quadratic, cubic, quadrato-cuhic, and higher Surds. 41 
It only remains, however, for the complete establishment 
of the last observation, to note, that any surds, contained in 
the denominator of any fractional quantity of an equation, 
which cannot be transferred to the numerator, by muitiplying 
both terms by a residual, as some have recommended to be 
done, may, by multiplying the whole equation by that deno- 
minator, be transferred to the other quantities, or numerators, 
of the equation. 
That observation will then hold of the surds therein named, 
however they be situated in the equation ; whether they be 
in the numerators, or in the denominators of fractions. 
P. S. Dr. Waking's method of taking away surds* is very 
ingenious. It is however evidently limited by the same pos- 
tulate, which restricts the application of my general method ; 
viz. to solve an equation of the dimension next lower than the 
index of the surd, being prime; for this must be effected in 
order to obtain the imaginary values of the surd as required 
by his method ; and this, and sometimes less than this, is suf- 
ficient in mine. 
e. g. To obtain the imaginary roots of the 5th power of 
unity, the biquadratic equation a* a 1 = 0 
must be solved. These roots are ^o-^2^/5 
4 
-1 + ^/5 ± __ — 10— 2v/5 ^ coefficients troublesome enough, espe- 
cially from their variety, as Waring himself has observed. 
* V. Meditat. Algebr. Ed. 3. p. 152, Prob. 26. 
MDCGCXIV. G 
