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EXTRACTION
REAL ROOTS
NUMERAL EQUATIONS
DENOMINATIONS.
BY WILLIAM HOYLE.
IHO
LONDON:
PRINTED FOR ALL THE BOOKSELLERS,
and
MAY BB HAD OF W. DEAN, MANCHESTER, AND J. WBSTALL; ROCHDALB.
18^6.
CAJORl
INTRODUCTION.
X HE following small treatise being chiefly intended fw:
those who have already made some advance in the science
of algebra, it will only be necessary to inform the general
reader, that the extraction of the roots of an algebraic
equation, and particularly the solution of the irreducible
case in cubic equations, has been assiduously sought after
during the last four centuries, by the most eminent mathe-
maticians of Europe ; amongst whom the following names
{with regard to the subject of the present work) stand pre-
eminent, viz. Scipio Ferreus, Nicholas Tartalea, Hierony-
mus Cardan, Lewis Ferrari, Raphael Bombelli, Vieta,
Albert Girard, Harriot, Oughtred, Descartes, Sir Isaac
Newton, Maclaurin, John and James Bernoulli, Fontaine,
Euler, Waring, Simpson, Legendre, and Lagrange,
To the more advanced mathematician, who is already
acquainted with the different methods employed by the above
mentioned eminent persons, it will be necessary to give a
demonstration of the system by which all the real roots
(either positive or negative) of a numeral equation, con-
taining only one unknown quantity, may be extracted.
iv INTRODUCTION.
In the following examples a number, as near the real
root as possible, is assumed, which is placed in the quotient
or root ; the said number is then involved to within one
power of the given equation, and the different powers multi-
plied by their proper coefficients, the products are then col-
lected into one sum, due regard being had to the signs + or
— , which sum being placed before the numeral (to which the
equation is equal), may be supposed to act as a divisor in
common arithmetic, the numeral as the dividend, and the
root which has been involved as the quotient ; multiply the
divisor by the quotient and it will be the same as though the
root had been involved to the proper powers of the given
equation, as may be seen by the following ; viz. (i?* — ^-i^'-J-
ix^-\-5x'\- 1 ) X A'^i-t'^ — 'Zx"^ +4^-3 -|-5j?^ -^x; the product of
the divisor and quotient is now subtracted from the dividend,
the remainder will be the difference between an equation of
the same order and coefficients, as the given equation, with the
assumed number for its root; to the remainder bring
down as many ciphers as there are units in the highest
power of the given equation; another figure must be
placed in the root, and managed according to the following
formula.
The first figure in the root having had its powers extract-
ed, may now be considered as known to be a part of the real
^'oot, and therefore may be represented by a, the second fi-
gure, which is not yet known to be a part of the real root,
by .r ; the first figure standing to the left may be consider-
ed as a tens figure and the second a unit; from whence the
present figures in the root may be represented by a'\'X; we
4nust new involve this binomial to the highest power of tiic
INTRODUCTION.
given equation ; thus, let the given equation be of the fourth
order, or y^-^by^ — cy^ — dyz=zO^ and the fourth power of a
+^=a*+4a'.r+6a^:i;^+'^^^''+^^; ^" which we may con-
sider a^ to have been already extracted, and as the sum will
have to be muhiplied by x at last, we shall have 4a3-f-6a*x
+4a2"^+a;', but as a may be considered to be ten times the
value of the figure by which it is represented, we have, in the
following examples, begun with a?', and retreated one figure
tothe left, it being the same way that is practised in common
numeral multiplication, in order to save the repetition of ci-
phers : now, as i/^ has only 1 for its co-efficient, the sum of
4a3 4-6a^a?4-'la^+^^ ™^y ^^ now left to represent t/*.
Then a^\^\^ =a^ -^-^a^ x-^-Zax^ -\-x^ where the same allow-
ances for a 3 and a lower power of :r will have to be made as
before, whence we shall have 3a^+3a^-f-;r*, which (as y* is
multiplied by 4-^) must be multiplied by i, and then 3a2Z>+
^ahx-^-hx^ will represent +^^^*
Then a'{-X{^=.a'^-{-2ax^x^ where, proceeding as be-
fore, we have — 2ac — ex to represent — ct/*.
Then aJ^x will be merely — d!, to represent — dy^ there-
fore we shall have Aa^ '\'Qa^ X'\'\ax^ -{-x^
2a^h+2ahx+hx''
— 2ae — ex
—■d
which, added together, will give the second divisor.
But in the addition of numerals it will have to be ob-
served, that a: is a decimal, and that therefore as many ci-
phers must be prefixed to the right of — d as there are units
▼1 INTRODUCTION.
within one in the highest power of the given equation ; i&
the above case it will be 4 — lz=3, the number of ciphers;
and generally let n represent the number of decimals in the
root and m the number of units in the highest power of the
given equation, then the number of ciphers to be added will
be 3/*=m— lX»
j/*=m — 2xn
y3=rm— SXWj&c.
whrch will be readily seen by adding together the root, square,
cube, &c. of a decimal, as .2 .02
.22=. 04 .02«=z:.0004
.2»=.008 .023=. 000008
.2*=. 001 6 .02*=. 000000 16
where, to make an equal quantity of decimals in each line, k
wrll be necessary to add 3 ciphers to .2 two to .04, 1
to .008, 6 to .02, 4 to .0004, and 2 to .000008.
When the second divisor has been multiplied by the last
figure in the root and the product subtracted, to the remain-
der as many ciphers must be brought down as before, and
every thing brought on in the same manner.
The two figures in the root will now be known to be a
part of the real root, and therefore will be a, the figure next
put in the root will be x.
After the first two or three divisors have been got it will
be easily seen what the next quotient figure will be, as the
significant figures, or those to the left in the divisors, will
not then vary much.
INTRODUCTION. VU
This treatise would not have been published in its present
form, if the author could have got it inserted in any of the
periodical publications.
_ThejoIul^^
case in cubic equations was sent to the editor of the Me-
chanics' Magazine, London, on the 1st of August ; its receipt
was acknowledged on the 13th of August, but, as it has ne-
ver yet appeared, what use the editor has made of it is not
known. The solution of a cubic equation was sent to the
editor of the Kaleidogigij^, Liverpool, on the 12th of Sep-
tember, and appeared in it on the 4th of October ; but the
editor, in a note, November 1st, declined inserting, for the
present, any thing more in mathematics.
OldhamyDec. hi, 1825.
Ml-.VK
SOLUTION
EQUATIONS,
CONTAINING ONLY ONE UNKNOWN QUANTITY.
Example I.
Given x^-^-l 5x^+63x^50=0.
1X1=1 1
+63 —15
49)50(1.028=1:
49
64 —15 357604)1000000
—15 715208
49 1st divisor. 35425744)284792000
283405952
1386048
2X2= 4
3x10x2= 60
3x10x10=300
202
—15
—3030
30604
+ 630000
+660604
—303000
357604 2nd divisor.
SOLUTION OF EQUATIONS,
8X8= 64 2048
3X102X8:^ 2448 —15
3x102x102=31212
3145744
+63000000
+66145744
—30720000
—30720
35426744 3rd divisor.
Solution 2. or' — 1 5x^ +6Sx=:50
9)50(6.576=^:
54
6X6=36 6 —
• 63 —15 —725;— 4000
_ >_ —3625
99 —90 :
—90 —49301 )— 375000
— —345107
9 1st divisor.
-4577004)— 29893000
—27462024
—2430976
5X5= 25 125
3x6x5— 90 —15
3x6x6=108
• —18750
11725 +18025
+6300
+ 18025
— 725 2nd divisor.
SOI.UTION OF EQUATIONS.
7X7-= 49
3x65x7= 1365
3x65x65—12675
1307
—15
—1960500
+ 1911199
—49301 3rd divisor.
1281199
+ 630000
+ 1911199
6x6=1 36
3x657x6;:i: 11826
3x657x657r=:1294947
13146
—15
1Q71Q0000
129612996
+ 63000000
+ 192612996
45177004 4fh divisor
+ 192612996
When A—6,576.
;i;3zzi+284.37 10709
— 15a;2=:--648.656640
+63:ir-+4 14.288
— 50.
76
+.002430976 the above remainder.
Solution 3. x^^\5x^+63x=50.
7xT— 49 7 7)50(7.395—^.
+63 —15 49
+ 112 —105 189)1000
—105 567
7 1st divisor. 44991)423000
404919
5172175)28081000
25860875
2220125
SOLUTION OF EQUATIONS.
3X3= 9
3X7X3= 63
3X7X7=147
livisor.
143
—15
-2145
15339
+ 6300
21639
—21450
189 2nd(
9X9= 81
3X73X9= 1971
3X73X73=15987
1469
—15
—22035
3rd divisor.
1618491
+ 630000
2248491
—2203500
44991
5x5= 25
3X739X5= 11085
3x739x739=1638363
14785
—15
—221775
163947175
+ 63000000
226947175
—221775000
5172175 4ih divisor.
SOLUTION OF EQUATIONS.
When x=7,295.
a:3=r+404.4031 54875
~.15:p2z=-^820.290375
+63;r=+465.885
+49.997779875
+ the above remainder = .002220125
50.0000000000
.028
The three values of x -^ +6.576
.395
C+1.0!
of a: ^+6.5
^+7.3i
, 1 . QQQ 5s6cond term with its
"*" * I sign changed.
Ex. 11. Given j:^—8.r» + 14;r»+4;p=:8; or to get the
negative root a;*+8.r' + I4^2 — 437=118.
7X7X7=343 7x7=49
8
392
7 10063)80000C.732
14 70441 — 732=:.r
9800 29753947)95590000
3920 89261841
343
30855146488)63281590000
14063 61710292976
—4000
10063 1st divisor.
1571297024
6 SOLUTION OF EQUATIONS,
3x3x3— 27 3x3= 9 143
4x7x3x3iz= 252 3x7x3:^ 63 14
6X7X7X3= 882 3x7x7.~147
4x7x7x7=1372 2002
15339
1462747 8
12271200
20020000 122712
33753947
— 4000000
29753947 2nd divisor.
2x2x2== 8 2x2= 4 1462
4x73x2x2= 1168 3x73x2= 438 14
6x73x73x2= 63948 3x73X73=15987
4x73x73x73=1556068 20468
1603084
1562474488 8
12824672000
20468000000 12824672
34855146488
- 4000000000
30855146488 3rd divisor.
Solution 2. x^—Sx^ + lix^+ix-^S.
4 2 +8) 8(2.732=ar
—8 14 16
—32 28 —10657)— 80000
8 —74599
28
4
+8 Ist divisor.
4 —16837253)— 54010000
—50511759
— 17057525512)-349824 10000
- 34115051024
—867358976
SOLUTION OF EQUATIONS.
7x7x7= 343
4x2x7x7= 392
6x2x2x7=168
4x2x2x2=32
7X7= 49
3x2x7= 42
3x2x2=12
IfifiQ
53063
—8
47
14
658
—133520
+ 122863
— 10657 2nd divisor.
4000
65800
53063
+ 122863
3x3x3= 27
4x27x3x3=: 972
6x27x27x3= 13122
4x27x27+27=78732
80053947
3x3= 9
3x27x3= 243
3x27x27=2187
221139
—8
—176911200
+ 160073947
543
14
7602
3rd divisor —16837253
4000000
76020000
80053947
160073947
8 SOLUTION OF EQUATIONS.
2x2x2= 8
4x273x2x2= 4368
6x273x273x2= 894348
4x273x273x273=81385668
81475146488
2x2= 4
3x273x2= 1638
3x273x273=223587
22375084
—8
^179000672000
+ 161943146488
5462
14
76468
—17057525512 4th divisor.
4000000000
76468000000
81475146488
161943146488
Solution 3.
343 49
—8
7
14
9800
4000
343
14143
-3920
10223)80000(.763=ar.
71561
—392
13201896)84390000
79211376
13171136187)51786240000
39513408561
12272831439
10223
SOLUTION OF EQUATIONS.
6X6X6= 216
4x7x6x6= 1008
6x7x7x6= 1764
4X7X7X7 = 1372
1558696
6xe=z 36
3x^X6= 126
3X7X7 = 147
146
14
20440000
15996
—8
4000000
1558696
— 127968
25998696
12796800
2nd divisor 13201896
3x3x3=
27
4X76X3X3= 2736
6x76x76x3= 103968
4x76x76x76=1755904
1766328187
3x3= 9
3x76x3= 684
3X76X76=17328
1739649
—8
-13917192
,1523
14
21322000000
4000000000
1766328187
27088328187
-13917192000
13171136187 3rd divisor.
c
10
SOLUTION OF EQUATIONS.
125
Solution 4.
X
25
—8
5
14
—200
70
4
125
-200
1st divisor
— 1
x^^Sx^ + \ix^+4x=S.
—1)8(5.236
-~5
53288)130000
106576
64626747)234240000
193880241
66509117736)403597590000
399054706416
4542883584
2x2x2= 8
4X5X2X2= 80
6x5x5x2= 300
4X5X5X5=500
530808
2x2= 4
3X5X2= 30
3x5x5=75
7804
^8
—62432
102
14
142800
4000
530808
677608
-624320
2nd divisor 53288
3x3x3= 27
. 4X52X3X3= J872
6x52x52X'^^ ^8672
4x52x52x52=562432
567317947
3x3= 9
3x52x3= 468
3x52x52=8112
815889
—8
-6527112
SOLUTION OF EQUATIONS.
u
1043
14
146020000
4000000
567317947
717337947
—652711200
64626747 3rd divisor.
6x6x6= 216
4x523x6x6= 75312
6x523x523x6= 9847044
4 X 523x 523 X 523=572222668
6xQ= 36
3x523x6— 9414
3x523x523=820587
82152876
—8
-657223008
573208125736
10466
14
146524000000
4000000000
573208125736
723732125736
—657223008000
4th divisor 66509117736
Four values of x
~ 1571297024
— 867358976
+ 12272831439
+ 4542883584
J • QQo ^second term with its
I sign changed.
- .732
) +2.732
+ .763
+5.236
12 SOLUTION OF EQUATIONS.
Ex. 11. Given ^*— 12:r2 + 12^-
roots.
8 2 12 —
—12 8
— —24
—24 — 1 1
—4 1st divisor.
—3=0 to find the four
-4) 3(2.858
—8
232)110000
89856
34388125)201440000
171940625
36*491110112)294993750000
291928880896
3064869104
8x8x8= 512
4x2x8x8= 512
6x2x2x8=192
4x2x2x2=132
48
— 12
—576
568.32
12000
—57600
11232 2nd divisor.
5x5x5= 125
4X28X5X5= 2800
6x28x28x5= 23520
4x28x28x28=87808
565
— 12
—6780
90188125
12000000
—67800000
84.388125 3rd divisor.
SOLUTION OF KQUATfONS.
13
8X8X8= 512
4X285X8X8= 72960
6 X 285 X 285 X 8== 3898800
4x285x285x285 = 92596500
929871 101 12
12000000000
—68496000000
5108
— 12
—68496
36491110112 4th divisor.
216
Solution 2.
^4_i2^2 + 12^-— 3ii:z0.
12000 5016)3.0000(.606=^,
-7200 30096
^ 216
—72 —1594953384;— 9600000000
5016 1st divisor. —9569720304
6
-12
30279696
6x6x6= 216
4x60x6x6= 8640
6x60x60x6= 129600
4 X 60x 60x 60=864000
1206 12000000000
—12 877046616
—14472000000
—14472
8T7046616
2nd divisor 1594953384
Solution 3,
64 4
—12
—4800
12000
64
c4__i 2:1:24.120;— 3=0.
7264)3.0000(.443=a;.
29056
2217024)9440000
8868096
. 1748236667)5719040000
7264 1st divisor. 52447i0001
474329999
14 SOLUTION OF EQUATIONS.
4x4X4= 64
4X4X4X4= 256
6X4x4x4= 384
4x4x4x4=256
84
—12
1008
297024
—10080000
12000000
2217024 2ik1 divisor.
3x3x3= 27
4X44X3X3= 1584
6x44x44x3= 34848
4x44x44x44=340736
883
12
10596
344236667
—10596000000
12000000000
1748236667 3rd divisor.
o;^— 12a;2 + I2^— 3=0,
or to tret a — root x^ — 12^* — 12i: — 3=0.
. ^ihcrefore -^3.907
12 —63
27 3 —21) 3(3.907 < •
^ ^ ^ IS a root.
— 36 72249(660000
— 12 650241
+27
— 132231584743)975900000000
—21 1st divisor. 925621093201
50278906799
SOLUTION OF EQUATIONS.
15
9x9x^- '^29
4x:^X9x9= 972
6x3x3x9zzi 486
4x3x3x3=108
69
-12
-828
167049
—82800
—12000
+72249 2nd divisor.
7x7x7= 343
4x390x7x7= 76440
6x390x390x7= 6388200
4 X 390 X 390 x 390=237276000
-93684
237915584743
—93684000000
—12000000000
132231584743 3d divisor.
2.858 +.003064869104
.606 —.000030279696
.443 +.000474329999
-3907 —.050278906799
16 SOLUTION OF EQUATIONS.
Ex. IV. Given ^*+6a;^— 10x3— 112x'2_207j:— 110=0.
4^=256 64 16 —175) 110(4.464 = ^;
6 —10 —700
+.384 —160 16676496)81000000
+256 66705984
—160 4
—448 —112 222439918096)1429401600000
_207 1334639508576
448
Istdiv— 175 2310222205747456)9476209142400000
9240888822989824
2353203194101T6
4X4X4X4= 256 4x4x4= 64
5x4x4x4x4= 1280 4x4x4x4= 256
10X4X4X4X4= 2560 6x4x4x4= 384
10X4X4X4X4= 2560 4x4X4X4=256
' A ^/S » Xf^'JIKJ
297024
17821440
15629056
15629056
33450496
—16774000
4x4= 16
3x4x4= 48
3x4x4 = 48
5296
— 10
2nd divisor 16676496
84
— 112
—9408000
—2070000
—5296000
—52960
— 16774000
SOLUTION OF KQUATIONS. IT
6x6x(3x6= 1296
5X44X6X6X6== 47520
10X44X44X6X6= 696960
10X44X44X44X6=1 51 J 1010
5X44X44X44X44 = 18740480
192586012496
6X6X6= 216
4X44X6X6= 6336
6X44X44X6= 69696
4X44X44X44=340736
347769176
208661505600
192586012496
401247518096
— 178807600000
3rd divisor 222439918096
6x6= 36 886
3x44x6= 792 —112
3x44x44=5808
--.99232
588756
—10
—58875600000
—99232000000
—20700000000
— 178807600000
18 SOLUTION OF EQUATIONS.
4X4X4X4— 256
5X446X4X4X4= 142720
10X446X446X4 X4zz: 31826560
10x416x446x446x4r=i: 3548661440
5 X 446 x446 X 446 X 446=197837875280
1981930598323456
4X4X4= 64
4X446X4X4= 28544
6X446X446X4= 4773984
4X446X446X446=354866144
355343827904
+6
2132062967424000
1981930598323456
4113993565747456
—1803771360000000
4th divisor 2310222205747456
4X4= 16 8924
3X446X4= 5352 ~I12
3X446X446=1596748
999488
59728336
—10
—597283360000000
—999488000000000
— 2O7OCO0OOO0O0OO
—1803771360000000
SOLUTION OF EQUATIONS.
[9
a;S^(\x^^]0xB—]l2x'-—201s—]l0—0. or to find a
-root 6x^—x^ + \0x'^—\l2x^ +207x~l\t^=zO ; on trying
this, — 2 will be found to be a root.
Dividing the equation by j;+2 we have x^-{-4x^ — \Sx^ —
76x — 55=0; or to find a — root ^''* — 4:r' — JSj7^+'^6«*'' —
55^:iO, whence — 1 is evidently a root; and again dividing
this equation by ^-)-^9 ^'^ ^^^^^ x^ -{-Sx^ — 21a; — 55:=:0 : or to
find a — root 3x^ — x^'^2\x — 55=0; by inspection — 5 is a
root, and again dividing, we have a; ^ — 2x — llnzO, we now
have four roots. The original equation will only admit of
one more, and by adding the roots together we find it must
be — , therefore a;*+2a? — 11 =0.
2
2
44
20
486 4924
200 2000
4
64
686 6924
The five roots
r+4.464
C -2.464
4)11(2.464 V^'''^°''r^-'^^^
^ g^ > IS a root.
64)300
256
■
6
686)4400
4U6
6924)28400
27696
704
(n^
>*f
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