48 Mr. W.G. Horner on Congeneric Surd Equations. 
lower will render its sides unequal, the results, as far as such 
values of 2 are concerned, are no longer coincident with 
the conditions of the axioms on which the management of 
equations is founded, but are illustrations of the opposite 
axioms, viz. that unequals added to, or subtracted from, or 
multiplied or divided by, equals, produce wnequal results, or 
in algebraic language, false equations. 
The reason why this inconvenience, in the use of the se- 
cond pair of axioms, cannot occur when all the terms are on 
one side and zero alone on the other, is very evident; al- 
though, by another of those paradoxes by which equations 
are beset, the complete truth appears at first sight to be the 
result of combining a truth with an error, and equals to re- 
sult from combining equals with uneguals. It is, however, 
easy to avoid all suspicion of error. ‘Thus, it was said, that 
the given equation is resolvable into a* + 3a + 2 = 0,and 
w2+5x+6=0. But as these statements are not s/multa- 
neously true, but, on the contrary, any value of 2 which satis- 
fies one of the quadratics will render the other = A, some 
numerical quantity differing from 0, we in fact collect the 
product of e — 32+ 2 = 0, or A; 
by ge ar + 12’*= Ay or ©, 
in finding (2? — 3x2 + 2) (a + 72 + 12) = 0; where the 
premises being strictly correct, the result is unexceptionable. 
And the same result arises, although not with equally clear 
evidence of its truth, when A is superseded by zero. 
The same test, of a hypothetical adjustment of one of the 
two proposed equations, would at once expose the fallacy of 
each of the conclusions attained in our imaginary experi- 
ment. 
The general propriety of keeping the zero-side of each 
equation in a. chain of argument clear from any transposed 
terms, is proved therefore by the liberty which it allows to 
the mind, of conceiving any zero, which happens to be pro 
tempore incorrect, to be superseded by the correct value, and 
of perceiving without any embarrassment or additional Jabour 
the exact conditions of the final result. But the especial pro- 
priety of adhering to this expedient, when surds are to be ex- 
tricated, appears in the necessity which it imposes of attending 
to the copula of the argument, the suppression of which in 
the vulgar process occasions all the obscurity that is com- 
plained of. ‘Thus, between the statements 
II 
a@a=Vf/nr or a4—-wVe (0) 
and Cm.” or a —) we ud 
