c 23 :] 
II. Methods of clearing Equations of quadratic^ cubic, quadrato^ 
cubic, and higher Surds. By William Allman, M. D, Com- 
municated by the Bight Hon, Sir Joseph Banks, K. B. P . R»S. 
Read July 8, 1813. 
Several years have elapsed, since my very highly esteemed 
friend, now Rev. Doctor Mooney, Fellow of Trinity College, 
Dublin, presented to the Royal Irish Academy a paper on 
the Extermination of Radicals from Equations. He has illus- 
trated, by sundry examples, the extermination of quadratic 
surds. As he has rightly observed, the method is universal. 
Any number of quadratic surds, independent, or dependent, on 
each other, may be removed from an equation; because, 1. 
Any quantity, or factor of a quantity, necessarily subjected to 
the radical sign, is but of one dimension. 2. This quantity or 
factor being brought to one side of the equation, while the 
quantities unaffected with it remain at the other, may, by 
squaring both sides, be freed from the radical sign. 3. By a 
repetition of these reductions for each remaining independent 
surd quantity, any number of surd quantities may be converted 
into rational. 
Examples, 
I. Let \/ a + -f- s/c = 0 : then 
(2;) v/^ + Vbz=z — \/c\ and, a -^-b SL\/ab = c 
(3’)^ + ^ — ^ = — 2v/ ab*, a b — c = 4^6 ; free from 
surds. 
