Chapter I: Front Matter (1)

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Transcriber's notes:

(1) Numbers following letters (without space) like C2 were originally
printed in subscript. Letter subscripts are preceded by an
underscore, like C_n.

(2) Characters following a carat (^) were printed in superscript.

(3) Side-notes were relocated to function as titles of their respective
paragraphs.

(4) Macrons and breves above letters and dots below letters were not
inserted.

(5) [root] stands for the root symbol; [oo] for infinity; [alpha],
[beta], etc. for greek letters.

(6) The following typographical errors have been corrected:

ARTICLE ESTEBANEZ CALDERON, SERAFIN: "His most interesting work,
Escenas andaluzas (1847), is in a curiously affected style ..."
'curiously' amended from 'curiouly'.

ARTICLE ESTERHAZY OF GALANTHA: "He was minister for foreign affairs
in the first responsible Hungarian ministry (1848), but resigned
his post in September because he could see no way of reconciling
the court with the nation." 'because' amended from 'bcause'.

ARTICLE ETHER: "The principal symptoms symptons of chronic
ether-drinking are a weakening of the activity of the special
senses ..." 'symptons' amended from 'symptons'.

ARTICLE ETHEREDGE, SIR GEORGE: "It is partly in rhymed rhymned
heroic verse, like the stilted tragedies of the Howards and
Killigrews ..." 'rhymned' amended from 'rhymned'.

ARTICLE ETHICS: "... or the glory of merely secular gifts and
acquirements, it is one aspect of the unworldliness which we have
already noticed ..." 'unworldliness' amended from 'unwordliness'.

ARTICLE ETHICS: "It will be seen that these changes, however
profoundly important, were, ethically considered, either negative
or quite general, relating to the tone and attitude of mind in
which all duty should be done." 'relating' amended from 'ralating'.

ENCYCLOPAEDIA BRITANNICA

A DICTIONARY OF ARTS, SCIENCES, LITERATURE
AND GENERAL INFORMATION

ELEVENTH EDITION

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.

_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.

I. _Simultaneous Equations._

Simultaneous equations which involve the second and higher powers of
the unknown may be impossible of solution. No general rules can be
given, and the solution of any particular problem will largely depend
upon the student's ingenuity. Here we shall only give a few typical
examples.

1. _Equations which may be reduced to linear equations.--Ex._ To solve
x(x - a) = yz, y(y - b) = zx, z(z - c)=xy. Multiply the equations by
y, z and x respectively, and divide the sum by xyz; then

a b c
-- + -- + -- = 0 ... (1).
z x y

Multiply by z, x and y, and divide the sum by xyz; then

a b c
-- + -- + -- = 0 ... (2).
y z x

From (1) and (2) by cross multiplication we obtain

1 1 1 1
----------- = ----------- = ----------- = -------- (suppose)(3).
y(b^2 - ac) z(c^2 - ab) x(a^2 - bc) [lambda]

Substituting for x, y and z in x(x - a) = yz we obtain

1 3abc - (a^3 + b^3 + c^3)
-------- = ------------------------------;
[lambda] (a^2 - bc)(b^2 - ac)(c^2 - ab)

and therefore x, y and z are known from (3). The same artifice solves
the equations x^2 - yz = a, y^2 - xz = b, z^2 - xy = c.

2. _Equations which are homogeneous and of the same degree._--These
equations can be solved by substituting y = mx. We proceed to explain
the method by an example.

_Ex._ To solve 3x^2 + xy + y^2 = 15, 31xy - 3x^2 - 5y^2 = 45.
Substituting y = mx in both these equations, and then dividing, we
obtain 31m - 3 - 5m^2 = 3(3 + m + m^2) or 8m^2 - 28m + 12 = 0. The
roots of this quadratic are m = 1/2 or 3, and therefore 2y = x, or y =
3x.

Taking 2y = x and substituting in 3x^2 + xy + y^2 = 0, we obtain
y^2(12 + 2 + 1) = 15; :. y^2 = 1, which gives y = [+-]1, x = [+-]2.
Taking the second value, y = 3x, and substituting for y, we obtain
x^2(3 + 3 + 9) = 15; :. x^2 = 1, which gives x = [+-]1, y = [+-]3.
Therefore the solutions are x = [+-]2, y = [+-]1 and x = [+-]1, y =
[+-]3. Other artifices have to be adopted to solve other forms of
simultaneous equations, for which the reader is referred to J.J.
Milne, _Companion to Weekly Problem Papers_.

II. _Indeterminate Equations._

1. When the number of unknown quantities exceeds the number of
equations, the equations will admit of innumerable solutions, and are
therefore said to be _indeterminate_. Thus if it be required to find
two numbers such that their sum be 10, we have two unknown quantities
x and y, and only one equation, viz. x + y = 10, which may evidently
be satisfied by innumerable different values of x and y, if fractional
solutions be admitted. It is, however, usual, in such questions as
this, to restrict values of the numbers sought to positive integers,
and therefore, in this case, we can have only these nine solutions,

x = 1, 2, 3, 4, 5, 6, 7, 8, 9;
y = 9, 8, 7, 6, 5, 4, 3, 2, 1;

which indeed may be reduced to five; for the first four become the
same as the last four, by simply changing x into y, and the contrary.
This branch of analysis was extensively studied by Diophantus, and is
sometimes termed the Diophantine Analysis.

2. Indeterminate problems are of different orders, according to the
dimensions of the equation which is obtained after all the unknown
quantities but two have been eliminated by means of the given
equations. Those of the first order lead always to equations of the
form

ax [+-] by = [+-]c,

where a, b, c denote given whole numbers, and x, y two numbers to be
found, so that both may be integers. That this condition may be
fulfilled, it is necessary that the coefficients a, b have no common
divisor which is not also a divisor of c; for if a = md and b = me,
then ax + by = mdx + mey = c, and dx + ey = c/m; but d, e, x, y are
supposed to be whole numbers, therefore c/m is a whole number; hence m
must be a divisor of c.

Of the four forms expressed by the equation ax [+-] by = [+-]c, it is
obvious that ax + by = -c can have no positive integral solutions.
Also ax - by = -c is equivalent to by - ax = c, and so we have only to
consider the forms ax [+-] by = c. Before proceeding to the general
solution of these equations we will give a numerical example.

To solve 2x + 3y = 25 in positive integers. From the given equation
we have x = (25 - 3y)/2 = 12 - y - (y - 1)/2. Now, since x must be a
whole number, it follows that (y - 1)/2 must be a whole number. Let us
assume (y - 1)/2 = z, then y = 1 + 2z; and x = 11 - 3z, where z might
be any whole number whatever, if there were no limitation as to the
signs of x and y. But since these quantities are required to be
positive, it is evident, from the value of y, that z must be either 0
or positive, and from the value of x, that it must be less than 4;
hence z may have these four values, 0, 1, 2, 3.

If z = 0, z = 1, z = 2, z = 3;

Then x = 11, x = 8, x = 5, x = 2,
y = 1, y = 3, y = 5, y = 7.

3. We shall now give the solution of the equation ax - by = c in
positive integers.

Convert a/b into a continued fraction, and let p/q be the convergent
immediately preceding a/b, then aq - bp = [+-]1 (see CONTINUED
FRACTION).

([alpha]) If aq - bp = 1, the given equation may be written

ax - by = c(aq - bp);
:. a(x - cq) = b(y - cp).

Since a and b are prime to one another, then x - cq must be divisible
by b and y - cp by a; hence

(x - cq) / b = (y - cq)/a = t.

That is, x = bt + cq and y = at + cp.

Positive integral solutions, unlimited in number, are obtained by
giving t any positive integral value, and any negative integral value,
so long as it is numerically less than the smaller of the quantities
cq/b, cp/a; t may also be zero.

([beta]) If aq - bp = -1, we obtain x = bt - cq, y = at - cp, from
which positive integral solutions, again unlimited in number, are
obtained by giving t any positive integral value which exceeds the
greater of the two quantities cq/b, cp/a.

If a or b is unity, a/b cannot be converted into a continued fraction
with unit numerators, and the above method fails. In this case the
solutions can be derived directly, for if b is unity, the equation may
be written y = ax - c, and solutions are obtained by giving x positive
integral values greater than c/a.

4. To solve ax + by = c in positive integers. Converting a/b into a
continued fraction and proceeding as before, we obtain, in the case of
aq - bp = 1,

x = cq - bt, y = at - cp.

Positive integral solutions are obtained by giving t positive integral
values not less than cp/a and not greater than cq/b.

In this case the number of solutions is limited. If aq - bp = -1 we
obtain the general solution x = bt - cq, y = cp - at, which is of the
same form as in the preceding case. For the determination of the
number of solutions the reader is referred to H.S. Hall and S.R.
Knight's _Higher Algebra_, G. Chrystal's _Algebra_, and other
text-books.

5. If an equation were proposed involving three unknown quantities, as
ax + by + cz = d, by transposition we have ax + by = d - cz, and,
putting d - cz = c', ax + by = c'. From this last equation we may find
values of x and y of this form,

x = mr + nc', y = mr + n'c',

or x = mr + n(d - cz), y = m'r + n'(d - cz);

where z and r may be taken at pleasure, except in so far as the values
of x, y, z may be required to be all positive; for from such
restriction the values of z and r may be confined within certain
limits to be determined from the given equation. For more advanced
treatment of linear indeterminate equations see COMBINATORIAL
ANALYSIS.

6. We proceed to indeterminate problems of the second degree: limiting
ourselves to the consideration of the formula y^2 = a + bx + cx^2,
where x is to be found, so that y may be a rational quantity. The
possibility of rendering the proposed formula a square depends
altogether upon the coefficients a, b, c; and there are four cases of
the problem, the solution of each of which is connected with some
peculiarity in its nature.

_Case_ 1. Let a be a square number; then, putting g^2 for a, we have
y^2 = g^2 + bx + cx^2. Suppose [root](g^2 + bx + cx^2) = g + mx; then
g^2 + bx + cx^2 = g^2 + 2gmx + m^2 x^2, or bx + cx^2 = 2gmx + m^2 x^2,
that is, b + cx = 2gm + m^2x; hence

2gm - b cg - bm + gm^2
x = --------, y = [root](g^2 + bx + cx^2) = --------------,
c - m^2 c - m^2

_Case_ 2. Let c be a square number = g^2; then, putting [root](a + bx
+ g^2 x^2) = m + gx, we find a + bx + g^2x^2 = m^2 + 2mgx + g^2 x^2,
or a + bx = m^2 + 2mgx; hence we find

m^2 - a bm - gm^2 - ag
x = -------, y = [root](a + bx + g^2 x^2) = --------------.
b - 2mg b - 2mg

_Case_ 3. When neither a nor c is a square number, yet if the
expression a + bx + cx^2 can be resolved into two simple factors, as f
+ gx and h + kx, the irrationality may be taken away as follows:--

Assume [root](a + bx + cx^2)=[root]{(f + gx)(h + kx)} = m(f + gx),
then (f + gx)(h + kx) = m^2(f + gx)^2, or h + kx = m^2(f + gx); hence
we find

fm^2 - h (fk - gh)m
x = --------, y = [root]{(f + gx)(h + kx)} = ----------;
k - gm^2 k - gm^2

and in all these formulae m may be taken at pleasure.

_Case_ 4. The expression a + bx + cx^2 may be transformed into a
square as often as it can be resolved into two parts, one of which is
a complete square, and the other a product of two simple factors; for
then it has this form, p^2 + qr, where p, q and r are quantities which
contain no power of x higher than the first. Let us assume [root](p^2
+ qr) = p + mq; thus we have p^2 + qr = p^2 + 2mpq + m^2q^2 and r =
2mp + m^2q, and as this equation involves only the first power of x,
we may by proper reduction obtain from it rational values of x and y,
as in the three foregoing cases.

The application of the preceding general methods of resolution to any
particular case is very easy; we shall therefore conclude with a
single example.

_Ex._ It is required to find two square numbers whose sum is a given
square number.

Let a^2 be the given square number, and x^2, y^2 the numbers required;
then, by the question, x^2 + y^2 = a^2, and y = [root](a^2 - x^2).
This equation is evidently of such a form as to be resolvable by the
method employed in case 1. Accordingly, by comparing [root](a^2 - x^2)
with the general expression [root](g^2 + bx + cx^2), we have g = a, b
= 0, c = -1, and substituting these values in the formulae, and also
-n for +m, we find

2an a(n^2 - 1)
x = -------, y = ----------.
n^2 + 1 n^2 + 1

If a = n^2 + 1, there results x = 2n, y = n^2 - 1, a = n^2 + 1. Hence
if r be an even number, the three sides of a rational right-angled
triangle are r, (1/2r)^2 - 1, (1/2r)^2 + 1. If r be an odd number,
they become (dividing by 2) r, 1/2(r^2 - 1), 1/2(r^2 + 1).

For example, if r = 4, 4, 4 - 1, 4 + 1, or 4, 3, 5, are the sides of a
right-angled triangle; if r = 7, 7, 24, 25 are the sides of a
right-angled triangle.

III. _Cubic Equations_.

1. Cubic equations, like all equations above the first degree, are
divided into two classes: they are said to be _pure_ when they contain
only one power of the unknown quantity; and _adfected_ when they
contain two or more powers of that quantity.

Pure cubic equations are therefore of the form x^3 = r; and hence it
appears that a value of the simple power of the unknown quantity may
always be found without difficulty, by extracting the cube root of
each side of the equation. Let us consider the equation x^3 - c^3 = 0
more fully. This is decomposable into the factors x - c = 0 and x^2 +
cx + c^2 = 0. The roots of this quadratic equation are 1/2(-1 [+-]
[root]-3)c, and we see that the equation x^3 = c^3 has three roots,
namely, one real root c, and two imaginary roots 1/2(-1 [+-]
[root]-3)c. By making c equal to unity, we observe that 1/2(-1 [+-]
[root]-3) are the imaginary cube roots of unity, which are generally
denoted by [omega] and [omega]^2, for it is easy to show that (1/2(-1
- [root]-3))^2 = 1/2(-1 + [root]-3).

2. Let us now consider such cubic equations as have all their terms,
and which are therefore of this form,

x^3 + Ax^2 + Bx + C = 0,

where A, B and C denote known quantities, either positive or negative.

This equation may be transformed into another in which the second term
is wanting by the substitution x = y - A/3. This transformation is a
particular case of a general theorem. Let x^n + Ax^(n - 1) + Bx^(n -
2) ... = 0. Substitute x = y + h; then (y + h)^n + A(y + h)^(n - 1)
... = 0. Expand each term by the binomial theorem, and let us fix our
attention on the coefficient of y^(n - 1). By this process we obtain 0
= y^n + y^(n - 1)(A + nh) + terms involving lower powers of y.

Now h can have any value, and if we choose it so that A + nh = 0, then
the second term of our derived equation vanishes.

Resuming, therefore, the equation y^3 + qy + r = 0, let us suppose y =
v + z; we then have y^3 = v^3 + z^3 + 3vz(v + z) = v^3 + z^3 + 3vzy,
and the original equation becomes v^3 + z^3 + (3vz + q)y + r = 0. Now
v and z are any two quantities subject to the relation y = v + z, and
if we suppose 3vz + q = 0, they are completely determined. This leads
to v^3 + z^3 + r = 0 and 3vz + q = 0. Therefore v^3 and z^3 are the
roots of the quadratic t^2 + rt - q^2/27 = 0. Therefore

v^3 = -1/2 r + [root][(1/27)q^3 + 1/4 r^2];
z^3 = -1/2 r - [root][(1/27)q^3 + 1/4 r^2];

v = [root 3]{-1/2 r + [root][(1/27)q^3 + 1/4 r^2]};
z = [root 3]{-1/2 r - [root][(1/27)q^3 + 1/4 r^2]};

and

y = v + z = [root 3]{-1/2 r + [root][(1/27)q^3 + 1/4 r^2]} +
[root 3]{-1/2 r - [root][(1/27)q^3 + 1/4 r^2]}.

Thus we have obtained a value of the unknown quantity y, in terms of
the known quantities q and r; therefore the equation is resolved.

3. But this is only one of three values which y may have. Let us, for
the sake of brevity, put

A = -1/2 r + [root]((1/27)q^3 + 1/4 r^2), B = -1/2 r -
[root]((1/27)q^3 + 1/4 r^2),

and put

[alpha] = 1/2(-1 + [root]-3),
[beta] = 1/2(-1 - [root]-3).

Then, from what has been shown (S 1), it is evident that v and z have
each these three values,

v = [root 3]A, v = [alpha][root 3]A, v = [beta][root 3]A;
z = [root 3]B, z = [alpha][root 3]B, z = [beta][root 3]B.

To determine the corresponding values of v and z, we must consider
that vz = -(1/3)q = [root 3](AB). Now if we observe that [alpha][beta]
= 1, it will immediately appear that v + z has these three values,

v + z = [root 3]A + [root 3]B,
v + z = [alpha][root 3]A + [beta][root 3]B,
v + z = [beta][root 3]A + [alpha][root 3]B,

which are therefore the three values of y.

The first of these formulae is commonly known by the name of Cardan's
rule (see ALGEBRA: _History_).

The formulae given above for the roots of a cubic equation may be put
under a different form, better adapted to the purposes of
arithmetical calculation, as follows:--Because vz = -(1/3)q, therefore
z = -(1/3)q X 1/v = -(1/3)q / [root 3]A; hence v + z = [root 3]A -
(1/3)q / [root 3]A; thus it appears that the three values of y may
also be expressed thus:

y = [root 3]A - (1/3)q / [root 3]A
y = [alpha][root 3]A - (1/3)q[beta] / [root 3]A
y = [beta][root 3]A - (1/3)q[alpha] / [root 3]A.

See below, _Theory of Equations_, SS 16 et seq.

IV. _Biquadratic Equations_.

1. When a biquadratic equation contains all its terms, it has this
form,

x^4 + Ax^3 + Bx^2 + Cx + D = 0,

where A, B, C, D denote known quantities.

We shall first consider pure biquadratics, or such as contain only the
first and last terms, and therefore are of this form, x^4 = b^4. In
this case it is evident that x may be readily had by two extractions
of the square root; by the first we find x^2 = b^2, and by the second
x = b. This, however, is only one of the values which x may have; for
since x^4 = b^4, therefore x^4 - b^4 = 0; but x^4 - b^4 may be
resolved into two factors x^2 - b^2 and x^2 + b^2, each of which
admits of a similar resolution; for x^2 - b^2 = (x - b)(x + b) and x^2
+ b^2 = (x - b[root]-1)(x + b[root]-1). Hence it appears that the
equation x^4 - b^4 = 0 may also be expressed thus,

(x - b)(x + b)(x - b[root]-1)(x + b[root]-1) = 0;

so that x may have these four values,

+b, -b, +b[root]-1, -b[root]-1,

two of which are real, and the others imaginary.

2. Next to pure biquadratic equations, in respect of easiness of
resolution, are such as want the second and fourth terms, and
therefore have this form,

x^4 + qx^2 + s = 0.

These may be resolved in the manner of quadratic equations; for if we
put y = x^2, we have

y^2 + qy + s = 0,

from which we find y = 1/2{-q [+-] [root](q^2 - 4s)}, and therefore

x = [+-][root]1/2{-q [+-] [root](q^2 - 4s)}.

3. When a biquadratic equation has all its terms, its resolution may
be always reduced to that of a cubic equation. There are various
methods by which such a reduction may be effected. The following was
first given by Leonhard Euler in the _Petersburg Commentaries_, and
afterwards explained more fully in his _Elements of Algebra_.

We have already explained how an equation which is complete in its
terms may be transformed into another of the same degree, but which
wants the second term; therefore any biquadratic equation may be
reduced to this form,

y^4 + py^2 + qy + r = 0,

where the second term is wanting, and where p, q, r denote any known
quantities whatever.

That we may form an equation similar to the above, let us assume y =
[root]a + [root]b + [root]c, and also suppose that the letters a, b, c
denote the roots of the cubic equation

z^3 + Pz^2 + Qz - R = 0;

then, from the theory of equations we have

a + b + c = -P, ab + ac + bc = Q, abc = R.

We square the assumed formula

y = [root]a + [root]b + [root]c,

and obtain

y^2 = a + b + c + 2([root]ab + [root]ac + [root]bc);

or, substituting -P for a + b + c, and transposing,

y^2 + P = 2([root]ab + [root]ac + [root]bc).

Let this equation be also squared, and we have

y^4 + 2Py^2 + P^2 = 4(ab + ac + bc) + 8([root]a^2 bc + [root]ab^2 c +
[root]abc^2);

and since

ab + ac + bc = Q,

and

[root]a^2 bc + [root]ab^2 c + [root]abc^ 2 = [root]abc([root]a +
[root]b + [root]c) = [root]R.y,

the same equation may be expressed thus:

y^4 + 2Py^2 + P^2 = 4Q + 8[root]R.y.

Thus we have the biquadratic equation

y^4 + 2Py^2 - 8[root]R.y + P^2 - 4Q = 0,

one of the roots of which is y = [root]a + [root]b + [root]c, while a,
b, c are the roots of the cubic equation z^3 + Pz^2 + Qz - R = 0.

4. In order to apply this resolution to the proposed equation y^4 +
py^2 + qy + r = 0, we must express the assumed coefficients P, Q, R by
means of p, q, r, the coefficients of that equation. For this purpose
let us compare the equations

y^4 + py^2 + qy + r = 0,
y^4 + 2Py^2 - 8[root]Ry + P^2 - 4Q = 0,

and it immediately appears that

2P = p, -8[root]R = q, P^2 - 4Q = r;

and from these equations we find

P = 1/2 p, Q = (1/16)(p^2 - 4r), R = (1/64)q^2.

Hence it follows that the roots of the proposed equation are generally
expressed by the formula

y = [root]a + [root]b + [root]c;

where a, b, c denote the roots of this cubic equation,

p p^2 - 4r q^2
z^3 + -- z^2 + -------- z - --- = 0.
2 16 64

But to find each particular root, we must consider, that as the square
root of a number may be either positive or negative, so each of the
quantities [root]a, [root]b, [root]c may have either the sign + or -
prefixed to it; and hence our formula will give eight different
expressions for the root. It is, however, to be observed, that as the
product of the three quantities [root]a, [root]b, [root]c must be
equal to [root]R or to -(1/8)q; when q is positive, their product must
be a negative quantity, and this can only be effected by making either
one or three of them negative; again, when q is negative, their
product must be a positive quantity; so that in this case they must
either be all positive, or two of them must be negative. These
considerations enable us to determine that four of the eight
expressions for the root belong to the case in which q is positive,
and the other four to that in which it is negative.

5. We shall now give the result of the preceding investigation in the
form of a practical rule; and as the coefficients of the cubic
equation which has been found involve fractions, we shall transform it
into another, in which the coefficients are integers, by supposing z =
1/4 v. Thus the equation

p p^2 - 4r q^2
z^3 + -- z^2 + -------- z - --- = 0
2 16 64

becomes, after reduction,

v^3 + 2pv^2 + (p^2 - 4r)v - q^2 = 0;

it also follows, that if the roots of the latter equation are a, b, c,
the roots of the former are 1/4 a, 1/4 b, 1/4 c, so that our rule may
now be expressed thus:

Let y^4 + py^2 + qy + r = 0 be any biquadratic equation wanting its
second term. Form this cubic equation

v^3 + 2pv^2 + (p^2 - 4r)v - q^2 = 0,

and find its roots, which let us denote by a, b, c.

Then the roots of the proposed biquadratic equation are,

when q is negative, when q is positive,

y = 1/2([root]a + [root]b + [root]c), y = 1/2(-[root]a - [root]b - [root]c),
y = 1/2([root]a - [root]b - [root]c), y = 1/2(-[root]a + [root]b + [root]c),
y = 1/2(-[root]a + [root]b - [root]c), y = 1/2([root]a - [root]b + [root]c),
y = 1/2(-[root]a - [root]b + [root]c), y = 1/2([root]a + [root]b - [root]c).

See also below, _Theory of Equations_, S 17 et seq. (X.)

V. _Theory of Equations_.

1. In the subject "Theory of Equations" the term _equation_ is used to denote an equation of the form x^n - p1x^(n - 1) ... [+-] p_n = 0, where p1, p2 ... p_n are regarded as known, and x as a quantity to be determined; for shortness the equation is written [f](x) = 0.

The equation may be _numerical_; that is, the coefficients p1, p2^n, ... p_n are then numbers--understanding by number a quantity of the form [alpha] + [beta]i ([alpha] and [beta] having any positive or negative real values whatever, or say each of these is regarded as susceptible of continuous variation from an indefinitely large negative to an indefinitely large positive value), and i denoting [root]-1.

Or the equation may be _algebraical_; that is, the coefficients are not then restricted to denote, or are not explicitly considered as denoting, numbers.

1. We consider first numerical equations. (Real theory, 2-6; Imaginary theory, 7-10.)

_Real Theory_.

2. Postponing all consideration of imaginaries, we take in the first instance the coefficients to be real, and attend only to the real roots (if any); that is, p1, p2, ... p_n are real positive or negative quantities, and a root a, if it exists, is a positive or negative quantity such that a^n - p1a^(n - 1) ... [+-] p_n = 0, or say, [f](a) = 0.

It is very useful to consider the curve y = [f](x),--or, what would come to the same, the curve Ay = [f](x),--but it is better to retain the first-mentioned form of equation, drawing, if need be, the ordinate y on a reduced scale. For instance, if the given equation be x^3 - 6x^2 + 11x -6.06 = 0,[1] then the curve y = x^3 - 6x^2 + 11x - 6.06 is as shown in fig. 1, without any reduction of scale for the ordinate.

It is clear that, in general, y is a continuous one-valued function of x, finite for every finite value of x, but becoming infinite when x is infinite; i.e., assuming throughout that the coefficient of x^n is +1, then when x = [oo], y = +[oo]; but when x = -[oo], then y = +[oo] or -[oo], according as n is even or odd; the curve cuts any line whatever, and in particular it cuts the axis (of x) in at most n points; and the value of x, at any point of intersection with the axis, is a root of the equation [f](x) = 0.

If [beta], [alpha] are any two values of x ([alpha] > [beta], that is, [alpha] nearer +[oo]), then if [f]([beta]), [f]([alpha]) have opposite signs, the curve cuts the axis an odd number of times, and therefore at least once, between the points x = [beta], x = [alpha]; but if [f]([beta]), [f]([alpha]) have the same sign, then between these points the curve cuts the axis an even number of times, or it may be not at all. That is, [f]([beta]), [f]([alpha]) having opposite signs, there are between the limits [beta], [alpha] an odd number of real roots, and therefore at least one real root; but [f]([beta]), [f]([alpha]) having the same sign, there are between these limits an even number of real roots, or it may be there is no real root. In particular, by giving to [beta], [alpha] the values -[oo], +[oo] (or, what is the same thing, any two values sufficiently near to these values respectively) it appears that an equation of an odd order has always an odd number of real roots, and therefore at least one real root; but that an equation of an even order has an even number of real roots, or it may be no real root.

If [alpha] be such that for x = or > a (that is, x nearer to +[oo]) [f](x) is always +, and [beta] be such that for x = or < [beta] (that is, x nearer to -[oo]) [f](x) is always -, then the real roots (if any) lie between these limits x = [beta], x = [alpha]; and it is easy to find by trial such two limits including between them all the real roots (if any).

3. Suppose that the positive value [delta] is an inferior limit to the difference between two real roots of the equation; or rather (since the foregoing expression would imply the existence of real roots) suppose that there are not two real roots such that their difference taken positively is = or < [delta]; then, [gamma] being any value whatever, there is clearly at most one real root between the limits [gamma] and [gamma] + [delta]; and by what precedes there is such real root or there is not such real root, according as [f]([gamma]), [f]([gamma] + [delta]) have opposite signs or have the same sign. And by dividing in this manner the interval [beta] to [alpha] into intervals each of which is = or < [delta], we should not only ascertain the number of the real roots (if any), but we should also separate the real roots, that is, find for each of them limits [gamma], [gamma] + [delta] between which there lies this one, and only this one, real root.

In particular cases it is frequently possible to ascertain the number
of the real roots, and to effect their separation by trial or
otherwise, without much difficulty; but the foregoing was the general
process as employed by Joseph Louis Lagrange even in the second
edition (1808) of the _Traite de la resolution des equations
numeriques_;[2] the determination of the limit [delta] had to be
effected by means of the "equation of differences" or equation of the
order 1/2 n(n - 1), the roots of which are the squares of the
differences of the roots of the given equation, and the process is a
cumbrous and unsatisfactory one.

4. The great step was effected by the theorem of J.C.F. Sturm (1835)--viz. here starting from the function [f](x), and its first derived function [f]'(x), we have (by a process which is a slight modification of that for obtaining the greatest common measure of these two functions) to form a series of functions

[f](x), [f]'(x), [f]2(x), ... [f]_n(x)

of the degrees n, n - 1, n - 2 ... 0 respectively,--the last term [f]_n(x) being thus an absolute constant. These lead to the immediate determination of the number of real roots (if any) between any two given limits [beta], [alpha]; viz. supposing [alpha] > [beta] (that is, [alpha] nearer to +[oo]), then substituting successively these two values in the series of functions, and attending only to the signs of the resulting values, the number of the changes of sign lost in passing from [beta] to [alpha] is the required number of real roots between the two limits. In particular, taking [beta], [alpha] = -[oo], +[oo] respectively, the signs of the several functions depend merely on the signs of the terms which contain the highest powers of x, and are seen by inspection, and the theorem thus gives at once the whole number of real roots.

And although theoretically, in order to complete by a finite number of operations the separation of the real roots, we still need to know the value of the before-mentioned limit [delta]; yet in any given case the separation may be effected by a limited number of repetitions of the process. The practical difficulty is when two or more roots are very near to each other. Suppose, for instance, that the theorem shows that there are two roots between 0 and 10; by giving to x the values 1, 2, 3, ... successively, it might appear that the two roots were between 5 and 6; then again that they were between 5.3 and 5.4, then between 5.34 and 5.35, and so on until we arrive at a separation; say it appears that between 5.346 and 5.347 there is one root, and between 5.348 and 5.349 the other root. But in the case in question [delta] would have a very small value, such as .002, and even supposing this value known, the direct application of the first-mentioned process would be still more laborious.

5. Supposing the separation once effected, the determination of the single real root which lies between the two given limits may be effected to any required degree of approximation either by the processes of W.G. Horner and Lagrange (which are in principle a carrying out of the method of Sturm's theorem), or by the process of Sir Isaac Newton, as perfected by Joseph Fourier (which requires to be separately considered).

First as to Horner and Lagrange. We know that between the limits
[beta], [alpha] there lies one, and only one, real root of the
equation; [f]([beta]) and [f]([alpha]) have therefore opposite signs.
Suppose any intermediate value is [theta]; in order to determine by
Sturm's theorem whether the root lies between [beta], [theta], or
between [theta], [alpha], it would be quite unnecessary to calculate
the signs of [f]([theta]),[f]'([theta]), [f]2([theta]) ...; only the
sign of [f]([theta]) is required; for, if this has the same sign as
[f]([beta]), then the root is between [beta], [theta]; if the same
sign as [f]([alpha]), then the root is between [theta], [alpha]. We
want to make [theta] increase from the inferior limit [beta], at which
[f]([theta]) has the sign of [f]([beta]), so long as [f]([theta])
retains this sign, and then to a value for which it assumes the
opposite sign; we have thus two nearer limits of the required root,
and the process may be repeated indefinitely.

Horner's method (1819) gives the root as a decimal, figure by figure;
thus if the equation be known to have one real root between 0 and 10,
it is in effect shown say that 5 is too small (that is, the root is
between 5 and 6); next that 5.4 is too small (that is, the root is
between 5.4 and 5.5); and so on to any number of decimals. Each figure
is obtained, _not_ by the successive trial of all the figures which
precede it, but (as in the ordinary process of the extraction of a
square root, which is in fact Horner's process applied to this
particular case) it is given presumptively as the first figure of a
quotient; such value may be too large, and then the next inferior
integer must be tried instead of it, or it may require to be further
diminished. And it is to be remarked that the process not only gives
the approximate value [alpha] of the root, but (as in the extraction
of a square root) it includes the calculation of the function
[f]([alpha]), which should be, and approximately is, = 0. The
arrangement of the calculations is very elegant, and forms an integral
part of the actual method. It is to be observed that after a certain
number of decimal places have been obtained, a good many more can be
found by a mere division. It is in the progress tacitly assumed that
the roots have been first separated.

Lagrange's method (1767) gives the root as a continued fraction a +
1/b + 1/c + ..., where a is a positive or negative integer (which may
be = 0), but b, c, ... are positive integers. Suppose the roots have
been separated; then (by trial if need be of consecutive integer
values) the limits may be made to be consecutive integer numbers: say
they are a, a + 1; the value of x is therefore = a + 1/y, where y is
positive and greater than 1; from the given equation for x, writing
therein x = a + 1/y, we form an equation of the same order for y, and
this equation will have one, and only one, positive root greater than
1; hence finding for it the limits b, b + 1 (where b is = or > 1), we
have y = b + 1/z, where z is positive and greater than 1; and so
on--that is, we thus obtain the successive denominators b, c, d ... of
the continued fraction. The method is theoretically very elegant, but
the disadvantage is that it gives the result in the form of a
continued fraction, which for the most part must ultimately be
converted into a decimal. There is one advantage in the method, that a
commensurable root (that is, a root equal to a rational fraction) is
found accurately, since, when such root exists, the continued fraction
terminates.

6. Newton's method (1711), as perfected by Fourier(1831), may be
roughly stated as follows. If x = [gamma] be an approximate value of
any root, and [gamma] + h the correct value, then [f]([gamma] + h) =
0, that is,

h h^2
[f]([gamma]) + -- [f]'([gamma]) + --- [f]"([gamma]) + ... = 0;
1 1.2