46 Mr. W.G. Horner on Congeneric Surd Equations. 
If equals be raised to powers denoted by equal ex-’) 
ponents, the powers are equal; and 
If of equal quantities roots be extracted, which SIII. 
are denoted by equal exponents, the roots are | 
equal. 
It has never been my chance, either to hear the validity of 
any of these principles called in question, or even any caution 
suggested as necessary in the application of them; and yet, 
when tested by the combined trial of their direct and reflex 
action, they will presently appear to be very susceptible of 
misuse in incautious hands. 
The first and second pair, abstractedly considered, afford 
such entire conviction, that in each of them, if either proposi- 
tion is granted, the other can be strictly demonstrated by 
means of it; and the second pair are truly corollaries to the 
first. No hesitation, no ambiguity, is felt. 
The fifth proposition, as a clear corollary to the third, is, 
in itself, equally satisfactory ; but quite otherwise in regard 
to its reflex effect, as described in the sixth. For, being aware 
that if wnegual quantities (+ a, —a) be raised to power, 
denoted by equal exponents, the powers may nevertheless be 
equal; we are assured that, conversely, if of equal quantities 
roots be extracted which are denoted by equal exponents, 
the roots may nevertheless be unequal. 
This remark furnishes a sufficient reason for rejecting the 
third pair of principles, and consequently the ordinary me- 
thod of clearing an equation from surds, For, in every in- 
stance in which this is effected by transposition and involution, 
in compliance with the fifth axiom, we tacitly assume that 
such step can be retraced with equal certainty by means of 
the sixth; whereas, in any such transit, the consequent equa- 
tion may be quite true, and yet the antecedent be quite false. 
If, however, we attribute the failure of the third pair of 
axioms to a special ambiguity peculiar to evolution, we shall 
remain under a delusion, and miss the cause and remedy of 
the evil. Involution is but a single instance of the erroneous 
application of the axioms of the third pair; but the use of any 
of the four unexceptionable axioms is liable to be frustrated by 
a similar cause, although in some cases the absurdity intro- 
duced is so palpable as to occasion a kind of instinctive uncon- 
scious avoidance. In other instances, however, even acute 
minds have failed to observe the fallacy. This I shall now 
point out, and prove that unless connected with the use of the 
first pair of axioms, it will be avoided, if no member of the 
equation is transposed to the zero side. 
The origin of the fallacy in question will be rendered more 
