Dr . Waring on, &c. 147 
will only apply in the moil Ample cafes, when not more than 
one root is to be extracted. In every complicate cafe ( viz. 
the extraction of roots of quantities which involve the roots of 
compound quantities) of irrational quantities, recourfe muft 
be had to the old methods of multiplication, divifion, and 
extraction of roots. 
4. If a root of a complicate irrational quantity be required 
by a feries proceeding according to the dimenfions of x ; firfh 
reduce all the irrational quantities contained under the root by- 
multiplication, divifion, and extraction of roots into feriefes 
proceeding according to the dimenfions of X , fo that the terms 
of the leaft dimenfions be conftituted firft, if an afcending 
feries be required, and fo on ; and the contrary, if a defend- 
ing ; then add the feveral fums together, and extraCt the root 
of the refulting fum by a feries which proceeds according to 
the dimenfions of x, and it will be the root required. 
Ex. Let the value of the quantity v- (x~ z + a i> -l- cx x 
(d+ ex + </(./+ S x l) + %/(^ + kx l) be required by an af- 
cending feries ; firft, extraCt the innermoft root (/+£**) *] 
— & c. which add to the quantity d+ex contained un- 
2/* 
der the fame root (jn), and place the terms of the fum 
according to the dimenfions of x, and it becomes {d +/*)+** + i 
1 L-.& c. of which extraCt the («) root, and there refults 
1 1 — m I-*” 1 
(d+fif+lxd+fi* xex + £x~xd+fi m 
m J 2 /* 
&c. multiply this feries into b + cx, and thence Is derived 
X 2 b x 
