302 A L G E B R A. 
p— O, and f=i, this progreflion becomes «4-r, r, — 
See. and in all cafes m is a divifor of the firft term of the 
equation. Let therefore i, o, — i, —2, &c. be fubftituted 
for * in the propofed equation, and the differences and 
Turns of the divifors of the refults, and m, o, m, 4 m, Sec. 
taken; then if all the arithmetical progreffions poflible be 
formed out of thefe-quantities, in order, amongft them, 
will be found the progrellion r , ■— n~fr, &c. there¬ 
fore, by trial, the divifor mx 2 -\-nx-\~r will be difeovered, 
if the equation admit one of a quadratic form. 
Let the propofed equation be 3 x 3 4-4jc 3 4-3x c — 2x4-4 
Sup. 
Ref. 
Divifors. 
Sq. 
Sums and Differences. 
Progrefs. 
I 
IO 
1, 2, 5, 10 
3 
1 
i 
j 
i 
OO 
Oj 
—2, 1 
X— 0 
2 
1, 2 
O 
—2, —1, 1, 2 
2, —1 
X — - 1 
6 
1* 2, 3, 6 
3 
3> ^5 4> 5> 9 
—3 
X— -2 
.34 
I» 2 > J 7> 34 
I 2 
— 22, —5, 10,11,13, 14,29,46 
JO, —5 
If an equation be of four or more dimenfions, though 
;t admit of no divifor of the form it may admit 
cue of the form ±mx 2 ±nx±r. To find when this is the 
cafe, let ±mx 2 -\-nx-\-ry. (7— o be the equation; and for 
.v fubftitute, fucceffively, p-\-e, p, p — e. Sc c. then 
it m-.p-\-<\ 2 -\-n.p-\-e-\-r,±.?np*-\-np-\-r,+m .p —ej* -\-n .p—c 4- r 
Sec. are divifors of the refuting quantities; and, if they 
be refpectively fubtraffed from, or added to, m.p^fe] 2 , 
m.p 2 , m .p —c)% 8e c. the remainders, or Aims, are n.p-\-e 
-pr, np-\-r , n.p — e-\-r, &c. which form a decreafing arith¬ 
metical progreflion whofe common difference is ne. When 
From the firft progreflion n— —4, r— 2 ; from the other, 
72— 2, and r ——1 ; therefore, fince m may either be poli¬ 
tive or negative, the divifors to be tried are ±32:'—4x4-2, 
and rt3x“4-2X—1 ; of which —3x“4' 2X “ I > or 3 *'— 2X 
+ J fucceeds; confequently the roots of the equation 
3.**—2x4-1=20, are two roots of the propofed biquadratic. 
The Method of Approximation. 
The mod ufeful and general method of difeovering the 
poflible roots of numeral equations, is approximation. 
Find by trial two numbers, which fubftituted for the un¬ 
known quantity give, one apofitive, and the other a nega¬ 
tive, refult; and-an odd number of roots lies between 
thefe two quantities, that is, one poflible root, at leaft, 
lies between them ; then, by increafing one of the limits, 
and diminilhing the other, an approximation may be made 
to the roof; fubftitute this approximate value, increafed 
or diminiflied by v , for the unknown quantity in the equa¬ 
tion, neglect all the powers of v above the firft, as being 
final! when compared with the other terms, and a Ample 
equation is obtained for determining v nearly; thus a 
nearer approximation is made to the root, and, by repeat¬ 
ing the operation, the approximation may be made to any 
required degree of exadtnefs. 
Ex. Let the roots of the equation^ 3 —374" I=0 be re - 
quired. When 1 is fubftituted for x the refult is —1, and 
when 2 is fubftituted, the refult is 4~3> therefore one root 
lies between 1 and 2; try 1.5, and the refult is—.123, or 
the root lies between 1.5 and 2. 
Let 1.54 -ll= >» then 
y= 3-37_5+<5-75»+4-5*M-» 3 ' 
—3y ——4-5 1 — 3 V 
“ f * 1 —— 1 - 
that is, —. 1 2.5 + 3-75w-H-5^+ ir,: = t >> and > neglecting the 
two laft terms, •—-1254-3-75^22:0, or v ——— = -°33 
3-75 
nearly, and y— 1 .54-12=1-533 nearly; again, fuppofe 
1.5334-21—.y; by proceeding as before we find .003686437 
, , a —-003686437 
4-4.oco267tc=o, and v— ---=—.0009101, See. 
4.050267 
hencej'=i.532089 nearly. The other roots may be found 
by the folution of a quadratic. 
If two roots rt-fnz, a-\-n, be nearly equal to each other 
and to <2, the quantities m and n, muff both be determined; 
therefore the third term of the equation mull be retained, 
or a quadratic, P — Qv-\-Rv 2 ~o, folved. 
When m and n are much lefs than r, s, See. and both po- 
P 1 m 
fitive or both negative, then -73=;-=2-, near- 
Q. 1 . 1 . rn 
— 4— 1 -— 
m n n 
]y, which is an approximation to in the lefs of the two; 
.but, if one of thefe quantities be politive and the other ne¬ 
gative, —4— may be either politive or negative, and 
lit 7) A at 
greater, equal to, or lefs, than -4 - p See. and confe- 
r s 
P 
quervtly — is not neceftarily an approximation to any of the 
quantities in, n, r, Sec. 
Let P— Qy-\-Rv 2 —o ; the roots of this equation will be 
m and n nearly. For if m, n, r, s, be the roots of the 
equation P — Qv-\-Rv 2 —Sw 3 -p &c. —0, P—mnrs, Q—mns 
-ymnr-\-mrs -\-nrs, R—mn-^mr^ms-^nr^ns-^rs, and 
fince m and n are fmall when compared with r and s, Q — 
mrs^-nrs nearly, and R—rs nearly; therefore the equation 
P — Ov-\-Rzj 2 =zo becomes mnrs — mrs-\-nrs.v-{-rsv 2 =z o-; 
hence, v 2 — m-\-n.v-\-w!—o, whofe roots are m and n. By 
the folution then of this quadratic a much nearer approxi¬ 
mation is made to the root a-\-m than by the former me¬ 
thod, and at the fame time an approximation is alfo made 
to the root a-pu ; but in this cafe, as in the former, m and 
n mull both be politive or both negative, otherwife mns 
and mnr do not nece-flarily vanilh with refpeft to mrs-\-nrs, 
w hich is fuppofed in determining the value of Q. 
On the Rever/ion of Series. 
If two quantities Tx-p5x 2 -pCx 3 -p &c. and ax-p£x 2 -p. 
c * 3 + Sec. be always equal, the invariable coefficients of 
the correfponding terms are equal. For if thefe equal 
quantities be divided by x, we have Sec. 
a-^bx-^cx 2 -\- Sec. or when x is indefinitely fmall, A—a, 
and A and a are invariable ; therefore, in all cafes, A=a. 
Hence alfo Bx4-f’x s 4-&c. =z6.v-prx 2 -p &c. or, dividing 
by x, 5-pCx-p Sec. —b^-cx-\- Sec. and when x is indefi¬ 
nitely fmall, 3— b, therefore in all cafes B—b. In the 
fame manner C—c, Sec. 
Cor. If Sec. —o in ali cafes, then 
M=o, 5—o, C=. o, Sec. 
Approximation may be made to a root of an equation by 
aftuming for it a feries, involving the powers of that quan¬ 
tity in terms of which it is fought, with indeterminate co¬ 
efficients; this feries being fubftituted for the unknown 
quantity in the propofed equation, the coefficients may be 
found by making each term equal to o, and thus the feries 
which expreftes the value of the unknown quantity may be 
determined. 
Ex. Letjy 3 —3jy4-*=°i required the value of y in terms 
of x. 
I.etjV2=ax4-^x 3 4-rx 5 4-ifx 7 4- Sec. then 
_y 3 — a :i x 2 f-s > a 2 bx 1 -\-T,a 2 cx' 1 f- See. 
4-3a^ a x 7 
-—3 y ——3 ax—3&X 3 — 3rx 5 — jdx 7 — &c. 
4*x 2=4“ x 
and, fuppoiing each term to vanilh, —3a-j-i=20, or 
3 
