with a Determination of the Limit beyond which it fails. 313 
mule for the solution of equations of the fifth degree. And it seems to be not 
undeserving of a passing notice that, as regards this latter important inquiry, as 
well as in reference to the more humble researches of this paper, ordinary arith- 
metic is found adequate to accomplish what is inherently impossible in algebraic 
symbols, whenever, by giving to those symbols a numerical interpretation, they 
are brought under the laws and operations of that science. All equations with 
numerical coefficients and numerical roots, may be solved by processes purely 
arithmetical; and, in like manner, the sixteen-square integral formula is always 
attainable, when the proposed factors are themselves composed of imtegral square 
numbers. This will at once appear from considering that, whatever be the pro- 
duct of these two sets of sixteen, we may separate it into four numerical parcels; 
and that each of these may be replaced by the sum of four squares.* Writings 
devoted to the theory of numbers are, in general, as much occupied with the 
relations, identities, decompositions, &c., of algebraical expressions as with those of 
pure numbers ; and I cannot help thinking that greater explicitness and preci- 
sion would be given to such writings, if the terms algebraic integer and numeri- 
cal integer were employed in their proper distinctive senses. A similar remark 
applies to the term fwnction, which, in some cases, may mean an algebraic or 
a purely symbolical function; and, in other cases, a strictly numerical function. 
The roots of an equation are functions of its coefficients, and, therefore, it has 
been inferred that the former are algebraically expressible in terms of the latter. 
The researches above adverted to have proved that this inference has been hastily 
made ; yet the nwmerical expressions for the roots, by aid of the coefficients,— 
whenever the case is brought within the limits of arithmetic,—are always attain- 
able. 
These preliminary observations will, I trust, not be considered as irrelevant; 
for I am anxious that it should be understood, throughout the following investi- 
gations, that the relations and identities discussed are purely algebraic, and thus 
admit of the widest interpretation ; and that when a property or relation is said 
not to have place, or to fail, it is to be understood that the algebraic conditions 
are not fulfilled in all their generality; and not that the analogous property may 
not hold in arithmetical integers. The integers spoken of here refer always to 
algebraic forms, and, of course, include those of arithmetic. 
* Legendre, Théorie des Nombres, p. 202. 
272 
