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.
_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 1/7th makes
19." Calling the unknown mass x, we have given x + (1/7)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 although they were presented and solved in
a geometrical form, the methods employed have no relation to the
generalized conception of algebraic geometry which represents a curve by
an equation and vice versa. The simplest quadratic arose in the
construction of a mean proportional (x) between two lines (a, b), or in
the construction of a square equal to a given rectangle; for we have the
proportion a:x = x:b; i.e. x^2 = ab. A more general equation, viz. x^2
-ax + a^2 = 0, is the algebraic equivalent of the problem to divide a
line in medial section; this is solved in _Euclid_, ii. 11. It is
possible that Diophantus was in possession of an algebraic solution of
quadratics; he recognized, however, only one root, the interpretation of
both being first effected by the Hindu Bhaskara. A simple cubic equation
was presented in the problem of finding two mean proportionals, x, y,
between two lines, one double the other. We have a:x = x:y = y:2a, which
gives x^2 = ay and xy = 2a^2; eliminating y we obtain x^3 = 2a^3, a
simple cubic. The Greeks could not solve this equation, which also arose
in the problems of duplicating a cube and trisecting an angle, by the
ruler and compasses, but only by mechanical curves such as the cissoid,
conchoid and quadratrix. Such solutions were much improved by the Arabs,
who also solved both cubics and biquadratics by means of intersecting
conics; at the same time, they developed methods, originated by
Diophantus and improved by the Hindus, for finding approximate roots of
numerical equations by algebraic processes. The algebraic solution of
the general cubic and biquadratic was effected in the 16th century by S.
Ferro, N. Tartaglia, H. Cardan and L. Ferrari (see ALGEBRA: _History_).
Many fruitless attempts were made to solve algebraically the quintic
equation until P. Ruffini and N.H. Abel proved the problem to be
impossible; a solution involving elliptic functions has been given by C.
Hermite and L. Kronecker, while F. Klein has given another solution.
In the geometric treatment of equations the Greeks and Arabs based their
constructions upon certain empirically deduced properties of the curves
and figures employed. Knowing various metrical relations, generally
expressed as proportions, it was found possible to solve particular
equations, but a general method was wanting. This lacuna was not filled
until the 17th century, when Descartes discovered the general theory
which explained the nature of such solutions, in particular those
wherein conics were employed, and, in addition, established the most
important facts that every equation represents a geometrical locus, and
conversely. To represent equations containing two unknowns, x, y, he
chose two axes of reference mutually perpendicular, and measured x along
the horizontal axis and y along the vertical. Then by the methods
described in the article GEOMETRY: _Analytical_, he showed that--(1) a
linear equation represents a straight line, and (2) a quadratic
represents a conic. If the equation be homogeneous or break up into
factors, it represents a number of straight lines in the first case, and
the loci corresponding to the factors in the second. The solution of
simultaneous equations is easily seen to be the values of x, y
corresponding to the intersections of the loci. It follows that there is
only one value of x, y which satisfies two linear equations, since two
lines intersect in one point only; two values which satisfy a linear and
quadratic, since a line intersects a conic in two points; and four
values which satisfy two quadratics, since two conics intersect in four
points. It may happen that the curves do not actually intersect in the
theoretical maximum number of points; the principle of continuity (see
GEOMETRICAL CONTINUITY) shows us that in such cases some of the roots
are imaginary. To represent equations involving three unknowns x, y, z,
a third axis is introduced, the z-axis, perpendicular to the plane xy
and passing through the intersection of the lines x, y. In this notation
a linear equation represents a plane, and two linear simultaneous
equations represent a line, i.e. the intersection of two planes; a
quadratic equation represents a surface of the second degree. In order
to graphically consider equations containing only one unknown, it is
convenient to equate the terms to y; i.e. if the equation be [f](x) = 0,
we take y = [f](x) and construct this curve on rectangular Cartesian
co-ordinates by determining the values of y which correspond to chosen
values of x, and describing a curve through the points so obtained. The
intersections of the curve with the axis of x gives the real roots of
the equation; imaginary roots are obviously not represented.
In this article we shall treat of: (1) Simultaneous equations, (2)
indeterminate equations, (3) cubic equations, (4) biquadratic equations,
(5) theory of equations. Simple, linear simultaneous and quadratic
equations are treated in the article ALGEBRA; for differential equations
see DIFFERENTIAL EQUATIONS.