2 Similarly, the solution of the cubic equation is made to depend upon that of the quadratic equa tion. and that of the biquadratic equation upon that of the cubic equation. These formulas, however, when applied to numerical equations often involve operations upon complex numbers not readily performed, and hence are of little value in such cases; e.g. in applying the general formula for the roots of the cubic equation, the cube root of a complex number is often required, in which case the methods of trigonometry are employed. The real roots of numerical equations of any degree may be calculated approximately by the methods of Newton, Lagrange, and llorner, the last being the most recent and gen erally preferred of the three.
Equations of the first degree were familiar to the Egyptians in the time of :Ames (q.v.), since a papyrus transcribed by him contains an equa tion in the following form: Heap (luau), its n, its its its whole, gives 37; that is, x + +x+x= 37.
The ancient Greeks knew little of linear equa tions except through proportion, but they treated in geometric form many quadratic and cubic equations. (See Crux.) Diophantus (c.300 A.D.), however, distinguished the coefficients ( Ouittor) of the unknown quantity, gave the equation a symbolic form, classified equations, and gave definite rules for reducing them to their simplest forms. His work was chiefly concerned with in determinate systems of equations, and his method of treatment is known as Diophantine analysis (q.v.).
The Chinese likewise solved quadratic equa tions geometrically, and Sun Tse (third century). like Diophantus, developed a method of solving linear indeterminate equations. The Hindus ad vanced the knowledge of the Greeks. l3haskara (twelfth century) used only one type of quad ratic equation, fi•= bx c =0. considered both signs of the square root, and distinguished the surd values. while the Greeks accepted only posi tive integers. The Arabs improved the methods of their predecessors. They developed quite an elaborate of synnbolisnn. The equations of Al Kalsadi (fifteenth century) are models of bret• ity, and this plan tor solving linear equation., a modified Hindu method, was what was later known as the ngfcla See FALSE PosITION.
The Europeans of the Middle Ages made little advance in the solving of equations until the discos cry by Ferro. Tartaglia, and Cm-dan (six teentb century) of the general solution of the cubic equation. The solution of the biquadratie equation ool followed, and the general quintic was attacked. But, although much was done to advance the general theory of the equation by Vanderinonde. Euler. Lagrange, Bezout, 11 aring, \lalfatti. and others, it was not until the begin ning of the nineteenth century that equations of a degree higher than the fourth received satis factory treatment. Bulni and Abel were the first to demonstrate that the solution, by alge braic methods, of a general equation of a degree higher than the fourth is impossible, and to direct investigation into new channels. \lathe matieians now sought to classify equations which could he solved algebraically, and to discover higher methods for those which could not. Gauss solved the cyelotomic group. Abel the group known as the Abelian equations, and Galois stated the necessary and sufficient condition for the algebraic solubility of any equation as fol lows: if the degree of an irreducible equation is a prime number, the equation is soluble by radicals alone, provided the roots of this equa tion can be expressed rationally in terms of any two of them. As to higher methods. Tschirn
hausen, Bring, and Hertnite have shown that the general equation of the fifth degree can be put in the form t — t — A - 0: llermite and Kro neeker solved the equation of the fifth degree by elliptic functions; and Klein has given the Sill lest solution by transcendental functions.
A few of the more important properties of equations are: (1) if r is a root of the equation f I x 1 = 0, then .r — r is a factor of f (x) : e.g. 2 being a root of 2x—R = 0, thenrr-2 isa factor of 2,r—S.
(2) If f (z) is divisible by .r— r, r is a root of = 0; e.g. in (x — (.rz + x + I) = n. a'-2 is a factor, hence .r —2 = (1 satisfies the equation. and ' = 2.
(3) Every equation of the nth degree has +a roots and no more Ithe fundara•ntal theorem of (•quations due to llarriut., or, iu its complete form, to D'_lembert) : e.g, x' — 1 = 0 has four roots, .r = - - 1, I. — i, nntl no more.
(4) The enetlieieut: of an equation are func tions of its roots. Thus. in + =0 if r„ r„ . . . .>' arc the roots, thin" (r +. . . r„), n r r , a, _ — (r, r. r, r, r, r, -- ... { n (5) 'rite nunther of positive of f (x) = 0 does not exceed the nunthor of changes of signs in II.rt. Ilh•.cartes's rule of sign;.) E.g. in .r' - :3.r' - '_'.r' + ' - - 1 - 0 there are 3 changes of signs, hence there can he no more than 3 positive root.
(nil The special functions :Issoeiated with the roots of an equation w]ueh serve to distinguish the nature of tbew• runts are called diserirni nnuts; e.g. the genera] form of the roots of (he quadratic equation, .r + pa' + q = 0, may be taken as The expression is the diseriminant, for if 4q> the roots are complex: if •tq = p= the roots are equal; if 4q p= the roots are real; and if p=-4q is a perfect square, the roots are rational. Similarly the dis criminant of the cubic z' + 3hx + g = 0 is 1/ 9 a + dh$ The discriminants of equations of higher degree are fully explained in works on the theory of equations.
A differential equation is an equation involv ing differential coefficients (see CALCULUS) ; e.g.
la — from which it is required to find the relation be tween J and x. 'rue theory of the solution of such equations is an extension of the integral calculus, and is a branch of study of the highest i flu po•tanee.
For the general theory of equations, consult: Burnside and Panton• Theory of Equations (4th ed., London, 1899.1901), the appendix to which contains valuable historical material; Peterson, Throric des equations algebriques (by Laurent, Paris, 1897) ; Salmon, Lessons Introductorit to Modern Iligher algebra (Dublin, 1859, and subsequent editions) ; Ferret, C'ours d'algebr•e superieure (3d ed.. 3'aris, 1806) ; ,Jordan, Traite des substitutions et des equations algebriques ( Pat is. 1870). An extensive work, covering both history and method, is Matthiescen. Grundriige der antih'•n scud nnoderncn algebra der literalcn Gleieltiiugcn (Leipzig. 1890).