Encyclopaedia Britannica, 11th Edition, "Equation" to "Ethics"
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Title: Encyclopaedia Britannica, 11th Edition, "Equation" to "Ethics"
Author: Various
Release date: February 25, 2011 [eBook #35398]
Most recently updated: January 7, 2021
Language: English
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Distributed Proofreading Team at https://www.pgdp.net
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
and then, if h be so small that the terms after the second may be
neglected, [f]([gamma]) + h[f]'([gamma]) = 0, that is, h =
{-[f]([gamma])/[f]'([gamma])}, or the new approximate value is x =
[gamma] - {[f]([gamma])/[f]'([gamma])}; and so on, as often as we
please. It will be observed that so far nothing has been assumed as to
the separation of the roots, or even as to the existence of a real
root; [gamma] has been taken as the approximate value of a root, but
no precise meaning has been attached to this expression. The question
arises, What are the conditions to be satisfied by [gamma] in order
that the process may by successive repetitions actually lead to a
certain real root of the equation; or that, [gamma] being an
approximate value of a certain real root, the new value [gamma] -
{[f]([gamma])/[f]'([gamma])} may be a more approximate value.
[Illustration: FIG. 1.]
Referring to fig. 1, it is easy to see that if OC represent the
assumed value [gamma], then, drawing the ordinate CP to meet the curve
in P, and the tangent PC' to meet the axis in C', we shall have OC' as
the new approximate value of the root. But observe that there is here
a real root OX, and that the curve beyond X is convex to the axis;
under these conditions the point C' is nearer to X than was C; and,
starting with C' instead of C, and proceeding in like manner to draw a
new ordinate and tangent, and so on as often as we please, we
approximate continually, and that with great rapidity, to the true
value OX. But if C had been taken on the other side of X, where the
curve is concave to the axis, the new point C' might or might not be
nearer to X than was the point C; and in this case the method, if it
succeeds at all, does so by accident only, i.e. it may happen that C'
or some subsequent point comes to be a point C, such that CO is a
_proper_ approximate value of the root, and then the subsequent
approximations proceed in the same manner as if this value had been
assumed in the first instance, all the preceding work being wasted. It
thus appears that for the proper application of the method we require
_more_ than the mere separation of the roots. In order to be able to
approximate to a certain root [alpha], =OX, we require to know that,
between OX and some value ON, the curve is always convex to the axis
(analytically, between the two values, [f](x) and [f]"(x) must have
always the same sign). When this is so, the point C may be taken
anywhere on the proper side of X, and within the portion XN of the
axis; and the process is then the one already explained. The
approximation is in general a very rapid one. If we know for the
required root OX the two limits OM, ON such that from M to X the curve
is always _concave_ to the axis, while from X to N it is always convex
to the axis,--then, taking D anywhere in the portion MX and (as
before) C in the portion XN, drawing the ordinates DQ, CP, and joining
the points P, Q by a line which meets the axis in D', also
constructing the point C' by means of the tangent at P as before, we
have for the required root the new limits OD', OC'; and proceeding in
like manner with the points D', C', and so on as often as we please,
we obtain at each step two limits approximating more and more nearly
to the required root OX. The process as to the point D', translated
into analysis, is the ordinate process of interpolation. Suppose OD =
[beta], OC = [alpha], we have approximately [f]([beta] + h) =
[f]([beta]) + h{[f]([alpha]) - [f]([beta])} / ([alpha] - [beta]),
whence if the root is [beta] + h then h = - ([alpha] -
[beta])[f]([beta]) / {[f]([alpha]) - [f]([beta])}.
Returning for a moment to Horner's method, it may be remarked that the
correction h, to an approximate value [alpha], is therein found as a
quotient the same or such as the quotient [f]([alpha]) / [f]'([alpha])
which presents itself in Newton's method. The difference is that with
Horner the integer part of this quotient is taken as the presumptive
value of h, and the figure is verified at each step. With Newton the
quotient itself, developed to the proper number of decimal places, is
taken as the value of h; if too many decimals are taken, there would
be a waste of work; but the error would correct itself at the next
step. Of course the calculation should be conducted without any such
waste of work.
_Imaginary Theory_.
7. It will be recollected that the expression _number_ and the
correlative epithet _numerical_ were at the outset used in a wide
sense, as extending to imaginaries. This extension arises out of the
theory of equations by a process analogous to that by which number, in
its original most restricted sense of positive integer number, was
extended to have the meaning of a real positive or negative magnitude
susceptible of continuous variation.
If for a moment number is understood in its most restricted sense as
meaning positive integer number, the solution of a simple equation leads
to an extension; ax - b = 0 gives x = (b/a), a positive fraction, and we
can in this manner represent, not accurately, but as nearly as we
please, any positive magnitude whatever; so an equation ax + b = 0 gives
x = -(b/a), which (approximately as before) represents any negative
magnitude. We thus arrive at the extended signification of number as a
continuously varying positive or negative magnitude. Such numbers may be
added or subtracted, multiplied or divided one by another, and the
result is always a number. Now from a quadric equation we derive, in
like manner, the notion of a complex or imaginary number such as is
spoken of above. The equation x^2 + 1 = 0 is not (in the foregoing
sense, number = real number) satisfied by any numerical value whatever
of x; but we assume that there is a number which we call i, satisfying
the equation i^2 + 1 = 0, and then taking a and b any real numbers, we
form an expression such as a + bi, and use the expression number in this
extended sense: any two such numbers may be added or subtracted,
multiplied or divided one by the other, and the result is always a
number. And if we consider first a quadric equation x^2 + px + q = 0
where p and q are real numbers, and next the like equation, where p and
q are any numbers whatever, it can be shown that there exists for x a
numerical value which satisfies the equation; or, in other words, it can
be shown that the equation has a numerical root. The like theorem, in
fact, holds good for an equation of any order whatever; but suppose for
a moment that this was not the case; say that there was a cubic equation
x^3 + px^2 + qx + r = 0, with numerical coefficients, not satisfied by
any numerical value of x, we should have to establish a new imaginary j
satisfying some such equation, and should then have to consider numbers
of the form a + bj, or perhaps a + bj + cj^2 (a, b, c numbers [alpha] +
[beta]i of the kind heretofore considered),--first we should be thrown
back on the quadric equation x^2 + px + q = 0, p and q being now numbers
of the last-mentioned extended form--_non constat_ that every such
equation has a numerical root--and if not, we might be led to _other_
imaginaries k, l, &c., and so on _ad infinitum_ in inextricable
confusion.
But in fact a numerical equation of any order whatever has always a
numerical root, and thus numbers (in the foregoing sense, number =
quantity of the form [alpha] + [beta]i) form (_what real numbers do
not_) a universe complete in itself, such that starting in it we are
never led out of it. There may very well be, and perhaps are, numbers in
a more general sense of the term (quaternions are not a case in point,
as the ordinary laws of combination are not adhered to), but in order to
have to do with such numbers (if any) we must start with them.
8. The capital theorem as regards numerical equations thus is, every
numerical equation has a numerical root; or for shortness (the meaning
being as before), every equation has a root. Of course the theorem is
the reverse of self-evident, and it requires proof; but provisionally
assuming it as true, we derive from it the general theory of numerical
equations. As the term root was introduced in the course of an
explanation, it will be convenient to give here the formal definition.
A number a such that substituted for x it makes the function x1^n -
p1x^(n - 1) ... [+-]p_n to be = 0, or say such that it satisfies the
equation [f](x) = 0, is said to be a root of the equation; that is, a
being a root, we have
a^n - p1a^(n - 1) ... [+-]p_n = 0, or say [f](a) = 0;
and it is then easily shown that x - a is a factor of the function
[f](x), viz. that we have [f](x) = (x - a)[f]1(x), where [f]1(x) is a
function x^(n - 1) - q1x^(n - 2) ... [+-]q_(n - 1) of the order n - 1,
with numerical coefficients q1, q2 ... q_(n - 1).
In general a is not a root of the equation [f]1(x) = 0, but it may be
so--i.e. [f]1(x) may contain the factor x - a; when this is so, [f](x)
will contain the factor (x - a)^2; writing then [f](x) = (x -
a)^2[f]2(x), and assuming that a is not a root of the equation [f]2(x)
= 0, x = a is then said to be a double root of the equation [f](x) =
0; and similarly [f](x) may contain the factor (x - a)^3 and no higher
power, and x = a is then a triple root; and so on.
Supposing in general that [f](x) = (x - a)^[alpha] F(x) ([alpha] being
a positive integer which may be = 1, (x - a)^[alpha] the highest power
of x - a which divides [f](x), and F(x) being of course of the order n
- [alpha]), then the equation F(x) = 0 will have a root b which will
be different from a; x - b will be a factor, in general a simple one,
but it may be a multiple one, of F(x), and [f](x) will in this case be
= (x - a)^[alpha] (x - b)^[beta] [Phi](x) ([beta] a positive integer
which may be = 1, (x-b)^[beta] the highest power of x - b in F(x) or
[f](x), and [Phi](x) being of course of the order n - [alpha] -
[beta]). The original equation [f](x) = 0 is in this case said to have
[alpha] roots each = a, [beta] roots each = b; and so on for any other
factors (x - c)^[gamma], &c.
We have thus the _theorem_--A numerical equation of the order n has in
every case n roots, viz. there exist n numbers, a, b, ... (in general
all distinct, but which may arrange themselves in any sets of equal
values), such that [f](x) = (x - a)(x - b)(x - c) ... identically.
If the equation has equal roots, these can in general be determined,
and the case is at any rate a special one which may be in the first
instance excluded from consideration. It is, therefore, in general
assumed that the equation [f](x) = 0 has all its roots unequal.
If the coefficients p1, p2, ... are all or any one or more of them
imaginary, then the equation [f](x) = 0, separating the real and
imaginary parts thereof, may be written F(x) + i[Phi](x) = 0, where
F(x), [Phi](x) are each of them a function with real coefficients; and
it thus appears that the equation [f](x) = 0, with imaginary
coefficients, has not in general any real root; supposing it to have a
real root a, this must be at once a root of each of the equations F(x)
= 0 and [Phi](x) = 0.
But an equation with real coefficients may have as well imaginary as
real roots, and we have further the _theorem_ that for any such
equation the imaginary roots enter in pairs, viz. [alpha] + [beta]i
being a root, then [alpha] - [beta]i will be also a root. It follows
that if the order be odd, there is always an odd number of real roots,
and therefore at least one real root.
9. In the case of an equation with real coefficients, the question of
the existence of real roots, and of their separation, has been already
considered. In the general case of an equation with imaginary (it may be
real) coefficients, the like question arises as to the situation of the
(real or imaginary) roots; thus, if for facility of conception we regard
the constituents [alpha], [beta] of a root [alpha] + [beta]i as the
co-ordinates of a point _in plano_, and accordingly represent the root
by such point, then drawing in the plane any closed curve or "contour,"
the question is how many roots lie within such contour.
This is solved theoretically by means of a theorem of A.L. Cauchy
(1837), viz. writing in the original equation x + iy in place of x,
the function [f](x + iy) becomes = P + iQ, where P and Q are each of
them a rational and integral function (with real coefficients) of (x,
y). Imagining the point (x, y) to travel along the contour, and
considering the number of changes of sign from - to + and from + to -
of the fraction corresponding to passages of the fraction through zero
(that is, to values for which P becomes = 0, disregarding those for
which Q becomes = 0), the difference of these numbers gives the number
of roots within the contour.
It is important to remark that the demonstration does not presuppose
the existence of any root; the contour may be the infinity of the
plane (such infinity regarded as a contour, or closed curve), and in
this case it can be shown (and that very easily) that the difference
of the numbers of changes of sign is = n; that is, there are within
the infinite contour, or (what is the same thing) there are in all n
roots; thus Cauchy's theorem contains really the proof of the
fundamental theorem that a numerical equation of the nth order (not
only has a numerical root, but) has precisely n roots. It would appear
that this proof of the fundamental theorem in its most complete form
is in principle identical with the last proof of K.F. Gauss (1849) of
the theorem, in the form--A numerical equation of the nth order has
always a root.[3]
But in the case of a finite contour, the actual determination of the
difference which gives the number of real roots can be effected only
in the case of a rectangular contour, by applying to each of its sides
separately a method such as that of Sturm's theorem; and thus the
actual determination ultimately depends on a method such as that of
Sturm's theorem.
Very little has been done in regard to the calculation of the
imaginary roots of an equation by approximation; and the question is
not here considered.
10. A class of numerical equations which needs to be considered is that
of the binomial equations x^n - a = 0 (a = [alpha] + [beta]i, a complex
number).
The foregoing conclusions apply, viz. there are always n roots, which,
it may be shown, are all unequal. And these can be found numerically
by the extraction of the square root, and of an nth root, of _real_
numbers, and by the aid of a table of natural sines and cosines.[4]
For writing
/ [alpha] [beta] \
[alpha] + [beta]i = [root]([alpha]^2 + [beta]^2) ( ---------------------------- + ----------------------------i ),
\[root]([alpha]^2 + [beta]^2) [root]([alpha]^2 + [beta]^2) /
there is always a real angle [lambda] (positive and less than 2[pi]),
such that its cosine and sine are = [alpha] / [root]([alpha]^2 +
[beta]^2) and [beta] / [root]([alpha]^2 + [beta]^2) respectively; that
is, writing for shortness [root]([alpha]^2 + [beta]^2) = [rho], we have
[alpha] + [beta]i = [rho](cos[lambda] + i sin[lambda]), or the
equation is x^n = [rho](cos[lambda] + i sin [lambda]); hence observing
that (cos [lambda]/n + i sin [lambda]/n )^n = cos[lambda] + i
sin[lambda], a value of x is = [root n][rho] (cos [lambda]/n + i sin
[lambda]/n). The formula really gives all the roots, for instead of
[lambda] we may write [lambda] + 2s[pi], s a positive or negative
integer, and then we have
/ [lambda] + 2s[pi] [lambda] + 2s[pi] \
x = [root n][rho] ( cos ----------------- + i sin ----------------- ),
\ n n /
which has the n values obtained by giving to s the values 0, 1, 2 ...
n - 1 in succession; the roots are, it is clear, represented by points
lying at equal intervals on a circle. But it is more convenient to
proceed somewhat differently; taking one of the roots to be [theta],
so that [theta]^n = a, then assuming x = [theta]y, the equation
becomes y^n - 1 = 0, which equation, like the original equation, has
precisely n roots (one of them being of course = 1). And the original
equation x^n - a = 0 is thus reduced to the more simple equation x^n -
1 = 0; and although the theory of this equation is included in the
preceding one, yet it is proper to state it separately.
The equation x^n - 1 = 0 has its several roots expressed in the form
1, [omega], [omega]^2, ... [omega]^(n - 1), where [omega] may be taken
= cos 2[pi]/n + i sin 2[pi]/n; in fact, [omega] having this value, any
integer power [omega]^k is = cos 2[pi]k/n + i sin 2[pi]k/n, and we
thence have ([omega]^k)^n = cos 2[pi]k + i sin 2[pi]k, = 1, that is,
[omega]^k is a root of the equation. The theory will be resumed
further on.
By what precedes, we are led to the notion (a numerical) of the
radical a^(1/n) regarded as an n-valued function; any one of these
being denoted by [root n]a, then the series of values is [root n]a,
[omega][root n]a, ... [omega]^(n - 1)[root n]a; or we may, if we
please, use [root n]a instead of a^(1/n) as a symbol to denote the
n-valued function.
As the coefficients of an algebraical equation may be numerical, all
which follows in regard to algebraical equations is (with, it may be,
some few modifications) applicable to numerical equations; and hence,
concluding for the present this subject, it will be convenient to pass
on to algebraical equations.
_Algebraical Equations._
11. The equation is
x^n - p1x^(n-1) + ... [+-]p_n = 0,
and we here _assume_ the existence of roots, viz. we assume that there
are n quantities a, b, c ... (in general all of them different, but
which in particular cases may become equal in sets in any manner), such
that
x^n - p1x^(n - 1) + ... [+-] p_n = 0;
or looking at the question in a different point of view, and starting
with the roots a, b, c ... as given, we express the product of the n
factors x - a, x - b, ... in the foregoing form, and thus arrive at an
equation of the order n having the n roots a, b, c.... In either case we
have
p1 = [Sigma]a, p2 = [Sigma]ab, ... p_n = abc ...;
i.e. regarding the coefficients p1, p2 ... p_n as given, then we assume
the existence of roots a, b, c, ... such that p1 = [Sigma]a, &c.; or,
regarding the roots as given, then we write p1, p2, &c., to denote the
functions [Sigma]a, [Sigma]ab, &c.
As already explained, the epithet algebraical is not used in
opposition to numerical; an algebraical equation is merely an equation
wherein the coefficients are not restricted to denote, or are not
explicitly considered as denoting, numbers. That the abstraction is
legitimate, appears by the simplest example; in saying that the
equation x^2 - px + q = 0 has a root x = 1/2{p + [root](p^2 - 4q)}, we
mean that writing this value for x the equation becomes an identity,
[1/2{p + [root](p^2 - 4q)}]^2 - p[1/2{p + [root](p^2 - 4q)}] + q = 0;
and the verification of this identity in nowise depends upon p and q
meaning numbers. But if it be asked what there is beyond numerical
equations included in the term algebraical equation, or, again, what
is the full extent of the meaning attributed to the term--the latter
question at any rate it would be very difficult to answer; as to the
former one, it may be said that the coefficients may, for instance, be
symbols of operation. As regards such equations, there is certainly no
proof that every equation has a root, or that an equation of the nth
order has n roots; nor is it in any wise clear what the precise
signification of the statement is. But it is found that the assumption
of the existence of the n roots can be made without contradictory
results; conclusions derived from it, if they involve the roots, rest
on the same ground as the original assumption; but the conclusion may
be independent of the roots altogether, and in this case it is
undoubtedly valid; the reasoning, although actually conducted by aid
of the assumption (and, it may be, most easily and elegantly in this
manner), is really independent of the assumption. In illustration, we
observe that it is allowable to express a function of p and q as
follows,--that is, by means of a rational symmetrical function of a
and b, this can, as a fact, be expressed as a rational function of a +
b and ab; and if we prescribe that a + b and ab shall then be changed
into p and q respectively, we have the required function of p, q. That
is, we have F([alpha], [beta]) as a representation of [f](p, q),
obtained as if we had p = a + b, q = ab, but without in any wise
assuming the existence of the a, b of these equations.
12. Starting from the equation
x^n - p1x^(n - 1) + ... = x - a.x - b. &c.
or the equivalent equations p1 = [Sigma]a, &c., we find
a^n - p1a^(n - 1) + ... = 0,
b^n - p1b^(n - 1) + ... = 0;
. . .
. . .
. . .
(it is as satisfying these equations that a, b ... are said to be the
roots of x^n - p1x^(n - 1) + ... = 0); and conversely from the
last-mentioned equations, assuming that a, b ... are all different, we
deduce
p1 = [Sigma]a, p2 = [Sigma]ab, &c.
and
x^n - p1x^(n - 1) + ... = x - a.x - b. &c.
Observe that if, for instance, a = b, then the equations a^n - p1a^(n -
1) + ... = 0, b^n - p1b^(n - 1) + ... = 0 would reduce themselves to a
single relation, which would not of itself express that a was a double
root,--that is, that (x - a)^2 was a factor of x^n - p1x^(n - 1) +, &c;
but by considering b as the limit of a + h, h indefinitely small, we
obtain a second equation
na^(n - 1) - (n - 1)p1a^(n - 2) + ... = 0,
which, with the first, expresses that a is a double root; and then the
whole system of equations leads as before to the equations p1 =
[Sigma]a, &c. But the existence of a double root implies a certain
relation between the coefficients; the general case is when the roots
are all unequal.
We have then the _theorem_ that every rational symmetrical function of
the roots is a rational function of the coefficients. This is an easy
consequence from the less general theorem, every rational and integral
symmetrical function of the roots is a rational and integral function of
the coefficients.
In particular, the sums of the powers [Sigma]a^2, [Sigma]a^3, &c., are
rational and integral functions of the coefficients.
The process originally employed for the expression of other functions
[Sigma]a^[alpha] b^[beta], &c., in terms of the coefficients is to
make them depend upon the sums of powers: for instance,
[Sigma]a^[alpha] b^[beta] = [Sigma]a^[alpha] [Sigma]a^[beta] -
[Sigma]a^([alpha] + [beta]); but this is very objectionable; the true
theory consists in showing that we have systems of equations
p1 = [Sigma]a,
p2 = [Sigma]ab,
p1^2 = [Sigma]a^2 + 2[Sigma]ab,
p3 = [Sigma]abc,
p1p2 = [Sigma]a^2 b + 3[Sigma]abc,
p1^3 = [Sigma]a^3 + 3[Sigma]a^2 b + 6[Sigma]abc,
where in each system there are precisely as many equations as there
are root-functions on the right-hand side--e.g. 3 equations and 3
functions [Sigma]abc, [Sigma]a^2 b, [Sigma]a^3. Hence in each system
the root-functions can be determined linearly in terms of the powers
and products of the coefficients:
[Sigma]ab = p2,
[Sigma]a^2 = p1^2 - 2p2,
[Sigma]abc = p3,
[Sigma]a^2 b = p1p2 - 3p3,
[Sigma]a^3 = p1^3 - 3p1p2 + 3p3,
and so on. The other process, if applied consistently, would derive
the originally assumed value [Sigma]ab = p2, from the two equations
[Sigma]a = p, [Sigma]a^2 = p1^2 - 2p2; i.e. we have 2[Sigma]ab =
[Sigma]a.[Sigma]a - [Sigma]a^2,= p1^2 - (p1^2 - 2p2), = 2p2.
13. It is convenient to mention here the theorem that, x being
determined as above by an equation of the order n, any rational and
integral function whatever of x, or more generally any rational function
which does not become infinite in virtue of the equation itself, can be
expressed as a rational and integral function of x, of the order n - 1,
the coefficients being rational functions of the coefficients of the
equation. Thus the equation gives x^n a function of the form in
question; multiplying each side by x, and on the right-hand side writing
for x^n its foregoing value, we have x^(n + 1), a function of the form
in question; and the like for any higher power of x, and therefore also
for any rational and integral function of x. The proof in the case of a
rational non-integral function is somewhat more complicated. The final
result is of the form [phi](x)/[psi](x) = I(x), or say [phi](x)
-[psi](x)I(x) = 0, where [phi], [psi], I are rational and integral
functions; in other words, this equation, being true if only [f](x) = 0,
can only be so by reason that the left-hand side contains [f](x) as a
factor, or we must have identically [phi](x) - [psi](x)I(x) =
M(x)[f](x). And it is, moreover, clear that the equation
[phi](x)/[psi](x) = I(x), being satisfied if only [f](x) = 0, must be
satisfied by each root of the equation.
From the theorem that a rational symmetrical function of the roots is
expressible in terms of the coefficients, it at once follows that it
is possible to determine an equation (of an assignable order) having
for its roots the several values of any given (unsymmetrical) function
of the roots of the given equation. For example, in the case of a
quartic equation, roots (a, b, c, d), it is possible to find an
equation having the roots ab, ac, ad, bc, bd, cd (being therefore a
sextic equation): viz. in the product
(y - ab)(y - ac)(y - ad)(y - bc)(y - bd)(y - cd)
the coefficients of the several powers of y will be symmetrical
functions of a, b, c, d and therefore rational and integral functions
of the coefficients of the quartic equation; hence, supposing the
product so expressed, and equating it to zero, we have the required
sextic equation. In the same manner can be found the sextic equation
having the roots (a - b)^2, (a - c)^2, (a - d)^2, (b - c)^2, (b -
d)^2, (c - d)^2, which is the equation of differences previously
referred to; and similarly we obtain the equation of differences for a
given equation of any order. Again, the equation sought for may be
that having for its n roots the given rational functions [phi](a),
[phi](b), ... of the several roots of the given equation. Any such
rational function can (as was shown) be expressed as a rational and
integral function of the order n - 1; and, retaining x in place of any
one of the roots, the problem is to find y from the equations x^n - p1
x^(n - 1) ... = 0, and y = M0x^(n - 1) + M1x^(n - 2) + ..., or, what
is the same thing, from these two equations to eliminate x. This is in
fact E.W. Tschirnhausen's transformation (1683).
14. In connexion with what precedes, the question arises as to the
number of values (obtained by permutations of the roots) of given
unsymmetrical functions of the roots, or say of a given set of letters:
for instance, with roots or letters (a, b, c, d) as before, how many
values are there of the function ab + cd, or better, how many functions
are there of this form? The answer is 3, viz. ab + cd, ac + bd, ad + bc;
or again we may ask whether, in the case of a given number of letters,
there exist functions with a given number of values, 3-valued, 4-valued
functions, &c.
It is at once seen that for any given number of letters there exist
2-valued functions; the product of the differences of the letters is
such a function; however the letters are interchanged, it alters only
its sign; or say the two values are [Delta] and -[Delta]. And if P, Q
are symmetrical functions of the letters, then the general form of
such a function is P + Q[Delta]; this has only the two values P +
Q[Delta], P - Q[Delta].
In the case of 4 letters there exist (as appears above) 3-valued
functions: but in the case of 5 letters there does not exist any
3-valued or 4-valued function; and the only 5-valued functions are
those which are symmetrical in regard to four of the letters, and can
thus be expressed in terms of one letter and of symmetrical functions
of all the letters. These last theorems present themselves in the
demonstration of the non-existence of a solution of a quintic equation
by radicals.
The theory is an extensive and important one, depending on the notions
of _substitutions_ and of _groups_ (q.v.).
15. Returning to equations, we have the very important theorem that,
given the value of any unsymmetrical function of the roots, e.g. in the
case of a quartic equation, the function ab + cd, it is in general
possible to determine rationally the value of any similar function, such
as (a + b)^3 + (c + d)^3.
The _a priori_ ground of this theorem may be illustrated by means of a
numerical equation. Suppose that the roots of a quartic equation are
1, 2, 3, 4, then if it is given that ab + cd = 14, this in effect
determines a, b to be 1, 2 and c, d to be 3, 4 (viz. a = 1, b = 2 or a
= 2, b = 1, and c = 3, d = 4 or c = 3, d = 4) or else a, b to be 3, 4
and c, d to be 1, 2; and it therefore in effect determines (a + b)^3 +
(c + d)^3 to be = 370, and not any other value; that is, (a + b)^3 + (c
+ d)^3, as having a single value, must be determinable rationally. And
we can in the same way account for cases of failure as regards
particular equations; thus, the roots being 1, 2, 3, 4 as before, a^2 b
= 2 determines a to be = 1 and b to be = 2, but if the roots had been
1, 2, 4, 16 then a^2 b = 16 does not uniquely determine a, b but only
makes them to be 1, 16 or 2, 4 respectively.
As to the _a posteriori_ proof, assume, for instance,
t1 = ab + cd, y1 = (a + b)^3 + (c + d)^3,
t2 = ac + bd, y2 = (a + c)^3 + (b + d)^3,
t3 = ad + bc, y3 = (a + d)^3 + (b + c)^3:
then y1 + y2 + y3, t1y1 + t2y2 + t3y3, t1^2 y1 + t2^2y2 + t3^2y3 will
be respectively symmetrical functions of the roots of the quartic, and
therefore rational and integral functions of the coefficients; that
is, they will be known.
Suppose for a moment that t1, t2, t3 are all known; then the equations
being linear in y1, y2, y3 these can be expressed rationally in terms
of the coefficients and of t1, t2, t3; that is, y1, y2, y3 will be
known. But observe further that y1 is obtained as a function of t1,
t2, t3 symmetrical as regards t2, t3; it can therefore be expressed as
a rational function of t1 and of t2 + t3, t2t3, and thence as a
rational function of t1 and of t1 + t2 + t3, t1t2 + t1t3 + t2t3,
t1t2t3; but these last are symmetrical functions of the roots, and as
such they are expressible rationally in terms of the coefficients;
that is, y1 will be expressed as a rational function of t1 and of the
coefficients; or t1 (alone, not t2 or t3) being known, y1 will be
rationally determined.
16. We now consider the question of the algebraical solution of
equations, or, more accurately, that of the _solution of equations by
radicals_.
In the case of a quadric equation x^2 - px + q = 0, we can by the
assistance of the sign [root]( ) or ( )^1/2 find an expression for x
as a 2-valued function of the coefficients p, q such that substituting
this value in the equation, the equation is thereby identically
satisfied; it has been found that this expression is
x = 1/2{p [+-] [root](p^2 - 4q)},
and the equation is on this account said to be algebraically solvable,
or more accurately solvable by radicals. Or we may by writing x =
-1/2 p + z reduce the equation to z^2 = 1/4(p^2 - 4q), viz. to an
equation of the form x^2 = a; and in virtue of its being thus
reducible we say that the original equation is solvable by radicals.
And the question for an equation of any higher order, say of the order
n, is, can we by means of radicals (that is, by aid of the sign [root
m]( ) or ( )^(1/m), using as many as we please of such signs and with
any values of m) find an n-valued function (or any function) of the
coefficients which substituted for x in the equation shall satisfy it
identically?
It will be observed that the coefficients p, q ... are not explicitly
considered as numbers, but even if they do denote numbers, the
question whether a numerical equation admits of solution by radicals
is wholly unconnected with the before-mentioned theorem of the
existence of the n roots of such an equation. It does not even follow
that in the case of a numerical equation solvable by radicals the
algebraical solution gives the numerical solution, but this requires
explanation. Consider first a numerical quadric equation with
imaginary coefficients. In the formula x = 1/2{p [+-] [root](p^2 -
4q)}, substituting for p, q their given numerical values, we obtain
for x an expression of the form x = [alpha] + [beta]i [+-]
[root]([gamma] + [delta]i), where [alpha], [beta], [gamma], [delta]
are real numbers. This expression substituted for x in the quadric
equation would satisfy it identically, and it is thus an algebraical
solution; but there is no obvious _a priori_ reason why
[root]([gamma]+[delta]i) should have a value = c + di, where c and d
are real numbers calculable by the extraction of a root or roots of
real numbers; however the case is (what there was no _a priori_ right
to expect) that [root]([gamma] + [delta]i) has such a value calculable
by means of the radical expressions [root]{[root]([gamma]^2 +
[delta]^2) [+-] [gamma]} : and hence the algebraical solution of a
numerical quadric equation does in every case give the numerical
solution. The case of a numerical cubic equation will be considered
presently.
17. A cubic equation can be solved by radicals.
Taking for greater simplicity the cubic in the reduced form x^3 + qx -
r = 0, and assuming x = a + b, this will be a solution if only 3ab = q
and a^3 + b^3 = r, equations which give (a^3 - b^3)^2 = r^2 -
(4/27)q^3, a quadric equation solvable by radicals, and giving a^3 -
b^3 = [root](r^2 - (4/27)q^3), a 2-valued function of the
coefficients: combining this with a^3 + b^3 = r, we have a^3 = 1/2{r +
[root](r^2 - (4/27)q^3)}, a 2-valued function: we then have a by means
of a cube root, viz.
a = [root 3][1/2{r + [root](r^2 - (4/27)q^3)}],
a 6-valued function of the coefficients; but then, writing q = b/3a,
we have, as may be shown, a + b a 3-valued function of the
coefficients; and x = a + b is the required solution by radicals. It
would have been wrong to complete the solution by writing
b = [root 3][1/2{r - [root](r^2 - (4/27)q^3)}],
for then a + b would have been given as a 9-valued function having
only 3 of its values roots, and the other 6 values being irrelevant.
Observe that in this last process we make no use of the equation 3ab
= q, in its original form, but use only the derived equation 27a^3 b^3
= q^3, implied in, but not implying, the original form.
An interesting variation of the solution is to write x = ab(a + b),
giving a^3 b^3(a^3 + b^3) = r and 3a^3 b^3 = q, or say a^3 + b^3 =
3r/q, a^3 b^3 = (1/3)q; and consequently
3/2 4 3/2 4
a^3 = --- {r + [root](r^2 - --q^3)}, b^3 = --- {r - [root](r^2 - --q^3)},
q 27 q 27
i.e. here a^3, b^3 are each of them a 2-valued function, but as the
only effect of altering the sign of the quadric radical is to
interchange a^3, b^3, they may be regarded as each of them 1-valued; a
and b are each of them 3-valued (for observe that here only a^3 b^3,
not ab, is given); and ab(a + b) thus is in appearance a 9-valued
function; but it can easily be shown that it is (as it ought to be)
only 3-valued.
In the case of a numerical cubic, even when the coefficients are real,
substituting their values in the expression
x = [root 3][1/2{r + [root](r^2 - (4/27)q^3)}] + (1/3)q /
[root 3][1/2{r + [root](r^2 - (4/27)q^3)}],
this may depend on an expression of the form [root 3]([gamma] +
[delta]i) where [gamma] and [delta] are real numbers (it will do so if
r^2 - (4/27)q^3 is a negative number), and then we _cannot_ by the
extraction of any root or roots of real positive numbers reduce [root
3]([gamma] + [delta]i) to the form c + di, c and d real numbers; hence
here the algebraical solution does not give the numerical solution,
and we have here the so-called "irreducible case" of a cubic equation.
By what precedes there is nothing in this that might not have been
expected; the algebraical solution makes the solution depend on the
extraction of the cube root of a number, and there was no reason for
expecting this to be a real number. It is well known that the case in
question is that wherein the three roots of the numerical cubic
equation are all real; if the roots are two imaginary, one real, then
contrariwise the quantity under the cube root is real; and the
algebraical solution gives the numerical one.
The irreducible case is solvable by a trigonometrical formula, but
this is not a solution by radicals: it consists in effect in reducing
the given numerical cubic (not to a cubic of the form z^3 = a, solvable
by the extraction of a cube root, but) to a cubic of the form 4x^3 - 3x
= a, corresponding to the equation 4 cos^3 [theta] - 3 cos[theta] = cos
3[theta] which serves to determine cos[theta] when cos 3[theta] is
known. The theory is applicable to an algebraical cubic equation; say
that such an equation, if it can be reduced to the form 4x^3 - 3x = a,
is solvable by "trisection"--then the general cubic equation is
solvable by trisection.
18. A quartic equation is solvable by radicals, and it is to be remarked
that the existence of such a solution depends on the existence of
3-valued functions such as ab + cd of the four roots (a, b, c, d): by
what precedes ab + cd is the root of a cubic equation, which equation is
solvable by radicals: hence ab + cd can be found by radicals; and since
abcd is a given function, ab and cd can then be found by radicals. But
by what precedes, if ab be known then any similar function, say a + b,
is obtainable rationally; and then from the values of a + b and ab we
may by radicals obtain the value of a or b, that is, an expression for
the root of the given quartic equation: the expression ultimately
obtained is 4-valued, corresponding to the different values of the
several radicals which enter therein, and we have thus the expression by
radicals of each of the four roots of the quartic equation. But when the
quartic is numerical the same thing happens as in the cubic, and the
algebraical solution does not in every case give the numerical one.
It will be understood from the foregoing explanation as to the quartic
how in the next following case, that of the quintic, the question of
the solvability by radicals depends on the existence or non-existence
of k-valued functions of the five roots (a, b, c, d, e); the
fundamental theorem is the one already stated, a rational function of
five letters, if it has less than 5, cannot have more than 2 values,
that is, there are no 3-valued or 4-valued functions of 5 letters: and
by reasoning depending in part upon this theorem, N.H. Abel (1824)
showed that a general quintic equation is not solvable by radicals;
and _a fortiori_ the general equation of any order higher than 5 is
not solvable by radicals.
19. The general theory of the solvability of an equation by radicals
depends fundamentally on A.T. Vandermonde's remark (1770) that,
supposing an equation is solvable by radicals, and that we have
therefore an algebraical expression of x in terms of the coefficients,
then substituting for the coefficients their values in terms of the
roots, the resulting expression must reduce itself to any one at
pleasure of the roots a, b, c ...; thus in the case of the quadric
equation, in the expression x = 1/2{p + [root](p^2 - 4q)},
substituting for p and q their values, and observing that (a + b)^2 -
4ab = (a - b)^2, this becomes x = 1/2{a + b + [root](a - b)^2}, the
value being a or b according as the radical is taken to be +(a - b) or
-(a - b).
So in the cubic equation x^3 - px^2 + qx - r = 0, if the roots are a,
b, c, and if [omega] is used to denote an imaginary cube root of
unity, [omega]^2 + [omega] + 1 = 0, then writing for shortness p = a +
b + c, L = a + [omega]b + [omega]^2 c, M = a + [omega]^2 b + [omega]c,
it is at once seen that LM, L^3 + M^3, and therefore also (L^3 -
M^3)^2 are symmetrical functions of the roots, and consequently
rational functions of the coefficients: hence
1/2{L^3 + M^3 + [root](L^3 - M^3)^2}
is a rational function of the coefficients, which when these are
replaced by their values as functions of the roots becomes, according
to the sign given to the quadric radical, = L^3 or M^3; taking it =
L^3, the cube root of the expression has the three values L, [omega]L,
[omega]^2 L; and LM divided by the same cube root has therefore the
values M, [omega]^2M, [omega]M; whence finally the expression
(1/3)[p + [root 3]{1/2(L^3 + M^3 + [root](L^3 - M^3)^2)} + LM /
[root 3]{1/2L^3 + M^3 + [root](L^3 - M^3)^2}]
has the three values
(1/3)(p + L + M), (1/3)(p + [omega]L + [omega]^2 M),
(1/3)(p + [omega]^2 L + [omega]M);
that is, these are = a, b, c respectively. If the value M^3 had been
taken instead of L^3, then the expression would have had the same
three values a, b, c. Comparing the solution given for the cubic x^3 +
qx - r = 0, it will readily be seen that the two solutions are
identical, and that the function r^2 - (4/27)q^3 under the radical
sign must (by aid of the relation p = 0 which subsists in this case)
reduce itself to (L^3 - M^3)^2; it is only by each radical being equal
to a rational function of the roots that the final expression _can_
become equal to the roots a, b, c respectively.
20. The formulae for the cubic were obtained by J.L. Lagrange
(1770-1771) from a different point of view. Upon examining and comparing
the principal known methods for the solution of algebraical equations,
he found that they all ultimately depended upon finding a "resolvent"
equation of which the root is a + [omega]b + [omega]^2 c + [omega]^3 d +
..., [omega] being an imaginary root of unity, of the same order as the
equation; e.g. for the cubic the root is a + [omega]b + [omega]^2 c,
[omega] an imaginary cube root of unity. Evidently the method gives for
L^3 a quadric equation, which is the "resolvent" equation in this
particular case.
For a quartic the formulae present themselves in a somewhat different
form, by reason that 4 is not a prime number. Attempting to apply it to
a quintic, we seek for the equation of which the root is (a + [omega]b +
[omega]^2 c + [omega]^3 d + [omega]^4 e), [omega] an imaginary fifth
root of unity, or rather the fifth power thereof (a + [omega]b +
[omega]^2 c + [omega]^3d + [omega]^4 e)^5; this is a 24-valued function,
but if we consider the four values corresponding to the roots of unity
[omega], [omega]^2, [omega]^3, [omega]^4, viz. the values
(a + [omega]b + [omega]^2 c + [omega]^3 d + [omega]^4 e)^5,
(a + [omega]^2 b + [omega]^4 c + [omega]d + [omega]^3e)^5,
(a + [omega]^3 b + [omega]c + [omega]^4 d + [omega]^2e)^5,
(a + [omega]^4 b + [omega]^3 c + [omega]^2 d + [omega]e)^5,
any symmetrical function of these, for instance their sum, is a 6-valued
function of the roots, and may therefore be determined by means of a
sextic equation, the coefficients whereof are rational functions of the
coefficients of the original quintic equation; the conclusion being that
the solution of an equation of the fifth order is made to depend upon
that of an equation of the sixth order. This is, of course, useless for
the solution of the quintic equation, which, as already mentioned, does
not admit of solution by radicals; but the equation of the sixth order,
Lagrange's resolvent sextic, is very important, and is intimately
connected with all the later investigations in the theory.
21. It is to be remarked, in regard to the question of solvability by
radicals, that not only the coefficients are taken to be arbitrary, but
it is assumed that they are represented each by a single letter, or say
rather that they are not so expressed in terms of other arbitrary
quantities as to make a solution possible. If the coefficients are not
all arbitrary, for instance, if some of them are zero, a sextic equation
might be of the form x^6 + bx^4 + cx^2 + d = 0, and so be solvable as a
cubic; or if the coefficients of the sextic are given functions of the
six arbitrary quantities a, b, c, d, e, f, such that the sextic is
really of the form (x^2 + ax + b)(x^4 + cx^3 + dx^2 + ex + f) = 0, then
it breaks up into the equations x^2 + ax + b = 0, x^4 + cx^3 + dx^2 + ex
+ f = 0, and is consequently solvable by radicals; so also if the form
is (x -a)(x - b)(x - c)(x - d)(x - e)(x - f) = 0, then the equation is
solvable by radicals,--in this extreme case rationally. Such cases of
solvability are self-evident; but they are enough to show that the
general theorem of the non-solvability by radicals of an equation of the
fifth or any higher order does not in any wise exclude for such orders
the existence of particular equations solvable by radicals, and there
are, in fact, extensive classes of equations which are thus solvable;
the binomial equations x^n - 1 = 0 present an instance.
22. It has already been shown how the several roots of the equation
x^n - 1 = 0 can be expressed in the form cos 2s[pi]/n + i sin
2s[pi]/n, but the question is now that of the algebraical solution (or
solution by radicals) of this equation. There is always a root = 1; if
[omega] be any other root, then obviously [omega], [omega]^2, ...
[omega]^(n - 1) are all of them roots; x^n - 1 contains the factor x -
1, and it thus appears that [omega], [omega]^2, ... [omega]^(n - 1) are
the n - 1 roots of the equation
x^(n - 1) + x^(n - 2) + ... x + 1 = 0;
we have, of course, [omega]^(n - 1) + [omega]^(n - 2) + ... + [omega]
+ 1 = 0.
It is proper to distinguish the cases n prime and n composite; and in
the latter case there is a distinction according as the prime factors
of n are simple or multiple. By way of illustration, suppose
successively n = 15 and n = 9; in the former case, if [alpha] be an
imaginary root of x^3 - 1 = 0 (or root of x^2 + x + 1 = 0), and [beta]
an imaginary root of x^5 - 1 = 0 (or root of x^4 + x^3 + x^2 + x + 1 =
0), then [omega] may be taken = [alpha][beta]; the successive powers
thereof, [alpha][beta], [alpha]^2 [beta]^2, [beta]^3, [alpha][beta]^4,
[alpha]^2, [beta], [alpha][beta]^2, [alpha]^2[beta]^3, [beta]^4,
[alpha], [alpha]^2 [beta], [beta]^2, [alpha][beta]^3, [alpha]^2
[beta]^4, are the roots of x^14 + x^13 + ... + x + 1 = 0; the solution
thus depends on the solution of the equations x^3 - 1 = 0 and x^5 - 1
= 0. In the latter case, if [alpha] be an imaginary root of x^3 - 1 =
0 (or root of x^2 + x + 1 = 0), then the equation x^9 - 1 = 0 gives
x^3 = 1, [alpha], or [alpha]^2; x^3 = 1 gives x = 1, [alpha], or
[alpha]^2; and the solution thus depends on the solution of the
equations x^3 - 1 = 0, x^3 - [alpha] = 0, x^3 - [alpha]^2 = 0. The
first equation has the roots 1, [alpha], [alpha]^2; if [beta] be a
root of either of the others, say if [beta]^3 = [alpha], then assuming
[omega] = [beta], the successive powers are [beta], [beta]^2, [alpha],
[alpha][beta], [alpha][beta]^2, [alpha]^2, [alpha]^2[beta], [alpha]^2
[beta]^2, which are the roots of the equation x^8 + x^7 + ... + x + 1
= 0.
It thus appears that the only case which need be considered is that of
n a prime number, and writing (as is more usual) r in place of
[omega], we have r, r^2, r^3, ... r^(n - 1) as the (n - 1) roots of
the reduced equation
x^(n - 1) + x^(n - 2) + ... + x + 1 = 0;
then not only r^n - 1 = 0, but also r^(n - 1) + r^(n - 2) + ... + r +
1 = 0.
23. The process of solution due to Karl Friedrich Gauss (1801) depends
essentially on the arrangement of the roots in a certain order, viz. not
as above, with the indices of r in arithmetical progression, but with
their indices in geometrical progression; the prime number n has a
certain number of prime roots g, which are such that g^(n - 1) is the
lowest power of g, which is [equivalent to] 1 to the modulus n; or, what
is the same thing, that the series of powers 1, g, g^2, ... g^(n - 2),
each divided by n, leave (in a different order) the remainders 1, 2, 3,
... n - 1; hence giving to r in succession the indices 1, g, g^2, ...
g^(n - 2), we have, in a different order, the whole series of roots r,
r^2, r^3, ... r^(n - 1).
In the most simple case, n = 5, the equation to be solved is x^4 + x^3
+ x^2 + x + 1 = 0; here 2 is a prime root of 5, and the order of the
roots is r, r^2, r^4, r^3. The Gaussian process consists in forming an
equation for determining the periods P1, P2, = r + r^4 and r^2 + r^3
respectively;--these being such that the symmetrical functions P1 +
P2, P1P2 are rationally determinable: in fact P1 + P2 = -1, P1P2 = (r
+ r^4)(r^2 + r^3), = r^3 + r^4 + r^6 + r^7, = r^3 + r^4 + r + r^2, =
-1. P1, P2 are thus the roots of u^2 + u - 1 = 0; and taking them to
be known, they are themselves broken up into subperiods, in the
present case single terms, r and r^4 for P1, r^2 and r^3 for P2; the
symmetrical functions of these are then rationally determined in terms
of P1 and P2; thus r + r^4 = P1, r.r^4 = 1, or r, r^4 are the roots of
u^2 - P1u + 1 = 0. The mode of division is more clearly seen for a
larger value of n; thus, for n = 7 a prime root is = 3, and the
arrangement of the roots is r, r^3, r^2, r^6, r^4, r^5. We may form
either 3 periods each of 2 terms, P1, P2, P3 = r + r^6, r^3 + r^4, r^2
+ r^5 respectively; or else 2 periods each of 3 terms, P1, P2 = r +
r^2 + r^4, r^3 + r^6 + r^5 respectively; in each ease the symmetrical
functions of the periods are rationally determinable: thus in the case
of the two periods P1 + P2 = -1, P1P2 = 3 + r + r^2 + r^3 + r^4 + r^5
+ r^6, = 2; and the periods being known the symmetrical functions of
the several terms of each period are rationally determined in terms of
the periods, thus r + r^2 + r^4 = P1, r.r^2 + r.r^4 + r^2.r^4 = P2,
r.r^2.r^4 = 1.
The theory was further developed by Lagrange (1808), who, applying his
general process to the equation in question, x^(n - 1) + x^(n - 2) + ...
+ x + 1 = 0 (the roots a, b, c ... being the several powers of r, the
indices in geometrical progression as above), showed that the function
(a + [omega]b + [omega]^2 c + ...)^(n - 1) was in this case a given
function of [omega] with integer coefficients.
Reverting to the before-mentioned particular equation x^4 + x^3 + x^2
+ x + 1 = 0, it is very interesting to compare the process of solution
with that for the solution of the general quartic the roots whereof
are a, b, c, d.
Take [omega], a root of the equation [omega]^4 - 1 = 0 (whence [omega]
is = 1, -1, i, or -i, at pleasure), and consider the expression
(a + [omega]b + [omega]^2 c + [omega]^3 d)^4,
the developed value of this is
= a^4 + b^4 + c^4 + d^4 + 6(a^2 c^2 + b^2 d^2) + 12(a^2 bd + b^2 ca + c^2 db + d^2ac)
+[omega] {4(a^3 b + b^3 c + c^3 + d^3 a) + 12(a^2 cd + b^2 da + c^2 ab + d^2 bc)}
+[omega]^2{6(a^2 b^2 + b^2 c^2 + c^2 d^2 + d^2 a^2) + 4(a^3 c + b^3 d + c^3 a + d^3 b) + 24abcd}
+[omega]^3{4(a^3 d + b^3 a + c^3 b + d^3 c) + 12(a^2 bc + b^2 cd + c^2 da + d^2 ab)}
that is, this is a 6-valued function of a, b, c, d, the root of a
sextic (which is, in fact, solvable by radicals; but this is not here
material).
If, however, a, b, c, d denote the roots r, r^2, r^4, r^3 of the
special equation, then the expression becomes
r^4 + r^3 + r + r^2 + 6(1 + 1)+12(r^2 + r^4 + r^3 + r)
+ [omega] {4(1 + 1 + 1 + 1) + 12(r^4 + r^3 + r + r^2)}
+ [omega]^2{6(r + r^2 + r^4 + r^3) + 4(r^2 + r^4 + r^3 + r)}
+ [omega]^3{4(r + r^2 + r^4 + r^3) + 12(r^3 + r + r^2 + r^4)}
viz. this is
= -1 + 4[omega] + 14[omega]^2 - 16[omega]^3,
a completely determined value. That is, we have
(r + [omega]r^2 + [omega]^2 r^4 + [omega]^3 r^3) = -1 + 4[omega] +
14[omega]^2 - 16[omega]^3,
which result contains the solution of the equation. If [omega] = 1, we
have (r + r^2 + r^4 + r^3)^4 = 1, which is right; if [omega] = -1,
then (r + r^4 - r^2 - r^3)^4 = 25; if [omega] = i, then we have {r -
r^4 + i(r^2 - r^3)}^4 = -15 + 20i; and if [omega] = -i, then {r - r^4
- i(r^2 - r^3)}^4 = -15 - 20i; the solution may be completed without
difficulty.
The result is perfectly general, thus:--n being a prime number, r a root
of the equation x^(n - 1) + x^(n - 2) + ... + x + 1 = 0, [omega] a root
of [omega]^(n - 1) - 1 = 0, and g a prime root of g^(n - 1) [equivalent]
1 (mod. n), then
{r + [omega]r^g + ... + [omega]^(n - 2) r^g^(n - 2)}^(n - 1)
is a given function M0 + M1[omega] ... + M_(n - 2)[omega]^(n - 2) with
integer coefficients, and by the extraction of (n - 1)th roots of this
and similar expressions we ultimately obtain r in terms of [omega],
which is taken to be known; the equation x^n - 1 = 0, n a prime number,
is thus solvable by radicals. In particular, if n - 1 be a power of 2,
the solution (by either process) requires the extraction of square roots
only; and it was thus that Gauss discovered that it was possible to
construct geometrically the regular polygons of 17 sides and 257 sides
respectively. Some interesting developments in regard to the theory were
obtained by C.G.J. Jacobi (1837); see the memoir "Ueber die
Kreistheilung, u.s.w.," _Crelle_, t. xxx. (1846).
The equation x^(n - 1) + ... + x + 1 = 0 has been considered for its own
sake, but it also serves as a specimen of a class of equations solvable
by radicals, considered by N.H. Abel (1828), and since called Abelian
equations, viz. for the Abelian equation of the order n, if x be any
root, the roots are x, [theta]x, [theta]^2 x, ... [theta]^(n - 1)x
([theta]x being a rational function of x, and [theta]^nx = x); the
theory is, in fact, very analogous to that of the above particular case.
A more general theorem obtained by Abel is as follows:--If the roots
of an equation of any order are connected together in such wise that
_all_ the roots can be expressed rationally in terms of any one of
them, say x; if, moreover, [theta]x, [theta]1x being any two of the
roots, we have [theta][theta]1x = [theta]1[theta]x, the equation will
be solvable algebraically. It is proper to refer also to Abel's
definition of an _irreducible_ equation:--an equation [phi]x = 0, the
coefficients of which are rational functions of a certain number of
known quantities a, b, c ..., is called irreducible when it is
impossible to express its roots by an equation of an inferior degree,
the coefficients of which are also rational functions of a, b, c ...
(or, what is the same thing, when [phi]x does not break up into
factors which are rational functions of a, b, c ...). Abel applied his
theory to the equations which present themselves in the division of
the elliptic functions, but not to the modular equations.
24. But the theory of the algebraical solution of equations in its most
complete form was established by Evariste Galois (born October 1811,
killed in a duel May 1832; see his collected works, _Liouville_, t. xl.,
1846). The definition of an irreducible equation resembles Abel's,--an
equation is reducible when it admits of a rational divisor, irreducible
in the contrary case; only the word _rational_ is used in this extended
sense that, in connexion with the coefficients of the given equation, or
with the irrational quantities (if any) whereof these are composed, he
considers any number of other irrational quantities called "adjoint
radicals," and he terms rational any rational function of the
coefficients (or the irrationals whereof they are composed) and of these
adjoint radicals; the epithet irreducible is thus taken either
absolutely or in a relative sense, according to the system of adjoint
radicals which are taken into account. For instance, the equation x^4 +
x^3 + x^2 + x + 1 = 0; the left hand side has here no rational divisor,
and the equation is irreducible; but this function is = (x^2 + 1/2 x +
1)^2 -(5/4)x^2, and it has thus the irrational divisors x^2 + 1/2(1 +
[root]5)x + 1, x^2 + 1/2(1 - [root]5)x + 1; and these, if we _adjoin_
the radical [root]5, are rational, and the equation is no longer
irreducible. In the case of a given equation, assumed to be irreducible,
the problem to solve the equation is, in fact, that of finding radicals
by the adjunction of which the equation becomes reducible; for instance,
the general quadric equation x^2 + px + q = 0 is irreducible, but it
becomes reducible, breaking up into rational linear factors, when we
adjoin the radical [root](1/4 p^2 - q).
The fundamental theorem is the Proposition I. of the "Memoire sur les
conditions de resolubilite des equations par radicaux"; viz. given an
equation of which a, b, c ... are the m roots, there is always a group
of permutations of the letters a, b, c ... possessed of the following
properties:--
1. Every function of the roots invariable by the substitutions of the
group is rationally known.
2. Reciprocally every rationally determinable function of the roots is
invariable by the substitutions of the group.
Here by an invariable function is meant not only a function of which
the form is invariable by the substitutions of the group, but further,
one of which the value is invariable by these substitutions: for
instance, if the equation be [phi](x) = 0, then [phi](x) is a function
of the roots invariable by any substitution whatever. And in saying
that a function is rationally known, it is meant that its value is
expressible rationally in terms of the coefficients and of the adjoint
quantities.
For instance in the case of a general equation, the group is simply
the system of the 1.2.3 ... n permutations of all the roots, since, in
this case, the only rationally determinable functions are the
symmetric functions of the roots.
In the case of the equation x^(n - 1) ... + x + 1 = 0, n a prime
number, a, b, c ... k = r, r^g, r^g^2 ... r^g^(n - 2), where g is a
prime root of n, then the group is the cyclical group abc ... k, bc
... ka, ... kab ... j, that is, in this particular case the number of
the permutations of the group is equal to the order of the equation.
This notion of the group of the original equation, or of the group of
the equation as varied by the adjunction of a series of radicals,
seems to be the fundamental one in Galois's theory. But the problem of
solution by radicals, instead of being the sole object of the theory,
appears as the first link of a long chain of questions relating to the
transformation and classification of irrationals.
Returning to the question of solution by radicals, it will be readily
understood that by the adjunction of a radical the group may be
diminished; for instance, in the case of the general cubic, where the
group is that of the six permutations, by the adjunction of the square
root which enters into the solution, the group is reduced to abc, bca,
cab; that is, it becomes possible to express rationally, in terms of
the coefficients and of the adjoint square root, any function such as
a^2 b + b^2 c + c^2 a which is not altered by the cyclical
substitution a into b, b into c, c into a. And hence, to determine
whether an equation of a given form is solvable by radicals, the
course of investigation is to inquire whether, by the successive
adjunction of radicals, it is possible to reduce the original group of
the equation so as to make it ultimately consist of a single
permutation.
The condition in order that an equation of a given prime order n may
be solvable by radicals was in this way obtained--in the first
instance in the form (scarcely intelligible without further
explanation) that every function of the roots x1, x2 ... x_n,
invariable by the substitutions x_(ak + b) for x_k, must be rationally
known; and then in the equivalent form that the resolvent equation of
the order 1.2 ... (n - 2) must have a rational root. In particular,
the condition in order that a quintic equation may be solvable is that
Lagrange's resolvent of the order 6 may have a rational factor, a
result obtained from a direct investigation in a valuable memoir by E.
Luther, _Crelle_, t. xxxiv. (1847).
Among other results demonstrated or announced by Galois may be
mentioned those relating to the modular equations in the theory of
elliptic functions; for the transformations of the orders 5, 7, 11,
the modular equations of the orders 6, 8, 12 are depressible to the
orders 5, 7, 11 respectively; but for the transformation, n a prime
number greater than 11, the depression is impossible.
The general theory of Galois in regard to the solution of equations
was completed, and some of the demonstrations supplied by E. Betti
(1852). See also J.A. Serret's _Cours d'algebre superieure_, 2nd ed.
(1854); 4th ed. (1877-1878).
25. Returning to quintic equations, George Birch Jerrard (1835)
established the theorem that the general quintic equation is by the
extraction of only square and cubic roots reducible to the form x^5 + ax
+ b = 0, or what is the same thing, to x^5 + x + b = 0. The actual
reduction by means of Tschirnhausen's theorem was effected by Charles
Hermite in connexion with his elliptic-function solution of the quintic
equation (1858) in a very elegant manner. It was shown by Sir James
Cockle and Robert Harley (1858-1859) in connexion with the Jerrardian
form, and by Arthur Cayley (1861), that Lagrange's resolvent equation of
the sixth order can be replaced by a more simple sextic equation
occupying a like place in the theory.
The theory of the modular equations, more particularly for the case n =
5, has been studied by C. Hermite, L. Kronecker and F. Brioschi. In the
case n = 5, the modular equation of the order 6 depends, as already
mentioned, on an equation of the order 5; and conversely the general
quintic equation may be made to depend upon this modular equation of the
order 6; that is, assuming the solution of this modular equation, we can
solve (not by radicals) the general quintic equation; this is Hermite's
solution of the general quintic equation by elliptic functions (1858);
it is analogous to the before-mentioned trigonometrical solution of the
cubic equation. The theory is reproduced and developed in Brioschi's
memoir, "Uber die Auflosung der Gleichungen vom funften Grade," _Math.
Annalen_, t. xiii. (1877-1878).
26. The modern work, reproducing the theories of Galois, and
exhibiting the theory of algebraic equations as a whole, is C.
Jordan's _Traite des substitutions et des equations algebriques_
(Paris, 1870). The work is divided into four books--book i.,
preliminary, relating to the theory of congruences; book ii. is in two
chapters, the first relating to substitutions in general, the second
to substitutions defined analytically, and chiefly to linear
substitutions; book iii. has four chapters, the first discussing the
principles of the general theory, the other three containing
applications to algebra, geometry, and the theory of transcendents;
lastly, book iv., divided into seven chapters, contains a
determination of the general types of equations solvable by radicals,
and a complete system of classification of these types. A glance
through the index will show the vast extent which the theory has
assumed, and the form of general conclusions arrived at; thus, in book
iii., the algebraical applications comprise Abelian equations,
equations of Galois; the geometrical ones comprise Q. Hesse's
equation, R.F.A. Clebsch's equations, lines on a quartic surface
having a nodal line, singular points of E.E. Kummer's surface, lines
on a cubic surface, problems of contact; the applications to the
theory of transcendents comprise circular functions, elliptic
functions (including division and the modular equation), hyperelliptic
functions, solution of equations by transcendents. And on this last
subject, solution of equations by transcendents, we may quote the
result--"the solution of the general equation of an order superior to
five cannot be made to depend upon that of the equations for the
division of the circular or elliptic functions"; and again (but with a
reference to a possible case of exception), "the general equation
cannot be solved by aid of the equations which give the division of
the hyperelliptic functions into an odd number of parts." (See also
GROUPS, THEORY OF.) (A. Ca.)
BIBLIOGRAPHY.--For the general theory see W.S. Burnside and A.W.
Panton, _The Theory of Equations_ (4th ed., 1899-1901); the Galoisian
theory is treated in G.B. Matthews, _Algebraic Equations_ (1907). See
also the _Ency. d. math. Wiss._ vol. ii.
FOOTNOTES:
[1] The coefficients were selected so that the roots might be nearly
1, 2, 3.
[2] The third edition (1826) is a reproduction of that of 1808; the
first edition has the date 1798, but a large part of the contents is
taken from memoirs of 1767-1768 and 1770-1771.
[3] The earlier demonstrations by Euler, Lagrange, &c, relate to the
case of a numerical equation with real coefficients; and they consist
in showing that such equation has always a real quadratic divisor,
furnishing two roots, which are either real or else conjugate
imaginaries [alpha] + [beta]i (see Lagrange's _Equations
numeriques_).
[4] The square root of [alpha] + [beta]i can be determined by the
extraction of square roots of positive real numbers, without the
trigonometrical tables.
EQUATION OF THE CENTRE, in astronomy, the angular distance, measured
around the centre of motion, by which a planet moving in an ellipse
deviates from the mean position which it would occupy if it moved
uniformly. Its amount is the correction which must be applied positively
or negatively to the mean anomaly in order to obtain the true anomaly.
It arises from the ellipticity of the orbit, is zero at pericentre and
apocentre, and reaches its greatest amount nearly midway between these
points. (See ANOMALY and ORBIT.)
EQUATION OF TIME, the difference between apparent time, determined by
the meridian passage of the real sun, and mean time, determined by the
passage of the mean sun. It goes through a double period in the course
of a year. Its amount varies a fraction of a minute for the same date,
from year to year and from one longitude to another, on the same day.
The following table shows an average value for any date and for the
Greenwich meridian for a number of years, from which the actual value
will seldom deviate more than 20 seconds until after 1950. The + sign
indicates that the real sun reaches the meridian _after_ mean noon; the
- sign _before_ mean noon.
_Table of the Equation of Time._
m. s. m. s. m. s.
Jan. 1 +3 26 Mar. 1 +12 39 May 1 -2 55
6 5 45 6 11 35 6 -3 27
11 7 51 11 10 20 11 -3 46
16 9 43 16 8 58 16 -3 51
21 11 19 21 7 30 21 -3 40
26 12 36 26 5 59 26 -3 16
Feb. 1 +13 42 Apr. 1 +4 9 June 1 -2 32
6 14 14 6 2 40 6 -1 44
11 14 25 11 +1 15 11 -0 48
16 14 17 16 -0 3 16 +0 14
21 13 52 21 -1 12 21 1 19
26 13 11 26 -2 10 26 2 24
July 1 +3 26 Sept. 1 +0 9 Nov. 1 -16 18
6 4 21 6 -1 28 6 -16 19
11 5 8 11 -3 10 11 -15 58
16 5 44 16 -4 55 16 -15 15
21 6 8 21 -6 41 21 -14 12
26 6 18 26 -8 25 26 -12 49
Aug. 1 +6 10 Oct. 1 -10 5 Dec. 1 -11 7
6 5 47 6 -11 38 6 - 9 9
11 5 9 11 -13 2 11 - 6 57
16 4 17 16 -14 14 16 - 4 35
21 3 12 21 -15 11 21 - 2 7
26 1 55 26 -15 52 26 + 0 23
EQUATOR (Late Lat. _aequator_, from _aequare_, to make equal), in
geography, that great circle of the earth, equidistant from the two
poles, which divides the northern from the southern hemisphere and lies
in a plane perpendicular to the axis of the earth; this is termed the
"geographical" or "terrestrial equator." In astronomy, the "celestial
equator" is the name given to the great circle in which the plane of the
terrestrial equator intersects the celestial sphere; it is consequently
equidistant from the celestial poles. The "magnetic equator" is an
imaginary line encircling the earth, along which the vertical component
of the earth's magnetic force is zero; it nearly coincides with the
terrestrial equator.
EQUERRY (from the Fr. _ecurie_, a stable, through its older form
_escurie_, from the Med. Lat. _scuria_, a word of Teutonic origin for a
stable or shed, cf. Ger. _Scheuer_; the modern spelling has confused the
word with the Lat. _equus_, a horse), a contracted form of "gentleman of
the equerry," an officer in charge of the stables of a royal household.
At the British court, equerries are officers attached to the department
of the master of the horse, the first of whom is called chief equerry
(see HOUSEHOLD, ROYAL).
EQUIDAE, the family of perissodactyle ungulate mammals typified by the
horse (_Equus caballus_); see HORSE. According to the older
classification this family was taken to include only the forms with
tall-crowned teeth, more or less closely allied to the typical genus
_Equus_. There is, however, such an almost complete graduation from the
former to earlier and more primitive mammals with short-crowned
cheek-teeth, at one time included in the family _Lophiodontidae_ (see
PERISSODACTYLA), that it has now become a very general practice to
include the whole "phylum" in the family _Equidae_. The _Equidae_, in
this extended sense, together with the extinct _Palaeotheriidae_, are
indeed now regarded as forming one of four main groups into which the
Perissodactyla are divided, the other groups being the Tapiroidea,
Rhinocerotoidea and Titanotheriide. For the horse-group the name
Hippoidea is employed. All four groups were closely connected in the
Lower Eocene, so that exact definition is almost impossible.
In the Hippoidea there is generally the full series of 44 teeth, but the
first premolar is often deciduous or wanting in the lower or in both
jaws. The incisors are chisel-shaped, and the canines tend to become
isolated so as in the now specialized forms to occupy nearly the middle
of a longer or shorter gap between the incisors and premolars. In the
upper molars the two outer columns of the primitive tubercular molar
coalesce to form an outer wall, from which proceed two crescentic
transverse crests; the connexion between the crests and the wall being
imperfect or slight, and the crests themselves sometimes tubercular.
Each of the lower molars carries two crescentic ridges. The number of
toes ranges from four to one in the fore-foot, and from three to one in
the hind-foot. The paroccipital, postglenoid and post-tympanic processes
of the skull are large, and the latter always distinct. Normally there
are no traces of horn-cores. The calcaneum lacks the facet for the
fibula found in the Titanotheroidea.
In the earlier _Equidae_ the teeth were short-crowned, with the
premolars simpler than the molars; but there is a gradual tendency to an
increase in the height of the crowns of the teeth, accompanied by
increasing complexity of structure and the filling up of the hollows
with cement. Similarly the gap on each side of the canine tooth in each
jaw continues to increase in length; while in all the later forms the
orbit is surrounded by a ring of bone. A third modification is the
increasing length of limb (as well as in general bodily size),
accompanied by a gradual reduction in the number of toes from three or
four to one.
[Illustration: FIG. 1.--a, Side view of second upper molar tooth of
_Anchitherium_ (brachyodont form); b, corresponding tooth of horse
(hypsidont form).]
All the existing members of the family, such as the domesticated horse
(_Equus caballus_) and its wild or half-wild relatives, the asses and
the zebras, are included in the typical genus. In all these the crowns
of the cheek-teeth are very tall (fig. 1, b) and only develop roots late
in life; while their grinding-surfaces (fig. 2, b and c) are very
complicated and have all the hollows filled with cement. The summits of
the incisors are infolded, producing, when partially worn, the "mark."
In the skull the orbit is surrounded by bone, and there is no distinct
depression in front of the same. Each limb terminates in one large toe;
the lateral digits being represented by the splint-bones, corresponding
to the lateral metacarpals and metatarsals of _Hipparion_. Not
unfrequently, however, the lower ends of the splint-bones carry a small
expansion, representing the phalanges.
Remains of horses indistinguishable from _E. caballus_ occur in the
Pleistocene deposits of Europe and Asia; and it is from them that the
dun-coloured small horses of northern Europe and Asia are probably
derived. The ancestor of these Pleistocene horses is probably _E.
stenonis_, of the Upper Pliocene of Europe, which has a small depression
in front of the orbit, while the skull is relatively larger, the feet
are rather shorter, and the splint-bones somewhat more developed. In
India a nearly allied species (_E. sivalensis_), occurs in the Lower
Pliocene, and may have been the ancestor of the Arab stock, which shows
traces of the depression in front of the orbit characteristic of the
earlier forms. In North America species of _Equus_ occur in the
Pleistocene and from that continent others reached South America during
the same epoch. In the latter country occurs _Hippidium_, in which the
cheek-teeth are shorter and simpler, and the nasal bones very long and
slender, with elongated slits at the side. The limbs, especially the
cannon-bones, are relatively short, and the splint-bones large. The
allied Argentine _Onohippidium_, which is also Pleistocene, has still
longer nasal bones and slits, and a deep double cavity in front of the
orbit, part of which probably contained a gland. _Onohippidium_ is
certainly off the direct line of descent of the modern horses, and, on
account of the length of the nasals and their slits, the same probably
holds good for _Hippidium_.
[Illustration: FIG. 2.--a, Grinding surface of unworn right upper molar
tooth of _Anchitherium_; b, corresponding surface of unworn molar of
young horse; c, the same tooth after it has been some time in use. The
uncoloured portions are the dentine or ivory, the shaded parts the
cement filling the cavities and surrounding the exterior. The black line
separating these two structures is the enamel or hardest constituent of
the tooth.]
Species from the Pliocene of Texas and the Upper Miocene (Loup Fork) of
Oregon were at one time assigned to _Hippidium_, but this is incorrect,
that genus being exclusively South American. The name _Pliohippus_ has
been applied to species from the same two formations on the supposition
that the foot-structure was similar to that of _Hippidium_, but Mr J.W.
Gidley is of opinion that the lateral digits may have been fully
developed.
Apparently there is here some gap in the line of descent of the horse,
and it may be suggested that the evolution took place, not as commonly
supposed, in North America, but in eastern central Asia, of which the
palaeontology is practically unknown; some support is given to this
theory by the fact that the earliest species with which we are
acquainted occur in northern India.
[Illustration: FIG. 3.--Successive stages of modification of the left
fore-feet of extinct forms of horse-like animals, showing gradual
reduction of the outer and enlargement of the middle toe (III).
a, _Hyracotherium_ (Eocene).
b, _Mesohippus_ (Oligocene).
c, _Anchitherium_ (Miocene).
d, _Hipparion_ (Pliocene).
e, _Equus_ (Pleistocene).]
Be this as it may, the next North American representatives of the family
constitute the genera _Protohippus_ and _Merychippus_ of the Miocene, in
both of which the lateral digits are fully developed and terminate in
small though perfect hoofs. In both the cheek-teeth have moderately tall
crowns, and in the first named of the two those of the milk-series are
nearly similar to their permanent successors. In _Merychippus_, on the
other hand, the milk-molars have short crowns, without any cement in the
hollows, thus resembling the permanent molars of the under-mentioned
genus _Anchitherium_. From the well-known _Hipparion_, or
_Hippotherium_, typically from the Lower Pliocene of Europe, but also
occurring in the corresponding formation in North Africa, Persia, India
and China, and represented in the Upper Miocene Loup Fork beds of the
United States by species which it has been proposed to separate
generically as _Neohipparion_, we reach small horses which are now
generally regarded as a lateral offshoot from the _Merychippus_ type.
The cheek-teeth, which have crowns of moderate height, differ from those
of all the foregoing in that the postero-internal pillar (the projection
on the right-hand top corner of c in fig. 2) is isolated in place of
being attached by a narrow neck to the adjacent crescent. The skull,
which is relatively short, has a large depression in front of the orbit,
commonly supposed to have contained a gland, but this may be doubtful.
In the typical, and also in the North American forms these were
complete, although small, lateral toes in both feet (fig. 3, d), but it
is possible that in _H. antilopinum_ of India the lateral toes had
disappeared. If this be so, we have the development of a monodactyle
foot in this genus independently of _Equus_.
The foregoing genera constitute the subfamily _Equinae_, or the
_Equidae_ as restricted by the older writers. In all the dentition is of
the hypsodont type, with the hollows of the cheek-teeth filled by
cement, the premolars molariform, and the first small and generally
deciduous. The orbit is surrounded by a bony ring; the ulna and radius
in the fore, and the tibia and fibula in the hind-limb are united, and
the feet are of the types described above. Between this subfamily and
the second subfamily, _Hyracotheriinae_, a partial connexion is formed
by the North American Upper Miocene genera _Desmatippus_ and _Anchippus_
or _Parahippus_. The characteristics of the group will be gathered from
the remarks on the leading genera; but it may be mentioned that the
orbit is open behind, the cheek-teeth are short-crowned and without
cement (fig. 1, a), the gap between the canine and the outermost
incisor is short, the bones of the middle part of the leg are separate,
and there are at least three toes to each foot.
The longest-known genus and the one containing the largest species is
_Anchitherium_, typically from the Middle Miocene of Europe, but also
represented by one species from the Upper Miocene of North America. The
European _A. aurelianense_ was of the size of an ordinary donkey. The
cheek-teeth are of the type shown in a of figs. 1 and 2; the premolars,
with the exception of the small first one, being molar-like; and the
lateral toes (fig. 3, c) were to some extent functional. The summits of
the incisors were infolded to a small extent. Nearly allied is the
American _Mesohippus_, ranging from the Lower Miocene to the Lower
Oligocene of the United States, of which the earliest species stood only
about 18 in. at the shoulder. The incisors were scarcely, if at all,
infolded, and there is a rudiment of the fifth metacarpal (fig. 3, b).
By some writers all the species of _Mesohippus_ are included in the
genus _Miohippus_, but others consider that the two genera are distinct.
_Mesohippus_ and _Miohippus_ are connected with the earliest and most
primitive mammal which it is possible to include in the family _Equidae_
by means of _Epihippus_ of the Uinta or Upper Eocene of North America,
and _Pachynolophus_, or _Orohippus_, of the Middle and Lower Eocene of
both halves of the northern hemisphere. The final stage, or rather the
initial stage, in the series is presented by _Hyracotherium_
(_Protorohippus_), a mammal no larger than a fox, common to the Lower
Eocene of Europe and North America. The general characteristics of this
progenitor of the horses are those given above as distinctive of the
group. The cheek-teeth are, however, much simpler than those of
_Anchitherium_; the transverse crests of the upper molars not being
fully connected with the outer wall, while the premolars in the upper
jaw are triangular, and thus unlike the molars. The incisors are small
and the canines scarcely enlarged; the latter having a gap on each side
in the lower, but only one on their hinder aspect in the upper jaw. The
fore-feet have four complete toes (fig. 3, a), but there are only three
hind-toes, with a rudiment of the fifth metatarsal. The vertebrae are
simpler in structure than in _Equus_. From _Hyracotherium_, which is
closely related to the Eocene representatives of the ancestral stocks of
the other three branches of the Perissodactyla, the transition is easy
to _Phenacodus_, the representative of the common ancestor of all the
Ungulata.
See also H.F. Osborn, "New Oligocene Horses," _Bull. Amer. Mus._ vol.
xx. p. 167 (1904); J.W. Gidley, _Proper Generic Names of Miocene
Horses_, p. 191; and the article PALAEONTOLOGY. (R. L.*)
EQUILIBRIUM (from the Lat. _aequus_, equal, and _libra_, a balance), a
condition of equal balance between opposite or counteracting forces. By
the "sense of equilibrium" is meant the sense, or sensations, by which
we have a feeling of security in standing, walking, and indeed in all
the movements by which the body is carried through space. Such a feeling
of security is necessary both for maintaining any posture, such as
standing, or for performing any movement. If this feeling is absent or
uncertain, or if there are contradictory sensations, then definite
muscular movements are inefficiently or irregularly performed, and the
body may stagger or fall. When we stand erect on a firm surface, like a
floor, there is a feeling of resistance, due to nervous impulses
reaching the brain from the soles of the feet and from the muscles of
the limbs and trunk. In walking or running, these feelings of resistance
seem to precede and guide the muscular movements necessary for the next
step. If these are absent or perverted or deficient, as is the case in
the disease known as locomotor ataxia, then, although there is no loss
of the power of voluntary movement, the patient staggers in walking,
especially if he is not allowed to look at his feet, or if he is
blind-folded. He misses the guiding sensations that come from the limbs;
and with a feeling that he is walking on a soft substance, offering
little or no resistance, he staggers, and his muscular movements become
irregular. Such a condition maybe artificially brought about by washing
the soles of the feet with chloroform or ether. And it has been observed
to exist partially after extensive destruction of the skin of the soles
of the feet by burns or scalds. This shows that tactile impulses from
the skin take a share in generating the guiding sensation. In the
disease above mentioned, however, tactile impressions may be nearly
normal, but the guiding sensation is weak and inefficient, owing to the
absence of impulses from the muscles. The disease is known to depend on
morbid changes in the posterior columns of the spinal cord, by which
impulses are not freely transmitted upwards to the brain. These facts
point to the existence of impulses coming from the muscles and tendons.
It is now known that there exist peculiar spindles, in muscle, and
rosettes or coils or loops of nerve fibres in close proximity to
tendons. These are the end organs of the sense. The transmission of
impulses gives rise to the _muscular sense_, and the guiding sensation
which precedes co-ordinated muscular movements depends on these
impulses. Thus from the limbs streams of nervous impulses pass to the
sensorium from the skin and from muscles and tendons; these may or may
not arouse consciousness, but they guide or evoke muscular movements of
a co-ordinated character, more especially of the limbs.
In animals whose limbs are not adapted for delicate touch nor for the
performance of complicated movements, such as some mammals and birds and
fishes, the guiding sensations depend largely on the sense of vision.
This sense in man, instead of assisting, sometimes disturbs the guiding
sensation. It is true that in locomotor ataxia visual sensations may
take the place of the tactile and muscular sensations that are
inefficient, and the man can walk without staggering if he is allowed to
look at the floor, and especially if he is guided by transverse straight
lines. On the other hand, the acrobat on the wire-rope dare not trust
his visual sensations in the maintenance of his equilibrium. He keeps
his eyes fixed on one point instead of allowing them to wander to
objects below him, and his muscular movements are regulated by the
impulses that come from the skin and muscles of his limbs. The feeling
of insecurity probably arises from a conception of height, and also from
the knowledge that by no muscular movements can a man avoid a
catastrophe if he should fall. A bird, on the other hand, depends
largely on visual impressions, and it knows by experience that if
launched into the air from a height it can fly. Here, probably, is an
explanation of the large size of the eyes of birds. Cover the head, as
in hooding a falcon, and the bird seems to be deprived of the power of
voluntary movement. Little effect will be produced if we attempt to
restrain the movements of a cat by covering its eyes. A fish also is
deprived of the power of motion if its eyes are covered. But both in the
bird and in the fish tactile and muscular impressions, especially the
latter, come into play in the mechanism of equilibrium. In flight the
large-winged birds, especially in soaring, can feel the most delicate
wind-pressures, both as regards direction and force, and they adapt the
position of their body so as to catch the pressure at the most efficient
angle. The same is true of the fish, especially of the flat-fishes. In
mammals the sense of equilibrium depends, then, on streams of tactile,
muscular and visual impressions pouring in on the sensorium, and calling
forth appropriate muscular movements. It has also been suggested that
impulses coming from the abdominal viscera may take part in the
mechanism. The presence in the mesentery of felines (cats, &c.) of large
numbers of Pacinian corpuscles, which are believed to be modified
tactile bodies, favours this supposition. Such animals are remarkable
for the delicacy of such muscular movements, as balancing and leaping.
There is another channel by which nervous impulses reach the sensorium
and play their part in the sense of equilibrium, namely, from the
semicircular canals, a portion of the internal ear. It is pointed out in
the article HEARING that the appreciation of sound is in reality an
appreciation of variations of pressure. The labyrinth consists of the
vestibule, the cochlea and the semicircular canals. The cochlea receives
the sound-waves (variations of pressure) that constitute musical tones.
This it accomplishes by the structures in the ductus cochlearis. In the
vestibule we find two sacs, the saccule next to and communicating with
the ductus cochlearis, and the utricle communicating with the
semicircular canals. The base of the stapes communicates pressures to
the utricle. The membranous portion of the semicircular canals consists
of a tube, dilated at one end into a swelling or pouch, termed the
ampulla, and each end communicates freely with the utricle. On the
posterior wall of both the saccule and of the utricle there is a ridge,
termed in each case the macula acustica, bearing a highly specialized
epithelium. A similar structure exists in each ampulla. This would
suggest that all three structures have to do with hearing; but, on the
other hand, there is experimental evidence that the utricle and the
canals may transmit impressions that have to do with equilibrium.
Pressure of the base of the stapes is exerted on the utricle. This will
compress the fluid in that cavity, and tend to drive the fluid into the
semicircular canals that communicate with that cavity by five openings.
Each canal is surrounded by a thin layer of perilymph, so that it may
yield a little to this pressure, and exert a pull or pressure on the
nerve-endings in each ampulla. Thus impulses may be generated in the
nerves of the ampullae.
The three semicircular canals lie in the three directions in space, and
it has been suggested that they have to do with our appreciation of the
direction of sound. But our appreciation of sound is very inaccurate: we
look with the eyes for the source of a sound, and instinctively direct
the ears or the head, or both, in the direction from which the sound
appears to proceed. But the relationship of the canals on the two sides
must have a physiological significance. Thus (1) the six canals are
parallel, two and two; or (2) the two horizontal canals are in the same
plane, while the superior canal on one side is nearly parallel with the
posterior canal of the other. These facts point to the two sets of
canals and ampullae acting as one organ, in a manner analogous to the
action of two retinae for single vision.
We have next to consider how the canals may possibly act in connexion
with the sense of equilibrium. In 1820 J. Purkinje studied the vertigo
that follows rapid rotation of the body in the erect position on a
vertical axis. On stopping the rotation there is a sense of rotation in
the opposite direction, and this may occur even when the eyes are
closed. Purkinje noticed that the position of the imaginary axis of
rotation depends on the axis around which the head revolves. In 1828
M.J.P. Flourens discovered that injury to the canals causes disturbance
to the equilibrium and loss of co-ordination, and that sections of the
canals produce a rotatory movement of a kind corresponding to the canal
that had been divided. Thus division of a membranous canal causes
rotatory movements round an axis at right angles to the plane of the
divided canal. The body of the animal always moves in the direction of
the cut canal. Many other observers have corroborated these experiments.
F. Goltz was the first who formulated the conditions necessary for
equilibration. He put the matter thus:--(1) A central co-ordinating
organ--in the brain; (2) centripetal fibres, with their peripheral
terminations--in the ampullae; and (3) centrifugal fibres, with their
terminal organs--in the muscular mechanisms. A lesion of any one of
these portions of the mechanism causes loss or impairment of balancing.
Cyon also investigated the subject, and concluded:--(1) To maintain
equilibrium, we must have an accurate notion of the position of the head
in space; (2) the function of the semicircular canals is to communicate
impressions that give a representation of this position--each canal
having a relation to one of the dimensions of space; (3) disturbance of
equilibrium follows section; (4) involuntary movements following section
are due to abnormal excitations; (5) abnormal movements occurring a few
days after the operation are caused by irritation of the cerebellum.
On theoretical considerations of a physical character, E. Mach,
Crum-Brown and Breuer have advanced theories based on the idea of the
canals being organs for sensations of acceleration of movement, or for
the sense of rotation. Mach first pointed out that Purkinje's phenomena,
already alluded to, were in all probability related to the semicircular
canals. "He showed that when the body is moved in space, in a straight
line, we are not conscious of the velocity of motion, but of variations
in this velocity. Similarly, if a body is rotated round a vertical axis,
we perceive only angular acceleration and not angular velocity. The
sensations produced by angular acceleration last longer than the
acceleration itself, and the position of the head during the movements
enables us to determine direction." Both Mach and Goltz state that
varying pressures of the fluid in the canals produced by angular
rotation produce sensations of movement (always in a direction opposite
to the rotation of the body), and that these, in turn, cause the vertigo
of Purkinje and the phenomena of Flourens. Mach, Crum-Brown and Breuer
advance hydrodynamical theories in which they assume that the fluids
move in the canals. Goltz, on the other hand, supports a hydrostatical
theory in which he assumes that the phenomena can be accounted for by
varying pressures. Crum-Brown differs from Mach and Breuer as
follows:--(1) In attributing movement or variation of pressure not
merely to the endolymph, but also to the walls of the membranous canals
and to the surrounding perilymph; and (2) in regarding the two
labyrinths as one organ, all the six canals being required to form a
true conception of the rotating motion of the head. He sums up the
matter thus: "We have two ways in which a relative motion can occur
between the endolymph and the walls of the cavity containing it--(1)
When the head begins to move, here the walls leave the fluid behind; (2)
when the head stops, here the fluid flows on. In both cases the
sensation of rotation is felt. In the first this sensation corresponds
to a real rotation, in the second it does not, but in both it
corresponds to a real acceleration (positive or negative) of rotation,
using the word acceleration in its technical kinematical sense."
Cyon states that the semicircular canals only indirectly assist in
giving a notion of spatial relations. "He holds that knowledge of the
position of bodies in space depends on nervous impulses coming from the
contracting ocular muscles; that the oculomotor centres are in intimate
physiological relationship with the centres receiving impulses from the
nerves of the semicircular canals; and that the oculomotor centres, thus
excited, produce the movements of the eyeballs, which then determine our
notions of spatial relations." These views are supported by experiments
of Lee on dog-fish. When the fish is rotated round different axes there
are compensating movements of the eyes and fins. "It was observed that
if the fish were rotated in the plane of one of the canals, exactly the
same movements of the eyes and fins occurred as were produced by
experimental operation and stimulation of the ampulla of that canal."
Sewall, in 1883, carried out experiments on young sharks and skates with
negative results. Lee returned to the subject in 1894, and, after
numerous experiments on dog-fish, in which the canals or the auditory
nerves were divided, obtained evidence that the ampullae contain
sense-organs connected with the sense of equilibrium.
It has been found by physicians and aurists that disease or injury of
the canals, occurring rapidly, produces giddiness, staggering, nystagmus
(a peculiar twitching movement of the muscles of the eyeballs),
vomiting, noises in the ear and more or less deafness. It is said,
however, that if pathological changes come on slowly, so that the canals
and vestibule are converted into a solid mass, none of these symptoms
may occur. On the whole, the evidence is in favour of the view that from
the semicircular canals nervous impulses are transmitted, which,
co-ordinated with impulses coming from the visual organs, from the
muscles and from the skin, form the bases of these guiding sensations on
which the sense of equilibrium depends. These impulses may not reach the
level of consciousness, but they call into action co-ordinated
mechanisms by which complicated muscular movements are effected.
Full bibliographical references are given in the article on "The Ear"
by J.G. McKendrick, in Schafer's _Textbook of Physiology_, vol. ii. p.
1194. (J. G. M.)
EQUINOX (from the Lat. _aequus_, equal, and _nox_, night), a term used
to express either the moment at which, or the point at which, the sun
apparently crosses the celestial equator. Since the sun moves in the
ecliptic, it is in the last-named sense the point of intersection of the
ecliptic and the celestial equator. This is the usual meaning of the
term in astronomy. There are two such points, opposite each other, at
one of which the sun crosses the equator toward the north and at the
other toward the south. They are called vernal and autumnal
respectively, from the relation of the corresponding times to the
seasons of the northern hemisphere. The line of the equinoxes is the
imaginary diameter of the celestial sphere which joins them.
The vernal equinox is the initial point from which the right ascensions
and the longitudes of the heavenly bodies are measured (see ASTRONOMY:
_Spherical_). It is affected by the motions of Precession and Nutation,
of which the former has been known since the time of Hipparchus. The
actual equinox is defined by first taking the conception of a fictitious
point called the Mean Equinox, which moves at a nearly uniform rate,
slow varying, however, from century to century. The true equinox then
moves around the mean equinox in a period equal to that of the moon's
nodes. These two motions are defined with greater detail in the articles
PRECESSION OF THE EQUINOXES and NUTATION.
_Equinoctial Gales._--At the time of the equinox it is commonly believed
that strong gales may be expected. This popular idea has no foundation
in fact, for continued observations have failed to show any unusual
prevalence of gales at this season. In one case observations taken for
fifty years show that during the five days from the 21st to the 25th of
March and September, there were fewer gales and storms than during the
preceding and succeeding five days.
EQUITES ("horsemen" or "knights," from _equus_, "horse"), in Roman
history, originally a division of the army, but subsequently a distinct
political order, which under the empire resumed its military character.
According to the traditional account, Romulus instituted a cavalry
corps, consisting of three _centuriae_ ("hundreds"), called after the
three tribes from which they were taken (Ramnes, Tities, Luceres),
divided into ten _turmae_ ("squadrons") of thirty men each. The
collective name for the corps was _celeres_ ("the swift," or possibly
from [Greek: keles], "a riding horse"); Livy, however, restricts the
term to a special body-guard of Romulus. The statements in ancient
authorities as to the changes in the number of the equites during the
regal period are very confusing; but it is regarded as certain that
Servius Tuillus found six centuries in existence, to which he added
twelve, making eighteen in all, a number which remained unchanged
throughout the republican period. A proposal by M. Porcius Cato the
elder to supplement the deficiency in the cavalry by the creation of
four additional centuries was not adopted. The earlier centuries were
called _sex suffragia_ ("the six votes"), and at first consisted
exclusively of patricians, while those of Servius Tullius were entirely
or for the most part plebeian. Until the reform of the comitia
centuriata (probably during the censorship of Gaius Flaminius in 220
B.C.; see COMITIA), the equites had voted first, but after that time
this privilege was transferred to one century selected by lot from the
centuries of the equites and the first class. The equites then voted
with the first class, the distinction between the _sex suffragia_ and
the other centuries being abolished.
Although the equites were selected from the wealthiest citizens, service
in the cavalry was so expensive that the state gave financial
assistance. A sum of money (_aes equestre_) was given to each eques for
the purchase of two horses (one for himself and one for his groom), and
a further sum for their keep (_aes hordearium_); hence the name _equites
equo publico_. In later times, pay was substituted for the _aes
hordearium_, three times as much as that of the infantry. If competent,
an eques could retain his horse and vote after the expiration of his ten
years' service, and (till 129 B.C.) even after entry into the senate.
As the demands upon the services of the cavalry increased, it was
decided to supplement the regulars by the enrolment of wealthy citizens
who kept horses of their own. The origin of these _equites equo privato_
dates back, according to Livy (v. 7), to the siege of Veii, when a
number of young men came forward and offered their services. According
to Mommsen, although the institution was not intended to be permanent,
in later times vacancies in the ranks were filled in this manner, with
the result that service in the cavalry, with either a public or a
private horse, became obligatory upon all Roman citizens possessed of a
certain income. These _equites equo privato_ had no vote in the
centuries, received pay in place of the _aes equestre_, and did not form
a distinct corps.
Thus, at a comparatively early period, three classes of equites may be
distinguished: (a) The patrician equites _equo publico_ of the _sex
suffragia_; (b) the plebeian equites in the twelve remaining centuries;
(c) the equites _equo privato_, both patrician and plebeian.
The equites were originally chosen by the curiae, then in succession by
the kings, the consuls, and (after 443 B.C.) by the censors, by whom
they were reviewed every five years in the Forum. Each eques, as his
name was called out, passed before the censors, leading his horse. Those
whose physique and character were satisfactory, and who had taken care
of their horses and equipments, were bidden to lead their horse on
(_traducere equum_), those who failed to pass the scrutiny were ordered
to sell it, in token of their expulsion from the corps. This inspection
(_recognitio_) must not be confounded with the full-dress procession
(_transvectio_) on the 15th of July from the temple of Mars or Honos to
the Capitol, instituted in 304 B.C. by the censor Q. Fabius Maximus
Rullianus to commemorate the miraculous intervention of Castor and
Pollux at the battle of Lake Regillus. Both inspection and procession
were discontinued before the end of the republic, but revived and in a
manner combined by Augustus.
In theory, the twelve plebeian centuries were open to all freeborn
youths of the age of seventeen, although in practice preference was
given to the members of the older families. Other requirements were
sound health, high moral character and an honourable calling. At the
beginning of the republican period, senators were included in the
equestrian centuries. The only definite information as to the amount of
fortune necessary refers to later republican and early imperial times,
when it is known to have been 400,000 sesterces (about L3500 to L4000).
The insignia of the equites were, at first, distinctly military--such as
the purple-edged, short military cloak (_trabea_) and decorations for
service in the field.
With the extension of the Roman dominions, the equites lost their
military character. Prolonged service abroad possessed little attraction
for the pick of the Roman youth, and recruiting for the cavalry from the
equestrian centuries was discontinued. The equites remained at home, or
only went out as members of the general's staff, their places being
taken by the _equites equo privato_, the cavalry of the allies and the
most skilled horsemen of the subject populations. The first gradually
disappeared, and Roman citizens were rarely found in the ranks of the
effective cavalry. In these circumstances there grew up in Rome a class
of wealthy men, whose sole occupation it was to amass large fortunes by
speculation, and who found a most lucrative field of enterprise in state
contracts and the farming of the public revenues. These tax-farmers (see
PUBLICANI) were already in existence at the time of the Second Punic
War; and their numbers and influence increased as the various provinces
were added to the Roman dominions. The change of the equites into a body
of financiers was further materially promoted (a) by the lex Claudia
(218 B.C.), which prohibited senators from engaging in commercial
pursuits, especially if (as seems probable) it included public contracts
(cf. FLAMINIUS, GAIUS); (b) by the enactment in the time of Gaius
Gracchus excluding members of the senate from the equestrian centuries.
These two measures definitely marked off the aristocracy of birth from
the aristocracy of wealth--the landed proprietor from the capitalist.
The term equites, originally confined to the purely military equestrian
centuries of Servius Tullius, now came to be applied to all who
possessed the property qualification of 400,000 sesterces.
As the equites practically monopolized the farming of the taxes, they
came to be regarded as identical with the _publicani_, not, as Pliny
remarks, because any particular rank was necessary to obtain the farming
of the taxes, but because such occupation was beyond the reach of all
except those who were possessed of considerable means. Thus, at the time
of the Gracchi, these _equites-publicani_ formed a close financial
corporation of about 30,000 members, holding an intermediate position
between the nobility and the lower classes, keenly alive to their own
interests, and ready to stand by one another when attacked. Although to
some extent looked down upon by the senate as following a dishonourable
occupation, they had as a rule sided with the latter, as being at least
less hostile to them than the democratic party. To obtain the support of
the capitalists, Gaius Gracchus conceived the plan of creating friction
between them and the senate, which he carried out by handing over to
them the control (a) of the jury-courts, and (b) of the revenues of
Asia.
(a) Hitherto, the list of jurymen for service in the majority of
processes, both civil and criminal, had been composed exclusively of
senators. The result was that charges of corruption and extortion
failed, when brought against members of that order, even in cases where
there was little doubt of their guilt. The popular indignation at such
scandalous miscarriages of justice rendered a change in the composition
of the courts imperative. Apparently Gracchus at first proposed to
create new senators from the equites and to select the jurymen from this
mixed body, but this moderate proposal was rejected in favour of one
more radical (see W.W. Fowler in _Classical Review_, July 1896). By the
lex Sempronia (123 B.C.) the list was to be drawn from persons of free
birth over thirty years of age, who must possess the equestrian census,
and must not be senators. Although this measure was bound to set
senators and equites at variance, it in no way improved the lot of those
chiefly concerned. In fact, it increased the burden of the luckless
provincials, whose only appeal lay to a body of men whose interests were
identical with those of the _publicani_. Provided he left the
tax-gatherer alone, the governor might squeeze what he could out of the
people, while on the other hand, if he were humanely disposed, it was
dangerous for him to remonstrate.
(b) The taxes of Asia had formerly been paid by the inhabitants
themselves in the shape of a fixed sum. Gracchus ordered that the taxes,
direct and indirect, should be increased, and that the farming of them
should be put up to auction at Rome. By this arrangement the provincials
were ignored, and everything was left in the hands of the capitalists.
From this time dates the existence of the equestrian order as an
officially recognized political instrument. When the control of the
courts passed into the hands of the property equites, all who were
summoned to undertake the duties of judices were called equites; the
_ordo judicum_ (the official title) and the _ordo equester_ were
regarded as identical. It is probable that certain privileges of the
equites were due to Gracchus; that of wearing the gold ring, hitherto
reserved for senators; that of special seats in the theatre,
subsequently withdrawn (probably by Sulla) and restored by the lex
Othonis (67 B.C.); the narrow band of purple on the tunic as
distinguished from the broad band worn by the senators.
Various attempts were made by the senate to regain control of the
courts, but without success. The lex Livia of M. Livius Drusus (q.v.),
passed with that object, but irregularly and by the aid of violence, was
annulled by the senate itself. In 82 Sulla restored the right of serving
as judices to the senate, to which he elevated 300 of the most
influential equites, whose support he thus hoped to secure; at the same
time he indirectly dealt a blow at the order generally, by abolishing
the office of the censor (immediately revived), in whom was vested the
right of bestowing the public horse. To this period Mommsen assigns the
regulation, generally attributed to Augustus, that the sons of senators
should be knights by right of birth. By the lex Aurelia (70 B.C.) the
judices were to be chosen in equal numbers from senators, equites and
tribuni aerarii (see AERARIUM), (the last-named being closely connected
with the equites), who thus practically commanded a majority. About this
time the influence of the equestrian order reached its height, and
Cicero's great object was to reconcile it with the senate. In this he
was successful at the time of the Catilinarian conspiracy, in the
suppression of which he was materially aided by the equites. But the
union did not last long; shortly afterwards the majority ranged
themselves on the side of Julius Caesar, who did away with the tribuni
aerarii as judices, and replaced them by equites.
Augustus undertook the thorough reorganization of the equestrian order
on a military basis. The _equites equo privato_ were abolished
(according to Herzog, not till the reign of Tiberius) and the term
equites was officially limited to the _equites equo publico_, although
all who possessed the property qualification were still considered to
belong to the "equestrian order." For the _equites equo publico_ high
moral character, good health and the equestrian fortune were necessary.
Although free birth was considered indispensable, the right of wearing
the gold ring (_jus anuli aurei_) was frequently bestowed by the emperor
upon freedmen, who thereby became _ingenui_ and eligible as equites.
Tiberius, however, insisted upon free birth on the father's side to the
third generation. Extreme youth was no bar; the emperor Marcus Aurelius
had been an eques at the age of six. The sons of senators were eligible
by right of birth, and appear to have been known as _equites illustres_.
The right of bestowing the _equus publicus_ was vested in the emperor;
once given, it was for life, and was only forfeitable through
degradation for some offence or the loss of the equestrian fortune.
Augustus divided the equites into six _turmae_ (regarded by Hirschfeld
as a continuation of the _sex suffragia_). Each was under the command of
a _sevir_ ([Greek: hilarchos]), who was appointed by the emperor and
changed every year. During their term of command the _seviri_ had to
exhibit games (_ludi sevirales_). Under these officers the equites
formed a kind of corporation, which, although not officially recognized,
had the right of passing resolutions, chiefly such as embodied acts of
homage to the imperial house. It is not known whether the _turmae_
contained a fixed number of equites; there is no doubt that, in
assigning the public horse, Augustus went far beyond the earlier figure
of 1800. Thus, Dionysius of Halicarnassus mentions 5000 equites as
taking part in a review at which he himself was present.
As before, the equites wore the narrow, purple-striped tunic, and the
gold ring, the latter now being considered the distinctive badge of
knighthood. The fourteen rows in the theatre were extended by Augustus
to seats in the circus.
The old _recognitio_ was replaced by the _probatio_, conducted by the
emperor in his censorial capacity, assisted by an advisory board of
specially selected senators. The ceremony was combined with a
procession, which, like the earlier _transvectio_, took place on the
15th of July, and at such other times as the emperor pleased. As in
earlier times, offenders were punished by expulsion.
In order to provide a supply of competent officers, each eques was
required to fill certain subordinate posts, called _militiae equestres_.
These were (1) the command of an auxiliary cohort; (2) the tribunate of
a legion; (3) the command of an auxiliary cavalry squadron, this order
being as a rule strictly adhered to. To these Septimius Severus added
the centurionship. Nomination to the _militiae equestres_ was in the
hands of the emperor. After the completion of their preliminary military
service, the equites were eligible for a number of civil posts, chiefly
those with which the emperor himself was closely concerned. Such were
various procuratorships; the prefectures of the corn supply, of the
fleet, of the watch, of the praetorian guards; the governorships of
recently acquired provinces (Egypt, Noricum), the others being reserved
for senators. At the same time, the abolition of the indirect method of
collecting the taxes in the provinces greatly reduced the political
influence of the equites. Certain religious functions of minor
importance were also reserved for them. In the jury courts, the equites,
thanks to Julius Caesar, already formed two-thirds of the judices;
Augustus, by excluding the senators altogether, virtually gave them the
sole control of the tribunals. One of the chief objects of the emperors
being to weaken the influence of the senate by the opposition of the
equestrian order, the practice was adopted of elevating those equites
who had reached a certain stage in their career to the rank of senator
by _adlectio_. Certain official posts, of which it would have been
inadvisable to deprive senators, could thus be bestowed upon the
promoted equites.
The control of the imperial correspondence and purse was at first in
the hands of freedmen and slaves. The emperor Claudius tentatively
entrusted certain posts connected with these to the equites; in the time
of Hadrian this became the regular custom. Thus a civil career was open
to the equites without the obligation of preliminary military service,
and the emperor was freed from the pernicious influence of freedmen.
After the reign of Marcus Aurelius (according to Mommsen) the equites
were divided into: (a) _viri eminentissimi_, the prefects of the
praetorian guard; (b) _viri perfectissimi_, the other prefects and the
heads of the financial and secretarial departments; (c) _viri egregii_,
first mentioned in the reign of Antoninus Pius, a title by right of the
procurators generally.
Under the empire the power of the equites was at its highest in the time
of Diocletian; in consequence of the transference of the capital to
Constantinople, they sank to the position of a mere city guard, under
the control of the prefect of the watch. Their history may be said to
end with the reign of Constantine the Great.
Mention may also be made of the _equites singulares Augusti_. The
body-guard of Augustus, consisting of foreign soldiers (chiefly Germans
and Batavians), abolished by Galba, was revived from the time of Trajan
or Hadrian under the above title. It was chiefly recruited from the pick
of the provincial cavalry, but contained some Roman citizens. It formed
the imperial "Swiss guard," and never left the city except to accompany
the emperor. In the time of Severus, these equites were divided into two
corps, each of which had its separate quarters, and was commanded by a
tribune under the orders of the prefect of the praetorian guard. They
were subsequently replaced by the _protectores Augusti_.
See further article ROME: _History_; also T. Mommsen, _Romisches
Staatsrecht_, iii.; J.N. Madvig, _Die Verfassung des romischen
Staates_, i.; R. Cagnat in Daremberg and Saglio's _Dictionnaire des
antiquites_, where full references to ancient authorities are given in
the footnotes; A.S. Wilkins in Smith's _Dictionary of Greek and Roman
Antiquities_ (3rd ed., 1891); E. Belot, _Histoire des chevaliers
romains_ (1866-1873); H.O. Hirschfeld, _Untersuchungen auf dem Gebiete
der romischen Verwaltungsgeschichte_ (Berlin, 1877); E. Herzog,
_Geschichte und System der romischen Staatsverfassung_ (Leipzig,
1884-1891); A.H. Friedlander, _Sittengeschichte Roms_, i. (1901);
A.H.J. Greenidge, _History of Rome_, i. (1904); J.B. Bury, _The
Student's Roman Empire_ (1893); T.M. Taylor, _Political and
Constitutional History of Rome_ (1899). For a concise summary of
different views of the _sex suffragia_ see A. Bouche-Leclercq's
_Manuel des antiquites romaines_, quoted in Daremberg and Saglio; and
on the _equites singulares_, T. Mommsen in _Hermes_, xvi. (1881), p.
458. (J. H. F.)
EQUITY (Lat. _aequitas_), a term which in its most general sense means
equality or justice; in its most technical sense it means a system of
law or a body of connected legal principles, which have superseded or
supplemented the common law on the ground of their intrinsic
superiority. Aristotle (_Ethics_, bk. v. c. 10) defines equity as a
better sort of justice, which corrects legal justice where the latter
errs through being expressed in a universal form and not taking account
of particular cases. When the law speaks universally, and something
happens which is not according to the common course of events, it is
right that the law should be modified in its application to that
particular case, as the lawgiver himself would have done, if the case
had been present to his mind. Accordingly the equitable man ([Greek:
epieikes]) is he who does not push the law to its extreme, but, having
legal justice on his side, is disposed to make allowances. Equity as
thus described would correspond rather to the judicial discretion which
modifies the administration of the law than to the antagonistic system
which claims to supersede the law.
The part played by equity in the development of law is admirably
illustrated in the well-known work of Sir Henry Maine on _Ancient Law_.
Positive law, at least in progressive societies, is constantly tending
to fall behind public opinion, and the expedients adopted for bringing
it into harmony therewith are three, viz. legal fictions, equity and
statutory legislation. Equity here is defined to mean "any body of rules
existing by the side of the original civil law, founded on distinct
principles, and claiming incidentally to supersede the civil law in
virtue of a superior sanctity inherent in those principles." It is thus
different from legal fiction, by which a new rule is introduced
surreptitiously, and under the pretence that no change has been made in
the law, and from statutory legislation, in which the obligatory force
of the rule is not supposed to depend upon its intrinsic fitness. The
source of Roman equity was the fertile theory of natural law, or the law
common to all nations. Even in the Institutes of Justinian the
distinction is carefully drawn in the laws of a country between those
which are peculiar to itself and those which natural reason appoints for
all mankind. The connexion in Roman law between the ideas of equity,
nature, natural law and the law common to all nations, and the influence
of the Stoical philosophy on their development, are fully discussed in
the third chapter of the work we have referred to. The agency by which
these principles were introduced was the edicts of the praetor, an
annual proclamation setting forth the manner in which the magistrate
intended to administer the law during his year of office. Each
successive praetor adopted the edict of his predecessor, and added new
equitable rules of his own, until the further growth of the irregular
code was stopped by the praetor Salvius Julianus in the reign of
Hadrian.
The place of the praetor was occupied in English jurisprudence by the
lord high chancellor. The real beginning of English equity is to be
found in the custom of handing over to that officer, for adjudication,
the complaints which were addressed to the king, praying for remedies
beyond the reach of the common law. Over and above the authority
delegated to the ordinary councils or courts, a reserve of judicial
power was believed to reside in the king, which was invoked as of grace
by the suitors who could not obtain relief from any inferior tribunal.
To the chancellor, as already the head of the judicial system, these
petitions were referred, although he was not at first the only officer
through whom the prerogative of grace was administered. In the reign of
Edward III. the equitable jurisdiction of the court appears to have been
established. Its constitutional origin was analogous to that of the star
chamber and the court of requests. The latter, in fact, was a minor
court of equity attached to the lord privy seal as the court of chancery
was to the chancellor. The successful assumption of extraordinary or
equitable jurisdiction by the chancellor caused similar pretensions to
be made by other officers and courts. "Not only the court of exchequer,
whose functions were in a peculiar manner connected with royal
authority, but the counties palatine of Chester, Lancaster and Durham,
the court of great session in Wales, the universities, the city of
London, the Cinque Ports and other places silently assumed extraordinary
jurisdiction similar to that exercised in the court of chancery." Even
private persons, lords and ladies, affected to establish in their
honours courts of equity.
English equity has one marked historical peculiarity, viz. that it
established itself in a set of independent tribunals which remained in
standing contrast to the ordinary courts for many hundred years. In
Roman law the judge gave the preference to the equitable rule; in
English law the equitable rule was enforced by a distinct set of judges.
One cause of this separation was the rigid adherence to precedent on the
part of the common law courts. Another was the jealousy prevailing in
England against the principles of the Roman law on which English equity
to a large extent was founded.
When a case of prerogative was referred to the chancellor in the reign
of Edward III., he was required to grant such remedy as should be
consonant to honesty (_honestas_). And honesty, conscience and equity
were said to be the fundamental principles of the court. The early
chancellors were ecclesiastics, and under their influence not only moral
principles, where these were not regarded by the common law, but also
the equitable principles of the Roman law were introduced into English
jurisprudence. Between this point and the time when equity became
settled as a portion of the legal system, having fixed principles of its
own, various views of its nature seem to have prevailed. For a long time
it was thought that precedents could have no place in equity, inasmuch
as it professed in each case to do that which was just; and we find this
view maintained by common lawyers after it had been abandoned by the
professors of equity themselves. G. Spence, in his book on the
_Equitable Jurisdiction of the Court of Chancery_, quotes a case in
the reign of Charles II., in which chief justice Vaughan said:
"I wonder to hear of citing of precedents in matter of equity, for if
there be equity in a case, that equity is an universal truth, and
there can be no precedent in it; so that in any precedent that can be
produced, if it be the same with this case, the reason and equity is
the same in itself; and if the precedent be not the same case with
this it is not to be cited."
But the lord keeper Bridgeman answered:
"Certainly precedents are very necessary and useful to us, for in them
we may find the reasons of the equity to guide us, and besides the
authority of those who made them is much to be regarded. We shall
suppose they did it upon great consideration and weighing of the
matter, and it would be very strange and very ill if we should disturb
and set aside what has been the course for a long series of times and
ages."
Selden's description is well known: "Equity is a roguish thing. 'Tis all
one as if they should make the standard for measure the chancellor's
foot." Lord Nottingham in 1676 reconciled the ancient theory and the
established practice by saying that the conscience which guided the
court was not the natural conscience of the man, but the civil and
political conscience of the judge. The same tendency of equity to settle
into a system of law is seen in the recognition of its limits--in the
fact that it did not attempt in all cases to give a remedy when the rule
of the common law was contrary to justice. Cases of hardship, which the
early chancellors would certainly have relieved, were passed over by
later judges, simply because no precedent could be found for their
interference. The point at which the introduction of new principles of
equity finally stopped is fixed by Sir Henry Maine in the chancellorship
of Lord Eldon, who held that the doctrines of the court ought to be as
well settled and made as uniform almost as those of the common law. From
that time certainly equity, like common law, has professed to take its
principles wholly from recorded decisions and statute law. The view
(traceable no doubt to the Aristotelian definition) that equity
mitigates the hardships of the law where the law errs through being
framed in universals, is to be found in some of the earlier writings.
Thus in the _Doctor and Student_ it is said:
"Law makers take heed to such things as may often come, and not to
every particular case, for they could not though they would;
therefore, in some cases it is necessary to leave the words of the law
and follow that reason and justice requireth, and to that intent
equity is ordained, that is to say, to temper and mitigate the rigour
of the law."
And Lord Ellesmere said:
"The cause why there is a chancery is for that men's actions are so
divers and infinite that it is impossible to make any general law
which shall aptly meet with every particular act and not fail in some
circumstances."
Modern equity, it need hardly be said, does not profess to soften the
rigour of the law, or to correct the errors into which it falls by
reason of its generality.
To give any account, even in outline, of the subject matter of equity
within the necessary limits of this article would be impossible. It will
be sufficient to say here that the classification generally adopted by
text-writers is based upon the relations of equity to the common law, of
which some explanation is given above. Thus equitable jurisdiction is
said to be exclusive, concurrent or auxiliary. Equity has _exclusive_
jurisdiction where it recognizes rights which are unknown to the common
law. The most important example is trusts. Equity has _concurrent_
jurisdiction in cases where the law recognized the right but did not
give adequate relief, or did not give relief without circuity of action
or some similar inconvenience. And equity has _auxiliary_ jurisdiction
when the machinery of the courts of law was unable to procure the
necessary evidence.
"The evils of this double system of judicature," says the report of the
judicature commission (1863-1867), "and the confusion and conflict of
jurisdiction to which it has led, have been long known and
acknowledged." A partial attempt to meet the difficulty was made by
several acts of parliament (passed after the reports of commissions
appointed in 1850 and 1851), which enabled courts of law and equity both
to exercise certain powers formerly peculiar to one or other of them. A
more complete remedy was introduced by the Judicature Act 1873, which
consolidated the courts of law and equity, and ordered that law and
equity should be administered concurrently according to the rules
contained in the 26th section of the act. At the same time many matters
of equitable jurisdiction are still left to the chancery division of the
High Court in the first instance. (See CHANCERY.)
AUTHORITIES.--The principles of equity as set out by the following
writers may be consulted: J. Story, J.W. Smith, H.A. Smith and W.
Ashburner; and for the history see G. Spence, _The Equitable
Jurisdiction of the Court of Chancery_ (2 vols., 1846-1849); D.M.
Kerly, _Historical Sketch of the Equitable Jurisdiction of the Court
of Chancery_ (1890).
EQUIVALENT, in chemistry, the proportion of an element which will
combine with or replace unit weight of hydrogen. When multiplied by the
valency it gives the atomic weight. The determination of equivalent
weights is treated in the article STOICHIOMETRY. (See also CHEMISTRY.)
In a more general sense the term "equivalent" is used to denote
quantities of substances which neutralize one another, as for example
NaOH, HCl, 1/2 H2SO4, 1/2 Ba(OH)2.
ERARD, SEBASTIEN (1752-1831), French manufacturer of musical
instruments, distinguished especially for the improvements he made upon
the harp and the pianoforte, was born at Strassburg on the 5th of April
1752. While a boy he showed great aptitude for practical geometry and
architectural drawing, and in the workshop of his father, who was an
upholsterer, he found opportunity for the early exercise of his
mechanical ingenuity. When he was sixteen his father died, and he
removed to Paris where he obtained employment with a harpsichord maker.
Here his remarkable constructive skill, though it speedily excited the
jealousy of his master and procured his dismissal, almost equally soon
attracted the notice of musicians and musical instrument makers of
eminence. Before he was twenty-five he set up in business for himself,
his first workshop being a room in the hotel of the duchesse de
Villeroi, who gave him warm encouragement. Here he constructed in 1780
his first pianoforte, which was also one of the first manufactured in
France. It quickly secured for its maker such a reputation that he was
soon overwhelmed with commissions, and finding assistance necessary, he
sent for his brother, Jean Baptiste, in conjunction with whom he
established in the rue de Bourbon, in the Faubourg St Germain, a piano
manufactory, which in a few years became one of the most celebrated in
Europe. On the outbreak of the Revolution he went to London where he
established a factory. Returning to Paris in 1796, he soon afterwards
introduced grand pianofortes, made in the English fashion, with
improvements of his own. In 1808 he again visited London, where, two
years later, he produced his first double-movement harp. He had
previously made various improvements in the manufacture of harps, but
the new instrument was an immense advance upon anything he had before
produced, and obtained such a reputation that for some time he devoted
himself exclusively to its manufacture. It has been said that in the
year following his invention he made harps to the value of L25,000. In
1812 he returned to Paris, and continued to devote himself to the
further perfecting of the two instruments with which his name is
associated. In 1823 he crowned his work by producing his model grand
pianoforte with the double escapement. Erard died at Passy, on the 5th
of August 1831. (See also HARP and PIANOFORTE.)
ERASMUS, DESIDERIUS (1466-1536), Dutch scholar and theologian, was born
on the night of the 27/28th of October, probably in 1466; but his
statements about his age are conflicting, and in view of his own
uncertainty (_Ep._ x. 29: 466) and the weakness of his memory for dates,
the year of his birth cannot be definitely fixed. His father's name
seems to have been Rogerius Gerardus. He himself was christened
Herasmus; but in 1503, when becoming familiar with Greek, he assimilated
the name to a fancied Greek original, which he had a few years before
Latinized into Desyderius. A contemporary authority states that he was
born at Gouda, his father's native town; but he adopted the style
_Rotterdammensis_ or _Roterodamus_, in accordance with a story to which
he himself gave credence. His first schooling was at Gouda under Peter
Winckel, who was afterwards vice-pastor of the church. In the dull round
of instruction in "grammar" he did not distinguish himself, and was
surpassed by his early friend and companion, William Herman, who was
Winckel's favourite pupil. From Gouda the two boys went to the school
attached to St Lebuin's church at Deventer, which was one of the first
in northern Europe to feel the influence of the Renaissance. Erasmus was
at Deventer from 1475 to 1484, and when he left, had learnt from
Johannes Sinthius (Syntheim) and Alexander Hegius, who had come as
headmaster in 1483, the love of letters which was the ruling passion of
his life. At some period, perhaps in an interval of his time at
Deventer, he was a chorister at Utrecht under the famous organist of the
cathedral, Jacob Obrecht.
About 1484 Erasmus' father died, leaving him and an elder brother Peter,
both born out of wedlock, to the care of guardians, their mother having
died shortly before. Erasmus was eager to go to a university, but the
guardians, acting under a perhaps genuine enthusiasm for the religious
life, sent the boys to another school at Hertogenbosch; and when they
returned after two or three years, prevailed on them to enter
monasteries. Peter went to Sion, near Delft; Erasmus after prolonged
reluctance became an Augustinian canon in St Gregory's at Steyn, a house
of the same Chapter near Gouda. There he found little religion and less
refinement; but no serious difficulty seems to have been made about his
reading the classics and the Fathers with his friends to his heart's
content. The monastery once entered, there was no drawing back; and
Erasmus passed through the various stages which culminated in his
ordination as priest on the 25th of April 1492.
But his ardent spirit could not long be content with monastic life. He
brought his attainments somehow to the notice of Henry of Bergen, bishop
of Cambrai, the leading prelate at the court of Brussels; and about 1494
permission was obtained for him to leave Steyn and become Latin
secretary to the bishop, who was then preparing for a visit to Rome. But
the journey was abandoned, and after some months Erasmus found that even
with occasional chances to read at Groenendael, the life of a court was
hardly more favourable to study than that of Steyn. At the suggestion of
a friend, James Batt, he applied to his patron for leave to go to Paris
University. The bishop consented and promised a small pension; and in
August 1495 Erasmus entered the "domus pauperum" of the college of
Montaigu, which was then under the somewhat rigid rule of the reformer
Jan Standonck. He at once introduced himself to the distinguished French
historian and diplomatist Robert Gaguin (1425-1502) and published a
small volume of poems; and he became intimate with Johann Mauburnus
(Mombaer), the leader of a mission summoned from Windesheim in 1496 to
reform the abbey of Chateau-Landon. But the life at Montaigu was too
hard for him. Every Lent he fell ill and had to return to Holland to
recover. He continued to read nevertheless for a degree in theology, and
at some time completed the requirements for the B.D. After a year or two
he left Montaigu and eked out his money from the bishop by taking
pupils. One of these, a young Englishman, William Blount, 4th Baron
Mountjoy (d. 1534), persuaded him to visit England in the spring of
1499.
Being without a benefice, he had no settled income to look to, and apart
from the precarious profits of teaching and writing books, could only
wait on the generosity of patrons to supply him with the leisure he
craved. The faithful Batt had sought a pension for him from his own
patroness, Anne of Borsselen, the Lady of Veere, who resided at the
castle of Tournehem near Calais, and whose son Batt was now teaching.
But as nothing promised at once, Erasmus accepted Mountjoy's offer, and
thus a tie was formed which led Mountjoy then or a few years later to
grant him a pension of L20 for life. Otherwise the visit to England gave
no hope of preferment; and in the summer Erasmus prepared to leave. He
was delayed, and used the interval to spend two or three months at
Oxford, where he found John Colet lecturing on the Epistle to the
Romans. Discussions between them on theological questions soon convinced
Colet of Erasmus' worth, and he sought to persuade him to stay and teach
at Oxford. But Erasmus could not be content with the Bible in Latin.
Oxford could teach him no Greek, so away he must go.
In January 1500 he returned to Paris, which though it could offer no
Greek teacher better than George Hermonymus, was at least a better
centre for buying and for printing books. The next few years were spent
still in preparation, supported by pupils' fees and the dedications of
books; the _Collectanea adagiorum_ in June 1500 to Mountjoy, and some
devotional and moral compositions to Batt's patroness and her son. When
the plague drove him from Paris, he went to Orleans or Tournehem or St
Omer, as the way opened. From 1502 to 1504 he was at Louvain, still
declining to teach publicly; among his friends being the future Pope
Adrian VI. In January 1504 the archduke Philip gave him fifty livres for
the Panegyric which "_ung religieux de l'ordre de St Augustin_" had
composed on his Spanish journey; and in October, ten more, for the
maintenance of his studies.
He had been working hard at Greek, of which he now felt himself master,
at the Fathers (above all at Jerome), and at the Epistles of St Paul,
fulfilling the promise made to Colet in Oxford, to give himself to
sacred learning. But the bent of his reading is shown by the manuscript
with which he returned to Paris at the close of 1504--Valla's
_Annotations on the New Testament_, which Badius printed for him in
1505.
Shortly afterwards Lord Mountjoy invited him again to England, and this
visit was more successful. He found in London a circle of learned
friends through whom he was introduced to William Warham, archbishop of
Canterbury, Richard Foxe, bishop of Winchester and other dignitaries.
John Fisher (bishop of Rochester), who was then superintending the
foundation of Christ's College for the Lady Margaret, took him down to
Cambridge for the king's visit; and at length the opportunity came to
fulfil his dream of seeing Italy. Baptista Boerio, the king's physician,
engaged him to accompany his two sons thither as supervisor of their
studies. In September 1506 he set foot on that sacred soil, and took his
D.D. at Turin. For a year he remained with his pupils at Bologna, and
then, his engagement completed, negotiated with Aldus Manutius for a new
edition of his _Adagia_ upon a very different scale. The volume of 1500
had been jejune, written when he knew nothing of Greek; 800 adages put
together with scanty elucidations. In 1508 he had conceived a work on
lines more to the taste of the learned world, full of apt and recondite
learning, and now and again relieved by telling comments or lively
anecdotes. Three thousand and more collected justified a new
title--_Chiliades adagiorum_; and the author's reputation was now
established. So secure in public favour did the book in time become,
that the council of Trent, unable to suppress it and not daring to
overlook it, ordered the preparation of a castrated edition.
To print the _Adagia_ he had gone to Venice, where he lived with Andrea
Torresano of Asola (Asulanus) and did the work of two men, writing and
correcting proof at the same time. When it was finished, with an ample
re-dedication to Mountjoy, a new pupil presented himself, Alexander
Stewart, natural son of James IV. of Scotland--perhaps through a
connexion formed in early days at Paris. They went together to Siena and
Rome and then on to Campania, thirsty under the summer sun. When they
returned to Rome, his pupil departed to Scotland, to fall a few years
later by his father's side at Flodden; Erasmus also found a summons to
call him northwards.
On the death of Henry VII. Lord Mountjoy, who had been companion to
Prince Henry in his studies, had become a person of influence. He wrote
to Erasmus of a land flowing with milk and honey under the "divine"
young king, and with Warham sent him L10 for journey money. At first
Erasmus hesitated. He had been disappointed in Italy, to find that he
had not much to learn from its famed scholarship; but he had made many
friends in Aldus's circle--Marcus Musurus, John Lascaris, Baptista
Egnatius, Paul Bombasius, Scipio Carteromachus; and his reception had
been flattering, especially in Rome, where cardinals had delighted to
honour him. But to remain in Rome was to sell himself. He might have the
leisure which was so indispensable, but at price of the freedom to read,
think, write what he liked. He decided, therefore, to go, though with
regrets; which returned upon him sometimes in after years, when the
English hopes had not borne fruit.
In the autumn he reached London, and in Thomas More's house in
Bucklersbury wrote the witty satire which Milton found "in every one's
hands" at Cambridge in 1628, and which is read to this day. The _Moriae
encomium_ was a sign of his decision. In it kings and princes, bishops
and popes alike are shown to be in bondage to Folly; and no class of men
is spared. Its author was willing to be beholden to any one for leisure;
but he would be no man's slave. For the next eighteen months he is
entirely lost to view; when he reappears in April 1511, he is leaving
More's house and taking the _Moria_ to be printed privily in Paris.
Wherever they were spent, these must have been months of hard work, as
were the years that followed. His time was now come. The long
preparation and training, bought by privation and uncongenial toil, was
over, and he was ready to apply himself to the scientific study of
sacred letters. His English patrons were liberal. Fisher sent him in
August 1511 to teach in Cambridge; Warham gave him a benefice, Aldington
in Kent, worth _L_33, 6_s._ 8_d._ a year, and in violation of his own
rule commuted it for a pension of L20 charged on the living; and the
dedications of his books were fruitful. In Cambridge he completed his
work on the New Testament, the Letters of Jerome, and Seneca; and then
in 1514, when there seemed no prospect of ampler preferment, he
determined to transfer himself to Basel and give the results of his
labours to the world.
The origin of Erasmus's connexion with Johann Froben is not clear. In
1511 he was preparing to reprint his _Adagia_ with Jodocus Badius, who
in the following year was to have also Seneca and Jerome. But in 1513
Froben, who had just reprinted the Aldine _Adagia_, acquired through a
bookseller-agent Erasmus' amended copy which had been destined for
Badius. That the agent was acting entirely on his own responsibility may
be doubted; for within a few months Erasmus had decided to betake
himself to Basel, bearing with him Seneca and Jerome, the latter to be
incorporated in the great edition which Johannes Amerbach and Froben had
had in hand since 1510. In Germany he was widely welcomed. The
Strassburg Literary Society feted him, and Johannes Sapidus, headmaster
of the Latin school at Schlettstadt, rode with him into Basel. Froben
received him with open arms, and the presses were soon busy with his
books. Through the winter of 1514-1515 Erasmus worked with the strength
of ten; and after a brief visit to England in the spring, the New
Testament was set up. Around him was a circle of students, some young,
some already distinguished--the three sons of Froben's partner, Johannes
Amerbach, who was now dead, Beatus Rhenanus, Wilhelm Nesen, Ludwig Ber,
Heinrich Glareanus, Nikolaus Gerbell, Johannes Oecolampadius--who looked
to him as their head and were proud to do him service.
Though from this time forward Basel became the centre of occupation and
interest for Erasmus, yet for the next few years he was mainly in the
Netherlands. On the completion of the New Testament in 1516 he returned
to his friends in England; but his appointment, then recent, as
councillor to the young king Charles, brought him back to Brussels in
the autumn. In the spring of 1517 he went for the last time to England,
about a dispensation from wearing his canonical dress, obtained
originally from Julius II. and recently confirmed by Leo X., and in May
1518 he journeyed to Basel for three months to set the second edition of
the New Testament in progress. But with these exceptions he remained in
proximity to the court, living much at Louvain, where he took great
interest in the foundation of Hieronymus Busleiden's Collegium
Trilingue. His circumstances had improved so much, by pensions, the
presents w
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