[ 46 J 
VIII. On the possibility of finding a Root, real or imaginary, of 
every Equation. By Professor CuaLts*. 
A S the proof of the proposition that every equation has a 
root is at this time attracting the attention of mathema- 
ticians, I am desirous of adding a few considerations to those 
contained in two articles on this subject, which 1 communicated 
to the Numbers of the Philosophical Magazine for February and 
April 1859. 
1. The proposition belongs to a branch of pure calculation, 
which is antecedent to, and altogether independent of, the rela- 
tions of space; and consequently the proof of it does not neces- 
sarily involve the consideration of either lines or angles. The 
use that has been made of geometry of two and of three dimen- 
sions in proofs that have been recently proposed, can only be 
regarded as an auxiliary means of exhibiting the variations of 
the value of a function corresponding to variations of its vari- 
ables, and not by any means as essential to the demonstration 
of the proposition. 
2. In all the proofs that I am acquainted with, as in that 
which I have given in the articles above referred to, the unknown 
quantity 2 is assumed to be represented by a function of the 
form z+y Vv —1, z and y being real quantities, positive or ne- 
gative. The reasons for this assumption, which are not usually 
much dwelt upon, appear to be such as follow. An equation 
may always be supposed to be formed according to the conditions 
of a proposed question ; and its object is to discover some un- 
known quantity which is the answer to the question. In the 
formation of the equation, the unknown quantity is brought into 
relation with certain known quantities by operations conducted 
in accordance with the given conditions, and by algebraic rules. 
The operations are necessarily algebraic, because the relative 
magnitudes of the given quantities and the quantity sought for 
are unknown ; and it is the essential principle of abstract algebra 
to furnish rules and symbols of operation which are proper for 
calculating independently of the knowledge of relative magni- 
tudes. On account of this necessary generality in algebraic 
operations, the final equation involves conditions not contained 
in the proposed question, and its dimensions are determined 
accordingly. When the equation is formed, the unknown quan- 
tity becomes an algebraic function of the given quantities, the 
exact form of which in certain cases may be actually found. In 
all other cases such a function can be obtained in the form of a 
series, by the following, or some equivalent method. 
Let the equation be of five dimensions; and if any terms be 
* Communicated by the Author. 
