Ax —
ete Bet on Cory oh 2 De ELS dee See or ie Be
2% 32 4% 5%
° c . I I
— Cx? — Dx? — Ext —F xs — &c (in which Bis =— A, or— x 1,
or —, and C is = ~—" B, and D is = ~—~ C, and E is = 2~— D, and F,
n 22 32 44
oe x 6re “ss
G, H, I, &c, are equal to ae a oF vs F, a ~G, and = - H, &c, re-
fpectively), be raifed to the wth power, or multiplied 2 — 1 times into itfelf,
the co-efficients of the third, fourth, fifth, fixth, and other following terms
of the compound feries which will be produced by fuch multiplication, and
which will be equal to the faid mth power of the feries t - Bx — Cwx* —
Dei — Ext —Fxs — &c, will, each of them, be equal to 0, or the fub-
tracted members of each of the faid co-efficients will be, all taken together,
equal to the other members of them, from which they are to be fubtrac-
ted; and confequently the whole of the faid compound feries will be equal
to its two firft terms; which will be 1 — x, becaufe the two firft terms of
the compound feries which is equal to the wth power of the feries 1 + Bx —-
Tell 4 Pi) Seto |
Cz + Dr? — E.x* 4+ Fwi — &c, or 1 + LAx — -Et Be --
Cxi — meg Dwt + ne Exs — &c (to which the compound feries that
is equal to the wth power of the feries 1 — Bw — Cx? — Dx? — Ew* — Fx’
— &c, has been fhewn to be analogous in the manner above-defcribed), are
Lr.
And, fince the wth power of the feries 1 — Bx — Cw* — Dv? — Ext —
I n— tI 22 — 1 32 —— I
Fxs — &c, or 1 — — Aw — |— Bx? — Cri — ~— Dx* —
2 22 32 4u
Ew’ — &c, will be equal to 1 — w, it follows that the faid feries itfelf
4n—1
gu
I
will be equal to 1 — x|*, or "(1 — «, or the ath root of the refidual quan-
tity I — x. 0. E. D;
Vous II. MeN 24. We
362 AOD LT stigh oh ulm ise FC ROANe CHER (N Fee G
Pc nm
24. We muft now proceed to confider the quantity 1 — x|#, or the mth
power of the wth root of the refidual quantity 1 — x.
mn
Of the feries which ts equal to the quantity 1 — x\7,
or the mth power of the nth root of the refidual
quantity 1 — x, when m and n are any whole num-
bers whatfoever.
25. It has been fhewn in the foregoing difcourfe concerning the binomial
theorem, art. 77, p. 269, that, ifm be any whole number whatfoever, and z
WW
any other whole number greater than m, the quantity I + x”, orthe mth power
of the zth root of the binonal quantity 1 + «w, will be equal to the feries 1 +
= Ax — Fee Bx? + =—2Cws — |2=* Des + P=" Ewxs — &e ad
Lymipe ; in which feries the fecond, and third, and fourth, Endl other following
terms, are alternately en with the fign + and the fign —. And it has been
fhewn in art, 104 of the faid difcourfe, page 289, that, when m is greater than
Wm
n, but lefs than 2”, the quantity 1 + *|*, or the mth power of aa nth root of
the binomial quantity 1 + x, will be equal to the feries 1 + — = Ax = —*
ee — Cus + a Teste fe Ews ++ &c; in which the three
firft terms of the feries are added together, and the fourth and fifth, and other
following terms of it are marked with the fign — and the fign + alternately, or
are alternately fubtracted from, and added to, the faid three firft terms.. And it
has been ‘fhewn in art. 115 of the faid diteourte: page 299, that, if m be of any
magnitude greater than 2%, -and pz is the greateft multiple of m that is lefs than
m,{o that m is greater than pz, but lefs ave? + 1] X #, or pz + n, the quan-
7 /
tity 1 + x\a , or the mth power of the zth root of the binomial quantity 1 + x,
will be equal to the feries 1 + = Ae eeepc eee Tae Cua hee
4a
th m— 4n 2- = ie Gitte 2 PP *.2 pu+2nu—m
Dx Fee a Eas 2 Fixe + &c, nate Cx eae |
4 prt3 “EE py ra pict: 4 1 Paria pbs _ [pet su—m 1 p+6
See n+ 4n sa pat bn pn + 6x sh
+- &c ad infinitum; in which feries all the terms after the firft term 1 are to be
added to the faid firft term, till we come to the oe Cig hie
> which
2 is
-
x
THkHER ARWEVSATKDW AL «THE OUR er Ms 363
pr+tn—m
pz + 22
are to be added to, and fubtracted from, the faid firft term alternately.
is to be fubtracted from it; and all the terms after the faid term
Ciyh +?
m1
Now the quantity 1 — x\*, or the mth power of the th root of the refidual
quantity 1 — x, will always be equal to a feries confifting of the very fame
« ™m
terms as the feries that is equal to the quantity 1 + x\ 7 , or the fame power of
_ the fame root of the binomial quantity 1 + *, but with the figns + and —
changed into their contraries in all the terms that involve «, w3, #5, “7, ¥°,
x**, and the other following odd powers of w, or in the fecond, and fourth, and
fixth, and eighth, and tenth, and other following even terms of the feries.
This I fhall now endeavour to demonttrate.
I
26. It has been fhewn above in art. 23, that 1 — «|, or the th root of the
. ° . : . I ae
refidual quantity 1 — x, is equal to the infinite feries 1 — — Aw — |—— Bx?
3 74 2%
— — iI — I .
> = yt “— Dx — |= = Ex’ — &c, which confifts of the
2
very fame terms, only differently connected together by the figns + and —, as
the feries 1 + ~ Ax — oe BNO BEC == Det + ts
Bf
3n An 5
Ex’ — &c, which is equal to r + x\”, or the wth root of the binomial quan-
tity 1 + *. And the difference between thefe two feriefes with refpec& to the
figns + and —, that are to be prefixed to their fecond and other following
I :
terms, is that in the feries which is equal to 1 — x|# the fign — is to be pre-
fixed to all the terms after the firft term, whereas in the feries which is equal to
I ‘
1 — x\# the fign — is to be prefixed only to its third, and fifth, and feventh,
and other following odd terms, and the fign + is to bé prefixed to its fecond,
and fourth, and fixth, and other following even terms.
ae eT
os : : : I 3 i At A
Now, fince 1 — x|* is equal to the infinite feries 1 — — Aw —|—— Bx
— 2% — — J - .
es — | yt — | Bt — &e, ‘it follows that the quan-
3n 42 G2 \
m I
tity 1 — x\”, or the mth power of 1 — «| will be equal to the mth power of
- ae : : I nt 2 — 1 a— I
the faid infinite feries 1 — — Aw — |—— Bx? — [= Ce? — 2 —— Des
7 22 3n 4a
— |" Exs — &c, or (as it will be convenient to denote it for thie fake of
Seba h :
brevity), of the feries 1 —Bw — Cx* — Dx} — Evt — Fx? — &c ad in-
finitum, We mutt therefore inquire what will be the terms of the feries that is
BAe equal
364 A Det Sve ONUTR & B “CHOMNECTESR W. Tew G
equal to the mth power of the feries 1 — Bw — Cx? — Dx? — Ext —Fxi =)
&e, and what will be the figns + and — that will be to be prefixed to them,
or what will be the terms of the feries that will be produced by multiplying the
faid feries 1 — Be — Cx* — Dx} — Ew* — Fx’ — &c m — 1 times_into
itfelf, and which of the figns + and — willbe to be prefixed to each of them:
and we mult compare the feries fo produced with the feries which is produced
F 3 sy : I —il 2nu— tI
by the multiplication of the feries 1 + = Ax — = By? + toe Cart. a.
read go gore 2h ee &c, OG + Bx Thy Cx? + D x3 Exe
I
F «xs — &c (which is equal to 1 + x)” ) m —1 times into itfelf, and which is
mM
therefore equal to the quantity 1 4+ «|*, or the mth power of the mth root of
the binomial quantity t + «.
27. Now, from what has been fhewn above in art. 11, 12, —-—-—-— 27,
it is evident, in the firft place, that if we raife both the feriefes 1 + B® — Cx?
+ Dx? — Ewt + xi — &c,.and 1 —Bxr — Cxr?'=— Dx? — Bvt
— &c, to the mth power (m being any whole number whatfoever), or muluply
each of them m — 1 times into itfelf, the produéts of the faid multiplications
will be two compound feriefes confifting of exaétly the fame terms, of which
the firft term will be 1; and, 2dly, that the figns + and —, that will be pre-
fixed to the members of the feveral terms of the faid compound feriefes, will be
the fame in all the terms which will involve the even powers of » in both feri-
efes, but will be different in thofe terms of the faid two feriefes which will in-
volve the odd powers of «. Therefore the compound {feries which is equal to
the mth power of the fimple feries 1 — Bx — Cx? — Dx? — Ext —F ys —
é&c, or 1 — a Av — — Be? — a Crs — — Det — oe
Ex’ — &c, may be derived from the compound feries which is equal to the
mth power of the fimple feries 1 + Bx — Cw* + Dx? — Ext 4 Fas —
&c, or 1 — Ax — lament ats caspeeenes Sid ee i Det j+ Ae
22 4 : n
Ex! — &c, by changing the figns of all thofe terms of the faid latter compound
feries which involve the odd powers of x, into their contraries. But the com-
pound feries which is equal to the mth power of the faid fimple feries 1 + Bx
5 / id
— Cx? + Dx? — Ext + Fx) — &c, or 1 +— Ax — ~Bu? 4
2a
ae NE | ua— I bd
ae Rice Aoi = Det + ag — &c, is equal to the mth power
z mm
of 1 + |”, orto the quantity 1 + +)", and confequently (by what is fhewn in
art. 115 of the foregoing difcourfe, page 299) is alfo equal to the fimple feries 1
“Bx? + po Diet 5 Be eee
E +a—m Cite
tn -+- 22
" m
Fi es
Ml — 24
—C
%
TlH EB.) ARAEOSIA Www A ke sTYHOESO:R BM. 365
jae ey Cp? b? os, Pes A Dy? t > car 8c. _ Therefore, if the figns +
+ 32
and — of thofe terms in this latk fimple feries which involve the odd powers of »,
that is, of the fecond, fourth, fixth, and other following even terms of it, be
changed into their contraries, the fimple feries thereby produced will be equal to the
compound feries produced by changing the figns of thofe terms of the compound
feries that is equal to the mth power of the fimple feries 1 + Bx — Cx? +
Dx? — Ext + Fx’ — &c, or 1 + ~ Ax _ o> Bx* + ae _
wee De+-+ “atm E «> — &c, which involve the odd-powers of «, or the
figns of its fecond, fourth, fixth, and other following even terms, into their:
contraries. But it has been juft now fhewn, that the compound feries pro-=
duced by changing the figns of thofe terms of the compound feries that is equal.
to the mth power of the fimple feries 1 + Bx — Cx* + Dx? — Ext 4+ Fas
— &c, orb + —Ax — a Bw? + Sop Ce — a Dxt + ee
E «> — &c, which involve the odd’ powers of «, into their contraries, is equal
to the compound feries, or rather is the compound feries, which is equal to the
mth power of the fimple feries 1 — Bx — Cx? — Dx? — Ex* — Fux’ — &c,
meee eA | Bix? — |—— Cx? — eee of 8 RE ah PE as po
7 2n n 2 n
&c. Therefore the fimple feries.that is produced by changing the figns of
thofe terms of the fimple feries 1 + - Ax + “—* Bx? eee oe Ce
n
putn—m cy pt2 a pr+2n—m
pu + 2n pr +32
Mm — 42
Dx +- Exs + &c, —
4n 7
Dae? t 3 _ ae E’x? 4 4. &c, which involve the odd powers of, or
the figns of the fecond, fourth, fixth, and other following even terms of it, into
their contraries,, will be equal to the compound feries which is equal to the mth
power of the fimple feries 1 — Bw — Cx* — Dx? — Ext — Fx’ — &e,
I Pars 24 — 1 A Sn wae:
m ze 3n 4n 5m
I
&c, and confequently will be equal to the mth power of 1 — x) (which is
equal to the faid laft-mentioned fimple feries), or to the mth power of the ath
root of the refidual quantity 1 — x.
On He ws
28. The reafonings in the foregoing article appear fomewhat perplexed in
confequence of the multitude of words which have been made ufe of in defcrib-
ing the feveral infinite feriefes mentioned in them. I will therefore, now re-
peat them in a concifer manner, which will, 1 hope, remove all obfcurity from
them; and for this purpofe 1 fhall denote all the feriefes that we fhall have oc-
cafion to confider, by fingle letters.
a0. bce
366 AUD MSOciOUUNR Sl El OO WICSEAR N PNG
29. Let the Greek capital letter P be put for the feries 1 -- An — = —
Be? + cane Cet at Det + — Exs — &c, sd infeoten ori +
By — trees + Dei — Evt + Fe! — Ke ad infinitum, which is equal to ©
1+ Be , or the wth root of the binomial quantity 1 + »; and let the Greek
re 24%— I Co x3
3%
Ew’ — &c ad infinitum, or 1 — Be — Cx? — Dx
I
— Ext — Fx’ — &c ad infinitum, which is equal to 1— x} , or the ath root
of the refidual quantity 1 — #. And let I™ f{tand for the compound feries which
is equal to the mth power (m being any whole number whatfoever) of the fimple
feries 1 +. By — Cx? + Dx? — Ex* 4+ Fx’ — &c, orI, or the produé which
- arifes by multiplying the faid fimple feries # — 1 times into itfelf; and let
A” {tand for the compound feries which is equal to the fame, or the mth, power
of the fimple feries 1 — Bx — Cx* — Dx? — Ex+ —Fx* — &e, or A, or
the product which arifes by multiplying it # — 1 times into itfelf. And let ‘the
“Bee +
Pe Na iy PE manne i Eile ‘ey *
4a
Greek capita letter A denote the fimple feries1 + = Aw 4°
m— 28 OL, “gel m= 3" D ys eo Eade ae Biyel)poigye ale (=e pt2 2
a put 2n
pet 2n—™m yy, Fes pega py, P+4 Ig ita Sy Fo PAE yn pa &c ad
put 32 pn + 42 pat su
™m
infinitum, which is equal to the quantity 1 + x), or the mth power of the ath
root of the binomia’ quantity 1-+*. And, laftly, let the Greek capital letter
II denote the fimple feries which is derived from the fimple feries A by chang-
ing thé figns of thofe terms in it which involve the odd powers of «, that i is, of
the fecond, fourth, fixth, eighth, and other following even terms of it, into
their contraries. With this notation the reafonings contained in the foregoing
article 27 will be as follows.
30. From what has been fhewn above, in art. 11, 12, &c, —-— — 21, it is
evident, in the 1ft place, that the compound feriefes I and A” will confit of
exactly the fame terms, of which the firft term will be 1; and, 2dly, that the
fizns + and — that will be prefixed to the members of the feveral terms of the
faid two compound feriefes will be the fame in all the quantities, or members of
the terms of the faid feriefes, which will involve the even powers of «, that is, in
the third, and fifth, and feventh, and other following odd terms of the faid fe-
riefes; but will be different in thofe terms of the {aid two feriefes which will in-
volve the odd powers of x, that is, in the fecond, and fourth, and fixth, and
other following even terms of the faid feriefes. Therefore the compound feries
A” may be derived from the compound feries I™ by changing the figns of all the
members of thofe terms of the feries F” which involve im them the odd powers
of
ToH B ROE VSI DV AL (Tuo Por HM, 307
of ¥, that is, the figns of all the members of the fecond, fourth, fixth, and other
following even terms of it, into their contraries. But, becaufe the fimple feries
: ji
I is equal to 1+ x\2 , the compound feries I will be equal to the mth power
13 LZ
of 1 + x\”, orto the quantity 1 + x)|*, and confequently (by what is thewn
in art. 115 of the foregoing difcourfe, page 299) will be alfo equal to the fimple
feries A. Therefore, if the figns + and — in thofe terms of this fimple feries
A. which involve the odd powers of «, that is, in the fecond, fourth, fixth, and
other following even terms of it, be changed -into their contraries, the fimple
feries thereby produced will be equal to the compound feries that is produced
by changing the figns + and — of all the members of thofe terms of the com-
pound feries I, which alfo involve the odd powers of x, or the figns of all
the members of the fecond, fourth, ‘fixth, and other following even terms of
the faid compound feries I, into their contraries. But it has been juft now
fhewn, that the compound feries which is produced by changing the figns +
and — of all the members of thofe terms of the compound feries T™ which
involve the odd powers of w, or of all the members of the fecond, fourth,
fixth, and other following even terms of it, is the compound feries A”. There-
fore, if the figns + and — in thofe terms of the fimple feries A which in-
volve the odd powers of «, or in the fecond, fourth, fixth, and other fol-
lowing.even terms of it, are changed into their contraries, the fimple feries
thereby produced will be equal to the compound feries A, and confequently
—
will be equal to the mth power of 1 — x\*, or of the mth root of the refidual
. quantity 1 —.*, which is equal to the fimple feries A ; that is, the fimple feries
; I
TT will be equal to the mth power of 1 — |”, or of the wth root of the refidual
™m
quantity 1 — x, or to the quantity 1 — ~|”. bee Eg
31. And, if fill greater brevity be defir’d, this demonftration of the equality
between the fimple feries II and the quantity 1 — «|» may be expreffed in the
manner following.
I
Since the fimple feries T is = 1 + x|*, it follows that the compound feries
a... i *
T will be = the mth power of 1 + w|#, or = 1 + x\
74
But (by the foregoing difcourfe, art 115, page 299) 1 + *\# is equal the
fimple feries A.
Therefore the compound feries F” is equal the fimple feries A.
Therefore, if we change the figns + and — of all the even terms, or terms
involving the odd powers of «, in both thefe feriefes 1” and A, into their con-
traries, the ferieles thereby produced will be equal to each other. ~
But
368 Aw D InSocwOuUTR 408 1CO NICs RM Ne Nes
But the feries produced by this change in the figns of the terms of the com-
pound feries I will be the compound feries A”, by what has been fhewn above
in art. 11, 12, 13, &c —— — 21. And the feries produced by this change
in the figns of the terms of the fimple feries A is the fimple feries TI.
Therefore the fimple feries II will be = the compound feries A”.
I
But becaufe the fimple feries A is = ng Inte it follows that the compound
feries A” muft be = the mth power sie aaa or = the qos catdaaae
Therefore the fimple feries IT will be equal the quantity esas y deren merne
32. We have now demonftrated in a manner that, I aie will be thought
fatisfactory, 1ft, that the quantity 1 — x\*, or the zth root of the refidual quan-
. . . . . — 22) —IT
tity 1 — x, is equal to the infinite feries 1 — — Aw — |- - By? — |
m 2% 32
ae ih
Cyr 422 Dx* — (A= yy — &c ad infinitum, in which all the terms
that come tae the firft term 1, are marked with the fign —, or fubtracted from
712
the faid firft term; and fecondly that the quantity 1 — x\#, or the mth power of
the nth root of the faid refidual quantity, is equal to a fimple feries (which we
have called II) confifting of the very fame terms as the feries (which we have
mM
called A) that 1s equal to the quantity 1 + «|z, or to the fame mth power of the
fame nth root of the binomial quantity 1 + », and derived from the faid feries A
by changing the figns of thofe terms of it which involve the odd powers of x,
or the figns of its fecond, fourth, fixth, and other following even terms, into
their contraries. Nothing therefore feems now to remain to be done with re-
fpect to this fubject, but to illuftrate thefe two feriefes by applying them to a
few particular cafes, or examples, in the fame manner as we illuftrated the fe-
if m
riefes which are equal to the quantities 1 + x|# and 1 + a” in the beginning of .
the foregoing difcourfe concerning the binomial theorem in art. 8, 9, 10, 11, 12,
&c. — — — ig of the faid difcourfe, pages 201, 202, 203, &c. — — — 209.
“Thefe examples may be as follows.
pila cee ala eat
Examples of the extraction of fome particular roots of the
refidual quantity 1 — x” by means of the feries given
above in art. 2.
ER seems
33. In the firft place we will extract the {quare root of the refidual quantity
. . . . . I
a — x by means of the feries given in art. 2, to wit, the feries 1 — —> Ax —
4—YI
22
HH RO sRaboselDW AL WHE OR BM. 369
fn 2u—~— 1 3n — 1 jn — 1 -
= Bee — ¥ Cre = lcqul? Det — a Ews — &c ad ine
Jinitum.
I
I
Now in this cafe 1 — x\s is = I—wl2, ormis = 2. Therefore 27 is =
29X29) ="4,-and 3” 1s (= 3 X 2) = 6, and 4” is (= 4 x 2) = 8, and
£2 6 £= 5.x; 2) aozjand confequently 2—1 is (= 2—1) = 41, and
24-1 is(= 4—1) = 3, and 3” —1 is ae Oper) Lome SANG AW mm 15
(= 8—1) = 7. We fhall therefore have 1 — ~Ax yaw a panes
PSV OF 1 34=* p ys SPR ee ay A te i a pes
42 52 2 4
3%
a pls ras GA, ce ee ae gut 7x5
= Cx 3 Dx ot Re ea : Ph ATE EE Se ae ie
&c.
Therefore, if the feries given above in art. 2 is really equal to 1 — x)”, the
I
quantity 1 — x«|2, or the fquare-root of the refidual quantity 1 — x, will be
; w “ x + ae 9x5
equal to the feries 1 -- —->—3Z— TB se6 &e. Q; E. I.
34. Now « that this feries is really equal to the fquare root of 1 — «,” will
appear by multiplying it into itfelf. For we fhall find that the produé of the
faid multiplication will be 1 — x. This multiplication may be performed in the
manner following.
2 3 4 5 5
aay TRI ra 56 Bec
(EHF E-§- Be
z 3 4 s
(En F- 5-8 Bk
i 4 St 4+ + &e
ae taste hase + &e
3 4 5
“Hie Figs + aa + Be
= a + &
— Ee + &e
It appears therefore that the feries 1 — a 7-2-3 &c, is
really and truly, as far as relates to the fix firft terms of it, the fquare-root of
the refidual quantity 1 —*; and confequently that the feries given above in
VoL, Il. 3 B art.
37° AS DF SAO O.UIR SF CAO INGCORSER WM Bee iG
I
art. 2 for the value of 1 —w, or /"[1 — w#, is true in the cafe of fquare-
roots.
I
35. In the next place we will inveftigate the value of 1 — x\3, or the cube-
root of the refidual quantity 1 — 4, by means of the fame feries 1 — + Aw —
a
— I 2u2— 1 n= T 1 — . .
— Bur — C3 — a7 Dat — (eo Ex’ — &c ad infinitum.
3n
Now in this cafe “is = 3; and confequently we fhall have 22 = 6, and
3n = 9, and 4m = 12, and 5u= 15, andu—1i = 2, 2n—1 = 5, 3n—1
= 8, and 4mn—1= 11. And therefore the feries 1 —~ ~Ax — |= By?
a 22
2u4— 1 3 3n— I ni vas 5 : : :
Sie Cwe ee Sey Dx Ps Ex’ — &c, will in this cafe be =
3
a §x* 10x4 2245 &
ees ees ee ee ee a Ce z ‘ cy
D gueht. 243) gag728 pee
36. Now “that this feries is really equal to the cube root of 1 —¥w,” will
appear by multiplying the faid feries twice into itfelf. For we fhall find that the
product of the faid multiplications will be equal to1——w. Thefe multiplica-
tions will be as follow.
&
&
wi
&
we
-_
[@) >
&
>
is}
N
S
wm
T ——— et ee ee Eee eee ss &
3 Laide weer hive ues, :
3 9 81 243 729
x “? gz? 10x4 2245
THE RESTDUAL THEOREM, 27%
2 2 3 a 4 aw S
ra nd lade. ee Sa
3 9 Be 7ath Zao we Linge
3 9 81 243 72
CO (ae NR ea OTE OPN oe Sa
3 om 81 243 729
2x X 4x 7x
—— + — + &c
2. + a ot 243 729 a
a we a 4x
hy sng Byers Sige &c
9 ai Pe ee a
5* 10% Sx 4
81 243 729 oer
ox* ox §
Oye 10% 20% we &c
243 iat)
22%
~ + &c
729 if ;
I—vx * Fi * * &c.
: 3 uv 2 3 4 a8
fap herchore tha he hia 1 Se SO eee
9 Br 243 729
&c, is really-and truly, as far as relates to the firft fix terms of it, the cube-root
of the refidual quantity 1 — +, and confequently that the feries given above
1
in art. 2 for the value of 1 — «|”, or /* 1 —w, is true in the cafe of cube-
roots as well as in that of {quare-roots.
Examples of the extraétion of the roots of Jome particular
powers of the refidual quantity 1 — x by means of the
Series given above in art. 25, and which in art. 29,
30, and 31 is denoted by the Greek capital letter Tl.
RS ee AE
2
_37. In the next place we will inveftigate the value of the quantity 1 — *|3,
or the cube-root of the fquare of the refidual quantity 1 —., or (which comes
to the fame thing) the fquare of its cube-root, by means of the feries given
above in art. 25, and denoted by the Greek capital letter IT in art. 29, 30, and
he
3 : 2
Now it is fhewn in the foregoing difcourfe, art. 12, page 203, that 1 + *\35
or the cube-root of the fquare of the binomial quantity 1 ++ *, is equal to the
3 4 5 . °
O44 ea ME _ Kc, ad infinitum. Therefore the
f{eries 1 Thea Sg eiaall
ee 9 81 243 729
2 ’
quantity 1 — x\3 muft, according to art. 25, be equal to a feries derived fom
aes this
~
37% ARS TD EMSFER OU TR ae CQ 'NLCLECR ® PNG
this feries by changing the figns of its fecond, fourth, fixth, and other follow-
. ; : . . (ori 26 x 4x3
ing even terms into their contraries, that 1s, to the feries 1 — ree
ward »$ 4 . ‘
x 14 __ &c, in which all the terms after the firft term 1 are marked with
243 i7y : ;
the fign —, or are fubtracted from the faid firft term. Therefore, if the propo-
fition contained in art. 25, is true, and the feries juft now fet down, and which,
”
in the foregoing articles, is denoted by 1, is really equal to 1 — x\”, the quan-
2
tity 1 — x\3, or the cube-root of the fquare of the refidual quantity 1 —w, will
; 2 3 sith der pres
be equal to the feries 1 —=* —~ —— F-— 7 Be,
38. Now “ that this feries is really equal to 1 — x\3, or to the cube root of
‘¢ the {quare of the refidual quantity 1 — «, or to the cube-root of the trinomial
“ quantity 1 — 2x” + xx,” (for both the quantities 1 + 2” + «x and 1— 24
+ xx are equally called ¢rinomial quantities, though in the latter quantity the
middle term 2 is marked with the fign —, or fubtracted from the fum of the
two other terms) will appear by multiplying the faid feries twice into- itfelf.
Thefe multiplications will be as follow. :
ee ee ee eet eee ee
ce eee ee eee ee
me . bes 2 ne Bea ht
"Haw how am Jas page
a9, 8 ee
ings + &
me ea
aoe HP Bec
Benya exci aahigg Ha tg ofa &c.
THF RLEVS'T DM AYL TRE OR i Me 373
nt! 20% 4x3 gue 8x5
por Donets et len es BT a ee a EO
gilp toy a: Sind 243 ui 729
BN ig ok BAS 7* 14a
, Toes a gts eal
3 9 81 243 “fT
4x pias x3 xt Sy 5
pe OER BSE as + 2 &c
Td . a3 h29,
2x z
ke a wey woe LORE el pth
3 9, ¥is 243 729
ad pani g li talah ala gaa Bp
9 27 81 92
nee = 16x4 Sx 5 “at oe
: 243 729
par: 7° 28%5 iy ae ;
‘ 243 ‘fe
14%
— c
729 +
I—2K” + XK * # % &ce.
. 2 3 4 5
It appears therefore that the feries 1 — aie Abe A 784 aE ae
81 243° 729
: des» o .
is really and truly the cube-root of the trinomial quantity 1 — 2% + xw, or of
the fquare of the refidual quantity 1 — x, and confequently that the propofition
contained in art. 25 is true, or that the feries II is really equal to the quantity
m
1 — x\”, in the cafe of the cube-root of the fquare of a refidual quantity, or
when m, the numerator of the fraCtion ~ (which is the index of the power of
mh
1 — x in the quantity 1 — «|*) 1s = 2, and #, the denominator of the faid
faction, 1s = 3.
1
39. As another example of the inveftigation of the value of 1 —x)* by
means of the feries II, we will fuppofe m to be = 3, and z to be = 5, or
= 3.
1 — x|# to be equal to 1 — «5, or to the fifth root of the cube of the refi-
dual quantity 1 —«, or to the fifth root of the quadrinomial quantity 1 — 3%
+ 3x7 — x7.
Now it appears by art. 14 of the foregoing difcourfe concerning the binomial
3 2
theorem, page 205, that the quantity 1 + x]5 is = the feries 1 + $ — 5
4 $s . .
+ 7H" 23") 4 3574" __ & , ~Therefore the feries II will be 1 — = — ote
125 625 15625 F 25
qa 2txt 35785
MEE Ose ic6a5 &c, and confequently, by art. 25, the quantity
3 A 2 3 4 G
7—x\5 will be equal to the feries 1 — = —.— Pb ype! wa a
5 25 125 625 15625
“? DLP S¥CTONUTR S* Ev CO WTC IER Ni EPG
3
40. Now * that this feries is really equal to the quantity 1 — x|5, or to the
«© fifth root of the cube of the refidual quantity 1 — x, or to the fifth root of the
«* guadrinomial quantity 1 — 3x + 3x* — #’,” will appear by raifing the faid
feries to the fifth power by multiplying it, firft, into itfelf, whereby we fhall
obtain its fquare, and afterwards multiplying the faid fquare into itfelf, whereby
we fhall obtain its fourth power, and, laftly, multiplying the faid fourth power
of it into the faid feries itfelf, whereby we fhall obtain its fifth power. For we
fhall find that the produét of thefe three multiplications will be the faid qua-
drinomial quantity 1 — 3x + 3** — «3. Thefe multiplications will be as fol-
low.
ahi La fi hs “e¥ spp Accolia Seah
5 25 125 625 15625
Paid at CY Slip ep tinea fae PY GIN RR! fs
4 Se ae at
;
ee.
— 3 + BE
spi atte
7 Sere Ae ease Gina ahs pA Oe ae
6x 3x 4x3 ox4 12645
ag aa 25 r res + 625 15625 8c
6x 3x 4x3 ox* 126«5
Ngee 5 + 25 126 625 15625 &c
‘ Ox i 30 ay) 4u3 4 gx+ RY 126% Soe
5 25 125 625 15625
_. O*. 36x 1873 2444 54x58 &e
5 25 125 625 3125
hy 1843 gx4 124%
25 125 625 mis 3125 ar Sc
4x3 24x 12x75 4+ &e
625 aia.
126%
&c
15625
12 42x pel 28x%3 __. 21«4 nt 168% 5 arc,
5 25 125 625 15625
This
y eS RS SL DW AL ,.8. 2.0.28 BM. 375
Pe)
This is thé fourth: power of the feries 1 — = — Xai Sia ee ASN. -OCFs
ee . 5 25 125 625 15625
: 12¥ 424% 28x 3 2t¥* 168%5
5 25 eT 625 15625 .
py cia cae ae LAA or 1. ae SRM ee STS ils &c
5 25 125 625 16625
; 124 420% 2843 21x04 16845
5 25 126 625 Ae 16625 wy
ax 36x? 126%3 8444 6345
5 25 125 625 rt 3125 bike
le Ge 36x 126x4 84x5
25 125 625 3125 plas
943 8404 2945
125 x cee ahaa + &e
I
3 le rie ts &c
625 arse
357x5
15625 7 ade
I— 3K + 3x7 — 8 % #
, 2 3 4 $
It appears therefore that the feries 1 — Be cls Sy ana a eal ala PR
26 126 625 15625
&c, is really and truly the fifth root of the quadrinomial quantity 1 — 3x +. 3x°
— «3, or of the cube of the refidual quantity 1 — x, and confequently that the
refidual theorem laid down in art. 25, to wit, that the quantity 1 — x\~ is
equal to the feries Tl, is true in the cafe of the fifth root of the cube of a refidual
quantity, or when m, the numerator of the fraction = (which is the index of
G
m
the power of 1 — # in the quantity 1 — x]*) is = 3, and #, the denominator
of the faid fraction, is = 5.
41. Inthe foregoing examples the denominator z of the fraction = (which ts
74
the index of the power of the refidual quantity 1 — in the quantity 1 — x\”)
has been greater than m, the numerator of the faid fraction. We will now give
m
an example, or two, of the expreffion of the value of the quantity 1 — x)» by
means of the feries II, when the denominator az of the fraction, or index, —, is
lefs than its numerator m.
3
And, firft, we will inveftigate the value of 1 — x|2, or of the fquare-root of
the cube of the refidual quantity 1 — « by means of the faid feries IT.
Now it appears by art. 16 of the foregoing difcourfe concerning the binomia]
. 7 theorem,
376 A DISCOURSE CONCERNING
x Ih . ’ a! 7
theorem, page 207, that the quantity 1 + x)? is equal tothe feries r + as
3 “! - . .
ee 4 3% __ 3*" 4 &c. =Therefore the feries II will be r — 3% = + sae
8 16 720 0Ne He 206
x3 3x4 3x5 f 4 ¢ 6 .
a3 a6 128 as Sy 256 + &c, and confequently, if the propofition Be ey: In
3
art. 25 is true, 1 —.«\2, or the fquare-root of the cube of the refidual quantity
I —w«, or the fquare-root of she obit Mane Tage I — 3x + 3x7 — x3,
2 4
= cae + &c.
will be equal to the feries 1 — = = + a2 hoe
=
42. Now “ that this feries is really equal to the {quare-root of the cube of
‘¢ 7 —w, or to the {quare-root of the quadrinomial quantity 1 — 3”. + 3«7 —
66 x3,” will appear by multiplying the faid feries into itfelf.. For we fhall find
that the produc of the faid multiplication will be the faid quadrinomial quantity.
This multiplication will be as follows.
Boe ae eo ee
ater aes hy ig 256 + &C
8 ‘
cla GAY siege a Es
Sica am 8 TaeiG abst sacha OF
3h «9 SN ae tghe deinen
nipie’ air iat iss Sud 2: 128 SAL Nagar
LN aS DIN at MS «IE
2 4 16 32 266 &e
ll aligh Sc Fy La
+ 3 76 toseqe ares ee e
eS poe
jst oss
128 per te ils 8
s
+ =e
256
Rare Re pee oh pers *% bd
hd
It appears therefore that the feries 1 — ae een ee + + oe + &&
is really and truly the {quare-root of the ie ett’ quantity 1 —= jx + 3x
— x°, or of the cube of the refidual quantity 1 —w, and confequently that the
m
refidual theorem laid down in art. 25, to wit, that the quantity 1 — «| is
equal to the feries II, is true in the cafe of the fquare-root of the cube of the
refidual quantity 1 — x, or when m, the numerator of the fraction — — (which is
mm
the index of the power-of 1 — w in the quantity 1 — x\7) is = 3, and #, the
denominator of the faid fra€tion, is = 2.
43. I
TOMKE IRIE Sel Dw AL. THE OR EM. 379
* . 43. I thall add one more example of the feries 1, in which the numerator
3 2 . :
of the index — fhall be greater than its denominator a,
Let it be required to find, by means of the faid feries II, the value of | — x)3,
or of the cube-root of the fifth power of the refidual quantity 1 —.~, or of the
‘ cube-root of the fextinomial quantity 1 — 5% + 1047 — 1048 + 5xut — x,
It appears by art, 18 of the foregoing difcourfe concerning the binomial
5
theorem, page 208, that the quantity 1 + x\3 is equal to the feries 1 +4. x a
3
io a a
——— — ——— — = Ge
9 Sr 243°. 729 = . : :
Therefore the feries I will be 1 — = + S ++ _ ae Ptah ay MT we &es
2]
+5
and confequently 1 — w|3, or the cube-root-of the fifth power of the refidual
quantity 1 — #, or the cube-root of the fextinomial quantity 1 — 5¥ + 10x*
— 10%% + 5x* — x*, will be equal to the feries 1 — 2 a he coer — ED Sade
bs 9 ey.
7*°
-+- + &c,
© 729
44. Now “ that this feries is really equal to the cube-root of the fifth power
« of 1 — w, or to the cube-root of the fextinomial quantity 1 — 5” + 10% ~
« 1ow%% + 5x* — x°,” will appear by multiplying the faid feries twice into
itfelf. For we fhall find that the product of the faid multiplications will be the
faid fextinomial quantity. ‘Thefe multiplications will be as follow.
5 xt 7x5
ne
LF pitica ot lant ie = + &e
Et Ee Bae
5a 5H 52 Tie: r
it 2 be b 5 fe ee 0 33
aecrbest (27, 243 729 abet?
) ee
Serr ag os
¥ 26x
mg 7p + Be
tei oC
OE EI 2g ae Gra
3 9 81 243 729
ae: -
| 3
374 A DISCOURSE CONCERNING THE RESIDUAL THEOREM,
. ae er ee Ut ) 3 243 129
kd a Sees 54 14
wo CE Wee TUE aan
10% 35.4% 140*3 ght L4«°
Sosa Fak 9 81 2.43 729 &e
esa if 50x” ay 175%3 700.«4 175% ee
3 iB af pa | 729
ba% 508% 175% 700%
T a a} 81 729 wx:
5x3 50x44 Tha &
— XC
a $1 243 720,
5 Sox pr
— Cc
w 243 a9 si
hae
T Syag eae
eT
i — 5x + 10x" — 10K? + 5x* — x? &c.
: 2 3 4 5
It appears therefore that the feries 1 — Sey 3 a + ne + = Occ.
is really and truly the cube-root of the fifth power of the refidual quantity 1 — «,
or the cube-root of thesfextinomial quantity I — 5” + 10%* — 104% - 5.x* —
x, and confequently that the refidual theorem laid down in art. 25, to wit, that
712.
the quantity 1 — «| is equal to the feries II, is true in the cafe of the cube-
root of the fifth power of the refidual quantity 1 — », or when m, the numera-
tor of the fraction — (which is the index of the power of 1 — # in the quantity
mM
1 —«\*) is = §, and#, the denominator of the faid fraction, is = 3.
45. Thefe examples will, I apprehend, be fufficient to illuftrate the two
theorems delivered above in art. 2 and 25 of this difcourfe concerning the
I 2
quantities 1 — x|* and 1 — |, or the mth root of the refidual quantity 1 — ~
and the mth power of the faid mth root. And therefore, having already given
demonftrations of thefe theorems in art. 11, 12, 13, &c....and 26, 27, 30;
31, 32, &c, I thall here. put an end to this difcourfe.
End of the difcourfe concerning the refidual theorem in the cafe of
Sraétional powers. ae
‘eel A Oe Silas Rell G ae Be
OF EXTENDING
fee LD) A Nis ROOD ULE
FOR RESOLVING THE CUBICK EQUATION
Keyan wr yt Par
TO THE RESOLUTION OF THE CUBICK EQUATION
3 rs P g*
VW—yr=r, when —- is of any magnitude le/s than %
4
I q3 : a/9
or — X +, or when ¢r ts lefs than f2 ——;
pies a5" = aes 3/3
By the Help of Sir Ifaac Newton’s binomial and refidual Theorems in the Cafe of
Roots, which have been demonftrated in the two preceeding Difcourfes.
By FRANCIS MASERES, Esa F.R.S.
CURSITOR BARON OF HIS MAJESTY’S COURT OF EXCHEQUER.
3C2
3 Duras TAR AO
x 3 | ~ ae
rhe 3 ; <4 ‘ " Pte 4 ie
ot") iat fT ty ; | sey) a Fal - iN Cf
& MALS SMM Diet ACMA Rh
j ( 4 i ; : Y wt uy Hes i, a .
| . ¥ i -_
CE PT a eee a
Fy Lys oer Us M% RW P| sep aa ie sia
~ = ey ps j ‘ ad ; 4 4 Sis in ‘ “es
it Oe aA or aa ey ET eG io Yew Oe ea
oe : . .
| ak cae aoe Se a Lins Ait wade ce
ils Bhat aah ee Wy) eh atria al ae whe Ch
ry | Sate, Seu, | spk” Deans
is weres E ’ r N : Ef ey i> pet
; | 3
4 ; oe Rr
Me AH GAA Re URGE Nom gt SHS aceon essa apa
) a % _ Db S Mabal as sav
., oe Asesia) gaameaany Men Ne wi romain vet
> a . yf ett 14 - : Pita we ring or
; oi ; a Nee ag : pO Lil ahr : roan LP _" ]
. SF x ry *¥ ’ 7 oe ie , 2 ‘ ' s _ \ A 5 as . > is ; my:
. ‘ vi -! hs wt si
he aa wn pat TATA diowhnag
a he ' i et a os '¢ ty ry 2 & ¥ te EAM. at 40 MOKA a iy te
) | ' 4 ; af pairs Wi f A he “4 rein Si ay { uf ane Sis *" 2M, et S0F ae Mi 2 .
ae Ee FREE DORIS 8 “ed
-
M E T H O D
OF EXTENDING
eR BA.” Nise fe Bol hb. eke.
ART. I. HE binomial and refidual theorems, which have been demon-
{trated in the preceeding tracts both with refpect to integral and
to fractional powers of the quantities 1 + » and 1 — w#, are of very extenfive ufe
in many other branches of the mathematicks as well as in the conftruction of the
logarithms, or meafures, of ratios. And, amongft other fubjedts, they may be
applied to the refolution of cubick equations in which the {quare of the unknown
queue is wanting, to wit, of the equations y* + gy =r, and y? — gy =r, and
i) — 7; fo as to enable us to find the values of y, in all the poffible ‘cafes
of thefe equations, or in all the different relative magnitudes of the co-efficient
g, and the abfolute term 7, that can be fuppofed, not excepting that cafe of the
fecond equation y? — gy = r in which ri is lefs than ans or rr is lefs than an,
or ris lefs than 242, and which (from its not being capable of a dire& and
immediate refolution by the help of Cardan’s rule for refolving the faid equation
3
y? — qy =7, in the firft cafe of it, or when ot is greater than at or 77 is
3 °
greater than ee or 7 is greater than 24v1) has obtained amongft algebraifts the
name of the irreducible cafe. Even in this cafe we may always, by the help of
Sir Ifaac Newton’s binomial and refidual theorems in the cafe of roots (which
have been inveftigated in the two preceeding difcourfes) derive from one or
other, of Cardan’s rules an expreffion of the true value of y in the faid equa-
tion y3 — gy =r. The method of doing this, when — is lefs than E, but
3 3 ; ;
greater than = . 27? OF Ve has been explained at confiderable length in a paper of
mine
982 A METHOD OF EXTENDING CARDAN’S FIRST RULE
»mine publifhed in the Philofophical TranfaCtions for the year 1778, which pro-
ceeds upon clear and intelligible principles, without any mention of impoffible
roots or impoffible Sa of any kind, or even of negative quantities. But when
2 is lefs than , or © orrris lefs than ( 4g"
4 2X27) 54 2
ye V4
ee the method explained in that paper will not enable us to find the value of
ag:
ot or) a? tT is lefs than /2
y in that equation; becaufe the feries obtained in that paper for the faid value
will not in that cafe be a converging feries. Nor had I, at the time of publifhing
that paper, difcovered any method of deriving from Cardan’s rules an expreffion
of the value of y in the faid equation y? — gy =~ 7 in this fecond branch of the
. ° . 3 3
irreducible cafe of it, or when "i was lefs than - - 3? OF ei , or 7 was lefs than
7 aK vs, But I have fince found out a way of doing it by the help of Car-
dan’s firlt rule, which gives us the value of y in the equation y* + gy == 7, or
qy +y* =r. For from the expreflion of this value of y in the faid equation
. 1 ?
qy +9? =r we may, in the cafe fuppofed, or when * is lefs than ae or
ae : A
_, or r is lefs than 4/2 X it, derive an expreffion of the value of the
leffer of the two roots of the equation gy — y? = 7, by the help of the two fe-
riefes for exprefling the cube-roots of the Gitiotad quantity 1 + » and the refi-
dual quantity 1 — » obtained by means of the binomial and refidual theorems :
and from the value of the faid leffer root of the equation gy —y? = 1, we may
afterwards derive the value of y in the equation y? — gy = r by the e refolution of
a quadratick equation. And this extenfion of Cardan’s firft rule Oy hick gives
uis the value of y in the cubick equation y? + gy = 1, or q) + 3 = r), to the
difcovery of the leffer root of the cubick equation gy — y* =r, when_r is lefs
G/4
than 4/2 x =“+, may be made in aclear and intelligible manner, without any
mention of impoifible roots, or other impoffible quantities, or even of negative
quantities, as well as the former extenfion of Cardan’s fecond rule (which gives
us ce value of y in the equation y? — gy =r in the firft cafe of it, or when
it) to the difcovery of the value of
g?
— is greater than 2 re or r is greater than
4
4; in the firft branch of the fecond cafe of that equation, or when - is lefs than,
3 3 . .
ore or-“. The manner of making this extenfion of
atte s firft rule to ae difcovery of the leffer Toot of the SAAR qgy—y=r,
A fhall now endeavour to explain.
, but greater than
The
FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. - 98¢
a I I
The values of the quantities 1 + x\ 3 and 1 — x\3, or 73
(i + w and-4/%% (x - — x, expreffed in infinite -Jeriefes by
means of the binomial and refidual theorems.
2. It has been fhewn above in the difcourfe concerning the binomial theorem
I é
in the cafe of fra€tional powers, art. 51, page 242, that 1 + «|3, or ft + x
or the cube-root of the binomial quantity 1 + «x, is equal to the infinite feries
I 2 5 8 TI 14 1
— v— — Bx — C x3 —_— — x4 faa oe ea wo uy 1}
gg hee Ax z 0 hie peed Te aeuer pnts a =
6 f 2
Gua — £2 Fx + 23° T 9 “3a Ki 2? at satuas — = Mx" + 35 Nx
24 27 3° 33 39 39
sO A ;
= Ox" he a Px — ag Qe" + f Rw? — 3 Sx* +. &c ad infinitum, or
ee eS ga MC rox.“ 22kse rege 37447 93548 21,5059
pe eg, ex 729 7 6561 + 19,683 =a
f 243 729 SMA BQ5093 592949 155945323
55,913x7° T47,4074'* —_ ¥4179,256.477 32174920019 8,61 7,640.r'4
45782,909 14,348,907 129,140,163 387,420,489 1,162,261,467
70,664,648 x75 1945327578247% , §379259,102477 — 13,431,479,050478
ana a — ee See | Ott Cc ad n-
10,460,353,203 — 31,381,059,609 = -944143,178,827 2541, 865,828,329
Jnitum. Therefore (by what is fhewn in the foregoing difcourfe concerning the
refidual theorem in the cafe of fractional powers, art. 23, page 360) the quantity
z
I — “3, or4/*f1 — x, or the cube-root of the refidual quantity 1 — x, will
be equal to a feries confifting of the very fame terms as the foregoing feries, but
with the fign — prefixed to every term after the firft term 1, inftead of every
3 ; I z 8
other term, that is, to the feries 1 — = A om = 1 Bn A aa il 2 al = Dx+
ae hc FOB! ES SE SIAR eee 22 Boke +23 iD +4 =
Ex = Fx is Gr oa Hx P Ix ie
eS Mars By Boge sper Hee Re
42 me
32 Sx* — &c ad infinitum, or 1 PER eRe we Rigs See ea EK Gh | 2A Lally GE
54 ; a ea 243 729 6561
3747 935%" 21,505%2 555913499 147,407.44 1,179,250"?
19,683 59,049 155942323 4,782,969 14,348,907 129,140,163
3174,920473 8,617,640.%14 70,664,648 «15 194,327,78247° 5379259,1024%7
387,420,439 1,162,261,467 10, 46073 53,203 31,381,059,009 94,143,178,827
1394315479050%1"
25541,865,828,329
The
~e.
384 A METHOD OF EXTENDING CARDAN’S FIRST RULE
The value of /3{i +x — V2[1 — x expreffed in an infinite feries.
~
3. Therefore the difference of the two quantities »/*/1 + wand o/*[{r — x,
or the excefs of »/*{1 + x, (or the cube-root of the binomial quantity 1 + x,)
above /{1 — «, (or the cube root of the refidual quantity 1 — wx, will be)
equal to the excefs of the former of thefe two feriefes above the latter, that is
(it, for the fake of brevity, we denote the co-efficients of the fecond, and third,
and fourth, and other following terms of thefe two feriefes by the fingle letters
B; C, D, E; F, G,'H, 1;K; L, M, N,'O,;P,; QR; S,'T, &c)Ptorthe exeefs
of the feries 1 + Bey — Cx? + Dx? — Ex* + Fei — Ge 4) Hy —
Iw? + Kw? — Lx? + Mat — Nx? 4+ Ox3 — Px + Qas — Rav? +
Sw? — Tx" + &c above the feries 1 — Be — Cx? — Dxs — Ext — Fy
— Gr — He — le? Ke = List? ~ Me" — Nx? Oa
Qx — Rat — Sx’? — Tx" — &c, and confequently to the feries 2 Br ++
2Dx3 + 2Fx + 2H»? + 2Kee + 2M" + 2 On + 2Qaus 4 2 Sx?
+ &c.
Of the root of the cubick equationy? + qy =r, orgy +yi? =P,
according to Cardan’s firft rule.
er RE LEE TR re NS
4. The firft of the two rules for the refolution of certain cubick equations ufu-
ally known by the name of Cardan’s rules was not invented by Cardan himfelf,
but, about 30 years before the publication of his book, by one Scipio Ferreus, of
Bononia, or Bologna, in aly ; as Cardan himfelf informs us. It relates to the re-
folution of the cubick equation y? ++ gy = 1, and is true :in all the poffible cafes
of that equation, or in all the different relative magnitudes of g (the co-efficient
of the unknown quantity y), and the abfolute term, and confequently when 7 is
b f ;
lefsthan 42 x 2%%, which is the cafe that we are about to confider. And it
gives us two different expreffions of the value of y in the faid equation, to wit,
eres.
i ate v[e+2 Us aac £ +4“, and, 2dly, 3
r q. orr 3 r a (ESS . Pee
g v {e a a ay: + /f SC ++ re (if, for the fake of bre-
vity, we:put ee = re ants £ + 7 it, fele+s 3% *[e + 5, and,
F 7 adly,
Lo)
FOR THE RESOLUTION OF CUBICK EQUATIONS, &c, 385
odty, lets — {(—e +s, or, rather, (becaufe ss is- greater than re
or e¢, and confequently s is greater than —, or ¢, and therefore ought to- be
af Ne 8
placed before it) 1ft, 1/2? +e 3/3 +e, and, adly; W3ls te — of?
(s —e. See my differtation on the ufe of the negative fign in Algebra, art. 208,
209, 210, 211, 212, pages 178, 179, 180.
It is the fecond of thefe two expreffions, to wit, the expreffion (/3{5 + e—
\/ 3s — ¢, that we fhall have occafion to confider in the courfe of the following
articles,
The value of the root of the faid cubick equation expreffed by an
infinite feries by means of the binomial and refidual the-
orems.
. e : A
Ge Nows eis =s x1 + =) and $ —\e¢ is = 5 x1 —-—. Therefore
‘ RY
VWs +e willbe = Vis x 31 +, and fis —e will be = /* 5 x
Vad —-; and confequently (ste — VW? [s—e will be = > 5
x Va 35 x Vit—t= Vis ae +in-yi{r-4
Therefore the root of the cubick equation)? + gy =7, orgy +y3 =7, will
ber 4/3's x? ppin-ys|p—s.
But, fince /*{1 + « — o/?{1 —« is equal to the feries 2Bx +2Dx* +
2Fxe> +2H«x? + 2K ae + 2Ma™ 4+:2 Ox8 + 2Qxa 4+ 25477 + &
ad infinitum (as is fhewn in art. 3) it follows (by fubftituting = inftead of w in
the terms of the faid equation) that veh +- = _ y>| I — < will be equal to
aco. 2 Be 2D 2Fes 2He7 2 K e? 2mMe™ 2063 2Q0¢5
the feries tse zt se + a 57 5? git 533 st of
il + &c ad infinitum. "Therefore Y?s xl¥?{1 + —-—v3{1 — ~ will be
iT A 5 5
2D ee 2F¢é 2He!7 2K¢ 2mMe™ 20e3
s3 $5 s7 59 ie i3
equal to 7? 5 x the feries — +
p 208% 4 28 | 8c ad infinitum; and confequently the root of the cubick
+ mage + &kca infinitum ; and confequently :
equation y? + gy =r, or gy +.y3 =F, will be equal to Y? s x the faid feries
2Be PEIY GS ol) 3 Bes 2He? 2Ke? 2mMei! 20¢'3 2 gets 2set7 &
ae + me ea? -- —— = ae ae = - -c
53 55 7 5? ches gt3 5 :
JS
Vor, Il. 3 D ad
386 A METHOD OF EXTENDING CARDAN’S FIRST RULE
D F 3 H Hf K e? Met
ad infinitum, or to 24/3 s X the feries ~ — = + = a tept oot oa Fae
0 33 aes $é
p38 pis
cents B, C, D, E, Fy G, Hy, bec) ay#s-x the ies 4. MOP la 2
or fer ie ie a the co-effi-
81 53 2958
374.67 SIT, COk 2S Se 1479407 et! 3,174,920e83 70,664,648 e%5
19,683 57 1,594,3 235? 14,348,907 s** 387,4.20,489573 10, 460,35 3,203 st5
,»162 e?7 Ty RY
eae a + &c ad infinitum. ;
? 2
We will now proceed to give an example of the refolution of a cubick equa-
tion of the faid form y? + 9y = 7, or gy + y? = 7, by means of this expreffion.
An example of the refolution of a cubick equation of the
foregoing form y* + ay =r, r qg+y3 = 7, by
means of the expreffion 2/35 xX the Series - oo ie ee
8153
we “aie 374¢e7 21,505 e9 eprehye:
7298° 1968357 1594532389 -14,348,9075"*
39174,929¢73 70,664,648 e'S §375259,162 677
3875420,4895'% ° — 10,460,353,2035"° 9.4, 1435178,827 577
+ &c ad infinitum,
6. Let the equation that is to be refolved by means of this expreffion, be
ye 15y = 4, or 15y + ys = 4 i
Here g is = 15, andris = 43 and confequently & IN Cee Sy = 5, and
. as 3 ; ; r\2 4
is C= +) te fe and 7 1s (= 4 i fe = 195, aga = is (= ae 2] )
+ 4.
We hall therefore have ss (= % _ ot oa = 125 -- 4) =) 129;nenelieeee
Y 129) = 11.357,816,691, and 3 s et 7? 11.357,010;091) = meee
835, and confequently 2 of? s (= 2 X 2.247,835) = 4.495,670. ~
And we fhall alfo have e (= =) = 2, and confequently = c=
2.000,000,000 )
11.357,816,691
= 0.176,090,181, and — {== _ 0.176,090,181|*) = c.opi congue.
and <> (ey = = 0.176,090,181 X 0.031,006,752) = 0.005,459,9845
5 3 2
and = om 7 x ~ = 0.005,459,984 X 0.031,006,752) = 0.000,169,296,
e7 e5 A Rona Ure
and > (= = X = = 9.000,169,296 X 0.031,006,752) = 0,000,005,249,
9 7 2
and > (= = X = = 0,000,005,249 X 0.031,006,752) = 0.000,00051625
7 and
FOR THE RESOLUTION OF CUBICKR RQVATIONS, &e. 38%
It e? et
and => (= -5 X = = 0.000,000,162 % 0.031,006;752) = 0.000;000;00%,
er3 eit e” ‘
and saz (= a X = 0.000,000,005 X 0.031,006,752) = 0.000,000,000,
é a I é Lk I be
and 4 (Hs ace i Rr OTE, Ogo; Tor), 7 0.058,696,727,
caer. /) 5 TEP 8 — 5X 0.005,459,984
and ¢> (= aq % FW = HX 0-005,459.984 = ees =
0.0275299,920, __ 2205 el ge ae Ar, See
a dys ee 036, and TOF (= Ly MO Fg % 01000:
22X0.000,169,296 __ 0.003,724,512 ‘ 74e7
169,296 == ———_—_—_——— = ete) = 0.000,002,1fo0, and BAe
9529 729 aes gO 519091 an ae
Lea 37 5 iste —_ 374%X0,000,005,249
EE rR sd reer ols
2.00 A 2 5
as = 19000,600,090s and ee (ae eee LO A Ln EOS
19,053 15943238 32592373 fe 1, 5945323
_. 21,505 X0,000,000,162 __ 0.003,483,810
X 0.000,000,162 = —-- ee SE " ) = 0.000,000,002
: ‘ a? Sh ha WA 755949323. Sie wa
47,4076 — 1475407 eH gs L447 _
RR 14,348,907 Uae aus ea
147,407 X 0,000,000,005 0.000, 737,035 ‘
eS a) = 0.000,000,;000. Therefore
142348,907 1493481997 ) soda pect
e@ Se 22¢ 374e 21,505¢ 147,407e"* :
35 ae 8153 72955 19,683 57 155943235? 14,439,907 5" + &c will be
(= 0.058,696,727 + 0.000,337,036 ++ 0.000,005,109 + 0.000,000,099
-F 0.000,000,002, + 0.000,000,000, + &c) = 0.059,038,973; and confe-
quently 2 /? s multiplied into the faid feries will be = 2 »/? s multiplied into
0.059,038;973 = 44953670 X 0:059,038,973 = 0,265,419,739- There-
fore y, or the root of the cubick equation y3 + 157 = 4, or 157 + y3 = 4,
will be = 0.265,419;739- Qs Es dy
47. This value of y approaches very nearly to the truth. For, if we fuppofe
y to be equal to 0.265,419,739, we fhall have yy = 0.070,447,637,850, and
3 = 0.018,698,193, and 15y (= 15 X 0.265,419,739) == 3-981,296,085,
and confequently y? + 15y (= 0.018,698,193 -- 3.981,296,085) = 3.999,
994,278 ;. which differs from the abfolute term, 4, of the propofed equation
y? + 15y = 4 by only 0.000,005,722, or lefs than 0.000,006, ORs rer 791 OF
>
6 millionth parts of an unit, or 6 four-millionth parts, or 3 two-millionth parts,
of the faid abfolute term 4 itfelf.
8. If we were to make the value of y already found, to wit, 0.265,419,7395
the bafis of a further approximation to its true value according to Mr. Raphfon’s
method of refolving equations, by fuppofing y to be equal to 0.265,419,739 +
z, and fubftituting this compound quantity inftead of y in the propofed equa-
tion y? + gy = 4, and then refolving the equation refulting from fuch fubfti-
tution in the fame manner as a fimple equation by omitting all the terms that
involve either the fquare or cube of z, we fhould find that z was equal to
3 Lys 0,000,005, 72 2,000,000,000
I§0211,342,913
388 _ A METHOD OF EXTENDING CARDAN’S FIRST RULE
0.000,005,722,000,000,000
150211,342,913
419,739 + 2, would be (= 0.265,419,739 + 0.000,000,376) = 0.265,420,
115. We may therefore confider this laft number, 0.265,420,115, as being
the true value of y as far as the faid true value can be expretied in nine places of
figures.
= 0.000,000,376, and confequently that y, or 0.265,
Pn en ete mare eee 2S
End of the refolution of the equation y? +- 159 = 4, orisy +y*? = 4.
re EET a email
g. It has been fhewn in the foregoing articles, that, if e be put = -, and ss
3 3 ° . °
beim = aa = or ss + e¢, the root yof the cubick equation y? + gy = r will
ale ge 22¢§ a7de? 21,506.69
3 _ ae - ee
be equal.to, 2 yi £25 ine AMG Se: iat tegen gaay 155943238
De Fe He
heer ee
53
147,407 €™* : : ‘ : Be
MA ABD c ad infinitum, or 2 S the feries —
¥4,348,9075** + & off My VA xX 5 +
K 6? Met o e%3 Qe § et7 : ;
4 ode Soma Hh cae phos Sia eh iG ad infinitum, the terms of which are
the fecond, fourth, fixth, eighth, tenth, and other following terms of the feries
I
. : aa e te. :
which is equal to 1 + — 35 or ¥3{t + —, or the cube-root of the binomial
quantity I + <. Now from this expreffion, which is equal to the value a J
in the equation y? + gy = 7, orgy + 9° = 7, we may, by a peculiar train of
reafoning, derive another expreffion, very much refembling the former, which
fhall be equal to the leffer of the two roots of the equation gv — x3 =r in
which the letters g and 7 denote the {ame known quantities as in the foregoing
equation y? + gy = 7, orgy +y3? = 7. The method of doing this I fhall now
endeavour to explain. ©3
Of the cubick equation qv —x? =r.
10. The equation gv — x? = r is not always poffible, whatever be the mag-
nitudes of g and 7, but only when 7 is equal to, or lefs than, the quantity
svt, or that value of the compound quantity gx — x? which refults from the
3 '
fubftitution of nae or ¥ fz, in its terms inftead of «. If r is equal to this
4
quantity, the equation gv — «3 =, will have only one root, to wit, vt, or
/ [z ; and, when the abfolute term 7 is lefs than the faid quantity 299 the
equation gv — #3 = r will have two roots, of which the leffer will be lefs than
vq
/3°
FOR THE RESOLUTION OF CUBICK EQUATIONS, &c, 389
v4. and the greater will be greater than “1, but lefs than 7. See my differtation
on the ufe of the negative fign in Algebra, art. 114, page 92. It is the leffer of
thefe two roots of the equation gv — x3 =r that I now propofe to find in thofe cafes
of the faid equation in which the abfolute term is lefs not only than tet (which
3’
is its greate{t pofible magnitude), but than 2 x vt, by means of an expreffion
hag
° . . 3
to be derived from the foregoing expreffion 2 Y? s x the feries = -} oa +
RK)
Fes He? K e? Me! o e%3 qets sei7 ; ‘ ‘ X
Mi BP. +. TA. ne -+- pyize + Ym + WW -1 &e ad infinitum, which is
equal to the root y of the equation y3 + gy = 7, or gy +93 =7, the letters ¢
and 7 being fuppofed to ftand for the fame quantities in both equations.
The value of the leffer root of the faid equation expreffed in an infinite feries.
J . 2q3 “, Fi
11. Now, when ris lefsthan ¥ 2 X i, or vr is lefs than Soe or i is lefs
2
3 3 3 3 .
than (ye os IS a al or) J. let oo be taken — 2 o> 2 and) let’ z. be
4 29 2% 27” 54 ; Rie? 4
fubftituted every where, inftead of 5, in the foregoing expreffion 2 ¥* 5 X the
oe BR Des Fe He? K @7 Me" 0 e's ig : : ‘
feries a + — “ie hearer ir ee gore Seggcae ath &c ad infinitum, whereby
the faid expreffion will be changed into the expreffion 2/3 z x the feries
Be De? Fe H e7 K e? M eT Gle*4 : :
nen “- PSs Ft 4 + + rr + cagnts a erry + &c ad infinitum ; and, laftly,
let the fign — be prefixed to the fecond, fourth, fixth, eighth, tenth, and
every following even term of this laft feries, inftead of the fign +. The new
. . . Be
expreffion thereby obtained, to wit, the expreffion 2 7? z X the ferles tone
pe Fe He? Kk ¢9 Me" oe%3
+ => — sr t+ ar — &c will be equal to the lefler root of
the equation gx — x? =r. This propofition we will now endeavour to demon-
ftrate.
ge RE OI TT ES TE ET eS SE
A proof that the infinite feries fet forth in the foregoing article is a converging fertes.
err nS I
12. In the firft place it will be proper to fhew that this feries will be a con-
verging feries. Now this may be {hewn in the manner following. -
Since —, or ¢@, is lefs than oe it follows that >, or 2¢¢, will be lefs than.
cn or che Therefore (fubtracting ¢e, or ane from both fides) 2¢e — ee will
54 2 | |
be lefs than = — “, that is ce will be lefsthan zz. Confequently ¢ will be lefs
2) a :
than
nae
4
39° . A METHOD OF EXTENDING CARDAN’S FIRST RULE
3
. ¢€ . 6
than z, and the a — will be lefs than 1. Therefore the fractions <, 5
es ¢7 e? |e
? 2
52 oF? 9? ott “as,
&c will form a acbnehlnis progreflion, and confequently
9 sags Be De Fe He? Ke? M eT
a fortiori, the feveral terms of the feries — — —— bth om ie hcl
z 23 2S 27 29 zit
4+ 25 — &c (of which the numeral co-efficients B, D, F, H, K, M, O, &c,
alfo Hee a decreafing progreffion) will alfo form 3 a decreafing progreffion, or a
converging feries. ; Git: De
: ae & Des Fe H e7 K ¢?
Therefore the expreffion 2 /? z X the feries — mon le ecg, OP oa ee a
2% 3 25 z7 2?
m e7* or
a a
" = — &c will converge toa certain aie magnitude, and ‘confe-
- 5
quently may be equal to the leffer root of the cubick equation qx—xi or, It
remains that we fhew that it is fo.
Preparations for demonftrating that the foregoing infinite feries
is equal to the leffer root of the equation gx —x* =r.
pes
Be
13. In order to demonftrate that the expreffion 2 /* z x the feries > — >>
Fé He? K e? M ex 0 e783 . ’
er eer din Miners emdesery muna re ree Ao mci ce the leffer root of ny equa
tion gx — «3 =r, it will be neceffary to ref{ume the confideration Ni the ex-
pe} F H e7 K e? mM ™
preffion 24/3 5 x the feries = + <> + > (Meera) awheret Le eto. of + &e
40 infinitum, from which it is derived.
oo Bie Des Fes He7 Ke
Now, fince the expreffion 2 35 x the feries — tingle + ae or
Ir
be vale niinkse as +- &c ad nage is equal to the root of the cubick equation
J
ykqg= a or gy +93 =¥F, it follows that, if we were, firft, to multiply the
De3 Fe 7 09 AE oes
{aid expreffion 2 /* s x the feries + RU + — + <> += += ay
+ &c ed infinitum into the oO ecient 2 ariel then to raife the faid expreffion
Met
20/35 the feries 22 + 25 4 22 Sahel oe aa Ne SS + Bec ad in-.
git
Jinitum to its cube, or third. sere and lafly, to add the faid cube to the faid
product, the fum hence abang would. be eaual to r, to whatever number of
ni ifs Seta
terms the faid feries = + = sf ee +=
continued. For, if this Pa were dick equal &0 a it HES not cg true that the
H : K €9 0 e73
expreflion 24/3 5 X the feries 425 re dhicarick op ote epee —— tar +
&c ad infinitum, was equal to ne root of the” ahh qgitye=en
Thefe operations may be performed in the following manner.
14. Since
FOR THE RESOLUTION OF CUBICK EQUATIONS, &c, 3gr
d : 3 rr:
14. Since = + = is = 85, or (becaufe = fa is == ¢e), 2 ~ ee is = 55, we
| : 3 aw as Bp ee
fhall lhc hapa 6, and 7?'S"47 X [ss — ee = 27 X $5 x1 color and
‘s..
I 2 cat
confequently g = 3 x 5s)3 xr 3 a3x59 x1 ——|3 =3s
a
aa oe
I
— 2
ee) 3 Bee ce D E¢€ F et° Gel?
_ 433 the feries:1 == —— ee =
: 3S pare 3 x 55 54 5° 58 10 siz
2
&c ad infinitum. ‘Therefore the product gy will be = 35 3 x the feries 1 —
2
Bee’ cet. De Fe F el? G elt
af So Spee tegere Some &c ad infinitum x y = 353 x the
(eri Bee ce pe Ee F et? Ge
eens s+) 1). aw O: stexiaghsPncogar
— &c ad infinitum x the ex-
I
— $5 Be pe F eS He? K e? Me re)
Prcinoar 0-5 —9<-the-fenes —— —- pnee—b pe e — + 4+ &c
; ; } Bee c é p Ee? F et? G et
ad infinitum = 6s x the feries 1 — Paes So Sao
Fes He Ke Met 0 e%3
Beclad sss x the feries — “try otter tees be ba
&c ad infinitum = 6 5 into the followitte compound feries, to wit,
Be n*e3 Bc eS BDe? BE €? BF e™? BG et %&
as hah 5 Mk eines tap Game ial, sl?
De BDe> cDe? D7e9 DE e*t DF et3 8
oF fe? MOTTE RE! OE ae Te Me aes S ear ae he st
Fe BF e? CF ec? DE ex* EF e'3 &
i wae he arp
He7 BH e9 cH ett DH et3 &
aA S68) SUT por noo: y
Re BK et CK e*3
* 59 Fy ATE ee oc
M et! BM ets
git gi3 - &c
0 ¢7s
4. Os
ecg |
or, if, for the fake of brevity, we denote this compound feries by the Greek ca~
pital letter I, we fhall have the product gy = 65 x the compound feries I.
15. In the foregoing compound feries T, which is the product of the multi-
Bee ca ee Ee® F et? vets e (of
Peeroncof the feries 1 me ma me ee ome
which all the terms after the firft term 1 are coco with he fign —, or fub-
ae ue? K e?
tracted from the faid firft term) into the feries = — = pS is ake 7 ay
13
pia = sar b &c (of which all the terms ire Hy fut term — are marked
e with
a see
492 A METHOD OF EXTENDING CARDAN’S FIRST RULE
with the fign +, or added to the faid firft term), it is evident that the firft term
of every horizontal row of terms wall be marked with the fign +-,-and that every
following term of the fame row will be marked with the fign —, to whatever
number of terms the faid horizontal rows of terms may be continued, and like-
wife that the firft term of every new herizontal row of terms is placed imme-
diately under the fecond term of the next preceeding horizontal row, becaufe
they both involve the fame power of the fraction <; whence it follows that all
the terms in every vertical column of terms, except the loweft term, will be
marked with the fign —, and that the faid loweft term will be marked with
the fign +. This will appear moft evidently upon performing the multiplica-
tion of the former of the faid feriefes into the latter.
16. If we multiply the foregoing compound feries called I into 65, the
product will be the following compound feries, to wit,
‘ 6 B2¢3 6Bces 6 BDe?7 6 BE e&? 6 BF elt 6 BG e%3 &
BE a ee ie re Cc
6 pe} 6 BpeS 6-cp e7 6 D269 6 DE et! 6 pF e?3
&
aa ah Lae Be IRS 510 —o 73 eed G
6 Fe 6 BF e7 6 cF e9 6 pFelt 6 EF e3
&
Day See ET EE So 2
6 He7 6 BH e? 6cH e¥ 6 pH e3 &
ot 6 Se aie oe ee c
6 Ke? 6 BK el! 6 cK e383 &
F g8 rs) Iz Cc
6 mM e®t 6 BmMet3
far Io T2 — &£¢
5 5
+ 6 o e!3 Se?
ell e
gi
Let this compound feries, for the fake of brevity, be denoted by the Greek
capital letter A. Then, fince the produ& qy is equal to 6s x the compound
feries T', it will alfo be equal to the compound feries A.
17. In the next place we mutt find the cube of the expreffion 2 /?5 x the
3 5 7 9 1 13 mn ite
feries — + == ale = st — at = uF — + a -+- &c ad infinitum, which 1s
equal to y.
j : . 3
Now the cube of this expreffion is = 8s x the cube of the feries “ + — +
orale wien ght =" Sees - ++ &c; which cube may be found by multi-
plying the faid feries twice into itfelf in the manner following.
The
y
tio
FOR THE RESOLUTION OF CUBICK EQUATIONS, &c. 393
The multiplication of the Series * _ = 4 and gor — #3 = 98 in the manner following.
In the egret 15¢%— x3 =r we have feen above that eis = 2, and 2 fs
= 11, and - is = 0.060,606,060,606, and ae i is = 0.000,005,996,277, and
Kee? Fen 3
—~ is = 0,000,000,002,928, and —- * is = 0.000,000,000,001, and 3 2 is
a
ae
Bt a/R R123 28 3500, eran &c. Therefore the feries - hee ae
K e?
a Pi aaa = cap &c will be (= 0.060,606,060,606 + o.cisctod ASME +
©.000,000,002,928 + 0. ciopehaaeer 000,001 + &c) = 0.060,612,059,812, &e,
and 4/3 X the faid feries = = + — = + = a + &c will be (54
2..223,980,090,569,361, &c X o. Beate, one a &e = 8. 895,920,362,277%5
444 X 0,060,612 gwas g ereater than ff, or cr. But the fiens prefixed to the faid
terms will be contrary to thofe which were prefixed to thém in the former
cafe.
Thus, for example, the cube of the difference ee —ffin the former cafe was
eo — 3e4*f? 4 26e7f* — f°; and the cube of the SCLC ASS ee in the pre-
fent cafe is f° — 3 fte? + 3 f%e* —e*®, which confifts of the fame terms (or
the fame powers, and products of ee and #) as are contained in the cube of
ee — ff, to wit, e°—getf* + 3e°*f* —f*; but they are placed ina contrary
order to that in which they fends in the former cafe, and the figns, which are
prefixed to them, are contrary in every term to what they were before. And
the fame thing is true of the fifth power of the difference ee — ff, to wit, e° —
[gw AS ep tke tare ae ar ibe *f* —f*°, and the fifth power of the oppofite
difference ff — ee, to wit, f° — 5f%e? + 10 f%et —10f%e® + 5f%e? — e%,
and of all the following odd powers of the faid oppofite diffetentess ee — ff and
if —¢.
3
145. It follows, therefore, that, if zz be put for fr — e or ff — ee, in the
/
; ; 3 ;
fecond cafe of the equation y* — gy = 7, in which — is lefs than f, or éé is
lefs than ff, the even powers of zz, to wit, zzl’, e@z\4, z2ls, zzl®, &c, or zt,
z*, 2, 27°, &c, will reprefent, or be equal to, the fame terms (or the fame
powers, products, and multiples of the two original quantities ee and r, or ¢é
and fF) in the prefent cafe, as were reprefented by the fame even powers of ss,
to wit, salt, ssl, ol, sie &c, or st, 5°, 37, 3”, &C,) 1m the tomer cares
when = or ee, was greater than - or ff, and ss was put for the difference p
;
ode Z, or ee — ff; and the feveral terms (or powers, products, and multiples, of
7
the original quantities ee and #/) reprefented by the i even ee of zz in the
fecond cafe of the equation y3 —gy =r, when — is lefs than 2 r, or e¢ is lefs
than ff, will have the fame figns ++ aig oo nrchced to them as iis had in the
former cafe of the faid equation, when — was greater than fi, or ¢é was greater
than f#; and the faid terms were reprefented by the even powers of ss; and
the only difference between the terms reprefented by the even powers of zz in
the latter cafe of the faid equation and the terms reprefented by the even powers
of ss in the former cafe of the faid equation will be in the order in which the
{aid terms will be placed in the two cafes, and in which the letters e and fin
the feveral middle terms of the values of the faid even powers of zz and ss
(which will involve both the quantities ce and #) will follow each other.
146, And.
. FOR RESOLVING THE CUBICK EQUATION &c, 537
146. And it likewife follows, in the fecond place, that the odd powers of zz,
to wit, zz itfelf and z2\?, zz)', Zz\7, zz\®, &c, or 2°, 2, 2 ‘ot &c, will
alfo reprefent, or be equal to, the fame terms (or the fame powers, products, and
° . . re e 3 =
multiples of the two original quantities — and ce or ee and ff) in the prefent
cafe as were reprefented by the fame odd powers of ss, to wit, ss itfelf, and s.)3,
= Yr
me ss\7, ss°, &c, or $%, 57°, 5%, 5°, &c, in the former cafe, or when —, or
> 3 3 4 >
ee, was greater than Lor ff; and. ss was put for the difference — — for
2 4 e7
ee— ff. But the figns + and —, that will be prefixed to the terms that are re-
prefented by the faid odd powers of zz, to wit, zz, 2°, 2°, x, 2", 8c, will
be refpectively contrary to thofe which were prefixed to the fame terms in the
former cafe, when they were reprefented by the fame odd powers of ss, to wit,
55) 5°, $*°, 5%, 5, 8c: and the terms reprefented by the faid odd powers of
zz will likewife differ from the terms reprefented by the fame odd powers of ss
in the order in which they will be placed, and in which the letters ¢ and / in
the feveral middle terms of each of the faid powers of ce — ff and ff — ee
(which middle terms will involve both the quantities ee and #) will follow each
other.
147. If therefore, in the fecond cafe of the equation y? — gy =r, in which
‘ 3 é 3 .
+ is lefs than 35 or e¢ is lefs than f, we put zz for - — vs or ff — ce, the fe-
é C 2B E x4 Gx 12° 1 cseae Nn 2 aes ;
en &c, ad infinitum, will re-
prefent, or be equal to, a fyftem of terms derived from the two original quan-
tities — and ©, or ce and ff, that will be the very fame in point of compofition,
4 2
that is, will be the very fame powers, products, and multiples of ee and ff, as the
terms of the complicated feries T', which was derived above, in art. 141, from
ei E s4 G 56 158 es Wis?
RS oe tae wore owner poamee is oe utc vs as &c, ad infinitum, by fub-
ftituting ee — ff in its terms inftead of ss in the former cafe of the equation y3 —
3 °
gy = 1, or when —, or ee, was greater than f, or f. But the terms of the faid
two complicated feriefes, or fyftems of terms, fo reprefented by the two fericfes
Cc 55 E s4 G 5° z38 Lee n si? ni C 2%
Pee to tas ee 8c, ad infinitum, and 1 — — =
pst Se ee _ XS _ kc, ad infinitum, will not all have the f
ae Oe ee CS infinitum, will not a// have the fame
figns + and — prefixed to them; but thofe terms only of the latter complicated
: ; : : C 2% Be
feries, or fy{tem of terms, which is reprefented by the feries 1 — ee ee
I 8 10 N xl2 3 .
kee — &c (which latter fyftem of terms we will, for the
e e ero ez
fake of brevity, denote by the Greek capital letter A), which correfpond to, or
are reprefented by, the terms which involve the even powers of zz, to wit, the
Vou. Il. 3 2 terms
538 A METHOD OF EXTENDING CARDAN’S RULE
Ext, 2°) Nz ‘ .
Aaah Baraat athe &c, will have the fame figns + and — prefixed to them
as were prefixed to them in the former fyftem.of terms, or complicated feries, —
~ : . F ‘ 4
called I (which was derived. above in art. 141 from the feries 1 — £2 — = —
: 2 é
G 56 re he B st? N si ;
= — =f as — ae — &c), when they correfponded to, or were repre-
boi es 4:
fented by, the terms which involved the even powers of ss, to wit, the terms >
:
Lie ee
—=—, -—_
& ek2 )
or are reprefented by, the terms which involve the odd powers of zz, to wit, the
Caz, G2 E 2° .
terms, 31, ate Ks will have contrary figns. prefixed to them to thofe
ee
which were prefixed to them in the former complicated feries T, when they
correfponded to, or were reprefented by, thofe terms of the ferieés 1 — [©
t ee
&c; and thofe terms of the complicated feries A which correfpond to,,
E 5* G 5° 158 I a N 51 : :
cial —z — &c, which involved the odd powers:
oe —
8 ero er
—— — —
e
et €
‘ Cle Gs hth ae ,
of ss, to wit, the terms Heyes 2 eds &c. And, laftly, the order in
which the feveral terms, or powers, products and multiples of the two original
quantities ee and fare placed in the complicated feries A, will be contrary to that
in which the fame terms are placed in the complicated feries F:. But this laft
circumftance will make no change in the magnitude of the terms, nor in that of
the whole feries that is compofed of them, and therefore need not be attended to.
any further.
148. It follows from the laft article that, if the terms of the complicated feries.
A be placed in the fame order. as the terms of the complicated feries T, the
faid complicated feries A will be as follows ; to wit, |
eee
2Ef E f*
Fcky bey Fo Maio
I + pn terig i, eo é
A gen tg cn og nad ae 8 Or is afr
ce et e? Ps elo
N oni 15 nf* 20 Nn f° r6nf? 6nf? y=
Sadat il red peace MOL as oo ee
+ &c, ad infinitum
eae : 4 6 ts Io T2:
449. Therefore, if in the feries 1 — — _ = —_ — _ ~~ — Me _ —
= &c, we change the figns of thofe terms which involve the odd powers of, zz,.
. czz Gz? rez z i
to wit, the terms vad Shy 3 ar &c, and in the terms of the new feries thereby
produced,,
»
FOR RESOLVING THE CUBICK EQUATION &c. 539
Cxzz E 24 1 2° a
produced, to wit, the feries 1 + —— — = EE. —s+ Soot
&c, we fubftitute ff — ee inftead of : 2%, and sce produce a third complicated
feries (which, for the fake of brevity, we will denote by the Greek capital letter A),
this third complicated feries A will confift of the fame terms in point of compofi-
tion (or the fame powers, products, and multiples, of the two original quantities .
ee and ff) as each of the two former complicated feriefes denoted by the Greek
capital letters T and A, and its terms will have the fame figns + and — prefixed
to them as are prefixed to the fame terms in the complicated feries I’, which is
fet down above, in art. 1413; and the only difference between the faid two com-
plicated feriefes T and A will be in the order in which their terms are ranged,
thofe of the feries proceeding according to the powers of the letter ¢, and
thofe of the feries A proceeding according to the powers of the letter f.
150. Now, fince this laft complicated feries A confifts of the fame terms in
point of compofition (or of the fame powers, products, and multiples of the ori-
ginal quantities ee and f#/) as the firft complicated feries T, which is fet down in
art. 141, and has every where the fame figns -- and — prefixed to its terms ;
and it has been fhewn above, in the fame arts 141, that if the faid feries P be
multiplied into 2 /?(e, or 24/3 (-, and the feries thereby produced, to wit, the
complicated feries 2 4/? (? x T, be cubed, or raifed to its third power, by mul-
tiplying it twice into itfelf, and alfo be multiplied into the co-efficient ¢, the faid
product, to wit, g x 2 +/3(e x TF, will beefs than the faid cube, to wit, 8e
x T°, and their difference, to wit, 8e x T? —g x 2y? fe » I, will be equal
to the abfolute term 7 ;—it will follow that, if the third complicated feries A
(which agrees fo entirely in all its terms, and the figns which are prefixed to
them, with the complicated feries T) be multiplied into 2 »/*(e, or 2 73 i.
and the complicated feries thereby produced, to wit, the complicated feries
2/'le x A, be cubed, or raifed to its third power by multiplying it twice in-
2 itfelf, and alfo be multiplied into the co-efficient g, the faid produé, to wit,
x 2 See x A, will be lefs than the faid cube, or 8¢ « A, and their dif-
foe to wit, 8¢ x Aig X 272 x A will be equal. to the abfolute
termirt, <
E 2*
11. But the complicated feries A is equal to the feries 1 + < poelgetrett
6 g 10 12
GAR, 4 RE AA a ge) ad infinitum, from which it was nme by fub-
e° e cn 3
ftituting ff — ¢e in its tert inftead of zz. Therefore 2 ?(e x the faid feries
6 8 Io 12
Tar +t Batata > i +i x= + &c will be equal to 2 3 {e x
ێ
the complicated feries A, or to the Spmetickita feries 2 Yate (ex A. And confe-
4 6
quently, if the quantity .2 vfe x the infinite feries 1 + —— — > +r oe see
3 Lb 2 : ra*
™~ rule
540 A METHOD OF EXTENDING CARDAN’S RULE
8 10 12
ays Jas a ~=— + &c (which is equal to the complicated feries 2 V/ife x
é é
A) be raifed to its ies 2 or third power, by multiplying it twice into itfelf, and
likewife be multiplied into the co-efficient g, the faid produét will be lefs than
the faid cube, and their difference will be equal to the abfolute term 7. There-
C 2% E x4 6 x
fore the faid aera 2/*(e x the infinite feries 1 + —~ — — i =
8 ' te) N
tel “> —X*" 4 &c, ad infinitum, is equal to the root y of the Nae equa-
é
tion y3 — gy = r in the fecond cafe of it, or when - ~ is lefs than 4 me or ¢¢ is lefs
: ‘ 3 rr i :
than ff, and zz is put for the difference a4 sale f— ee; or, if in this fe~
cond cafe of the equation y3— gy = 7, we fubftitute ¢ inftead of r for the ab-
folute term of the equation ; and put g, inftead of e, for half the abfolute term ¢ ;
and denote the root of the equation by the letter x inftead of the letter y, the:
. ° ° . + 6
tranfcendental quantity 2 «/?/[g x the infinite feries 1 + a — <= et
& &
+ &c, ad infinitum, will be equal to the root x of the cubick.
323 L zt0 N 2?
“ge gt
equation x3 — gv = 7. Q@ Es Ik.
End of the Inveftigation of the Tranfcendental Exprefio on 2 Va fe Xx the infinite
C Zz E24 G 2x8 1 x8 Le
erie a CE ee Ne + &e which is: equal te
f te pt ss s g ; rhs Ne
the root x of the cubick equation x? —-qx = t, in sobich t is lefs than — we, Ly 3 le
Ae
reater than f2 x 1¥4
g of ia
A Remark on the foregoing Inveftigation.
152. The foregoing inveftigation i is very abftrufe and difficult, and therefore:
has been fet forth at great length, in order to make the reafonings ufed in it as:
clear as poffible: and I hope the attentive reader will have found them, in ge-
neral, intelligible and fatisfactory. There is, however, one part of the deduc=.
tion which is more fubtle than the reft, and may therefore require fome further
elucidation ; I mean that part of it which is contained in art. 150.. In that ar-
ticle the reafoning is as follows: ‘* Since the complicated feries A confifts of
the very fame terms in point of compofition (or the very fame powers, produtts,.
and multiples of the two original quantities ee and #) as the complicated feries.
I, and with the fame figns + and — prefixed to them; and fince it has been
fhewn in art, 141, that, if the expreffion 2 /? (e x the complicated feries Tbe:
cubed, or raifed to its third power by multiplying it twice into itfelf, and alfo. be
multiplied into the co-efficient g, the faid produ@t g x 2? (e x T will be lefs.
than the faid cube, or 8¢ x I°*, and their difference will be equal to the abfo-
lute:
FOR RESOLVING THE CUBICK EQUATION &c. 541
lute term r, or 2 ¢ ;—it will follow that, if the expreffion 2/3 (¢ x the compli-
cated feries A be cubed, or raifed to its third power by multiplying it twice into
itfelf, and alfo be multiplied into the co-efficient g, the faid product g X 2 / 3(e
x A muft in like manner be lefs than the faid cube, or 8e x A3, and their
difference muft be equal to the abfolute term 7, or 2 ¢.”
An Oljeétion to the Conclufion ftated in the laft Article.
a ta near
153. Now to this conclufion it may be objected, “¢ That the letters 7 and e
do not denote the fame quantities in the two expreffions 2 ,/3 (e x T and
2 /*fe x A, and in the two cafes of the equation y? —gy = r which corre-
fpond to thofe expreffions ; but that in the Pas of the equation y3 —gy=r
294/49
’ BA 3s |
preflion 2 ¥* (e x F the letter e fignifies a quantity greater than f, and that in
the fecond cafe of the equation y3 —qy =r the letter r fignifies a quantity lefs
than aang and in the correfponding expreffion 2 4/3 (e x A the letter ¢ figni-
fies a quantity lefs than f; and that confequently the fuppofed refemblance be-
tween the expreffion 2 f?fe x I and 2/3 fe x Ais only apparent, and not
real, and therefore will not warrant the conclufion drawn from it.”
the letter 7 fignifies a quantity greater than , and in the correfponding ex-
An Anfwer to the faid Objettion.
154. In anfwer to this objection we muft obferve that it never has been af-
ferted, that the expreffion 2 /* (e x the feries A was equal to the expreffion
avi(e x the feries T. For that would not be true; becaufe 2 ¥* fe x Ais
always lefs than 2? (e x I. But it was only faid that thefe two expreffions
confifted of terms compofed in the fame manner of the two original quantities ee
and ff, and that therefore, fince the cube of 2 ¥*{e x T’ was greater than the
rodu& of the multiplication of 2 3 (e x T into the co-efficient g, and their
difference was equal to 7, or 2¢, that is, to the greater value of r or 2e, which
belongs to the firft cafe of the equation y3 —gy = 7, it followed that the cube
of 2+/2(e X A muft be greater than the product of the multiplication of
2./?(e x A into the co-efficient g, and that their difference mult be equal to
the correfponding, or leffer, value of 7, or 2¢, which belongs to the fecond
cafe of the faid equation y* —qgy =r. And this conclufion is moft certainly
juft and true, notwithftanding the inequality of the two values of y in the two
different cafes of the equation y? — gy = 7, and of the two values of ¢ in the
two expreffions 2 of? {¢ x T and 2 /3lex A.
A fur-
542 A METHOD OF EXTENDING CARDAN’S RULE
A further Proof of the Conclufion ftated in Art. 152.
15s. But that the truth of this. conclufion may be made as plain as poflible,
let us fubftitute the letter g inftead of e in the expreffion 2 /* fe x the feries A.
Then will this expreffion become 2 ./*(g x the feries A, or 2 o/° (¢ x the
following complicated feries, to wit, ;
— C
I + o
A ee ci
88 Ee sg
8 g -
jy 4 61/4 41f iss
Bp GER ea! ER ae
L IO L 10 L L Lo
ee Rear eo 7m
6 “4 8 °. [2
“ah | -N4 SMP ey ey Te
— &c, ad infinitum.
We mutt therefore endeavour to prove that, fince the cube of the expreffion
2 4? {e x the complicated feries I (which is fet down above in art. 141) is
greater than the product of the multiplication of the faid expreffion 2 3 (xr
‘nto the co-efficient g, and their difference is equal to 7, or 2¢, it muft follow
that the cube of the other expreflion 2 /: {¢ x the complicated feries A (which
has been juft now fet down) will be greater than the produét of the multiplica-
tion of the faid expreffion 2 4/3 (g x 3
(which has been explained in art. 139, 140, 141, &c..... 157), was that by
which I difcovered this expreffion to be equal to the root x of the equation
— gx = ¢, after having feen it afferted to be fo by Monfieur Clairaut in his Elé-
mens d’ Algébre. But, as the propofition appeared to me a very curious one, and
worthy to be eftablifhed by more than one method of proof, I afterwards fought
for, and difcovered, the long fynthetical demontftration of it which has taken up
fo great a part of this difcourfe, and which, I apprehend, will have confirmed
the truth of it beyond any poffibility of doubt.
161. I will now proceed to give a few examples of the refolution of cubick
equations of the aforefaid form x* — qx = #, in which ¢ is fuppofed to be lefs
. 3
than sit, but greater than /2X iv, or ¢é is f{uppofed to be lefs than ea
aggre taat lefs than £, b 4
coe is e lefs than ~, but greater than
but greater than eben ds fuppofed to b See ae rere)
or —, or £, by means of the foregoing tran{cendental expreffion 2 /?(g x
4
r
2X27 F
d : ‘ C 2% E zt G2 128 mime WN xt? 1 phe
the infinite feries 1 + ——— tees Ot er es +s«CKS, cd infini--
tum, in order to confirm the truth of the reafonings by which the faid feries has
been obtained.
Wot. Il. 4A Examples,
546 A METHOD OF EXTENDING CARDAN’S RULE
' Examples of the refolution of cubick equations of the forego=
ing form, x3 —-qx==t, when the abfolute term t is
les than ws but greater than he 2x v4 or — is
Vid
mn +, but greater th £
lefs than 4 = but g an rr or Fi by means of
the tranfcendental exprefion 2 »/3(g Xx the infinite fe-
Ds C Uz E 24 G 2° 12° Bate N zi
Pes Nate ar ae Serge OW paloOT gia
+. &c, ad infinitum.
EX -A.-M..P-E -E I,
162. Let it be required to refolve the equation #3 — 50” = 120 by means
of the faid tranfcendental expreffion.
. . if . tt . .-
Tere g is = 503 418 = 120; —, or g, is = 60; 70 0! SEs TS ae 3600; g3 Is
ne 125,000
= 124,000; and iis 3
i)
= 4629.629,629,629,629, &c, which is
4]
tt . 4 ;
greater than 3600, or ve Therefore this equation cannot be refolved by Car-
dan’s rule, but may by the setae 24/3 fg x the infinite feries 1 + <= —
&&
: of 6 x3 Le N
ye ee ae = + &c, provided that the faid feries con--
verges.
Now, fince 5 is = 4629.629,629,629,629, &c, and a is 3600, we fhall
27
have 2z (= 4 a -—-i = = 4629.629,629,629,629, &c — it 1029.6295
629,629,629, &c, which is much lefs than 3600, or gg; and confequently the
feries will converge.
.629,629,629,6
163. We fhall therefore have — a Si Bee ay Bes = 0.286,008,2 30,4,
and confequently 5 (= 0:286,008,230,4 |") — 0.001,000;707,00
and = ome = x ae = sania eiubabeiosbeiie x Sikes fo akg = 0,023,395
6
x8 25 Sa (os 756
and Fb Gee PLT Hanae: X 0.286,008 ina = 0.006,691,
i : 355973 7
and pet i a x a = 0.006,691,355.7 X 0.286,008,230,4) = 0,001,913,
7828 3
i Joyand
FOR RESOLVING THE CUBICK EQUATION &c. 547
Fa Cad ay
and ge (=p x aa 0-001,913,782,8 X 0.286,008,230,4) = 0.000,547,
357205
=e 4 at aod
and pes X 0.000,547)357,6 X 0.286,008,230,4) = 0,000,156,
543573
zd zit %
and (= aa x pate fone X 0.286,008,230,4) = O.000,044,
77452
x8 zi6 zz
and a Cas ge X a = 2900 pO NY X 0.286,008,230,4) = 0.000,012,
80557.
= CX 0.286,008,230,4 = — x 0.286,
86,008,
pene ey ie = ae tay ore. ane
Berk 10 '
and — will be (= E x 0.081,800,707,8 = Fe X 0.081,800,707,8 =
10 X 0.081,800,707,8 _ 0.818,007,078,0 aw ;
; ie. Paes RL, = 0.003,366,284,2;
G2e 154
De A © TX 0,0235395,075,0 = a. XK 0.025,995)67 5:6
154 X 0.02 3,395,675,6 3:602,934,042,4,
Zea GELS te oe = abel 5) aaa ') = 0.000,549,144,05
and will be (= I x 0.006,691,355,7 = 85. x 0.006,691,355,7 =
935 X0.006,691,355,7 __ — 6-256s41757995
ee ee IID) — 6.000310 2
59,049 504g fia Mids
— 55,913 Hae
=" LX 6200739137782 50 orn X 0.001,913,782,8 =
6 696
$5913 X 0,001,913,782,8 __ 107,005,337,69 2°94) = 9,000,022,372,1 ;
4,782,969 4,782,969 ;
meee e197 240:
and + ro — will be (=n NS ‘0,000, 54453597,6 = 129,146,164 7+ OO25 47 35736
— 15179,256 X 0,000, 547535755 __ 645-4745733294596) 0.000,00 8,2;
129,140,163 rey 129,140,163 ) ioe 4599 3© 9
ie 1 8.67'7,620 6 8
and - = P X.9.000)1 §6,54857 :=— tedjnsis6u, oF X 0-000,150,545,7
:080, 68,
zk _ 8,617,640 x 0.000.19654847 = 1:349:080,33906810) — 4, 5400,001,16059 ;
1,162,261,467 1,162,261,467
E will be (= R a 10413270782 yg 000;0
and = or will be (= X 0.000,044,774,2 = pies 059,609 9944577452
— 194,327,782 xo. 000044577492 _. SRT TIOrERAIE) == _0.000,000,277,2
31,381, 059,009.” 31,381,059,609
a a — _131431:4791050
and ro will be (= T X 0.000,012,805,7 = ; EA OOS, Edn2G SOSRE2,
— 1354315479,050 X 0.000, 012,805,7 __ —_ 371,999+4915270.585,0 = 0.000.000
805.7 = 25541,905,928,329 2,541,865,828,329 ) ; :
067,6 ;
ye And
548 A METHOD OF EXTENDING CARDAN’S RULE
‘ C zz E 2x4 G2 12° L2to N xt
And confequently the feries 1 + — — ee 4+ 4 ee
Bott R2to Tene 5 e > 5 ms ‘i
——- — —_ — — &c will be
ol4 gto 28
C7 fay 4
== 1.000,000,000,0
+- 0,031,778,692,2 — 0.003,366,284,2
+ 0.000,549,144,;0 — 0.000,105,952,9
+ 0.000,022,372,1 — 0.000,004,998,2
-- 0.000,001,160,7 — 0.000,000,277,2
+ 0.000,000,067,6 — &c
= 1.032,351,436,6 — 0.003,477,512,5
1.028,873,924,1.
Further, fince g is = 60, we fhall have 3 [g = ? (60 = 3-914,867,641,1
and confequently 2 s/?(g (= 2 X 3.914,867,641,1) = 7.829,735,282,26
Therefore the tranfcendental expreffion 2 /?(g x the infinite feries 1 + on oes
I]
E v4 G x8 12° rete N xi Pp zit R 215
ro
282,2 X 1.028,873,924,1 = 8.055,810,464,4. Therefore the root # of the
propofed equation x*—— 50%” = 120 is = 8.055,810,454,4. ee
zr8 ‘
“L ae — &c is = 7,829,735,
6. pee 8 TO' Linke, Iz ere 16
& & «
164. This value of the root x in the equation x3 — 50” = 120 is exact in
the firft feven figures 8.055,810, its more accurate value being 8.055,810,345,
702, as may eafily be found by Mr. Raphfon’s method of approximation.
N. B. This equation «7 — 50% = 120 expreffes the relation between the
diameter of a circle and three chords in it that lie contiguous to each other, and
together take up the arch of a femicircle, and form a trapezium of which the
diameter of the circle is the fourth fide. For, if the three chords are called 4, k,
and ¢, and the diameter of the circle is called x, the relation between them will
be expreffed by the cubick equation «7 — 24 |
—kkp X * = 2bkt3 which, if the
— tt
numbers 3, 4, and 5 are fubftituted inftead of the letters 4, k, and ¢, will be-
come 7— 50% =120. See Sir Ifaac Newton’s drithmetica Univerfalis, edit.
2d, A. D. 1722, page 101.
Exe Avi ee Leer IT.
165. Let it be required to find by means of the fame tranfcendental expref-
Vo ; : . C 2% E xt Gx 12° L xie N23
fion 2,/3j{¢ xX the infinite feries 1 -- —- — -—— pha I tel es ee
Vig ag gh te ge hee ee
+ &c, the root of the equation «7 — «= ~~
In-
FOR RESOLVING THE CUBICK EQUATION &c. 549
. . . . I Zr °
In this equation g is = 1, and fis = ri and confequently —, or g, is = -"
it : : rar Te
x x = = and ie or gg, 1s = 563 and A is = = and - Se =, which is
greater than 6 or i Therefore this equation cannot be refolved by Cardan’s
CUZ
rule, but may by the tranfcendental expreffion 2 4/?(g x the feries 1 + eam
E x4 eer ko Vee 2 nett s ;
= +> = + &c, in cafe that feries is a converging one.
roy
Te bhi 3 ti
Now, fince {is = —, and — is = —, we fhall have zz, or 2. — = = —
; 27 aa 4 36 27 4 27
ee 7) = = = , which is lefs than , or —
Ba 29 X30 F327 X36 37 RG BHO 1 K10k 36” 4?
C ZZ E xt
or gg, in the proportion of 1 to 3. Confequently the feries 1 + rice ps, +
x : ; 3 :
a: a = +- &c will be a converging feries, and the expreffion 2 /?([g x the
faid feries will be equal to the root of the equation x3 — x = —.
3
. . t ° I ZB I
166. Since zz is = ——, and —, or gg, is = —, we thall have = =— =
3 x 36 rots 307 re pins,
} 9-3335333233353 3
and confequently = (= = x = = 0.37253333,3 ee 2:33 99333933303
Bere Nee spy gg 3335333333393 % = ;
Sleds deb] pL oT
2 x4 a I O.EVIgl ii, titst
Coe 0.5 1d Ei sius — 22. 0102702
eH F7%*e ital TAs tas 5 j 3720373
, O350 5
z $ Ig es" 1.050374049,503 7,0
and = (= = x = = 0.037,037,037,0 X ~ = 3234 2 wees =+0,0125
& & && 3 3
: 345207930 3 :
x z I 0.012,345,679,0
and — (= 7 x Fe = 0.012,345,679,0 X -— = Sao ee ==4.0,0045
2 .
b Pigs22 0545
x* ae ae 1) 79'6.004, 1155220 bs
and = = re pes 0.004,11§,226,3 X es ac ope = 0,001,
3715742515
Mh a Ls eae TY __ 0.001,371,742,1
and — (= —- X — = 0.001,371,742,1 X — = ——**— 0.000
gt git x ge 937 274 2 3 3 ) b
: 457924723 3
at a a IT __ 0.000,457,24753, __
Mase saa x Bika 0:000,4.57:247,3 X at SD : ~—:) = 0.000,
1525415573
wr8 16 Sei ee, Bn 0.000, 18 AT 7. 8 Oe
and — (= = X Fe = 0:000;152,41557 X = BALE GY ey ry
050;80552.
Therefore will be (= C X 0.333,333,33323 => X .0-3335333:93393
5 a aa, = 0.037,0373037303
and
350 A METHOD OF EXTENDING CARDAN’S RULE ~
4 Pe
and => will be (=" Hx corre yet, era 6s re X O.1LI,III,III,1 =
TO XO.1LLI PIE TL, 2 UF TEV,ELIGIII,T :
i EE Te OUST 2 47 Si
GRP) se ike sek teas he
and ~~ will be’ (= G K°6169730375037,0 “<= 6561 % 0:03720372037,0 =
15 4 X 0.037,03 7203759 __ 5-793,703,698,0) __
SA \6.500 deere cela PRIOR A ae a 0.000,869,344,5 5
8 °.
and > will be (= EASE 0.01 2,345,679,0 = SRG x 0.012,345,679,0 =
is >
935 X0.012,345,679,0 __ I1.543,209,865,0
pe Meshal Sic bs A BLN GR ae Ure EA LE oe 85,23
593049 snaps 1F 95:40 599
552913 ona
an hls oe sere De F52,966 X 0.004,1153226,.3 =
55913 X0.004,115,226,3 _ 230.094,648,111,9, __
at hn LNA ls ui INA ak tnd ah Pd | See is aie ote) 6 ;
4,782,969 4,782,969 J Roe
Wa. ate _._- 1,179,256
and sar Will Deis ses 0.001,3715742,1 = TERME: X 0.001,371,742,8
pat 1,179,256 X 0.001,371,74251 pee 1617. 635,101,377,6 37756 oe gates oa =" aye 00O, OI2 526, i yo 129,540,163
Pk F i ye Ma 8,617,640
and — wil be. (= BP. x"0.660,457-247 53-6 ican ie X 0.000,457,24.753
— 8,617,640 X 0.000,457247)3 __ 3940-392,622,37240
1,162,261,467 ~~ -15162,261,467
Retin. i —— 19453275782 _
and 76 will be (= R X 0.000,152,415,7 = UAT RRNA
x (2927782 X0.0R0. 1S 41507 ge 298 TES s92 29977 eo a900, 08m, eee
one . 3 > b Dea)
) = 0:000,003,390,2 ;
X 0.000,152,415,7
a 31,381,059,609 ~~ - 31,381,059,609
is — _13,431479,050_ |
and = ~ will bé (==-T. X 0:000,050; 805520 = Finite; 86-435 X 0.000,050,
__ 139431,479,050 X 0.000,050,805,2 _ 682,388.979,431,060,0. __
aoe ra 2.541,805,828,329 ~ —-2,.641,865,828,329 7% O:00G; ORs
268,45
: CUR E x4 G x6 128 L zt N x72
and confequently the feries 1 + elon BLL ty ee "e 2, =
Pp z4 R wd sez . =
porn ree aor — &c will be
== 1,000,000,000,0
++ 0-037;037;037,0 —= 0.004,672,473,7
+ 0.000,869,334,5 —- 0.000,195,485,2
++ 0.000,048,107,0 — 0.000,012,526,1
+ 0.000,003,390,2 — 0.000,000,943,8
+ 0.000,000,268,4 — &c
= 1.037,958,137,1 == 0.004,781,428,8
SS 1.03 331 705;708,3.
Further, fince g is = = we fhall have Vile =a [é = ie WAG °% = ane
an
!
FOR RESOLVING THE CUBICK EQUATION &c. 551
; ,
. Therefore the tranfcendental expreffion
and confequently 2fif{e=
1,817,121
; : : C 2% E x4 G 2° i L x0 N x2
2/’\g x the infinite feries 1 onl a ee as he ee ae
(g + £2 a ae 2 pe iad
p zit rz T zt8 . a ee 2
= peo -- ee — WXc wil cd 1,817,121 x 1:033,176,708,3 —
2.066,353,416,6 _
en 1.137,157,853,8, &c. Therefore the root x of the propofed
equation «7 —x* = = will be equal to 1.137,157,853,8, &c, @: Es. De
*
167. This value of the root x of the equation x? — x = — is exact in the firft
fix figures of it, to wit, 1.137,15, the more accurate value of the faid root being
1.137,158,104, as may eafily be found by Mr. Raphfon’s method of approxi-
mation.
Eee acm Nie Esau III.
168. Let it be required to find the root of the equation *7 — 5x = 4 by
means of the fame tranfcendental expreffion:2 /*(g x the infinite feries 1 +
C 2% E x4 Gz 128 ue N x12
-+- &c.
5 a i eee
meeherg is*—)6 53°F 15'=> 4s =, Oe; terezie sy or gg, is = 43 9° is = 125,
and £ Ey cw os = 4.629,629,629,629, &c, which is greater than 4, or =
Therefore this equation cannot be refolved by Cardan’s rule, but may by means
of the tranfcendental expreffion 2 3 (g x the infinite feries 1 + a —=
G2 rz List? N.wi? ; : oy ee y
+ &c, in cafe the faid feries is a converging
ne Cy eee
2
& &
eee twit
one. |
2 tt
Now, fince = is = 4.629,629,629,629, &c, and mi or gg, 18 = 4, we thal
3 , . . .
have = — a Or ZZ, = 0.629,629,629,629, &c, which is lefs than 4, or gg, in
the proportion of about 6 to 40, which is a pretty large proportion of minority,
and much larger than the proportion of zz to gg in either of the former ex-
E24 G 2 rx
amples; and confequently the faid feries 1 + es) =~ + est + &c
will converge with a greater degree of {wiftnefs than in either of thofe examples.
Therefore the aid tranfcendental expreffion 2 4: (g x the infinite feries 1 +
E zt G 6 I 8 * ¢
oe — — - + &C will be equal to the root « of the equation «3? —
& Pe ce £
5*=4. This expreflion may be computed as follows.
169. Since
aga A METHOD OF EXTENDING CARDAN’s RULE.
169. Since gg is = 4, and - — a Or Z%, 1s = 0.629,629,629,629, &c, we
% 0.629,629,629,629 __
fall have = = EOE = 0.157,407,407,45
BZ
x4 LB
and confequently m3 eee X Ty = 9-157,407,40754 XK 0.157,407,407,4)
$5 &&
=01024,7997,091595
26 net zt BB _
and <5 (= Ge X ge = 1024277720919 X 0-1575407:407,4) = 0.003,900,
09773
ee Zs
and (cer e x ap > PR ASR BO ye 4 X 0.157;407,407,4) = 0.000,613,
, 90452 5
zr? 4 ZL P
and = (3 ra x ae 0.000,613,904,2 X 0.157,407,407,4) = 0.000,096,
653,03
i wre LN
and a — ~~ x ee = 0.000,096,633,0 X 0.157,407,407,4) == 0.000,
“: O15,240,75
and <; (= a x i = 0.000,015,210,7 X 0.157,407,407,4) == 0.000,
002394525
zi6 zit Dad
and =z (= aa X je, =. 9:000,002539432 X 0.157,407,407,4) == 0.000,
roy
000, 376,8 ;
zi8 zi6 8 F¥
and a (= ae XZ = 0.000,000,376,8 X. 0.157,407,407,4) = 0.090,
0000593.
Therefore will be (= C X 0.157,407,407,4 = - X 0.157,407,407,4,
a 9915 79497249704,
She tat) 5 OT ARG ti ag
9
ack,
and —— wil be. (=e AF: x 0,024,7.77,001,0 “== et X 0,024,777,091,9 =
f=)
10 X0.024,797;091,9 ___ 0-24.7,770,919,0, __ ,
and <= wil be (= G.& -0.003,900,097,7 = rat X 0.003,900,09737 =
154 X 0.003,900,097;7 _ 0.600,615,045,8, __ :
To ngnGsGing Deak widane his CEOS a 0.000,091,54352 $
s P j
and al be (=-1,:% 50,000,613,9004;2)— 935 x 0.000,613,904,2 =
& 592049 |
935 X 0.000,613,904,2 __ .0.6'74;000,427,0, __
A tla ai ciate Seti (ee Tinea aval day PE Mg alone lee el
59,049 39,049 PMA ae
Laie ia — _55:913 Nate
and re will be (= L. X 0.000,096,633,0 = RA PRR 0.000,096,633,0 =
«403,040.92 30
we —— ©,000,001.1 29,6 :
and
FOR RESOLVING THE CUBICK EQUATION &c. 553
I, 151 795256_
POP CED X 0,000,015,210,7
IMs 4;
and a will be (= N X 0.000,015,210,7 =
_ 1,179,256 X 0.000,016,210,7 &917-93 75309523952
Pan 129,140,163 Wp. £20;840;003 ) = 0,000,000,1 38,8 5
8 1617,640 _
1,162,261,467
_ 8,617,640 X 0.000,002,394,2 __ 20.632,353,688,0, __
wT 1,162,261,467 = tpit, 162,267,467 ) = 0.000,000,017,7 5
= R X 0.000,000,376,8 = praee yt
31381 059,609
— 194,327) 782 X 0.000,000,376,8 _— 73.222,708,257,6, __ ,
31,381,059,009 “Hs, 9353827059;609 727 Akg Weneas Bekah CX
te
and will be (== \Ts X0.000,000,059,3 = sts oot
Raith _.
and aes will be (= P X 0.000,002,394,2 = X 0.000,002,394,2
X 0.000,000,376,8
2,541,865,828,3 29 X 0.009,000,
13,43 154.79:050 X 0-900,000,05953 __ — 796.486, 707,66550
©5993 = 2,541, 865,828,329 _ ENTE ERE ib ig soe cient
4 8 Io Iz
and confequently the feries 1 + = — = 4 i sal i ip rl tke
&& & & g ae ee
P uk Bat Ta Mas
5.9 ole Na ls &c will be f
= 1.000,000,000,0
+ 0.017,489,711,9 — 0.001,019,63 3,4
+ 0,.000,091,543,2 — 0.000,009,720,7
+ 0.000,001,129,6 — 0.000,000,138,8
-- 0.000,000,017,7 —- 0.000,000,002,3
+ 0.000,000,000,3 == 0.000,000,000,0
= 1.017,582,402,7 —= 0.001,029,495,2
=e HOLG, 5.52,90755.
Further, fince g is = 2, we fhall have 4/3 (¢ = f3 (2 == TZ 605021,049;05
and confequently 2 /3 (g = 2 X 1.259,921,049,8 = 2.519,842,099,6. There-
fore the tranfcendental expreffion 2 3 (g X the agate feries 1 + caleaeen +}.
6 8 10 12 xl4 16
ee — &c will be equal to 2.519,
§ g & & &
842,099,6 X 1.016,552,907,5 = Senta tied Therefore the root of the
propofed equation #3 — 57 = 4. is = 2.561,552,812,7. Q, Ev Ie
170. This number 2.561,552,812,7 is true to ten places of figures, and errs
only in the laft figure, which pent to bean 8 inftead of a 7. For the accurate
ett ot et 1+4.123,106,625,6 , OF gees Yt We
value of w in this equation is ;
2.561,552,812,8. For, if we fubftitute at V7 inftead of w in the compound
quantity #3 ts 5x, we fhall find that the faid s@uasiticy will be equal to 4, which is
the abfolute term of the equation ¥* —5x—= 4, For, if wis = wt, we
Vou. II. 48 fhall
554 A METHOD OF EXTENDING CARDAN’S RULE
TH ZX UX SITAR ZX EX ITAA UAB IT EA XITEIDAIY
: Ce . vie! a Eh Ee cca CS ON dk A RM
fhall have «3 (= ———— 7 oa ‘
— IAB SITES IT IZ 52+ 20/17 au ee baba = _ 139+5,/ 17
was a 8 m4 amo opal” "3 ee
y and 5x
(= 5% att) ae SAE ated confequently #3 — 54 (= tS
a el 9 ot
5 ae fa sosGm 3p or oe
171. Thefe examples fufficiently prove that the expreffion 2 4/* (g x the in-
C2 E 24 G-2x° 1.28 Lat0 N zi? P ait Rzt6 T zt
finite feries I - eg . gt i ¢ "eS o® gt? Ses: ar f pers Y a o gi8
z roy ~
— &c (which we derived from the former expreffion 2 (e x the infinite fe-
Cc ss E s4# G 56 15° L 5% nisi? _. pst4 R 516 T zr8 &
The Ke GOT gag gh ee ee ge ae
by the peculiar train of reafoning ufed in the inyeftigation fet forth in art. 139,
T40,° 447, Sey. 4158) B hie the true Foot of the equation #3 — gx =f,
29Vv'9
3/3 Sep
be refolved by Cardan’s rule ; stided that ¢ (though lefs than Pry: is greater
when ¢ is lefs than = or — is lefs than 4 f or when the faid equation cannot
than 4/72 X if, or that = (though lefs than f) is greater than 2,
I will, however, fubjoin one more example to the fame purpofe; which fhall
be that of the equation x? — 63" = 162, which both Dr. Wallis and Mr. De
Moivre have refolved by extracting what they call the impoffible cube-roots of
the impoffible binomial quantities 81 + / — 2700 and 81 — — 2700.
Now this equation may be refolved by the foregoing expreffion 2/3 (¢ x the
C 2% E 24 G 28 12° Lie N zi”
infinite feries 1 + er er Ci oS oe
gt ty Sabie EF
+ &c, in the man-
ner following.
ED Xr Ae MP bi EB IV.
172. Let it be required to find the root of the equation v3 —634 = 162
by means of the tranfcendental expreffion 24/3 (g¢ x the infinite feries 1 -+
C 2% E.2* Gz 128 Toh N 2i% Pp zi4 RxI5 T x8
ee os a ey —- — —-— Ht ee — XC. aa
ge ‘* ise gt ve fn pe Pas Pine :
°o
infinitum. ,
Tete 71s. 02 sf 18 ce LO a =, or g, is = 813, or gg; is = 65615 —
. Ee : : tt :
iS 21 3 and = is = 9261, which is greater than 6561, or oi, Therefore this
equation cannot be refolved by Cardan’s rule, but may by the expreffion
C 2x E24 G28 128 so Nizt
273(¢ X the infinite feries t + eeipiaiersty 7 por isifueiges ar
&c, in cafe that feries is a converging one.
FOR RESOLVING THE CUBICK EQUATION &C. 555
Now, fince 2 =— is = 9261, and - or gf, is = 6561, we fhall have 2 “ ees
(= 9261 — 661) = = 2700, that is, zz will be = 2700, which is lefs than
6561, or gg, in the proportion of 100 to cS, or) 243. Therefore the feries
C 22 E z* G 26 12° L zt Nz
Efe = tH eae to a —— +. &c will be a converging feries,
and the expreflion 2 f(g x the faid feries will be equal to the root of the.
equation w* — 637= peal
%% ” | 2700
(ee -—
173. Since 2% is = 2700, and = is = 6561, we fhall have — = sor
a ee 5ag) = Or42 14522563 NN oe
4 %
and confequently a C= a x a mar Org. t 1552250535 June ak =
00 XO. 633, Te 26 :
a age) = 041690350,878,05
x6 za BB 100 100 X 0.169,350,878,0
and — (= — a ny BO 0,878,0 So C88 eee oe
wapemry rama = 0.069,691,719,35
28 26 B® : 100 100 X 0.069,691,719,3
and — (= — — = 0.069;691,71 es Ce terreno ee EES
s ( 2° x g : 9> 9 27 953 x 243 243
171,
$9691719392) = 9,028,6795719585
are x8 RR 100 100 X 0.028,679,719,8
and — (= -— — = 0.028,6 19,9 Ed 2 fn tte et a Re EARL,
gio ( 2 x 8 , 79971930 X 243 243
2 ?
pn) F 0.0115802,353,8 5
eR ae i . 100 __ 100X0.011,802,353,8 __
and os (= ——°xX Bx Sent ah Rese! 9 sees ere rr rom
ee) = 0.004544554130 5
zis axe Bw 100 100 X 0,004,445,413,0 __
and — (= — — = 0.00 13,0 — 3. eS
ope 45445,413,0 X so re
eee = 0.001 pM
Uy Bind as BR ___ 100 ¥ 0.001,829,388,0
and =e (= Fa X a, ti 829,388,0 x = ——— =
toc: ee way ee) | 2a = 0.000,7 32,894,655
py sea ae BX 100 __ 100 X 0.000,752,934,5 __
and ro i ~ x in — ine 000,752, 834,55 om aT ped oe ey L: =
ee = 0.000,309,808,4.
Therefore as Will De"(== CO Oty, 221039, 7 ne X 0.411, 522,63357
411562250335
me ESET) = 0.045,724573720 5
4B2 and
556 A METHOD OF EXTENDING CARDAN’S RULE
I |
and == will be (= E x 0.169,350,878,0 = ro X 0.169, 350,878,0° ==
gt
10 X0.169,350,878,0 ___ 1.693,508,780,0 Sey
243 dy 243
dS will be & G x 0.069,6 = a4 6
and —- will be (= G x 0.069,691,719,3 = gs; X. 0.069,691,719,3 =
ea ESP hile
= 0.006,969,1 71393
= 0.001,635,806,2 ;
6561
Les a a ap
and ~F will be Se I x Mee eae a5,08h 0.028,679,719,8 =.
935 X 0.025,079,710, 26.015,530,013,0
safe le LX Bar AK SION eae Lae at, tae WR ah 5 129,45
‘ eye | CARE 24545123,43
LZ ; Se.Q1
and —— will be (= L X 0.011,802,353,8 = aaRe X 0.011,802,353,8 =
& >
555913 X 0.011,802,353,8 _ 659.905,008,019,4
22 eS I *) = 0.000, 1 6
4,782,969 4,782,469 cay : 27379999973
z eee 117025 .
and - “ay ~ will be (= NX> 0.0045445 545 350 = Pid ae 0.004,445,413,0
Lia ey ecern, fore a — 52242-2791952,728,0, __ :
129,140,163 129,140,163 ) = 0,000,040) 59357 3
8,617,6 )
(=P x 0.001,829,388,0 = Mita NEY, 0.001,829,388,0
14 aR pniae
__ 8,617,640 X 0.001,829,388,0 _ 15765.007,204,320,0
Hd 1,162,261,467 ve 1,162,261, 467 has 9.000;01. 955 Ose 2
_ 194,327,782
and ~ a = will be (= R X 0.000,7 52,834,5 = casa cee
3
a, adage 27,782 X 0.000,752,834,5 _ 146,296.658,598,079,0, __ Y
3.1,381,059,009 3.1,381,059,009 ) = 0.000,004,661,93
ana _1314315479,050
and — ae = will be (= T X o. 000,309,808,4 = afar, 2,541,305,828,329
X 0.000,7 52,834, 5.
X 0.000;3095
13)431,479)050 X 0.000 000,309,808,4 __ 4,161,185.034,114,020,0 oa
saa Reaa bec bekees a atbeeatg me —) 0.000,001,,
637,0;
C xz E 24 G2 128 B Zio N 27
and confequently the feries 1 + —— — — + ett eee
Pp 24 R z16 ptt - = s > 5 *
Cambie 7 CET &c will be
é
I.000,000,000,0
+ 0.045,724,737,0 —= 0.006,969,171,9
+ 0.001,635,806,2 —- 0.000,454,123,4.
+ 0.000,137,969,7 — 0.000,040,59357
++ 0.000,013,564,0 — 0.000,004,661,9
+ 0.000,001,637,0 —- &c
= 1.047,5135743,9 — 0.007,468, 55059
ext 3.040,04 6410750.
Further, fince gis = 81, we fhall have /?{g = /?[8 (81 = 4.326,749, and
confequently 2 /?{g (= 2 X 4.326,749) = 8.653,498. Therefore the tranf-
cendental
= >
FOR RESOLVING THE CUBICK EQUATION &c. oy
, ate ke vers ve ‘ E 4 G <6 s
cendéntal expreffion 2 4/?(g x the infinite feries 1 + <= — oc pose AEs [ha
&& & &
Ls Pee ib, “ad <2 ;
Bee Crt eae ae Tree pe ae &e will be = 8.653.498 x 1.040,
045,103,0 = 9.000,028,737,9, and confequently the root of the propofed
€quation x? — 63 x = 162 is equal to 9.000,028,73739. Q: Ew Te
174. This number 9.000,028,737,9 1s true to five places of figures, the true
value of « in the equation «7 — 63* = 162 being the whole number 9, as will
appear by fubfiituting 9 inftead of x in the compound quantity x3 — 63 «. For,
if we fuppofe « to be equal to 9, we fhall have «3 = 729, and 63 x(= 63 X 9)
= 567, and conilequently «* — 63 (= 729 — 567) = 162; and confe-
quently g is equal to the value of x in the equation «* — 63 x = 162.
Q. E. Ds
seeentenenemmnanemennammmenininiat ESS SCTE TIT ET ne ny
A S SC ere oz ols Uns
175. This refolution of the equation x* — 63 x = 162 anfwers to Dr. Wallis’s
refolution of it by extracting the cube-roots of the impoffible binomial quanti-
ties 81 + y — 2700 and 81 — 4“ — 2700, in as much as both refolutions are
originally derived from Cardan’s rule. But the difference between them is, that
the method here delivered is intelligible in every ftep of it, whereas Dr. Wallis’s
method treats of impoffible quantities, or quantities of which no clear idea can
be formed, in the whole courfe of the procefs by which the value of w is invefti-
gated, though it concludes with a refult that is intelligible, to wit, that w is
equal to the fum of the two impoffible quantities — as - X / — 3 and 2 pe
— x ¥ — 3; of which quantities the impoffible members + = xX ¥ — 3 and
—— x VY —3 are equal to each other, and are marked with the contrary figns
2
+ and —, and therefore (when added together in order to obtain the fum of the
faid two impoffible quantities ~ + V — 3 and 2 — a — 3) will deftroy
each other, and leave us only the two poffible members of the faid two quanti-
ties, to wit, 2 and 2, of which the fum is the whole number 9. Doctor Wallis’s
method of finding “ -- = V¥ — 3 and aa _ -. / — 3 to be the cube-roots of
the impoffible binomial quantities $1 +4 /— 2700 and 81 — / — 2700, is
confidered by both Profeffor Saunderfon and Mr. De Moivre as only sentative,
and not likely to fucceed in equations of which the roots are incommenfurable
to unity, which is the cafe with ninety-nine equations out of a hundred, when
the equations are taken at random, and not framed on purpofe with rational
numbers for their roots). But Mr. De Moivre has fupplied this defect, and
given a certain method of finding the cube-roots of fuch impoffible binomial
quantities : but not without the trifection of an angle, or finding (by the help of
a table
558 A METHOD OF EXTENDING CARDAN’S RULE
a table of fines, or otherwife) the cofine of the third part of a circular are of
which the cofine is given; by means of which trifection it is well known (inde-
pendently both of Cardan’s rule, and of Mr. De Moivre’s procefs) that the fe-
cond cafe of the cubick equation y? — gy = r (in which = is lefs than £) may
be refolved. So that Mr. De Moivre’s method of doing this bufinefs, though
more perfect than Dr. Wallis’s, does not feem to be of much ufe in the refolu-
tion of thefe equations. And both methods are equally liable to the objection
above-mentioned, of exhibiting to our eyes during the whole courfe of the pro-
ceffes, a parcel of algebraick quantities, of which our underftandings cannot
form any idea ; though, by means of the ultimate exclufion of thofe quantities,
the refults become intelligible and are true. It is by the introduction of fuch
needlefs difficulties and myfteries into algebra (which, for the moft part, take
their rife from the fuppofition of the exiftence of negative quantities, or quanti-
ties lefs than nothing, or of the poffibility of fubtracting a greater quantity from
a leffer) that the otherwife clear and elegant fcience of algebra has been
clouded and obfcured, and rendered difgufting to numbers of men of learning,
who are poflefled of a juft tafte for reafoning, and could therefore, if they
pleafed, make great advances in the mathematical {ciences, but who are apt to
complain of this branch of them, and defpife it on that account. And, doubt-
lefs, they have too much reafon to do fo; and to fay, in the words of the famous
French mathematician and philofopher, Monfieur Des Cartes, in his diflertation
De Methodo, page 11—Algebram vero, ut folet doceri, animadverti certis regulis et
numerandi formulis ita effe contentam, ut videatur potiis ars quedam confufa, cujus
ufa ingenium quodam modo turbatur et obfcuratur, quam fcientia, qué excolatur et per-
fpicacius reddatur. If this complaint was juft in Des Cartes’s time, there is cer-
tainly much more reafon for it now.
_ 1496. The paflage above alluded to in Dr, Wallis’s Algebra, is in the 48th
chapter, pages 179, 180, of the folio edition printed at London in 1685. And
Mr. De Moivre’s method of extracting the cube-root of an impoffible binomial
quantity, as 81 + / — 2700, or a ++ / —4, is publifhed in the appendix to
the fecond volume of Profeifor Saunderfon’s Algebra, pages 744, 745, 746, 74.7.
It is very ingenious, and fhews that author’s great fkill in the ufe and manage-
ment of algebraick quantities. See alfo on this fubject Monfieur Clairaut’s
Elémens d’ Algébre, and a paper of Mr. Nicole in the Memoirs of the French
Academy of Sciences for the year 1738, pages 99 and too. See alfo Mac
Laurin’s Algebra, part 1, the fupplement-to the 14th chapter, pages 127, 128,
129, 1303; and the Philofophical Tranfactions, No. 451.
Another
q
A ee ct
FOR RESOLVING THE CUBICK EQUATION &c, 559
Another expreffion of the value of the root of the equation
RIMS: but greater
3 3
x? — gx —=t, when t is lefs than
than /2 xX asd or — ; is lefs than + a, om greater than
373
ey os i
ree tA than pr derived from the foregoing expref-
Sion of it.
177- But there is another expreffion for the value of the 1 root # of the equa-
tion «* — gx = ¢ in the cafe here fuppofed, which, as it may be derived from
C2
the foregoing expreffion of it, to wit, 2 /” [¢ x the infinite feries 1 + Lae:
12° L zr N zl? p zis R to a
. - e 5
think, to be omitted. This expreffion does not confift seri or an infinite fe.
ries (as the foregoing expreffion does), but partly of a finite algebraick expref-
fion, and partly of an infinite feries; and fewer terms of the infinite feries are ne-
ceflary to be computed and added together in order to obtain the value of the
: ; . 4
feries to any propofed degree of exactnefs, than of the feries 1 + ri -—- #3 “|.
= ie
follows, to wit, /; erties 4-4/3 [g—z+4av? (gx the infinite feries © = =
< — “o a =f a + &c, ad infinitum; of which expreffion the firft
part, to wit, ‘ist th V3 Ae — 2, is algebraick ; and the latter part, to wit,
AS Pp zi4 et
47% (¢ x the feries — a AR a a - pe 4. =r + &c is tranfcen-
dental.
Bees! Pp zt4 A
. CUZ
The terms of the feries — + —> + —>
ge a eo gio
Lr N 2
++ &c, contained in the foregoing expreffion. It is as
+ &c, are taken
¢ I
from the feries that is equal tolr + =)3 , or the cube root of the binomial quan-
: z : : BZ Cz DZ E 2* FS Gz
tity I + +; to wit, the feries 1 + = — =~ on ee ae arf
H27 128 4 K 29 Tear? M 2?! N xi? 0 233 P zi4 Q2ts nia a $ x7
g! 2 2? gr pr? yt 23 git gs gre gi
18 Ca
— —_ + &c, ad infinitum, by beginning with the third term, a —,*and ae
aay “fourth term reckoned from it. sie Si eee v3
x6 L ze ate as oie
Toe Ps Tay 7s +
ee may be deriv os eee the foregoing tranfcendental expreffion 2 4/3 (fz. x the
4
E 2 Gz 1S, sh v2 p xi4 x16
Pmenteicreas | —s — tt ee ee — — <5
oY 88 g g z g g git — “gis
—- — &c, in the manner following.
ie .
The
560 A METHOD OF EXTENDING CARDAN’S RULE
The derivation of the mixt expreffion of the value of the root
of the cubick equation x3 — gx = t, given in the pre-
ceeding article 177, from the former tranfcendental ex-
preffion of it.
178. By the binomial theorem in the cafe of roots we have /3 [1 4+-— =
: ; . B& C zz D 23 E x4 F 25 Gxt H 7 rae
the infinite (ents 7 Tye tea te tes yer le ee
K 29 i ztO Mit N 31? o 213 Pp t4 Qzts R xt6 $zt7 T 218
9? fae Pei i ee g3 a git gis 7% “gio gt — ge
&c; and, by the refidual theorem in the cafe of roots, we have /3 ae =
| are t BS Cz D x3 Ext F 25 G26 H 27 ref
the in nite EW Miceekivertie wien Ty 2G EAP TT io: vee ae ae
K 2? Li zr° M zit N xt 0 213 Pzt4 Q2i5 AE Sule s zt? Ts
vy fle giidiheag ool a ea ee toe ae
&c. Therefore, if we add /? F446 BY ae = and add the latter of the
two foregoing feriefes (which is det toa F =) to the former feries (which
is equal to 4/3 jt + =) the fums Se a obtained wall be equal to each other ;
that is, /3 (1 tee + /3{1 — 2 will be equal to the infinite feries 2 — ==
&S
2E x4 26% 212° Bae 2 Nght 2 P xi4 ee 2723
ue sgt nt 2 a
I & I z , .
and confequently = x /* F -- , ta X Ve f ae will be equal to the in-
ae C zz E x4 G 2 128 1. Re N x72 P zi4 R 216
finite alo Peete th een pep ee ee
18 2 C2% S 7 as z: 14 E oe
Z . . : CZ 2 & ZB & 272
“= — &c, Let the infinite feries toe +e tH ———
2 ge g g g g
+ &c be added to both fides of the laft equation; and we fhall have = x
% 1 z . ‘ Taz CE 262° sp eonl
Vir bot xv [: — — + the infinite feries eee
Thats 2 ey aay one
14 18 ° . . iz 26 8 To
SO EAeiel) Oem ae OC = te ee feries 1 + —— — — 4 — 4
4 4 6 8 ro
gh ne mts ne && & & & &
NZ zt
—3 we tak ae &c. Therefore (multiplying both fides by
Rem Span Zp awe xt x
2625 ane Therefore, it «3 — gx, or 7,
is lefs than
sea or — is lefs than 4 _ the foregoing method of refolving the
cubick equation #3 — Tee =r will fe PA Cas but, if #3 — gx =r, orr,
1s greater than SWE or re is greater than 4 o it will be practicable.
Art. 10. It now only remains to be proved, that while » increafes, from be-
ing equal to 4, ad infinitum, the compound quantity «3 — gx will likewife in-
creafe from 0 ad infinitum, without ever decreafing. Now this may be demon-
{trated as follows.
Art. 11. It is evident that while w increafes from being equal to ~/q ad infi-
nitum, both the quantities x3 and gx will increafe ad infinitum likewife. But it
does not therefore follow, that the excefs of * above gx will continually increafe
at the fame time. This will depend upon the relation of the contemporary in-
crements of #3 and gw: if the increment of x3 in any given time is equal to the
contemporary increment of gx, the compound quantity «? — gw will neither in-
creafe nor deécreafe, but continue always of the fame magnitude during the faid
time, notwithftanding the increafe of w; if the former increment is lefs than the
2 latter,
THE INVENTION OF CARDAN’S RULES, &é. 585
latter, the faid compound quantity will decreafe; and if it is greater, it will in-
creafe. We mutt therefore enquire whether the increment of x? in any given
time is greater or lefs than the contemporary increment of gx.
Art. 12. Now, if % be put for the increment which « receives ‘in any given
time, the increment of #3 in the fame time will be the excefs of « + x]? above
x3, that is, the excefs of ** + 3v7W + 3xX* + 43 above x3; and the incre-
ment of gx in the fame time will be the excefs of g x * + X, or gx + gx, above
gx ; that is, the increment of x? will be 3¥*4 + 3xv? + 43, and that of gw will
be gx. Now in the equation x3 — gx = r it is evident that wv mutt be greater
than g; for otherwife «* would not be greater than gx, as it is fuppofed to be.
Confequently, «« x % muft be greater than gx’; and, @ fortiori, 3x°X + 3x”
++ %3 (which is more than triple of ~*~) muft be greater than gv; that is, the in-
crement of «3 will be greater than the contemporary increment of gx. Therefore
the excefs of «3 above gx, or the compound quantity «3 — gx, will increafe con-
tinually, without decreafing, while « increafes from /q ad infinitum. QE. D.
Art. 13. It follows, therefore, upon the whole of thefe enquiries, that, if the
compound quantity x? — qx, or, its equal, the abfolute term 7, is lefs than
2409 , or — is lefs than f, it will be impoffible for yz to be equal to, and
3/3 4 :
confequently the foregoing method of refolving the equation «3 — qx = 7 will
be impracticable ; but, if «3 — qv, or r, is greater than 2909" or = is greater
3/3
than r it will be poffible for yz to be equal to 2, and confequently, the fore-
going method of refolving the equation v3 — gx =r will be praéticable. And
thus we fee in what manner it is probable that Carp awn’s rule for refolving the
cubick equation w? — gx =r in the firft cafe of it, or when r is greater than
. 3 . . . .
=i, or oa is greater than = together with the reftrition of it to that firft
cafe, may have been difcovered.
Of the Equation qx — x? =r.
ee ee RR IE IR a ae aes
Art. 14. In the third equation gx — x? = r the terms x? and gw have diffe-
rent figns, as well as in the fecond equation «? — gx =r; and therefore it feems:
to have been natural for the inventor of CarpDAwn’s rules to try both the fubfti-
tutions of yy — z and y + 2 inftead of « in this equation, as well as in that fe-
cond equation, in hopes of an extermination of equal terms that are marked with
contrary figns, and a confequent reduction of the equation to another which,
though of double the dimenfions of the equation gx—w«3= 7, fhould have
been of a fimpler form, and more eafy to.be refolved. But it will be found, upon
trial, that neither of thefe fubftitutions will anfwer the end propofed.
Vou. Il. 4 F Art. 15..
586 A conjecturr, &e,
Art. 15. For, in the firft place, let us fuppofe «tobe =y—z. Then we.
fhall have w= y? — 3972 + gyz—2oy? — 39e X ¥—Z— 23, and gx
=9¢ xX y—z and confequently gy —x?7 = 9X y—%—)' + 392 X fy —Zz
+23, Therefore, gX y—z—y? + 39% X¥ (Y¥—2+ 2? willbe =r. Now
in this equation it is evident, the terms g x [y —z and 39z X [y —z have the
fame figns, and therefore can never deftroy each other. Therefore, no fuch me-
thod of refolving this equation gx—wx> =r as was found above for refolv-
ing the two former equations x? + qx = r and «3? —qv = 7, can be obtained by
fubftituting the difference y — z in it inftead of w.
Art. 16. We will now try the fubftitution of y + z inftead of w in the terms
of this equation.
Now, if « be fuppofed to be = y + z, we fhall have «$ = y3 + 3yyz + 3yzz
+2 =y + 372 X yt e2t2, andgoqyx yt 2, and confequently gx
—ogxy+2—y—3yeX yy +z—2*: Therefore,g x y+ z—y*
—372 X Yt Z—2 will be = 7. |
In this equation it is true indeed that the terms g X y + zand 39% X y— 2
have different figns. But they cannot be equal to each other: for, fince the
three terms y? and 3yz x y—2z and 2? areall marked with the fign —, or are
to be fubtracted from the firft term g x y + 2, and the remainder is = 7, it is
evident that ¢g X y + z mutt be greater than the fum of all the three terms y3,
3yz X y + 2, and z3, taken together, and therefore, @ fortiori, greater than
392 X y + zalone. Therefore, no fuch extermination of equal terms marked
with contrary figns as took place in the transformed equations derived from the
two former equations v3 -+-- gv—=r and «3— gx = 7, can take place in this
transformed equation derived from the equation gx — x3 = 7 by fubftituting
y + 2 in its terms inftead of «; and confequently no fuch method of refolving
the equation gx — «3 = r as has been found for the refolution of the equations
#3 4+ gx = rand «* —gx =7, can be obtained by means of that fubftitution.
Art. 17. Thefe are the methods of inveftigation by which I conceive it to be
probable that Carpan’s rules for the refolution of the cubick equations x3 ++
gx =rand «3 —gx =r, together with the limitation of the rule relating to
the latter of thofe equations, and their inapplicability to the third equation ga
—— *> =r, may have been difcovered by the firft inventors of them.
AN
AN
re Pes GB NG: Dee X
TO THE
Brig ned rs g's West SAT
CONTAINED IN THE FOREGOING PART OF THIS SECOND VOLUME OF
MATHEMATICAL TRACTS, IN PAGES 153, 154, 155, &C, To 169;
INTITLED,
€¢ A DEMONSTRATION OF SIR ISAAC NEWTON’s BINOMIAL THEOREM
IN THE CASE OF INTEGRAL POWERS, OR POWERS OF WHICH
THE INDEXES ARE WHOLE NUMBERS:”
Containing an Inveftigation of the Law by which the co-
efficients of the third and fourth and other following terms
of the feries which is equal to any integral power of a
binomial quantity, are derived from the co-efficient of the
Second term of the faid feries, grounded on a probable in-
duction from particular examples.
By FRANCIS MASERES, Esa. F.R.S.
€URSITOR BARON OF HIS MAJESTY’S COURT OF EXCHEQUER,
Art. 1. WT is fhewn in art. 5 of the foregoing tract, pages 155, 156, that, in:
all integral powers whatfoever of the binomial quantity ¢ + 4, the
literal parts of the terms of the feries which is equal to @ + 4)” (in which the
letter m denotes any whole number whatfoever), will always be P toda Shean |
a” *b, a” 33, &c, of which every term is generated from the next be-
fore it by the multiplication of the fraction a And it is-alfo fhewn in art. 6
of the faid tract, page 156, that the numeral co-efficient of the firft term of
the feries that is equal to a + 2)” muft always be 1, or that the firft term of the
faid feries will always be a”, and that the numeral co-efficient of the fecond
term of the faid feries will always be m, or the index of the power to which a + d
is raifed, or that the fecond term of the faid feries will always be m x a” ‘6
to whatever whole number the lettgr m may be fuppofed to be equal.
3
)
Art. 2. And it is further fhewn in the faid tra& that the numeral co-efficients of
the third, and fourth, and fifth, and other following terms of the feries which is:
Ar equal:
308 ON THE BINOMIAL THEOREM.
equal to @ + 2 ke may always be derived from m, the co-efficient of the fecond
term of the faid feries, by the continual multiplication of the following fractions,
M—l m—2 M—-% mM—4 M—SF
. . ° Mm
to wit, ——; ae aa ase Coes &c, till we come to the fraction ree
which is = 0, or till the faid feries of fraftions is terminated or exhaufted ;.
which fra¢tions, ma ee ara —. &c, are therefore called she ge=
nerating fractions of the co-efficients of the third and other following terms of
the feries which is equal to a + 4)”.
Art. 3. And the method by which it is fhewn in the faid tra& that the faid frace
Fotis eel ae mn 4 M3 &c, are in all cafes, or when m is fuppofed
2
to reprefent any whole number whatfoever, the generating fractions of the co-
efficients of the terms of the feries that is equal to a + b)”; or the fractions by
which the co-efficients of the third and fourth, and other following terms of the
{aid feries, are derived from m, the co-efficient of the fecond term mx a” ‘3,
confifts of the three following parts; to wit, firft, of a demonftration that, if it
be true that thefe are the generating fractions of the co-efficients of the third and
other following terms of the feries that is equal to a + 4|” when m is equal to
any one whole number whatfoever, it will alfo be true that they will be the gene-
rating fractions of the co-efficients of the third and other following terms of the
feries that is equal to a + b\” when m is equal to any other whole number greater
than the former; and, fecondly, of a proof, by actual trials of the co-efficients
of the terms of the feveral feriefes that are equal to a + d|*, 2+ 43, a+ 4,
and a@ + 4}5, that in thefe four feriefes, or when m is equal either to the number 2,
or the number 3, or the number 4, or the number 5, the faid fractions —
ma? 273, and =>, are the generating fractions by which the co-efficients of
3° 4 6 s
the third and other following terms of the faid feriefes (that are equal to 2 + dl*,
a+ 3, a + d+, and a+ d)5) are derived from the co-efficients of the fecond
terms of the faid feriefes refpectively ; and, thirdly, of a conclufion evidently
following from the former two propofitions, to wit, that, to whatever whole
number the index m be fuppofed to be equal, it will always be true that the
3
. : 42 — I _ _ we — ° . .
faid fractions “—, ==", 223, 7=4 2—5 wc, will be the generating fractions
2 4
by the continual multiplication of which the co-efficients of the third and other
following terms of the feries that is equa! to @ + 4}” will be derived from m,
the co-efficient of the fecond term. ‘Thefe reafonings I take to be juft and
clear, and fuch as muft give every reader full fatisfaction as to the truth of the
propofition, or conclufion, obtained by means of them.
Alt ay
ON THE BINOMIAL THEOREM. 58g
Art. 4. But it may be afked, ‘* How came it to be fufpected that the fractions
oe =3, &c, were the generating fractions by which the
S 4 5
“¢ co-efficients of the third and other following terms of the feries that is equal
7 . ° °
£©to a + b| are derived from m, the co-efficient of the fecond term, in any
“¢ one value of the index m, fince it is by no means apparent from the mere
*¢ infpection of the terms of the feriefes that are equal to a + 2\” when m is
** equal to the {mall numbers 2, 3, 4, and 5?”
This is a very natural and reafonable queftion, and well deferves to be con-
fidered ; more efpecially if we recollect that Dr. Wallis informs us that he had
fought for thefe generating fractions without being able to difcover them. And
till a perfon had firft fufpected, and then found upon trial, that thefe are the ge-
nerating fractions of the co-efficients of the terms of the feries that is equal to
a + 2)” in fome of the lower values of m, he could never think of fhewing, in
the method above defcribed, that the fame generating fractions would enable
us to find the co-efficients of the terms of the like feriefes in all other integral
values of m.
Art. 5. Now, inanfwer to this queftion it may be obferved, that thefe fractions
will occur to our notice as the generating fractions of the co-efficients of the
third and other following terms of the feriefes that are equal to a + 2)” in fome
of the lower values of m, if we divide the feveral fucceffive terms of thefe fe-
riefes by the terms next before them, in order to difcover the generating frac-
tions by which they are derived one from another, and then reduce the gene-
rating fractions fo obtained to their loweft deneminations. Thus, for example,
the fixth power of the binomial quantity ¢ + dis = 4° + 64a'b + 15a*d*
+ 204353 + 154a°b* + 6ab5 + O°; 29 eas une) 4 we divide a fecond
term 645d by the firft term 2°, prOLeOr x a and, if
we divide in like manner the third term by the conc: and the fourth term by
the third, and the fifth term by the fourth, and the fixth term by the fifth, and
ash
the feventh, or laft, term by the fixth, the a will be ———— a rere r 2 x =
20 a353 20 15 a7b4 ue ay 6 abs _ ‘
d se 5 x ~ and ——3 0 x and TET Bh read and
> or— X- Ee And confequently, if we muldaly the firft term a° by the
a
quotient i x eo we fhall thereby produce the fecond term 6454; and, if we
multiply the fecond term-6 4°d by “3 x i) we fhall thereby produce the third
term 152d; and, if we multiply the third term 15 4*d* by = x a we
fhall thereby prague i fourth term 2047453 ; and, if we UNICENTS the ame
term 20473 by 4 potas 5, we fhall thereby produce the fifth term 15 470+ 3 and,
2 if
590 ON THE BINOMIAL THEOREM,
if we multiply the fifth term 15 2° by = Ss =, we fhall thereby produce the
fixth term 6 a)‘ ; and, if we multiply the fixth term 6 bs by = — X 4 we hall
thereby produce the feventh, or laft, term 4°. Therefore es generating frac-
tions, by the peaual multiplication of which the fecond, and other following
terms of the feries 2° 4+ 645d + 15a*b” + 204363 + 15307b* + 6455
+ 6° (which is equal to a + 4°) are pan from the fins term @°, are ©
x se 3 sf = x 4 - une hi Klee and ess and confequently
the batcipati fractions, by the eit frlatuiplicatial bf which the numeral
co-efficients of the fecond and other following terms of the faid feries 4° +
6a°b + 154a*b* + 2043b3 + 1507b* 4+. 6 abs + 6° (independently of the
literal parts of the faid yep are derived from 1, oe numeral co-efficient of
the firft term a°, will be ut | Hil 28) Py and +. Now let thefe co-effi-
Bi 140.° 6 67) 207/915 6
: : : 6
cients be reduced to their loweft terms; and they will then be —, 4, 2) ,
6". ;
pat = of which the numerator of the firft term — Is the index 6 of the power
of the binomial quantity to which the faid feries of terms 1s equal, and: the nu-
. . 2 I . ‘ acy
merators of the following fractions 3, Eero ree are derived from the faid
index 6 by the continual fubtraction of 1, and the denominators of the faid frac-
tions are the natural numbers 1, 2, 3, 4, 5, 6, which begin with an unit, and
increafe by the continual addition of 1. This obfervation on the increafe and.
decreafe of the denominators and numerators of the fractions + 2 al - and
6 ; ;
> and. their derivation from the index 6, or oa in the cafe of the feries which.
is equal to a + 4%, is fufficient to have induced the perfon who. fhould have
made it, to conjecture, that pofibly, when the index m was equal to any other
number (fuch as 5, or 4, or 7, or 8), the generating fractions whereby the nu-
meral co-efficients of the third and fourth and other following terms of the feries
that was equal to @ + 4\”, were derived from the numeral co-efficient m of the
fecond term, and from each other, might likewife, when properly reduced, be
found to confit of numerators and denominators that did in like manner de-
creafe from the index m by the continual fubtraétion. of an unit, and increafe
from 1 by the continual addition of an unit; or, in other words, might be equal.
wi las ee as “=, &c. And this conjeCture might have pro-
5
duced a trial whether this ae et place in fome particular examples, and more
efpecially in the feriefes that were equal to @ + Alaa ah 135 a+%l+,a+43,
a+ d\°, a+", anda -+ d\%, and perhaps a few more of the lower inte-
gral powers of a + 4; after which trials, and’ the fuccefs that would have at-
tended them, it would have become fo highly probable that the fame rule would
take-
ON THE BINOMIAL THEOREM. 59or
take place in the feriefes that were equal to any other integral powers of a + 3,
that it would have been almoft impoffible to doubt of it. And then it would
have been natural to endeavour to find fome general demonftration of the truth
of the rule in all integral powers of the binomial quantity @ + 2 whatfoever,
’ which might have led to fuch a demontftration as that which is given in the tract
above-mentioned, which is contained in pages 153,154, 155, &c.... 169,
of this volume. ;
N. B. This method of difcovering (by a conjecture grounded on atrial or two,
in fome particular examples), that the generating fractions by which the co-effi-
cients of the third and fourth and other following terms of the feries that is equal
to a + 4)” (or any integral power of the binomial quantity 2 + 4) are derived
from m (the index of the power to which the faid binomial quantity is raifed)
or from the co-efficient of the fecond term of the faid feries (which is equal to
the faid index) are =, se on HRS a &c, is fuggefted by profeffor
Saunderfon, in the fecond volume of his Algebra, in the chapter on the bino-
mial theorem, where the reader will find a good explanation and illuftration of
the faid celebrated theorem, by a variety of examples, both in the cafe of inte-
gral powers and in the cafe of roots, and other fractional powers, and even in
the cafe of negative powers, and of powers that are both fractional and negative ;
but no demonftration of it in any cafe, not even in that of integral and affirmas
tive powers.
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