44° Mr. W.G. Horner on Congeneric Surd Equations. 
exercises, because, no doubt, the compilers had not thought 
the matter out, we who use their collections, being as in- 
dolent as they, have contented ourselves with the general 
probability of, at least, partial success. In the mean time 
even classical writers have spoken of clearing an equation 
from radicals, in order to its solution, as a process of course, 
and which would not in any way affect the conditions. The 
consequence is, that a habit prevails of talking about equations 
without any regard to this peculiar case, and therefore in 
language which when applied to it becomes quite incorrect. 
The term root of an equation passes for synonymous with any 
quantity which, being substituted for the unknown, satisfies 
the conditions; and it is affirmed, and demonstrated, that 
every equation has at least one root; and that, having one, it 
must have as many roots as there are units in the greatest 
index attached to the unknown. It is therefore ‘quite start- 
ling, when we are reminded that equations may be proposed 
ad libitum, whose conditions cannot be satisfied by any quan- 
tity, positive, negative, or imaginary; that notwithstanding 
this, the roots obtained from such equations may be real quan- 
tities. Nor is the enigma solved by discovering that the roots 
obtained from one equation are sure to satisfy the conditions 
of another, not much unlike it: on the contrary, one is quite 
displeased at this kind of thimble-rig shuffling, where we were 
assured of finding truth, the whole truth, and nothing but the 
truth. A logician of the old school would settle the business 
by crying ‘ distinguo” ; but we should still reply, that it is a 
lame distinction which clears up only one half of the pre- 
mises: we know that these are surd equations we are now 
speaking of, and that just before we were speaking of rational 
equations, or equations cleared of surds; but the difficulty 
remains unexplained. If he really knew a little of the sub- 
ject, he would, perhaps, next try the pass-word ambiguity : 
** There is always a certain ambiguity adhering to surd ex- 
pressions.” When, however, the most is said that can be said 
to that purport, it amounts in short to this, that in the reading 
of formulz, when we meet with a radical, we ought not to 
use the definite but the indefinite article. We have a knack 
of saying “the”, where we ought to say “a”, that is all; 
and if we did but read a square root, a cube root, and so on, 
we should be certain of finding one that would satisfy the ex- 
isting conditions. ‘This sounds plausibly, and at least ninety- 
nine out of every hundred of algebraists would inquire no 
further; but you would perhaps object, that at this rate 
+ ¥«ex= — ¥ x might be a good equation, unless, with 
