3 °° 
A L G E 
R Ac 
pleting the fquare, c 6 -f ra s 4 - ——--—, anda 3 4-- = 
4 4 27 2 
——» therefore, « 3 =— -±V -—, and a~ 
4 27 2 4 27 
/-t±v £ 
27 
Alfo, fince a 3 -|-i 3 -}-r=o, b 3 — 
r ,r^~7 „ nA _^ V / ^ ZZZ ; 
.-ipt/-—, and £2 
2 4 2 1 
J'l -2_; there- 
2 4 27 
fore 
* = a-\-b ~ 3 -J — 
1 J r --q=y' l i-£ 
2 *4 27 
<r_ 
4 27 
+ 
We may obferve that when the fign of y' —— 
4 27 
in one part of the expreflion is pofitive, it is negative 
. 3 / t fr* 
in the other, that is x = v --+v -- ’ Y -+ 
27 
W—'-v — 
4 27' 
Since b~~, the value of ar is alfo 
3 a 
•4 - W 4 
a 2 
27 
4 - 
i ’j-;± v 'zn 
■a 5 and the values of the cube root of P are b, 
-l±±zl b> 
-1—V'-—3 
V“ 
"t +V —3 2 
2 
—■>—t/- 
■0+ ■ 
2 
-I— V— 3 ; 
In the operation we affume 3 ab~q, that is, the produft 
of the correfponding values of a and b is fuppofed to be 
poffible. This confideration excludes the fecond, third, 
fourth, fifth, feventh, and ninth, values of a-\-b, or x; 
therefore, the three roots of the equation are a-\-b , 
—1— sj— 3 
"t-hV—3 , — 1 — V~ 
- aA - 
2 
-I+V : 
and 
- £L 
4 . 
Cor. 3. This folution only extends to thofe cafes in 
which the cubic has two impoiTible roots. For it the 
roots be m —372 and —2 m, then •— q , (the 
fum of the produfts of every two with their figns chan¬ 
ged,) 2=—3 ml —3 n, and --=.rP-\-n ; alfo r, the product of 
3 
all the roots with their figns changed, 2=2 m 3 —6 mn, and 
r 
imn. And, by involution, 
W 
2 
27 
Hence 
4 27 
rw 6 — 
x 
4 27 
Ex. Let * 3 -}-6.v—202=0-, here q=. —6, r=—20, a-=2 
3 V / io 4 \ / io 8 - 4 - 3 '^io— y/108=22.73—.73=22. 
Cor. 1. Having obtained one value of „r, the equation 
may be deprelfed to a quadratic, and the other roots found. 
Cor. 2. The poffible values of a and b being difco- 
vered, the other roots are kno^n without the folution of 
a quadratic. 
The values of the cube roots of a 3 are a , — — — -a, 
Hence it appears that there are nine values of a-\-b, 
three only of which can anfwer the conditions of the 
equation, the others having been introduced by involution. 
Thefe values are, 
1. a-\- b 
*. „+=LtlS 4 
2 
3 . «+— 
2 
—* 4 V—3 , , 
4 ^ ■ —*—ciAfb 
2 
=HV = 3« + =t+Vri i 
2 2 
2 2 
-9m« 4-6»2V—?r=2 
— nXyP —and - ~ — yJ—n X 30* 2 — n, 
4 27 
a quantity manifefily impoiTible, unlefs n is negative ; that 
is, unlefs two roots of the propofed cubic are impoiTible. 
Solution of a Biquadratic by Des Cartes’s Method. 
Any biquadratic may be reduced to the form x^-^-qx* 
-J-ra.--)- 52 =o, by taking away the fecond term. Suppofe 
this to be made up of the two quadratics, x*-\-ex-\f— o, 
and .v 2 — exAfg— o, where -{-£ and—rare made the coeffi¬ 
cients of the fecond terms, becaufe the fecond term of 
the biquadratic is wanting, that is, the fum of the roots 
is o. By multiplying thefe quadratics together, we have 
— £2 -* 2 -\-cg — e f x -\~fg—°> which equation is 
made to coincide with the former by equating their coeffi¬ 
cients, or making g-\f—e 2 = q, eg—ef—r , andy%=j; 
T 
hence g^f^zq-^e 2 , alfo and, by taking the fum 
T 
and difference of thefe equations, lg—q-fc 2 ^-, and 2 f 
6 
T r 2 ~ 
+ 5 therefore fg~q 2 -\-2qe 2 -}-£ 4 ——=245, and 
multiplying by e 2 , and arranging the terms according to the 
dimenfionsof e, e 6 -\-aqe‘ l -\.q^ —4sxr 2 —r 2 2=o; or, making 
y~e 2 , y 3 -\- 2 C iy 2 -\-q 2 —4S.7— r 2 ~o. 
By the folution of this cubic, a value of y, and there¬ 
fore of y'j, or e, is obtained; alfo f and g , which are 
refpeftively equal to 
2_ 
and 
q+e 2 + - 
f,areknown: 
2 2 
the biquadratic is thus refolved into two quadratics, whofe 
roots may be found. It may be obferved, that whichever 
value of y is ufed, the fame values of x are obtained. 
This folution can only be applied to'thofe cafes, in 
which two roots of the biquadratic are poffible, and two 
impoiTible. Let the roots be a, b, c, —<z-p^-pc; then, fince e, 
the coefficient of the fecond term of one of the reducing 
3 quadratics. 
