t 
quadratic, cubic, quadrate- cubic, and higher Surds, 25 
n' 
V +2” ’ HH ’4“ 
-\-ab—zn .1 ^ — zn. 
-\-ac-\~za V 4 - 2 &. 
( 6 ;) XtZ"c\ ^ab = X""oa\^bc 
+ be 
n*-\- ab -|- ac-\- tc^-\- ^ab . w + 156' — 4 . n + 1 56* . /2.“4“ 4“ ^ + be 
. y/^6=4a<: . 7z-|- i 56‘+4^^-^+^5'^ +86’ . w+ 156. w+i5<2 . + ab 
+ iS c.}i^-\-a b -\-ac~{-bc . 
(7;) «•+ ab + ac + fc’ = 4 • +2C . t 'M'+^^S“i>- 
— 4&c.«+iSa +*5^' 
• , _i_ A*/'* -\-^ 6 ^ubc 
i.e.n —zac.n — i^oaocn _„ „ „ — ^ « +i5ac* .<yab, 
_898a6V +3‘*‘^- +*S4‘^* 
4-902a6c^ 
which being squared, an equation will result of 8 dimensions, 
free from surds. 
In like manner may surds of the 16th, 32d, 64th, &c. powers 
be taken away from any equation. 
The number which may be taken away is unlimited, as the 
removal of each surd quantity or factor, in all these cases, 
depends on the principles which direct the solution of simple 
equations. 
In the case of cubic surds, the quantity or factor necessarily 
subjected to the radical sign may be of one, or of two dimen- 
sions, but not higher : since then, an universal method is 
known for solving quadratic equations, any number of cubic 
surds, independent, or dependent, on each other, may be re- 
moved from an equation. 
Let *v/a +*^6 +*+ c = 0: then (2;) ^^ + ^ + 3 ^\/a*b + 
g'\/ab* = — c: therefore, (3;) ^^a^b +*+^6*= 
3 
E 
MDCCCXIV. 
