Mr. W. G. Horner on Congeneric Surd Equations. 49 
the copula a+ /r=A has been lost sight of. ‘The com- 
plete chain is 
We I FS OOr Fe 
a+ /x=A, or 0 
Sc Ga Ch OQ. 
You are well aware that this copula will, in all cases of surd 
equations, consist of all the variations that can be made of 
the given formula by varying the affection of each radical it 
contains in all possible ways. You also can refer, more readily 
than myself, to various authors in whose works the method 
of forming the continued product of a formula and all such 
variations of it (for the sake of a convenient term I would ven- 
ture to say, its congeners) has been simplified. You see, that 
by retaining the entire set of congeneric equations, all doubt 
respecting the constancy of every symbol employed, whether 
letter or radical sign, is entirely cleared away. Uncertainty, 
indeed, still remains attached to the results of the solution of 
the final equation; namely, uncertainty as to which of the 
congeneric formule will be reduced to zero, by the resulting 
values of x; but this doubt is unconnected with any perplexity 
respecting the general theory. 
A very unnecessary ambiguity is admitted in the current 
acceptation of the word root; and great advantage would ac- 
crue from restricting it to its only legitimate signification of 
“such a value of the unknown in any linear divisor of the 
equation, as will cause that divisor to vanish.” 
The sum of the whole matter, respecting surd equations, 
I conceive to be this. We know that the continued product 
of a surd formula and all its congeners will produce a ra- 
tional formula; and that such rational formula, being equated 
to zero, may be solved by as many roots as it has dimensions. 
We are also certain that each of these roots will cause one of 
the congeneric surd formule to vanish ; otherwise the product 
of all would not be = 0 as assumed. But, is the value of x 
which effects this, to be called a root of the surd formula? 
No, it is a root of the rational combination only.— Have irra- 
tional equations, then, no roots? None at all.— What have 
they, then, in the place of roots ? An equitable chance, in com- 
mon with each formula in the congeneric society, of solution 
by means of the solution of the stock-equation.—But if an 
equation has no root, nor even a certainty of solution, in what 
form can it be intelligibly proposed? A note of interrogation 
subjoined might serve to intimate that the equation is pro- 
posed either for solution or correction—To what order can 
. ‘ F m , 
surd equations be assigned ? To the fractional order >? when 
’ 
Third Series. Vol. 8. No. 43. Jan, 1836. H 
