quadratic, cubic, quadrato^cubic , and higher Surds, 43 
The preceding biquadratic equation is the difference of two 
squares ^ ■“ * consequently, its 
quadratic factors will be the sum and difference of the roots 
of these squares, viz. a^ Xws. ^+1=0, and a^ ^ + 
j:*® 4- ax^ + bx * -f cx '^ -f dx* -f- mx ^ -f dx^ + cx^ -\-ax-\-i~o 
by the combination of corresponding terms, which has, in the preceding sheets, beeu 
repeatedly exemplified, and by division of the roots by x, may be obtained. 
.r±T — ila. -r -+- 3^- ;e ± JL = o. 
1 ^ 4 - 6 . 
X 
-H 5 - 
1 "f e. 
'a 4 * d.' 
4 -m 
+5 j: 4-2tf 
If this equation could be solved, the root being called n, the solution of the qua- 
dratic, x^ — nx ± 1 — 0, would give the root of the proposed equation, which, when 
every coefficient is unity, yields, x 4 — ^ 4- 4- 
•2T 
4.X4-- — 3.^4_± 
X X 
+ 3 -^ 4 -^ 4-1 = 0 - 
If this equation could be solved, then would Dr. Warinc’s method serve for the 
universal extermination of surds of the 11th power. 
I may also observe, that my method universally holds for exterminating quadrato- 
cubic surds, without the solution of any higher equation than a quadratic, as may 
appear from a former example, though the observation was not before made ; and, in 
like manner, that it universally holds for exterminating surds of the 7th power, with- 
in the condition of solving a cubic equation. For the resulting equation of six dimen- 
sions may be reduced to one of three, independently on the simplicity or composition 
of the third quantity of a given combination of surds. Thus, if ^v^a 4 -^v' 64 -’v/<^ = 
4- 3^\/aS6*4- s''V^*b^-¥ 3^/a*6*4- ^y/ah^ — — — -i then after 
multiplying by i*, may be obtained this cubic equation, 4- 4- 
b'^y/ab^'^ 4- h* . ' y/ 4- ab^ ~ . h*. Here, whatever 
may be the value of r, a and b may be freed from the radical sign of the 7th power, 
by the solution, first of a cubic, then of a quadratic, equation, and afterwards invo- 
lution. 
So that both Wa r i n g*s method and mine universally hold, until we arrive at surds 
of the iith power; and according to mine, three such may be exterminated. 
