EQUATION. 
= s / —n X ji# 2 — «, a quantity manifest- 
ly impossible, unless n be negative, that is, 
unless two roots of the proposed cubic be 
impossible. 
Equations, biquadratic, solution of, by 
Des Carte’s method. Any biquadratic may 
be reduced to the form x 4 -|- q x 2 -f- rx + 
s = 0, by taking away the second term. 
Suppose this to be made up of the two qua- 
dratics, x 2 -|- ex -\-f— 0, and x 2 — ex -f- 
g = 0, where -}- e and — e are made the 
coefficients of the second terms, because 
the second term of the biquadratic is want- 
ing, that is, the sum of its roots is 0. By 
multiplying these quadratics together we 
haveixt -)- g -)- f — e 2 . x 2 -j- eg — ef. x + 
fg — 0, which equation is made to coincide 
with the former by equating their coeffi- 
cients, or making g — e 2 =zq, eg — ef 
= r, and fg = s ; hence, g +/= 9 + 
also g — f— -, and by taking the sum and 
difference of these equations, 2g — q -}- & 
-f- ~, and 2f=q-j-e 2 — therefore 4 fg 
\» r 2 
= 9 2 + 2 9 e 2 + e 4 — = 4 s, and multi- 
plying by e 2 , and arranging the terms ac- 
cording to the dimensions of e, 
-| - q 2 — 4 s X e 2 — r 2 = 0 ; or, making 
y=.e 2 , y l -\-Zq y 2 -\-q 2 — 4 s . y — -r 2 = 0. 
By the solution of this cubic, a value of y, 
and therefore of \/ y, or e, is obtained; 
als-o f and g, which are respectively equal 
g-j-e 2 — - § e 2 -|-U 
to - and f L, are known ; 
2 2 
the biquadratic is thus resolved into two 
quadratics, whose roots may be found. 
It may be observed, that which ever 
value of y is used, the same values of x are 
obtained. 
This solution can only be applied to those 
cases, in which two roots of the biquadratic 
are possible and two impossible. 
Let the roots be a, b, c, — a-f-6-j-c; 
then since e, the coefficient of the second 
term of one of the reducing quadratics, is 
the sum of two roots, its different values are 
a b, a -J- c, b -|— c, — a -j- b, — a -|— c , 
— b c, and the values of e 2 , or y, are 
a -j- b) 2 , a -J- c| 2 , 6 — c| 2 j all of which be- 
ing possible, the cubic cannot be solved by 
any direct method. Suppose the roots of 
the biquadratic to be a -|- b \/ — i; a — h 
\/ — T; — — l; — a- — c\/ — 1; 
the values of e are 2 a, b + cV-1. 
b- c . \/ 1, , b — c . • — 1, — b — J- c . 
\/ — 1 and - — -2a; and the three values 
of yare — 6 -[-cl 2 , — b — cl 2 , which 
are all possible, as in the preceding case. 
But if the roots of the biquadratic be a -J- b 
if — 1, a — -6 \f — 1, — a — c, — a — c, 
the values of y are i"al 2 , c -\-h ll 2 , 
c — b — ll 2 , two of which are impossi- 
ble; therefore the cubic may be solved by 
Cardan’s rule. 
Equation, annual, of the mean motion 
of the sun and moon’s apogee and nodes. 
The annual equation of the sun’s mean mo- 
tion depends upon the excentricity of the 
earth’s orbit round him, and is 16 -U suc ij 
parts, of which the mean distance between 
the sun and the earth is 1000; whence some 
have called it the equation of the centre, 
which, when greatest, is 1° 56' 20". 
The equation of the moon’s mean motion 
is 1 1 ' 40" ; of the apogee, 20' ; and of its 
node, 9' 30". 
These four annual equations are always 
mutually proportionable to each other ; so 
that when any of them is at the greatest, the 
three others will also he greatest; and 
when one diminishes, the rest diminish in 
the same ratio. Wherefore the annual equa- 
tion of the centre of the sun being given, 
the other three corresponding equations will 
be given, so that one table of the central 
equations will serve for all. 
Equation of a curve, is an equation 
shewing the nature of a curve by expressing 
the relation between any absciss and its 
corresponding ordinate, or else the relation 
of their fluxions, &c. Thus, the equation to 
the circle is ax — x 2 = if, where a is its 
diameter, x any absciss or part of that 
diameter, and y the ordinate at that point 
of the diameter; the meaning being that 
whatever qbsciss is denoted by x, thenthe 
square of its corresponding ordinate will be 
ax — x 2 . In like manner the equation 
of the ellipse is £ . ax — x 2 — y 2 
a 
p 
of the hyperbola is - • a x -)- x 2 — if, 
of the parabola is px — y 2 . 
Where a is an axis, and p the parameter. 
And in like manner for any other curves. 
This method of expressing the nature of 
curves by algebraical equations, was first 
