VOLUME IX, SLICE VII✦

Equation to Ethics

ARTICLES IN THIS SLICE:

EQUATION ESCHEAT EQUATION OF THE CENTRE ESCHENBURG, JOHANN JOACHIM EQUATION OF TIME ESCHENMAYER, ADAM KARL AUGUST VON EQUATOR ESCHER VON DER LINTH, ARNOLD EQUERRY ESCHSCHOLTZ, JOHANN FRIEDRICH EQUIDAE ESCHWEGE EQUILIBRIUM ESCHWEILER EQUINOX ESCOBAR Y MENDOZA, ANTONIO EQUITES ESCOIQUIZ, JUAN EQUITY ESCOMBE, HARRY EQUIVALENT ESCORIAL ERARD, SEBASTIEN ESCOVEDO, JUAN DE ERASMUS, DESIDERIUS ESCUINTLA ERASTUS, THOMAS ESCUTCHEON ERATOSTHENES OF ALEXANDRIA ESHER, WILLIAM BALIOL BRETT ERBACH ESHER ERBIUM ESKER ERCILLA Y ZUNIGA, ALONSO DE ESKILSTUNA ERCKMANN-CHATRIAN ESKIMO ERDELYI, JANOS ESKI-SHEHR ERDMANN, JOHANN EDUARD ESMARCH, JOHANNES FRIEDRICH AUGUST VON ERDMANN, OTTO LINNE ESNA EREBUS ESOTERIC ERECH ESPAGNOLS SUR MER, LES ERECHTHEUM ESPALIER ERECHTHEUS ESPARTERO, BALDOMERO ERESHKIGAL ESPARTO ERETRIA ESPERANCE ERETRIAN SCHOOL OF PHILOSOPHY ESPERANTO ERFURT ESPINAY, TIMOLEON D' ERGOT ESPINEL, VICENTE MARTINEZ ERIC XIV ESPIRITO SANTO ERICACEAE ESPRONCEDA, JOSE IGNACIO ENCARNACION DE ERICHSEN, SIR JOHN ERIC ESQUIRE ERICHT, LOCH ESQUIROL, JEAN ETIENNE DOMINIQUE ERICSSON, JOHN ESQUIROS, HENRI FRANCOIS ALPHONSE ERIDANUS ESS, JOHANN HEINRICH VAN ERIDU ESSAY, ESSAYIST ERIE (lake) ESSEG ERIE (city) ESSEN ERIGENA, JOHANNES SCOTUS ESSENES ERIGONE ESSENTUKI ERIN ESSEQUIBO ERINNA ESSEX, EARLS OF ERINYES ESSEX, ARTHUR CAPEL ERIPHYLE ESSEX, ROBERT DEVEREUX ERIS ESSEX, ROBERT DEVEREUX ERITH ESSEX, WALTER DEVEREUX ERITREA ESSEX ERIVAN (government of Russia) ESSEX, KINGDOM OF ERIVAN (town of Russia) ESSLINGEN ERLANGEN ESTABLISHMENT ERLE, SIR WILLIAM ESTABLISHMENT OF A PORT ERLKONIG ESTAING, CHARLES HECTOR ERMAN, PAUL ESTATE ERMANARIC ESTATE AND HOUSE AGENTS ERMELAND ESTATE DUTY ERMELO ESTCOURT, RICHARD ERMINE ESTE (family) ERMINE STREET ESTE (town) ERMOLDUS NIGELLUS ESTEBANEZ CALDERON, SERAFIN ERNE ESTELLA ERNEST I ESTERHAZY OF GALANTHA ERNEST II ESTERS ERNEST AUGUSTUS ESTHER ERNESTI, JOHANN AUGUST ESTHONIA ERNESTI, JOHANN GOTTLIEB ESTIENNE ERNST, HEINRICH WILHELM ESTON ERODE ESTOPPEL EROS (planet) ESTOUTEVILLE, GUILLAUME D' EROS (god of love) ESTOVERS ERPENIUS, THOMAS ESTRADA, LA ERROLL, FRANCIS HAY ESTRADE ERROR ESTRADES, GODEFROI ERSCH, JOHANN SAMUEL ESTREAT ERSKINE, EBENEZER ESTREES, GABRIELLE D' ERSKINE, HENRY ESTREMADURA ERSKINE, JOHN (Scottish divine) ESTREMOZ ERSKINE, JOHN (of Carnock) ESTUARY ERSKINE, JOHN (of Dun) ESZTERGOM ERSKINE, RALPH ETAGERE ERSKINE, THOMAS (of Linlathen) ETAH ERSKINE, THOMAS ERSKINE ETAMPES, ANNE DE PISSELEU D'HEILLY ERUBESCITE ETAMPES ERYSIPELAS ETAPLES ERYTHRAE ETAWAH ERYTHRITE ETCHING ERZERUM ETEOCLES ERZGEBIRGE ETESIAN WIND ERZINGAN ETEX, ANTOINE ESAR-HADDON ETHER ESAU ETHEREDGE, SIR GEORGE ESBJERG ETHERIDGE, JOHN WESLEY ESCANABA ETHERIDGE, ROBERT ESCAPE ETHERS ESCHATOLOGY ETHICS

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:--

(1) ax^2 + 2bx + c = 0, (2) xy^2 + 4a^2x = 8a^3, (3) ax^2 + 2hxy + by^2 = 0.

In (1) the unknown is x, and the knowns a, b, c; the coefficients of x^2 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 xy^2. (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 ax^2 + by^2 = 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 x^2 - a^2 = (x - a)(x + a) is an identity, for on reduction it gives 0 = 0. It is usual to employ the sign [Identical to] 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.