40 Dr. Allman’s Methods of clearing Equations oj 
This, involved to the nth power, will yield an equation, 
which shall have no other surds than quadratic and cubic; and, 
since these may be removed, whatever be their number, it is 
evident, that an equation may be at length deduced free from 
all surds : But the accomplishment of this would require so 
great labour, that it may at present suffice, to have shewn the 
possibility, and pointed out the method, of removing all surds 
from an equation consisting of 3 surds of the nth power. 
Far greater would be the labour to exterminate 3 surds of 
the 13th power. 
Surds of the 12th power, it must already have sufficiently 
appeared, may be taken away in any number, according to the 
principles of extermination of cubic and quadratic surds. 
It is also sufficiently manifest, that, if an equation, consisting 
of 3 surds of a certain power [v. g. the 7th), may be cleared 
of surds, an equation containing 2 such surds, together with 
any number of other surds whose extermination is unlimited, 
may be also cleared of surds ; and that surds, whose extermi- 
nation, as to their number, is unlimited, may be exterminated 
from any equation containing them, however diverse they be 
from each other. 
V 
Thus, has been pointed out, the extermination from equa- 
tions, of surds whose indices do not exceed the number 6 , or 
of any combinations of such surds, in any number ; of three 
surds, whose common index is either of the prime numbers be- 
tween 6 and 12, or whose indices are either of these multiplied 
by any numbers, or powers of any numbers under 6, provided 
the equation contain no other quantity; also, of two surds, 
whose common index is, or, whose indices are, as last described,- 
with an indefinite number of surds of the former description. 
