are the most important of the historical epochs. The creation has formed the foundation of various chronologies, the chief of which are: (1) The epoch adopted by Bossuet, Ussher, and other Catholic and Protestant divines, which places the creation in 4004 B.C. (2) The Era of Constantinople (adopted by Russia), which places it in 5508 B.C. (3) The Era of Antioch, used till A.D. 284, placed the creation 5502 B.C. (4) The Era of Alexandria, which made the creation 5492 B.C. This is also the Abyssinian Era. (5) The Jewish Era, which places the creation in 3760 B.C. The Greeks computed their time by periods of four years, called Olympiads, from the occurrence every fourth year of the Olympic games. The first Olympiad, being the year in which Corœbus was victor in the Olympic games, was in the year 776 B.C. The Romans dated from the supposed era of the foundation of their city (Ab Urbe Condita, A.U.C.), the 21st of April, in the third year of the sixth Olympiad, or 753 B.C. (according to some authorities, 752 B.C.). The Christian Era, or mode of computing from the birth of Christ as a starting-point, was first introduced in the sixth century, and was generally adopted by the year 1000. This event is believed to have taken place earlier, perhaps by four years, than the received date. The Julian epoch, based on the coincidence of the solar, lunar, and indictional periods, is fixed at 4713 B.C., and is the only epoch established on an astronomical basis. The Mohammedan Era, or Hejĭra, commences on 16th July, 622, and the years are computed by lunar months. The Chinese reckon their time by cycles of 60 years. Instead of numbering them as we do, they give a different name to every year in the cycle.
Epping, a village of England, in Essex (giving name to a parliamentary division), 17 miles from London, in the midst of an ancient royal forest which one time covered nearly the whole of Essex. The unenclosed portion (5600 acres) was bought by the Corporation of London in 1882, and secured to the public as a free place of recreation. Pop. 4253.
Epsom, a town in the county of Surrey, England, 15 miles S.W. of London, formerly celebrated for a mineral spring, from the water of which the well-known Epsom salts were manufactured. The principal attraction Epsom can now boast of is the grand race-meeting held on the Downs, the chief races being the Derby and Oaks. Epsom gives name to one of the seven parliamentary divisions of the county. Pop. 19,156.
Epsom Salts, sulphate of magnesium (MgSO47H2O), a cathartic salt which appears in capillary fibres or acicular crystals. It is found covering crevices of rocks, in mineral springs, &c.; but is commonly prepared by artificial processes from magnesian limestone by treating it with sulphuric acid, or by dissolving the mineral kieserite (MgSO4H2O) in boiling water, allowing the insoluble matter to settle, and crystallizing out the Epsom salts from the clear solution. It is employed in medicine as a purgative, and in the arts. The name is derived from its having been first procured from the mineral waters at Epsom.
Epworth, a small town of N. Lincolnshire, 9 miles N. of Gainsborough, the birth-place of John Wesley, the founder of Methodism. Pop. 1836.
Equation, in algebra, a statement that two expressions have the same numerical value. An equation may be either identical or conditional. An example of an identical equation, or identity, is (x + y)(x - y) = x2 - y2. The left side here can be transformed into the right side, simply by applying the laws of algebra so as to carry out the operations indicated, without taking account in any way of the numerical values of x and y. An identical equation is, therefore, true for all values of the variables which appear in it. A conditional equation is not true unless certain special values are assigned to the variables. Thus the equation 4x + 7 = 15 is not true for any value of x except 2. This value 2 is called a root, or solution, of the equation. An equation may have more than one root, e.g. x2 + 6x = 7 has two roots, 1 and -7; and 2x3 + 3x2 = 2x + 3 has three roots, 1, -1, 3/2.
Rational Integral Equations (one Variable).—The three equations just given are special cases of the class of rational integral algebraic equations. The general form of these is axn + bxn-1 + ... + kx + l = 0, where n is a positive integer, and a, b, ... k, l are given numbers. This equation is said to be of degree n. The branch of mathematics called the Theory of Equations is conventionally restricted to equations of this type. The fundamental result in this subject is that every rational integral equation has a root, a theorem which it is by no means easy to prove. It follows without difficulty that an equation of degree n has exactly n roots, real or imaginary. Two or more of the roots, however, may be equal to each other. To solve an equation is to find its roots. The general equation of degree n can always be solved to any degree of approximation desired, when the numerical values of the coefficients a, b, ... are assigned. Graphical methods of solution are often the best (see Graph). When the coefficients a, b, ... are arbitrary, the general equation can be solved algebraically if n does not exceed 4, but not for greater values of n. It is not that the algebraical solution, or algebraic formula for the roots, when n is greater than 4, has not been discovered; it does not exist. This was proved more than
a hundred years ago by Abel and Galois, two mathematicians of the highest distinction, who both died before they were thirty. For n = 2, the roots of the quadratic equation ax2 + bx + c = 0 are (-b ± √( b2 - 4ac))/2a. For n = 3, the cubic equation ax3 + bx2 + cx + d = 0 is reduced to the form z3 + pz + q = 0 by putting x = z - b/3a ; the solution of z3 + pz + q = 0 can be verified to be z = u - p/3u, where u is any one of the three cube roots of the quantity ½q + √(¼q2 + 1/27p3). For n = 4, the biquadratic equation is solved with the help of the solution of the cubic. The cubic was first solved by the Italian mathematician Tartaglia, who communicated the solution to Cardan, after binding him to keep it a secret. Cardan, however, gave the solution in his Algebra, published at Nürnberg in 1545.
Equations with more than one Variable.—A solution of an equation which contains more than one variable is a set of values of the variables making the equation true. Thus the equation x2 - y2 = 9 has solutions (x = 5, y = 4), (x = 3, y = 0), (x = 5, y = -4), and an unlimited number of others. When several variables occur, there are usually also several simultaneous equations connecting them; a solution of these is a set of values of the variables making all the equations true. When the number of equations is equal to the number of variables, there is in general a limited number of solutions of the system. Thus, e.g. the system of equations x2 - y2 = 9, 2x - y = 6 has two solutions (x = 3, y = 0) and (x = 5, y = 4), and no others. A useful rule is that the number of solutions of a system of this type is equal to the product of the degrees of the equations. Exceptions may arise when two solutions coalesce, or when infinite values of the variables occur.
Equations are of great importance in applied mathematics. The data of a problem generally lead to an equation, or a set of equations, among the quantities concerned. In practice a certain number of these quantities are known in any given case; the unknown quantities are then found by the solution of an equation or equations. Non-algebraic equations occur frequently—equations involving trigonometrical functions, for example. For a modern practical method of solving equations of many types, see Nomography.—Bibliography: A. E. Layng, Elementary Algebra; C. Smith, Algebra. More advanced works are: G. Chrystal, Algebra; W. S. Burnside and A. W. Panton, Theory of Equations.