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X. On the Theory of Congeneric Surd Equations. By 
W.G. Horner, Esq. (Ina Letter addressed to T. S. Davies, 
Esq., F.RS. L. & E.)* 
[If those mathematicians who have met with a quadratic equation + whose 
“roots” either under a real or imaginary forin could not be exhibited, 
will recall to memory the surprise with which they viewed the circum- 
stance, and the attempts which they made to solve the mystery, they 
will read with no ordinary gratification the following discussion of the 
general question of which this formsa part. The general theory of such 
equations, very happily named by Mr. Horner “ Congeneric Equations,” 
is here laid dewn with great clearness, and, so far as I know, for the 
first time,—as it is, indeed, nearly the first time the formation of any 
general and philosophic views respecting them has been attempted. 
The following letter was drawn up in answer to some passages in one 
which I had a short time previvusly addressed to Mr. Horner, and was 
a private and friendly communication; yet I have sincere pleasure in 
having obtained his permission to publish it in the Phil. Mag. I do so 
under the conviction that it will furnish the same satisfaction to others 
that it has done to me. J shall only add in conclusion my hope that the 
inquiry which, in the close of his letter, he has assigned to me, will 
be pursued by himself, as I know no man to whom such researches can 
be so safely and successfully referred. 
Royal Military Academy, Nov. 15, 1835. aS: D:] 
My pear Sir, 
I AGREE with you in thinking that the properties of irra- 
tional equations have not received that degree or kind of 
attention from writers on the elements of algebra, which was 
due either to the importance of the subject, or to a considera- 
tion for the comfort of young students. This appears the 
more extraordinary, because the methods of clearing an 
equation from irrational expressions, whether involving the 
unknown or not, have been so fully discussed, that really very 
little remained to be done for rendering the state of the whole 
case very intelligible. Waring (Med. Alg., Prob. 26.) may 
be cited as a case in point. But “a miss is as good as a 
mile.” In solving equations involving radicals every one has 
experienced the necessity of putting his results to the proof 
before he could venture to decide which of them, or whether 
any of them, could be trusted; but as the latter alternative, 
or the failure of every result, is of rare occurrence in books of 
* Communicated by Mr. Davies. 
4+ For instance, 27 + ,/x*—7 = 5, the “roots” of which are 4 and 8 
as determined by the common process; neither of which substituted in 
the equation reduces it to zero, These are the roots of its congeneric surd 
equation 2a— f/er—7 =5, . 
G2 
