EQUATION. 
circle may be one, and the parabola is usu- 
ally assumed for the other. See Simpson’s 
and Maclaurin’s algebra. 
Equations, nature of. Any equation in- 
volving the powers of one unknown quantity 
may be reduced to the form z" — pz' _1 - {- 
q s' 1 ~ 2 , &c. = 0 : here the whole expression 
is equal to nothing, and the terms are ar- 
ranged according to the dimensions of the 
unknown quantity, the coefficient of the 
highest dimension is unity, understood, and 
the coefficients p, q, r, and are affected with 
the proper signs. An equation, where the in- 
dex is of the highest power of the unknown 
quantity is n, is said to be of n dimensions, 
and in speaking simply of an equation of n di- 
mensions, we understand one reduced to the 
above form. Any quantity z m — p z " — 1 -f- 
&c.-f Pz — Q may be supposed 
to arise from the multiplication of z — a X 
z — Sxz — c, Ac. to n factors. For by 
actually multiplying the factors together, 
we obtain a quantity of n dimensions simi- 
lar to the proposed quantity, z n — p z n ~ 1 -|- 
qz n ~ i , &c.; and if a, b, c, Ac. can be so 
assumed that the coefficients of the corre- 
sponding terms in the two quantities become 
equal, the whole expressions coincide. And 
these coefficients may be made equal, be- 
cause these will be n equations, to determine 
n quantities, a, b , c, &c, If then the quanti- 
ties a, b, c, &c. be properly assumed, the 
equation z n — p z” — 1 -j -q z”~ 2 , &c. == 0, is 
the same with z — a )(t ‘ — b X a — c, Ac. 
= 0. The quantities a, b, c, d, Ac. are 
called roots of the equation, or values of z; 
because, if any one of them be substituted 
for z, the whole expression becomes nothing, 
which is the condition proposed by the 
equation. 
Every equation has as many roots as it has 
dimensions. If z n — p z m ~ 1 -J- p z ” — 2 , &c. = 
0, or z — a X z — b x z — c, &c. to n factors 
£= 0, there are n quantities, a, b, c, &c. 
each of which when substituted for z makes 
the whole = 0, because in each case one of 
the factors becomes = 0; but any given 
quantity , different from these, as e when 
substituted for z, gives the product e — a X 
e — b x e — -c, &c. which does not vanish, 
because none of the factors vanish, that is, 
e will not answer the condition which the 
equation requires. 
When one of the roots, a, is obtained, the 
equation z — a X s — b X * — c, &c. = 0, 
z n — p z”- 1 q z n ~ 2 , &c, =0 is divisible 
by ar- — a without a remainder, and is thus 
reducible to z — b X « — c, &c. = 0, an 
equation one dimension lower, whose roots 
are b and c. 
Ex. One root of x l 1 = 0, or x -f- 1 = 
0, and the equation may be depressed to a 
quadratic in the following manner : 
x -j- 1 ).r- 3 — {— 1 ( x 2 — x -|- 1 
X 3 -(- X 1 
— 
— ,r 2 — OC 
+ ■*+ 1 
T-J-l 
Hence the other two roots are the roots of 
the quadratic x 2 — x -f- 1 = 0 . If two 
roots, a and b, be obtained, the equation is 
divisible by x — a x x — b, and may be re- 
duced in the same manner two dimensions 
lower, 
Ex. Two roots of the equation z s — 1 — 
0, are -)- 1 and — 1, or z — 1 = 0, and 
z — j -1 — 0 ; therefore it may be depressed 
to a biquadratic by dividing by z — l x 
z4- i == z 2 — l. 
— 1>S? — 1 _J_ ! 
z 6 — z 4 
-j-2 2 — 1 
Hence the equation z 4 -j- z 2 -}- 1 = 0 con- 
tains the other four roots of the proposed 
equation. 
Conversely, if the equation be divisible 
by x — a Without a remainder, a is a root; 
if by x — a x x — b, a and b are both roots. 
Let Q be the quotient arising from the di- 
vision, then the equation is x — a X x — b 
X Q = 0, in which, if a or b be substituted 
for x the whole vanishes. 
Equations, cubic, solution of, by Cardan's 
rule. Let the equation be reduced to the 
form .x 1 — q x -(- r ~ 0, where q and r ntay 
be positive or negative. 
Assume a: = a -j- b, then the equation be- 
comes a-f- b\ 3 — q X « b -f- r z= 0, or a 3 
— £> 3 — )— 3 a & X a-\-b — q X a-\-b -}-»■== 0 ; 
and since we have two unknown quantities, 
a and 6, and have made only one supposi- 
tion respecting them, viz. that « -| -b — x, 
we are at liberty to make another ; let 3 a ft 
