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Rh about 1618. For some time after their discovery the town enjoyed a wonderful degree of prosperity. After the Restoration it was often visited by Charles II., and when Queen Anne came to the throne, her husband, Prince George of Denmark, made it his frequent resort. Epsom gradually lost its celebrity as a spa, but the annual races held on its downs arrested the decay of the town. Races appear to have been established here as early as James I’s residence at Nonsuch, but they did not assume a permanent character until 1730. The principal races—the Derby and Oaks—are named after one of the earls of Derby and his seat, the Oaks, which is in the neighbourhood. The latter race was established in 1779, and the former in the following year. The spring races are held on a Thursday and Friday towards the close of April; and the great Epsom meeting takes place on the Tuesday and three following days immediately before Whitsuntide,—the Derby on the Wednesday, and the Oaks on the Friday (see ). The grand stand was erected in 1829, and subsequently enlarged; and there are numerous training stables in the vicinity. Close to the town are the extensive buildings of the Royal Medical Benevolent College, commonly called Epsom College, founded in 1855. Scholars on the foundation must be the sons of medical men, but in other respects the school is open. In the neighbourhood is the Durdans, a seat of the earl of Rosebery.

 EPSOM SALTS, heptohydrated magnesium sulphate, MgSO4·7H2O, the magnesii sulphas of pharmacy (Ger. Bittersalz). It occurs dissolved in sea water and in most mineral waters, especially in those at Epsom (from which place it takes its name), Seidlitz, Saidschutz and Pullna. It also occurs in nature in fibrous excrescences, constituting the mineral epsomite or hair-salt; and as compact masses (reichardite), as in the Stassfurt mines. It is also found associated with limestone, as in the Mammoth Caves, Kentucky, and with gypsum, as at Montmartre. Epsom salts crystallizes in the orthorhombic system, being isomorphous with the corresponding zinc and nickel sulphates, and also with magnesium chromate. Occasionally monoclinic crystals are obtained by crystallizing from a strong solution. It is used in the arts for weighting cotton fabrics, as a top-dressing for clover hay in agriculture, and in dyeing. In medicine it is frequently employed as a hydragogue purgative, specially valuable in febrile diseases, in congestion of the portal system, and in the obstinate constipation of painters’ colic. In the last case it is combined with potassium iodide, the two salts being exceedingly effective in causing the elimination of lead from the system. It is also very useful as a supplement to mercury, which needs a saline aperient to complete its action. The salt should be given a few hours after the mercury, e.g. in the early morning, the mercury having been given at night. It possesses the advantage of exercising but little irritant effect upon the bowels. Its nauseous bitter taste may to some extent be concealed by acidifying the solution with dilute sulphuric acid, and in some cases where full doses have failed the repeated administration of small ones has proved effectual.

For the manufacture of Epsom salts and for other hydrated magnesium sulphates see.

 EQUATION (from Lat. aequatio, aequare, to equalize), an expression or statement of the equality of two quantities. Mathematical equivalence is denoted by the sign =, a symbol invented by Robert Recorde (1510–1558), who considered that nothing could be more equal than two equal and parallel straight lines. An equation states an equality existing between two classes of quantities, distinguished as known and unknown; these correspond to the data of a problem and the thing sought. It is the purpose of the mathematician to state the unknowns separately in terms of the knowns; this is called solving the equation, and the values of the unknowns so obtained are called the roots or solutions. The unknowns are usually denoted by the terminal letters, x, y, z, of the alphabet, and the knowns are either actual numbers or are represented by the literals a, b, c, &c., i.e. the introductory letters of the alphabet. Any number or literal which expresses what multiple of term occurs in an equation is called the coefficient of that term; and the term which does not contain an unknown is called the absolute term. The degree of an equation is equal to the greatest index of an unknown in the equation, or to the greatest sum of the indices of products of unknowns. If each term has the sum of its indices the same, the equation is said to be homogeneous. These definitions are exemplified in the equations:— In (1) the unknown is x, and the knowns a, b, c; the coefficients of x2 and x are a and 2b; the absolute term is c, and the degree is 2. In (2) the unknowns are x and y, and the known a; the degree is 3, i.e. the sum of the indices in the term xy2. (3) is a homogeneous equation of the second degree in x and y. Equations of the first degree are called simple or linear; of the second, quadratic; of the third, cubic; of the fourth, biquadratic; of the fifth, quintic, and so on. Of equations containing only one unknown the number of roots equals the degree of the equation; thus a simple equation has one root, a quadratic two, a cubic three, and so on. If one equation be given containing two unknowns, as for example ax + by = c or ax2 + by2 = c, it is seen that there are an infinite number of roots, for we can give x, say, any value and then determine the corresponding value of y; such an equation is called indeterminate; of the examples chosen the first is a linear and the second a quadratic indeterminate equation. In general, an indeterminate equation results when the number of unknowns exceeds by unity the number of equations. If, on the other hand, we have two equations connecting two unknowns, it is possible to solve the equations separately for one unknown, and then if we equate these values we obtain an equation in one unknown, which is soluble if its degree does not exceed the fourth. By substituting these values the corresponding values of the other unknown are determined. Such equations are called simultaneous; and a simultaneous system is a series of equations equal in number to the number of unknowns. Such a system is not always soluble, for it may happen that one equation is implied by the others; when this occurs the system is called porismatic or poristic. An identity differs from an equation inasmuch as it cannot be solved, the terms mutually cancelling; for example, the expression x2 − a2 = (x − a)(x + a) is an identity, for on reduction it gives 0 = 0. It is usual to employ the sign ≡ to express this relation.

An equation admits of description in two ways:—(1) It may be regarded purely as an algebraic expression, or (2) as a geometrical locus. In the first case there is obviously no limit to the number of unknowns and to the degree of the equation; and, consequently, this aspect is the most general. In the second case the number of unknowns is limited to three, corresponding to the three dimensions of space; the degree is unlimited as before. It must be noticed, however, that by the introduction of appropriate hyperspaces, i.e. of degree equal to the number of unknowns, any equation theoretically admits of geometrical visualization, in other words, every equation may be represented by a geometrical figure and every geometrical figure by an equation. Corresponding to these two aspects, there are two typical methods by which equations can be solved, viz. the algebraic and geometric. The former leads to exact results, or, by methods of approximation, to results correct to any required degree of accuracy. The latter can only yield approximate values: when theoretically exact constructions are available there is a source of error in the draughtsmanship, and when the constructions are only approximate, the accuracy of the results is more problematical. The geometric aspect, however, is of considerable value in discussing the theory of equations.

History.—There is little doubt that the earliest solutions of equations are given, in the Rhind papyrus, a hieratic document written some 2000 years before our era. The problems solved were of an arithmetical nature, assuming such forms as “a mass and its th makes 19.” Calling the unknown mass x, we have given x + x = 19, which is a simple equation. Arithmetical problems also gave origin to equations involving two unknowns; the early Greeks were familiar with and solved simultaneous linear equations, but indeterminate equations, such, for instance, as the system given in the “cattle problem” of Archimedes, were not seriously studied until Diophantus solved many particular problems. Quadratic equations arose in the Greek investigations in the doctrine of proportion, and