THEORY OF EQUATIONS. Under this term is expressed all that part of algebra which treats of the properties of rational and inte gral functions of a single variable, such as ax+b, +e, and so on ; a, b, c, &c., being any algebraical quanti ties, positive or negative, whole or fractional, real or imaginary. Unless however the contrary be specified, it is usual to suppose these co.effmcients real, not imaginary.
The great question of the earlier algebraists was the finding of a value for the variable which should make the expression equal to a given number or fraction : as what must x bo no that 2x may be 11, or may be 40, and so on. In modern form it would be asked what value of x will make 2x-11 s0, or 6x-90 = 0, and so on. To find values of a variable which should make an expression vanish, or become equal to nothing, was then the first desi deratum; and these values are now called roots of the expressions. Later algebraists made the finding of these roots subservient to the discovery of other properties of the expressions.
The Hindu algebraists communicated to the Arabs, and through them to the Italians, the complete solution of equations of the first and second degrees. The Italians added the solution of equations of the third degree, and of the fourth imperfectly. These last two degrees have been completed in more recent times, so that it may be now said that the equations of the first four degrees have been com pletely conquered: that is to say, having given the equation as' + cal + ex 0, an algebraical expression can be found, hav ing four values, and four values only, and being n function of a, b, c, e, f, which being substituted for x on the first side of the equation, shall make that first side vanish. But the student would look in vain through the books of algebra to see this expression : it is both com plicated and useless, and it is more desirable to indicate how it is to be found, than to fad it.
The equation of the fifth degree was attempted in all quarters, with out mses's : means were found of approximating to the arithmetical value of one or another root in any one given equation; but never a definite function of the co-efficionts which would apply in all cases.
A proof was given by Abel, in Crelle's Journal (reprinted in his works), that such an expression was impossible, but this proof was not gene rally received : it was admitted by Sir W. Hamilton, who illustrated the argument at great length in the Transactions' of the Royal Irish Academy, vol. xviii., part ii.; lint the singular complexity of the reasoning will probably prevent most persons from attendin? to the subject. We do not mean in this article to cuter into the history of the theory of equations, but only to place its general state before the reader by an exhibition of the principal theorems, without proof. For works on the subject we may refer m follows :—Ilutton, • Tract's' voL IL, tract 33, which contains a full ae,count of the earlier algebntiata ; Peacock, "Report on certain Parts of Analysis," in the ' Report of the Third 'Meeting of the British Associatiou ; or the recent works of Murphy, Young, or llymera ; all of which are good, and written on such different plans that any ono who makes a particular study of the subject will find it advantageous to consult them all. In French the standard works are those of Budan, Lagrange, and Fourier, which however all treat of particular topics : the algebraical treatises of Bourdon and Lefebvre de Fourcy treat it more generally, The particular points relative to equations of the first four degrees are as follows : 1. The expression of the first degree can be reduced to the form ax+b ; it vanishes when x = : a, and has only this one root. And is of the same sign as a or not, according as x Is greater or less than the root.
2. Tho expression of the second degree is more important. It can always be reduced to the form bx + e, and its properties are best developed by transforming the preceding into (2ax + + 4ac — 4a There are three distinct mom, according as is greater than, equal to, or leas than, 4ac.