Encyclopaedia Britannica, 11th Edition, "Equation" to "Ethics" (part 1)

The Encyclopædia Britannica

A Dictionary of Arts, Sciences, Literature and General Information

Eleventh Edition

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

ÉRARD, SÉBASTIEN ESCOVEDO, JUAN DE

ERASMUS, DESIDERIUS ESCUINTLA

ERASTUS, THOMAS ESCUTCHEON

ERATOSTHENES OF ALEXANDRIA ESHER, WILLIAM BALIOL BRETT

ERBACH ESHER

ERBIUM ESKER

ERCILLA Y ZÚNIGA, ALONSO DE ESKILSTUNA

ERCKMANN-CHATRIAN ESKIMO

ERDÉLYI, JÁNOS ESKI-SHEHR

ERDMANN, JOHANN EDUARD ESMARCH, JOHANNES FRIEDRICH AUGUST VON

ERDMANN, OTTO LINNÉ ESNA

EREBUS ESOTERIC

ERECH ESPAGNOLS SUR MER, LES

ERECHTHEUM ESPALIER

ERECHTHEUS ESPARTERO, BALDOMERO

ERESHKIGAL ESPARTO

ERETRIA ESPERANCE

ERETRIAN SCHOOL OF PHILOSOPHY ESPERANTO

ERFURT ESPINAY, TIMOLÉON D’

ERGOT ESPINEL, VICENTE MARTINEZ

ERIC XIV ESPIRITO SANTO

ERICACEAE ESPRONCEDA, JOSÉ IGNACIO JAVIER ORIOL ENCARNACIÓN DE

ERICHSEN, SIR JOHN ERIC ESQUIRE

ERICHT, LOCH ESQUIROL, JEAN ÉTIENNE DOMINIQUE

ERICSSON, JOHN ESQUIROS, HENRI FRANÇOIS 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

ERLKÖNIG 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 ESTÉBANEZ CALDERÓN, SERAFÍN

ERNE ESTELLA

ERNEST I ESTERHÁZY OF GALÁNTHA

ERNEST II ESTERS

ERNEST AUGUSTUS ESTHER

ERNESTI, JOHANN AUGUST ESTHONIA

ERNESTI, JOHANN CHRISTIAN 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 ESTRÉES, GABRIELLE D’

ERSKINE, HENRY ESTREMADURA

ERSKINE, JOHN (Scottish divine) ESTREMOZ

ERSKINE, JOHN (of Carnock) ESTUARY

ERSKINE, JOHN (of Dun) ESZTERGOM

ERSKINE, RALPH ÉTAGÈRE

ERSKINE, THOMAS (of Linlathen) ETAH

ERSKINE, THOMAS ERSKINE ÉTAMPES, ANNE DE PISSELEU D’HEILLY

ERUBESCITE ÉTAMPES

ERYSIPELAS ÉTAPLES

ERYTHRAE ETAWAH

ERYTHRITE ETCHING

ERZERUM ETEOCLES

ERZGEBIRGE ETESIAN WIND

ERZINGAN ÉTEX, 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) ax² + 2bx + c = 0,

(2) xy² + 4a²x = 8a³,

(3) ax² + 2hxy + by² = 0.

In (1) the unknown is x, and the knowns a, b, c; the coefficients of x² and x are a and 2b; the absolute term is c, and the degree is 2. In (2) the unknowns are x and y, and the known a; the degree is 3, i.e. the sum of the indices in the term xy². (3) is a homogeneous equation of the second degree in x and y. Equations of the first degree are called simple or linear; of the second, quadratic; of the third, cubic; of the fourth, biquadratic; of the fifth, quintic, and so on. Of equations containing only one unknown the number of roots equals the degree of the equation; thus a simple equation has one root, a quadratic two, a cubic three, and so on. If one equation be given containing two unknowns, as for example ax + by = c or ax² + by² = c, it is seen that there are an infinite number of roots, for we can give x, say, any value and then determine the corresponding value of y; such an equation is called indeterminate; of the examples chosen the first is a linear and the second a quadratic indeterminate equation. In general, an indeterminate equation results when the number of unknowns exceeds by unity the number of equations. If, on the other hand, we have two equations connecting two unknowns, it is possible to solve the equations separately for one unknown, and then if we equate these values we obtain an equation in one unknown, which is soluble if its degree does not exceed the fourth. By substituting these values the corresponding values of the other unknown are determined. Such equations are called simultaneous; and a simultaneous system is a series of equations equal in number to the number of unknowns. Such a system is not always soluble, for it may happen that one equation is implied by the others; when this occurs the system is called porismatic or poristic. An identity differs from an equation inasmuch as it cannot be solved, the terms mutually cancelling; for example, the expression x² − a² = (x − a)(x + a) is an identity, for on reduction it gives 0 = 0. It is usual to employ the sign ≡ to express this relation.

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

History.—There is little doubt that the earliest solutions of equations are given, in the Rhind papyrus, a hieratic document written some 2000 years before our era. The problems solved were of an arithmetical nature, assuming such forms as “a mass and its 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. x² = ab. A more general equation, viz. x² − ax + a² = 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 x² = ay and xy = 2a²; eliminating y we obtain x³ = 2a³, 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 ƒ(x) = 0, we take y = ƒ(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

z x y

(1).

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

a + b + c = 0

y z x

(2).

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

1 = 1 = 1 = 1 (suppose)

y (b² − ac) z (c² − ab) x (a² − bc) λ

(3).

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

1 = 3abc − (a³ + b³ + c³) ;

λ (a² − bc) (b² − ac) (c² − ab)

and therefore x, y and z are known from (3). The same artifice solves the equations x² − yz = a, y² − xz = b, z² − 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 3x² + xy + y² = 15, 31xy − 3x² − 5y² = 45. Substituting y = mx in both these equations, and then dividing, we obtain 31m − 3 − 5m² = 3 (3 + m + m²) or 8m² − 28m + 12 = 0. The roots of this quadratic are m = ½ or 3, and therefore 2y = x, or y = 3x.

Taking 2y = x and substituting in 3x² + xy + y² = 0, we obtain y² (12 + 2 + 1) = 15; ∴ y² = 1, which gives y = ±1, x = ±2. Taking the second value, y = 3x, and substituting for y, we obtain x² (3 + 3 + 9) = 15; ∴ x² = 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).

(α) 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.

(β) 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 y² = a + bx + cx², 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 g² for a, we have y² = g² + bx + cx². Suppose √(g² + bx + cx²) = g + mx; then g² + bx + cx² = g² + 2gmx + m²x², or bx + cx² = 2gmx + m²x², that is, b + cx = 2gm + m²x; hence

x = 2gm − b , y = √(g² + bx + cx²)= cg − bm + gm² .

c − m² c − m²

Case 2. Let c be a square number = g²; then, putting √(a + bx + g²x²) = m + gx, we find a + bx + g²x² = m² + 2mgx + g²x², or a + bx = m² + 2mgx; hence we find

x = m² − a , y = √(a + bx + g²x²) = bm − gm² − ag .

b − 2mg b − 2mg

Case 3. When neither a nor c is a square number, yet if the expression a + bx + cx² can be resolved into two simple factors, as f + gx and h + kx, the irrationality may be taken away as follows:—

Assume √(a + bx + cx²) = √{ (f + gx) (h + kx) } = m (f + gx), then (f + gx) (h + kx) = m² (f + gx)², or h + kx = m² (f + gx); hence we find

x = fm² − h , y = √{ (f + gx) (h + kx) } = (fk − gh) m ;

k − gm² k − gm²

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

Case 4. The expression a + bx + cx² 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, p² + qr, where p, q and r are quantities which contain no power of x higher than the first. Let us assume √(p² + qr) = p + mq; thus we have p² + qr = p² + 2mpq + m²q² and r = 2mp + m²q, 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 a² be the given square number, and x², y² the numbers required; then, by the question, x² + y² = a², and y = √(a² − x²). This equation is evidently of such a form as to be resolvable by the method employed in case 1. Accordingly, by comparing √(a² − x²) with the general expression √(g² + bx + cx²), we have g = a, b = 0, c = −1, and substituting these values in the formulae, and also −n for +m, we find

x = 2an , y = a (n² − 1) .

n² + 1 n² + 1

If a = n² + 1, there results x = 2n, y = n² − 1, a = n² + 1. Hence if r be an even number, the three sides of a rational right-angled triangle are r, (½ r)² − 1, (½ r)² + 1. If r be an odd number, they become (dividing by 2) r, ½ (r² − 1), ½ (r² + 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 x³ = 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 x³ − c³ = 0 more fully. This is decomposable into the factors x − c = 0 and x² + cx + c² = 0. The roots of this quadratic equation are ½ (−1 ± √−3) c, and we see that the equation x³ = c³ has three roots, namely, one real root c, and two imaginary roots ½ (−1 ± √−3) c. By making c equal to unity, we observe that ½ (−1 ± √−3) are the imaginary cube roots of unity, which are generally denoted by ω and ω², for it is easy to show that (½ (−1 − √−3))² = ½ (−1 + √−3).

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

x³ + Ax² + 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 xn + Axn−1 + Bxn−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 yn−1. By this process we obtain 0 = yn + yn−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 y³ + qy + r = 0, let us suppose y = v + z; we then have y³ = v³ + z³ + 3vz (v + z) = v³ + z³ + 3vzy, and the original equation becomes v³ + z³ + (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 v³ + z³ + r = 0 and 3vz + q = 0. Therefore v³ and z³ are the roots of the quadratic t² + rt − q²/27 = 0. Therefore

v³ = −½ r + √(1⁄27 q³ + ¼ r²); z³ = −½ r − √(1⁄27 q³ + ¼r²);

v = 3√{−½ r + √(1⁄27 q³ + ¼ r²) }; z = 3√{ (−½ r − √(1⁄27 q³ + ¼ r²) };

and y = v + z = 3√{−½ r + √(1⁄27q³ + ¼ r²) } + 3√{−½ r − √(1⁄27 q³ + ¼ r²) }.

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 = −½ r + √(1⁄27 q³ + ¼ r²), B = −½ r − √(1⁄27 q³ + ¼ r²),

and put α = ½ (−1 + √−3),

β = ½ (−1 − √−3).

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

v = 3√A, v = α3√A, v = β3√A;
z = 3√B, z = α3√B, z = β3√B.

To determine the corresponding values of v and z, we must consider that vz = −1⁄3 q = 3√(AB). Now if we observe that αβ = 1, it will immediately appear that v + z has these three values,

v + z = 3√A + 3√B,
v + z = α3√A + β3√B,
v + z = β3√A + α3√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 = −1⁄3q × 1/v = −1⁄3 q / 3√A; hence v + z = 3√A − 1⁄3 q / 3√A: thus it appears that the three values of y may also be expressed thus:

y = 3√A − 1⁄3 q / 3√A
y = α3√A − 1⁄3 qβ / 3√A
y = β3√A − 1⁄3 qα / 3√A.

See below, Theory of Equations, §§ 16 et seq.

IV. Biquadratic Equations.

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

x4 + Ax³ + Bx² + 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, x4 = b4. In this case it is evident that x may be readily had by two extractions of the square root; by the first we find x² = b², and by the second x = b. This, however, is only one of the values which x may have; for since x4 = b4, therefore x4 − b4 = 0; but x4 − b4 may be resolved into two factors x² − b² and x² + b², each of which admits of a similar resolution; for x² − b² = (x − b)(x + b) and x² + b² = (x − b√−1)(x + b√−1). Hence it appears that the equation x4 − b4 = 0 may also be expressed thus,

(x − b) (x + b) (x − b√−1) (x + b√−1) = 0;

so that x may have these four values,

+b, −b, +b√−1, −b√−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,

x4 + qx² + s = 0.

These may be resolved in the manner of quadratic equations; for if we put y = x², we have

y² + qy + s = 0,

from which we find y = ½ {−q ± √(q² − 4s) }, and therefore

x = ±√½ {−q ± √(q² − 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,

y4 + py² + 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 = √a + √b + √c, and also suppose that the letters a, b, c denote the roots of the cubic equation

z³ + Pz² + 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 = √a + √b + √c,

and obtain y² = a + b + c + 2(√ab + √ac + √bc);

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

y² + P = 2(√ab + √ac + √bc).

Let this equation be also squared, and we have

y4 + 2Py² + P² = 4 (ab + ac + bc) + 8 (√a²bc + √ab²c + √abc²);

and since ab + ac + bc = Q,

and √a²bc + √ab²c + √abc² = √abc (√a + √b + √c) = √R·y,

the same equation may be expressed thus:

y4 + 2Py² + P² = 4Q + 8√R·y.

Thus we have the biquadratic equation

y4 + 2Py² − 8√R·y + P² − 4Q = 0,

one of the roots of which is y = √a + √b + √c, while a, b, c are the roots of the cubic equation z³ + Pz² + Qz − R = 0.

4. In order to apply this resolution to the proposed equation y4 + py² + 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

y4 + py² + qy + r = 0,
y4 + 2Py² − 8√Ry + P² − 4Q = 0,

and it immediately appears that

2P = p, −8√R = q, P² − 4Q = r;

and from these equations we find

P = ½ p, Q = 1⁄16 (p² − 4r), R = 1⁄64 q².

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

y = √a + √b + √c;

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

z³ + p z² + p² − 4r z − q² = 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 √a, √b, √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 √a, √b, √c must be equal to √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 = ¼ v. Thus the equation

z³ + p z² + p² − 4r z − q² = 0

2 16 64

becomes, after reduction,

v³ + 2pv² + (p² − 4r) v − q² = 0;

it also follows, that if the roots of the latter equation are a, b, c, the roots of the former are ¼ a, ¼ b, ¼ c, so that our rule may now be expressed thus:

Let y4 + py² + qy + r = 0 be any biquadratic equation wanting its second term. Form this cubic equation

v³ + 2pv² + (p² − 4r) v − q² = 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 = ½ (√a + √b + √c), y = ½ (−√a − √b − √c),

y = ½ (√a − √b − √c), y = ½ (−√a + √b + √c),

y = ½ (−√a + √b − √c), y = ½ (√a − √b + √c),

y = ½ (−√a − √b + √c), y = ½ (√a + √b − √c).

See also below, Theory of Equations, § 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 xn − p1xn−1 ... ± pn = 0, where p1, p2 ... pn are regarded as known, and x as a quantity to be determined; for shortness the equation is written ƒ(x) = 0.

The equation may be numerical; that is, the coefficients p1, p2n, ... pn are then numbers—understanding by number a quantity of the form α + βi (α and β 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 √−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, ... pn are real positive or negative quantities, and a root a, if it exists, is a positive or negative quantity such that an − p1an−1 ... ± pn = 0, or say, ƒ(a) = 0.

It is very useful to consider the curve y = ƒ(x),—or, what would come to the same, the curve Ay = ƒ(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 x³ − 6x² + 11x − 6.06 = 0, then the curve y = x³ − 6x² + 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 xn is +1, then when x = ∞, y = +∞; but when x = −∞, then y = +∞ or −∞, 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 ƒ(x) = 0.

If β, α are any two values of x (α > β, that is, α nearer +∞), then if ƒ(β), ƒ(α) have opposite signs, the curve cuts the axis an odd number of times, and therefore at least once, between the points x = β, x = α; but if ƒ(β), ƒ(α) 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, ƒ(β), ƒ(α) having opposite signs, there are between the limits β, α an odd number of real roots, and therefore at least one real root; but ƒ(β), ƒ(α) 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 β, α the values -∞, +∞ (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 α be such that for x = or > a (that is, x nearer to +∞) ƒ(x) is always +, and β be such that for x = or < β (that is, x nearer to −∞) ƒ(x) is always −, then the real roots (if any) lie between these limits x = β, x = α; 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 δ 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 < δ; then, γ being any value whatever, there is clearly at most one real root between the limits γ and γ + δ; and by what precedes there is such real root or there is not such real root, according as ƒ(γ), ƒ(γ + δ) have opposite signs or have the same sign. And by dividing in this manner the interval β to α into intervals each of which is = or < δ, 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 γ, γ + δ 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 Traité de la résolution des équations numériques; the determination of the limit δ had to be effected by means of the “equation of differences” or equation of the order ½ 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 ƒ(x), and its first derived function ƒ′(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

ƒ(x), ƒ′(x), ƒ2(x), ... ƒn(x)

of the degrees n, n − 1, n − 2 ... 0 respectively,—the last term ƒn(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 β, α; viz. supposing α > β (that is, α nearer to +∞), 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 β to α is the required number of real roots between the two limits. In particular, taking β, α = −∞, +∞ 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 δ; 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 δ 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 β, α there lies one, and only one, real root of the equation; ƒ(β) and ƒ(α) have therefore opposite signs. Suppose any intermediate value is θ; in order to determine by Sturm’s theorem whether the root lies between β, θ, or between θ, α, it would be quite unnecessary to calculate the signs of ƒ(θ),ƒ′(θ), ƒ2(θ) ...; only the sign of ƒ(θ) is required; for, if this has the same sign as ƒ(β), then the root is between β, θ; if the same sign as ƒ(α), then the root is between θ, α. We want to make θ increase from the inferior limit β, at which ƒ(θ) has the sign of ƒ(β), so long as ƒ(θ) 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 α of the root, but (as in the extraction of a square root) it includes the calculation of the function ƒ(α), 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 = γ be an approximate value of any root, and γ + h the correct value, then ƒ(γ + h) = 0, that is,

ƒ(γ) + h ƒ′(γ) + h² ƒ″(γ) + ... = 0;

1 1·2

and then, if h be so small that the terms after the second may be neglected, ƒ(γ) + hƒ′(γ) = 0, that is, h = {−ƒ(γ)/ƒ′(γ) }, or the new approximate value is x = γ − {ƒ(γ)/ƒ′(γ) }; 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; γ 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 γ in order that the process may by successive repetitions actually lead to a certain real root of the equation; or that, γ being an approximate value of a certain real root, the new value γ − {ƒ(γ)/ƒ′(γ) } may be a more approximate value.

Referring to fig. 1, it is easy to see that if OC represent the assumed value γ, 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 α, = 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, ƒ(x) and ƒ″(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 = β, OC = α, we have approximately ƒ(β + h) = ƒ(β) + h{ƒ(α) − ƒ(β) } / (α − β), whence if the root is β + h then h = − (α − β)ƒ(β) / {ƒ(α) − ƒ(β) }.

Returning for a moment to Horner’s method, it may be remarked that the correction h, to an approximate value α, is therein found as a quotient the same or such as the quotient ƒ(α) ÷ ƒ′(α) 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 x² + 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 i² + 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 x² + 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 x³ + px² + 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 + cj² (a, b, c numbers α + βi of the kind heretofore considered),—first we should be thrown back on the quadric equation x² + 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 α + βi) 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 x1n − p1xn−1 ... ± pn to be = 0, or say such that it satisfies the equation ƒ(x) = 0, is said to be a root of the equation; that is, a being a root, we have

an − p1an−1 ... ± pn = 0, or say ƒ(a) = 0;

and it is then easily shown that x − a is a factor of the function ƒ(x), viz. that we have ƒ(x) = (x − a)ƒ1(x), where ƒ1(x) is a function xn−1 − q1xn−2 ... ± qn−1 of the order n − 1, with numerical coefficients q1, q2 ... qn−1.

In general a is not a root of the equation ƒ1(x) = 0, but it may be so—i.e. ƒ1(x) may contain the factor x − a; when this is so, ƒ(x) will contain the factor (x − a)²; writing then ƒ(x) = (x − a)²ƒ2(x), and assuming that a is not a root of the equation ƒ2(x) = 0, x = a is then said to be a double root of the equation ƒ(x) = 0; and similarly ƒ(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 ƒ(x) = (x − a)αF(x) (α being a positive integer which may be = 1, (x − a)α the highest power of x − a which divides ƒ(x), and F(x) being of course of the order n − α), 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 ƒ(x) will in this case be = (x − a)α (x − b)β Φ(x) (β a positive integer which may be = 1, (x − b)β the highest power of x − b in F(x) or ƒ(x), and Φ(x) being of course of the order n − α − β). The original equation ƒ(x) = 0 is in this case said to have α roots each = a, β roots each = b; and so on for any other factors (x − c)γ, &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 ƒ(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 ƒ(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 ƒ(x) = 0, separating the real and imaginary parts thereof, may be written F(x) + iΦ(x) = 0, where F(x), Φ(x) are each of them a function with real coefficients; and it thus appears that the equation ƒ(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 Φ(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. α + βi being a root, then α − βi 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 α, β of a root α + βi 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 ƒ(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.

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 xn − a = 0 (a = α + βi, 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. For writing

α + βi = √(α² + β²) { α + β i },

√(α² + β²) √(α² + β²)

there is always a real angle λ (positive and less than 2π), such that its cosine and sine are = α / √(α² + β²) and β / √(α² + β²) respectively; that is, writing for shortness √(α² + β²) = ρ, we have α + βi = ρ (cos λ + i sin λ), or the equation is xn = ρ (cos λ + i sin λ); hence observing that (cos λ/n + i sin λ/n )n = cos λ + i sin λ, a value of x is = n√ρ (cos λ/n + i sin λ/n). The formula really gives all the roots, for instead of λ we may write λ + 2sπ, s a positive or negative integer, and then we have

x = n√ρ ( cos λ + 2sπ + i sin λ + 2sπ ),

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 θ, so that θn = a, then assuming x = θy, the equation becomes yn − 1 = 0, which equation, like the original equation, has precisely n roots (one of them being of course = 1). And the original equation xn − a = 0 is thus reduced to the more simple equation xn − 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 xn − 1 = 0 has its several roots expressed in the form 1, ω, ω², ... ωn−1, where ω may be taken = cos 2π/n + i sin 2π/n; in fact, ω having this value, any integer power ωk is = cos 2πk/n + i sin 2πk/n, and we thence have (ωk)n = cos 2πk + i sin 2πk, = 1, that is, ωk 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 a1/n regarded as an n-valued function; any one of these being denoted by n√a, then the series of values is n√a, ωn√a, ... ωn−1 n√a; or we may, if we please, use n√a instead of a1/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

xn − p1xn−1 + ... ± pn = 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

xn − p1xn−1 + ... ± pn = 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 = Σa, p2 = Σab, ... pn = abc...;

i.e. regarding the coefficients p1, p2 ... pn as given, then we assume the existence of roots a, b, c, ... such that p1 = Σa, &c.; or, regarding the roots as given, then we write p1, p2, &c., to denote the functions Σa, Σab, &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 x² − px + q = 0 has a root x = ½ {p + √(p² − 4q) }, we mean that writing this value for x the equation becomes an identity, [½ {p + √(p² − 4q) }]² − p[½ {p + √(p² − 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(α, β) as a representation of ƒ(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

xn − p1xn−1 + ... = x − a·x − b. &c.

or the equivalent equations p1 = Σa, &c., we find

an − p1an−1 + ... = 0,
bn − p1bn−1 + ... = 0;
· · ·
· · ·
· · ·

(it is as satisfying these equations that a, b ... are said to be the roots of xn − p1xn−1 + ... = 0); and conversely from the last-mentioned equations, assuming that a, b ... are all different, we deduce

p1 = Σa, p2 = Σab, &c.

and

xn − p1xn−1 + ... = x − a·x − b. &c.

Observe that if, for instance, a = b, then the equations an − p1an−1 + ... = 0, bn − p1bn−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 xn − p1xn−1 +, &c; but by considering b as the limit of a + h, h indefinitely small, we obtain a second equation

nan−1 − (n − 1) p1an−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 = Σa, &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 Σa², Σa³, &c., are rational and integral functions of the coefficients.

The process originally employed for the expression of other functions Σaαbβ, &c., in terms of the coefficients is to make them depend upon the sums of powers: for instance, Σaαbβ = ΣaαΣaβ − Σaα+β; but this is very objectionable; the true theory consists in showing that we have systems of equations

p1 = Σa,

p2 = Σab,

p1² = Σa² + 2Σab,

p3 = Σabc,

p1p2 = Σa²b + 3Σabc,

p1³ = Σa³ + 3Σa²b + 6Σabc,

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 Σabc, Σa²b, Σa³. Hence in each system the root-functions can be determined linearly in terms of the powers and products of the coefficients:

Σab = p2,

Σa² = p1² − 2p2,

Σabc = p3,

Σa²b = p1p2 − 3p3,

Σa³ = p1³ − 3p1p2 + 3p3,

and so on. The other process, if applied consistently, would derive the originally assumed value Σab = p2, from the two equations Σa = p, Σa² = p1² − 2p2; i.e. we have 2Σab = Σa·Σa − Σa²,= p1² − (p1² − 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 xn a function of the form in question; multiplying each side by x, and on the right-hand side writing for xn its foregoing value, we have xn+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 φ(x)/ψ(x) = I(x), or say φ(x) − ψ(x)I(x) = 0, where φ, ψ, I are rational and integral functions; in other words, this equation, being true if only ƒ(x) = 0, can only be so by reason that the left-hand side contains ƒ(x) as a factor, or we must have identically φ(x) − ψ(x)I(x) = M(x)ƒ(x). And it is, moreover, clear that the equation φ(x)/ψ(x) = I(x), being satisfied if only ƒ(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 φ(a), φ(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 xn − p1xn−1 ... = 0, and y = M0xn−1 + M1xn−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 Δ and −Δ. And if P, Q are symmetrical functions of the letters, then the general form of such a function is P + QΔ; this has only the two values P + QΔ, P − QΔ.

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, a²b = 2 determines a to be = 1 and b to be = 2, but if the roots had been 1, 2, 4, 16 then a²b = 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, t1²y1 + t2²y2 + t3²y3 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 x² − px + q = 0, we can by the assistance of the sign √ 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 = ½ {p ± √(p² − 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 = −½ p + z reduce the equation to z² = ¼ (p² − 4q), viz. to an equation of the form x² = 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 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 = ½ {p ± √(p² − 4q) }, substituting for p, q their given numerical values, we obtain for x an expression of the form x = α + βi ± √(γ + δi), where α, β, γ, δ 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 √(γ + δi) 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 √(γ + δi) has such a value calculable by means of the radical expressions √{√(γ² + δ²) ± γ}; 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 x³ + qx − r = 0, and assuming x = a + b, this will be a solution if only 3ab = q and a³ + b³ = r, equations which give (a³ − b³)² = r² − 4⁄27 q³, a quadric equation solvable by radicals, and giving a³ − b³ = √(r² − 4⁄27 q³), a 2-valued function of the coefficients: combining this with a³ + b³ = r, we have a³ = ½ {r + √(r² − 4⁄27 q³) }, a 2-valued function: we then have a by means of a cube root, viz.

a = 3√[½ {r + √(r² − 4⁄27 q³) }],

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 = 3√[½ {r − √(r² − 4⁄27 q³) } ],

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 27a³b³ = q³, implied in, but not implying, the original form.

An interesting variation of the solution is to write x = ab(a + b), giving a³b³ (a³ + b³) = r and 3a³b³ = q, or say a³ + b³ = 3r/q, a³b³ = 1⁄3 q; and consequently

a³ = 3⁄2 {r + √(r² − 4⁄27 q³) }, b³ = 3⁄2 {r − √(r² − 4⁄27 q³) },

q q

i.e. here a³, b³ are each of them a 2-valued function, but as the only effect of altering the sign of the quadric radical is to interchange a³, b³, they may be regarded as each of them 1-valued; a and b are each of them 3-valued (for observe that here only a³b³, 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 = 3√[½ {r + √(r² − 4⁄27 q³) }] + 1⁄3 q ÷ 3√[½ {r + √(r² − 4⁄27 q³) }],

this may depend on an expression of the form 3√(γ + δi) where γ and δ are real numbers (it will do so if r² − 4⁄27 q³ is a negative number), and then we cannot by the extraction of any root or roots of real positive numbers reduce 3√(γ + δi) 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 z³ = a, solvable by the extraction of a cube root, but) to a cubic of the form 4x³ − 3x = a, corresponding to the equation 4 cos³ θ − 3 cos θ = cos 3θ which serves to determine cosθ when cos 3θ is known. The theory is applicable to an algebraical cubic equation; say that such an equation, if it can be reduced to the form 4x³ − 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 = ½ {p + √(p² − 4q) }, substituting for p and q their values, and observing that (a + b)² − 4ab = (a − b)², this becomes x = ½ {a + b + √(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 x³ − px² + qx − r = 0, if the roots are a, b, c, and if ω is used to denote an imaginary cube root of unity, ω² + ω + 1 = 0, then writing for shortness p = a + b + c, L = a + ωb + ω²c, M = a + ω²b + ωc, it is at once seen that LM, L³ + M³, and therefore also (L³ − M³)² are symmetrical functions of the roots, and consequently rational functions of the coefficients; hence

½ {L³ + M³ + √(L³ − M³)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, = L³ or M³; taking it = L³, the cube root of the expression has the three values L, ωL, ω²L; and LM divided by the same cube root has therefore the values M, ω²M, ωM; whence finally the expression

1⁄3 [p + 3√{½ (L³ + M³ + √(L³ − M³)²) } + LM ÷ 3√{½ L³ + M³ + √(L³ − M³)²) }]

has the three values

1⁄3 (p + L + M), 1⁄3 (p + ωL + ω²M), 1⁄3 (p + ω²L + ωM);

that is, these are = a, b, c respectively. If the value M³ had been taken instead of L³, then the expression would have had the same three values a, b, c. Comparing the solution given for the cubic x³ + qx − r = 0, it will readily be seen that the two solutions are identical, and that the function r² − 4⁄27 q³ under the radical sign must (by aid of the relation p = 0 which subsists in this case) reduce itself to (L³ − M³)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 + ωb + ω²c + ω³d + ..., ω being an imaginary root of unity, of the same order as the equation; e.g. for the cubic the root is a + ωb + ω²c, ω an imaginary cube root of unity. Evidently the method gives for L³ 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 + ωb + ω²c + ω³d + ω4e), ω an imaginary fifth root of unity, or rather the fifth power thereof (a + ωb + ω²c + ω³d + ω4e)5; this is a 24-valued function, but if we consider the four values corresponding to the roots of unity ω, ω², ω³, ω4, viz. the values

(a + ω b + ω²c + ω³d + ω4e)5,

(a + ω²b + ω4c + ω d + ω³e)5,

(a + ω³b + ω c + ω4d + ω²e)5,

(a + ω4b + ω³c + ω²d + ω 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 x6 + bx4 + cx² + 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 (x² + ax + b)(x4 + cx³ + dx² + ex + f) = 0, then it breaks up into the equations x² + ax + b = 0, x4 + cx³ + dx² + 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 xn − 1 = 0 present an instance.

22. It has already been shown how the several roots of the equation xn − 1 = 0 can be expressed in the form cos 2sπ/n + i sin 2sπ/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 ω be any other root, then obviously ω, ω², ... ωn−1 are all of them roots; xn − 1 contains the factor x − 1, and it thus appears that ω, ω², ... ωn−1 are the n-1 roots of the equation

xn−1 + xn−2 + ... x + 1 = 0;

we have, of course, ωn−1 + ωn−2 + ... + ω + 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 α be an imaginary root of x³ − 1 = 0 (or root of x² + x + 1 = 0), and β an imaginary root of x5 − 1 = 0 (or root of x4 + x³ + x² + x + 1 = 0), then ω may be taken = αβ; the successive powers thereof, αβ, α²β², β³, αβ4, α², β, αβ², α²β³, β4, α, α²β, β², αβ³, α²β4, are the roots of x14 + x13 + ... + x + 1 = 0; the solution thus depends on the solution of the equations x³ − 1 = 0 and x5 − 1 = 0. In the latter case, if α be an imaginary root of x³ − 1 = 0 (or root of x² + x + 1 = 0), then the equation x9 − 1 = 0 gives x³ = 1, α, or α²; x³ = 1 gives x = 1, α, or α²; and the solution thus depends on the solution of the equations x³ − 1 = 0, x³ − α = 0, x³ − α² = 0. The first equation has the roots 1, α, α²; if β be a root of either of the others, say if β³ = α, then assuming ω = β, the successive powers are β, β², α, αβ, αβ², α², α²β, α²β², which are the roots of the equation x8 + x7 + ... + 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 ω, we have r, r², r³,...rn−1 as the (n − 1) roots of the reduced equation

xn−1 + xn−2 + ... + x + 1 = 0;

then not only rn − 1 = 0, but also rn−1 + rn−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 gn−1 is the lowest power of g, which is ≡ 1 to the modulus n; or, what is the same thing, that the series of powers 1, g, g², ... gn−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, g²,...gn−2, we have, in a different order, the whole series of roots r, r², r³,...rn−1.

In the most simple case, n = 5, the equation to be solved is x4 + x³ + x² + x + 1 = 0; here 2 is a prime root of 5, and the order of the roots is r, r², r4, r³. The Gaussian process consists in forming an equation for determining the periods P1, P2, = r + r4 and r² + r³ respectively;—these being such that the symmetrical functions P1 + P2, P1P2 are rationally determinable: in fact P1 + P2 = −1, P1P2 = (r + r4) (r² + r³), = r³ + r4 + r6 + r7, = r³ + r4 + r + r², = −1. P1, P2 are thus the roots of u² + u − 1 = 0; and taking them to be known, they are themselves broken up into subperiods, in the present case single terms, r and r4 for P1, r² and r³ for P2; the symmetrical functions of these are then rationally determined in terms of P1 and P2; thus r + r4 = P1, r·r4 = 1, or r, r4 are the roots of u² − 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, r³, r², r6, r4, r5. We may form either 3 periods each of 2 terms, P1, P2, P3 = r + r6, r³ + r4, r² + r5 respectively; or else 2 periods each of 3 terms, P1, P2 = r + r² + r4, r³ + r6 + r5 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 + r² + r³ + r4 + r5 + r6, = 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 + r² + r4 = P1, r·r² + r·r4 + r²·r4 = P2, r·r²·r4 = 1.

The theory was further developed by Lagrange (1808), who, applying his general process to the equation in question, xn−1 + xn−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 + ωb + ω²c + ...)n−1 was in this case a given function of ω with integer coefficients.

Reverting to the before-mentioned particular equation x4 + x³ + x² + 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 ω, a root of the equation ω4 − 1 = 0 (whence ω is = 1, −1, i, or −i, at pleasure), and consider the expression

(a + ωb + ω²c + ω³d)4,

the developed value of this is

= a4 + b4 + c4 + d4 + 6 (a²c² + b²d²) + 12 (a²bd + b²ca + c²db + d²ac)

+ω {4 (a³b + b³c + c³ + d³a) + 12 (a²cd + b²da + c²ab + d²bc) }

+ω² {6 (a²b² + b²c² + c²d² + d²a²) + 4 (a³c + b³d + c³a + d³b) + 24abcd}

+ω³ {4 (a³d + b³a + c³b + d³c) + 12 (a²bc + b²cd + c²da + d²ab) }

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, r², r4, r³ of the special equation, then the expression becomes

r4 + r³ + r + r² + 6 (1 + 1) + 12 (r² + r4 + r³ + r)

+ ω {4 (1 + 1 + 1 + 1) + 12 (r4 + r³ + r + r²) }

+ ω²{6 (r + r² + r4 + r³) + 4 (r² + r4 + r³ + r) }

+ ω³{4 (r + r² + r4 + r³) + 12 (r³ + r + r² + r4) }

viz. this is

= −1 + 4ω + 14ω² − 16ω³,

a completely determined value. That is, we have

(r + ωr² + ω²r4 + ω³r³) = −1 + 4ω + 14ω² − 16ω³,

which result contains the solution of the equation. If ω = 1, we have (r + r² + r4 + r³)4 = 1, which is right; if ω = −1, then (r + r4 − r² − r³)4 = 25; if ω = i, then we have {r − r4 + i(r² − r³) }4 = −15 + 20i; and if ω = −i, then {r − r4 − i (r² − r³) }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 xn−1 + xn−2 + ... + x + 1 = 0, ω a root of ωn−1 − 1 = 0, and g a prime root of gn−1 ≡ 1 (mod. n), then

(r + ωr g + ... + ωn − 2r g n−2) n−1

is a given function M0 + M1ω ... + Mn−2ωn−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 ω, which is taken to be known; the equation xn − 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 xn−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, θx, θ²x, ... θn−1x (θx being a rational function of x, and θnx = 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, θx, θ1x being any two of the roots, we have θθ1x = θ1θx, the equation will be solvable algebraically. It is proper to refer also to Abel’s definition of an irreducible equation:—an equation φx = 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 φx 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 x4 + x³ + x² + x + 1 = 0; the left hand side has here no rational divisor, and the equation is irreducible; but this function is = (x² + ½ x + 1)² − 5⁄4 x², and it has thus the irrational divisors x² + ½ (1 + √5)x + 1, x² + ½ (1 − √5)x + 1; and these, if we adjoin the radical √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 x² + px + q = 0 is irreducible, but it becomes reducible, breaking up into rational linear factors, when we adjoin the radical √(¼ p² − q).

The fundamental theorem is the Proposition I. of the “Mémoire sur les conditions de résolubilité des équations 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 φ(x) = 0, then φ(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 xn−1 ... + x + 1 = 0, n a prime number, a, b, c ... k = r, r g, r g² ... 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 a²b + b²c + c²a 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 ... xn, invariable by the substitutions xak + b for xk, 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’algèbre supérieure, 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 x5 + ax + b = 0, or what is the same thing, to x5 + 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, “Über die Auflösung der Gleichungen vom fünften 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 Traité des substitutions et des équations algébriques (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.

The coefficients were selected so that the roots might be nearly 1, 2, 3.

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.

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 α + βi (see Lagrange’s Équations numériques).

The square root of α + βi 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. écurie, 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.

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.

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.

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 Schäfer’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 κέλης, “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 £3500 to £4000). 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 (ἴλαρχος), 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, Römisches Staatsrecht, iii.; J.N. Madvig, Die Verfassung des römischen Staates, i.; R. Cagnat in Daremberg and Saglio’s Dictionnaire des antiquités, 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 römischen Verwaltungsgeschichte (Berlin, 1877); E. Herzog, Geschichte und System der römischen Staatsverfassung (Leipzig, 1884-1891); A.H. Friedländer, 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. Bouché-Leclercq’s Manuel des antiquités 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 (ἐπιεικής) 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, ½H2SO4, ½Ba(OH)2.

ÉRARD, SÉBASTIEN (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 £25,000. 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. Érard 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 Château-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 £20 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 £10 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 £33, 6s. 8d. a year, and in violation of his own rule commuted it for a pension of £20 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 fêted 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 which were showered upon him, and the sale of his books, that he was now in a position to refuse all proposals which would have interfered with his cherished independence. The general ardour for the restoration of the arts and of learning created an aristocratic public, of which Erasmus was supreme pontiff. Luther spoke to the people and the ignorant; Erasmus had the ear of the educated class. His friends and admirers were distributed over all the countries of Europe, and presents were continually arriving from small as well as great, from a donation of 200 florins, made by Pope Clement VII., down to sweetmeats and comfits contributed by the nuns of Cologne (Ep. 666). From England, in particular, he continued to receive supplies of money. In the last year of his life Thomas Cromwell sent him 20 angels, and Archbishop Cranmer 18. Though Erasmus led a very hard-working and far from luxurious life, and had no extravagant habits, yet he could not live upon little. The excessive delicacy of his constitution, not pampered appetite, exacted some unusual indulgences. He could not bear the stoves of Germany, and required an open fireplace in the room in which he worked. He was afflicted with the stone, and obliged to be particular as to what he drank. Beer he could not touch. The white wines of Baden or the Rhine did not suit him; he could only drink those of Burgundy or Franche-Comté. He could neither eat, nor bear the smell of, fish. “His heart,” he said, “was Catholic, but his stomach was Lutheran.” For his constant journeys he required two horses, one for himself and one for his attendant. And though he was almost always found in horse-flesh by his friends, the keep had to be paid for. For his literary labours and his extensive correspondence he required one or more amanuenses. He often had occasion, on his own business, or on that of Froben’s press, to send special couriers to a distance, employing them by the way in collecting the free gifts of his tributaries.

Precarious as these means of subsistence seem, he preferred the independence thus obtained to an assured position which would have involved obligations to a patron or professional duties which his weak health would have made onerous. The duke of Bavaria offered to dispense with teaching, if he would only reside, and would have named him on these terms to a chair in his new university of Ingolstadt, with a salary of 200 ducats, and the reversion of one or more prebendal stalls. The archduke Ferdinand offered a pension of 400 florins, if he would only come to reside at Vienna. Adrian VI. offered him a deanery, but the offer seems to have been of a possible and not an actual deanery. Offers, flattering but equally vague, were made from France, on the part of the bishop of Bayeux, and even of Francis I. “Invitor amplissimis conditionibus; offeruntur dignitates et episcopatus; plane rex essem, si juvenis essem” (Ep. xix. 106; 735). Erasmus declined all, and in November 1521 settled permanently at Basel, in the capacity of general editor and literary adviser of Froben’s press. As a subject of the emperor, and attached to his court by a pension, it would have been convenient to him to have fixed his residence in Louvain. But the bigotry of the Flemish clergy, and the monkish atmosphere of the university of Louvain, overrun with Dominicans and Franciscans, united for once in their enmity to the new classical learning, inclined Erasmus to seek a more congenial home in Basel. To Froben his arrival was the advent of the very man whom he had long wanted. Froben’s enterprise, united with Erasmus’s editorial skill, raised the press of Basel, for a time, to be the most important in Europe. The death of Froben in 1527, the final separation of Basel from the Empire, the wreck of learning in the religious disputes, and the cheap paper and scamped work of the Frankfort presses, gradually withdrew the trade from Basel. But during the years of Erasmus’s co-operation the Froben press took the lead of all the presses in Europe, both in the standard value of the works published and in style of typographical execution. Like some other publishers who preferred reputation to returns in money, Froben died poor, and his impressions never reached the splendour afterwards attained by those of the Estiennes, or of Plantin. The series of the Fathers alone contains Jerome (1516), Cyprian (1520), Pseudo-Arnobius (1522), Hilarius (1523), Irenaeus (Latin, 1526), Ambrose (1527), Augustine (1528), Chrysostom (Latin, 1530), Basil (Greek, 1532, the first Greek author printed in Germany), and Origen (Latin, 1536). In these editions, partly texts, partly translations, it is impossible to determine the respective shares of Erasmus and his many helpers. The prefaces and dedications are all written by him, and some of them, as that to the Hilarius, are of importance for the history as well of the times as of Erasmus himself. Of his most important edition, that of the Greek text of the New Testament, something will be said farther on.

In this “mill,” as he calls it, Erasmus continued to grind incessantly for eight years. Besides his work as editor, he was always writing himself some book or pamphlet called for by the event of the day, some general fray in which he was compelled to mingle, or some personal assault which it was necessary to repel. But though painfully conscious how much his reputation as a writer was damaged by this extempore production, he was unable to resist the fatal facility of print. He was the object of those solicitations which always beset the author whose name upon the title page assures the sale of a book. He was besieged for dedications, and as every dedication meant a present proportioned to the circumstances of the dedicatee, there was a natural temptation to be lavish of them. Add to this a correspondence so extensive as to require him at times to write forty letters in one day. “I receive daily,” he writes, “letters from remote parts, from kings, princes, prelates and men of learning, and even from persons of whose existence I was ignorant.” His day was thus one of incessant mental activity; but hard work was so far from breeding a distaste for his occupation, that reading and writing grew ever more delightful to him (literarum assiduitas non modo mihi fastidium non parit, sed voluptatem; crescit scribendo scribendi studium).

Shortly after Froben’s death the disturbances at Basel, occasioned by the zealots for the religious revolution which was in progress throughout Switzerland, began to make Erasmus desirous of changing his residence. He selected Freiburg in the Breisgau, as a city which was still in the dominion of the emperor, and was free from religious dissension. Thither he removed in April 1529. He was received with public marks of respect by the authorities, who granted him the use of an unfinished residence which had been begun to be built for the late emperor Maximilian. Erasmus proposed only to remain at Freiburg for a few months, but found the place so suited to his habits that he bought a house of his own, and remained there six years. A desire for change of air—he fancied Freiburg was damp—rumours of a new war with France, and the necessity of seeing his Ecclesiastes through the press, took him back to Basel in 1535. He lived now a very retired life, and saw only a small circle of intimate friends. A last attempt was made by the papal court to enlist him in some public way against the Reformation. On the election of Paul III. in 1534, he had, as usual, sent the new pope a congratulatory letter. After his arrival in Basel, he received a complimentary answer, together with the nomination to the deanery of Deventer, the income of which was reckoned at 600 ducats. This nomination was accompanied with an intimation that more was in store for him, and that steps would be taken to provide for him the income, viz., 3000 ducats, which was necessary to qualify for the cardinal’s hat. But Erasmus was even less disposed now than he had been before to barter his reputation for honours. His health had been for some years gradually declining, and disease in the shape of gout gaining upon him. In the winter of 1535-1536 he was confined entirely to his chamber, many days to his bed. Though thus afflicted he never ceased his literary activity, dictating his tract On the Purity of the Church, and revising the sheets of a translation of Origen which was passing through the Froben press. His last letter is dated the 28th of June 1536, and subscribed “Eras. Rot. aegra manu.” “I have never been so ill in my life before as I am now,—for many days unable even to read.” Dysentery setting in carried him off on the 12th of July 1536, in his 70th year.

By his will, made on the 12th of February 1536, he left what he had to leave, with the exception of some legacies, to Bonifazius Amerbach, partly for himself, partly in trust for the benefit of the aged and the infirm, or to be spent in portioning young girls, and in educating young men of promise. He left none of the usual legacies for masses or other clerical purposes, and was not attended by any priest or confessor in his last moments.

Erasmus’s features are familiar to all, from Holbein’s many portraits or their copies. Beatus Rhenanus, “summus Erasmi observator,” as he is called by de Thou, describes his person thus: “In stature not tall, but not noticeably short; in figure well built and graceful; of an extremely delicate constitution, sensitive to the slightest changes of climate, food or drink. After middle life he suffered from the stone, not to mention the common plague of studious men, an irritable mucous membrane. His complexion was fair; light blue eyes, and yellowish hair. Though his voice was weak, his enunciation was distinct; the expression of his face cheerful; his manner and conversation polished, affable, even charming.” His highly nervous organization made his feelings acute, and his brain incessantly active. Through his ready sympathy with all forms of life and character, his attention was always alive. The active movement of his spirit spent itself, not in following out its own trains of thought, but in outward observation. No man was ever less introspective, and though he talks much of himself, his egotism is the genial egotism which takes the world into its confidence, not the selfish egotism which feels no interest but in its own woes. He says of himself, and justly, “that he was incapable of dissimulation” (Ep. xxvi. 19; 1152). There is nothing behind, no pose, no scenic effect. It may be said of his letters that in them “tota patet vita senis.” His nature was flexible without being faultily weak. He has many moods and each mood imprints itself in turn on his words. Hence, on a superficial view, Erasmus is set down as the most inconsistent of men. Further acquaintance makes us feel a unity of character underlying this susceptibility to the impressions of the moment. His seeming inconsistencies are reconciled to apprehension, not by a formula of the intellect, but by the many-sidedness of a highly impressible nature. In the words of J. Nisard, Erasmus was one of those “dont la gloire a été de beaucoup comprendre et d’affirmer peu.”

This equal openness to every vibration of his environment is the key to all Erasmus’s acts and words, and among them to the middle attitude which he took up towards the great religious conflict of his time. The reproaches of party assailed him in his lifetime, and have continued to be heaped upon his memory. He was loudly accused by the Catholics of collusion with the enemies of the faith. His powerful friends, the pope, Wolsey, Henry VIII., the emperor, called upon him to declare against Luther. Theological historians from that time forward have perpetuated the indictment that Erasmus sided with neither party in the struggle for religious truth. The most moderate form of the censure presents him in the odious light of a trimmer; the vulgar and venomous assailant is sure that Erasmus was a Protestant at heart, but withheld the avowal that he might not forfeit the worldly advantages he enjoyed as a Catholic. When by study of his writings we come to know Erasmus intimately, there is revealed to us one of those natures to which partisanship is an impossibility. It was not timidity or weakness which kept Erasmus neutral, but the reasonableness of his nature. It was not only that his intellect revolted against the narrowness of party, his whole being repudiated its clamorous and vulgar excesses. As he loathed fish, so he loathed clerical fanaticism. Himself a Catholic priest—“the glory of the priesthood and the shame”—the tone of the orthodox clergy was distasteful to him; the ignorant hostility to classical learning which reigned in their colleges and convents disgusted him. In common with all the learned men of his age, he wished to see the power of the clergy broken, as that of an obscurantist army arrayed against light. He had employed all his resources of wit and satire against the priests and monks, and the superstitions in which they traded, long before Luther’s name was heard of. The motto which was already current in his lifetime, “that Erasmus laid the egg and Luther hatched it,” is so far true, and no more. Erasmus would have suppressed the monasteries, put an end to the domination of the clergy, and swept away scandalous and profitable abuses, but to attack the church or re-mould received theology was far from his thoughts. And when out of Luther’s revolt there arose a new fanaticism—that of evangelism, Erasmus recoiled from the violence of the new preachers. “Is it for this,” he writes to Melanchthon (Ep. xix. 113; 703), “that we have shaken off bishops and popes, that we may come under the yoke of such madmen as Otto and Farel?” Passages have been collected, and it is an easy task, from the writings of Erasmus to prove that he shared the doctrines of the Reformers. Passages equally strong might be culled to show that he repudiated them. The truth is that theological questions in themselves had no attraction for him. And when a theological position was emphasized by party passion it became odious to him. In the words of Drummond: “Erasmus was in his own age the apostle of common sense and of rational religion. He did not care for dogma, and accordingly the dogmas of Rome, which had the consent of the Christian world, were in his eyes preferable to the dogmas of Protestantism.... From the beginning to the end of his career he remained true to the purpose of his life, which was to fight the battle of sound learning and plain common sense against the powers of ignorance and superstition, and amid all the convulsions of that period he never once lost his mental balance.”

Erasmus is accused of indifference. But he was far from indifferent to the progress of the revolution. He was keenly alive to its pernicious influence on the cherished interest of his life, the cause of learning. “I abhor the evangelics, because it is through them that literature is everywhere declining, and upon the point of perishing.” He had been born with the hopes of the Renaissance, with its anticipation of a new Augustan age, and had seen this fair promise blighted by the irruption of a new horde of theological polemics, worse than the old scholastics, inasmuch as they were revolutionary instead of conservative. Erasmus never flouted at religion nor even at theology as such, but only at blind and intemperate theologians.

In the mind of Erasmus there was no metaphysical inclination; he was a man of letters, with a general tendency to rational views on every subject which came under his pen. His was not the mind to originate, like Calvin, a new scheme of Christian thought. He is at his weakest in defending free will against Luther, and indeed he can hardly be said to enter on the metaphysical question. He treats the dispute entirely from the outside. It is impossible in reading Erasmus not to be reminded of the rationalist of the 18th century. Erasmus has been called the “Voltaire of the Renaissance.” But there is a vast difference in the relations in which they respectively stood to the church and to Christianity. Voltaire, though he did not originate, yet adopted a moral and religious scheme which he sought to substitute for the church tradition. He waged war, not only against the clergy, but against the church and its sovereigns. Erasmus drew the line at the first of these. He was not an anticipation of the 18th century; he was the man of his age, as Voltaire of his; though Erasmus did not intend it, he undoubtedly shook the ecclesiastical edifice in all its parts; and, as Melchior Adam says of him, “pontifici Romano plus nocuit jocando quam Lutherus stomachando.”

But if Erasmus was unlike the 18th century rationalist in that he did not declare war against the church, but remained a Catholic and mourned the disruption, he was yet a true rationalist in principle. The principle that reason is the one only guide of life, the supreme arbiter of all questions, politics and religion included, has its earliest and most complete exemplar in Erasmus. He does not dogmatically denounce the rights of reason, but he practically exercises them. Along with the charm of style, the great attraction of the writings of Erasmus is this unconscious freedom by which they are pervaded.

It must excite our surprise that one who used his pen so freely should have escaped the pains and penalties which invariably overtook minor offenders in the same kind. For it was not only against the clergy and the monks that he kept up a ceaseless stream of satiric raillery; he treated nobles, princes and kings with equal freedom. No 18th century republican has used stronger language than has this pensioner of Charles V. “The people build cities, princes pull them down; the industry of the citizens creates wealth for rapacious lords to plunder; plebeian magistrates pass good laws for kings to violate; the people love peace, and their rulers stir up war.” Such outbursts are frequent in the Adagia. These freedoms are part cause of Erasmus’s popularity. He was here in sympathy with the secret sore of his age, and gave utterance to what all felt but none dared to whisper but he. It marks the difference between 1513 and 1669 that, in a reprint of the Julius Exclusus published in 1669 at Oxford, it was thought necessary to leave out a sentence in which the writer of that dialogue, supposed by the editor to be Erasmus, asserts the right of states to deprive and punish bad kings. It is difficult to say to what we are to ascribe his immunity from painful consequences. We have to remember that he was removed from the scene early in the reaction, before force was fully organized for the suppression of the revolution. And his popular works, the Adagia, and the Colloquia (1524), had established themselves as standard books in the more easy going age, when power, secure in its unchallenged strength, could afford to laugh with the laughers at itself. At the date of his death the Catholic revival, with its fell antipathy to art and letters, was only in its infancy; and when times became dangerous, Erasmus cautiously declined to venture out of the protection of the Empire, refusing repeated invitations to Italy and to France. “I had thought of going to Besançon,” he said, “ne non essem in ditione Caesaris” (Ep. xxx. 74; 1299). In Italy a Bembo and a Sadoleto wrote a purer Latin than Erasmus, but contented themselves with pretty phrases, and were careful to touch no living chord of feeling. In France it was necessary for a Rabelais to hide his free-thinking under a disguise of revolting and unintelligible jargon. It was only in the Empire that such liberty of speech as Erasmus used was practicable, and in the Empire Erasmus passed for a moderate man. Upon the strength of an established character for moderation he enjoyed an exceptional licence for the utterance of unwelcome truths; and in spite of his flings at the rich and powerful, he remained through life a privileged person with them.

But though the men of the keys and the sword let him go his way unmolested, it was otherwise with his brethren of the pen. A man who is always launching opinions must expect to be retorted on. And when these judgments were winged by epigram, and weighted by the name of Erasmus, who stood at the head of letters, a widespread exasperation was the consequence. Disraeli has not noticed Erasmus in his Quarrels of Authors, perhaps because Erasmus’s quarrels would require a volume to themselves. “So thin-skinned that a fly would draw blood,” as the prince of Carpi expressed it, he could not himself restrain his pen from sarcasm. He forgot that though it is safe to lash the dunces, he could not with equal impunity sneer at those who, though they might not have the ear of the public as he had, could yet contradict and call names. And when literary jealousy was complicated with theological differences, as in the case of the free-thinkers, or with French vanity, as in that of Budaeus, the cause of the enemy was espoused by a party and a nation. The quarrel with Budaeus was strictly a national one. Cosmopolitan as Erasmus was, to the French literati he was still the Teuton. Étienne Dolet calls him “enemy of Cicero, and jealous detractor of the French name.” The only contemporary name which could approach to a rivalry with his was that of Budaeus (Budé), who was exactly contemporary, having been born in the same year as Erasmus. Rivals in fame, they were unlike in accomplishment, each having the quality which the other wanted. Budaeus, though a Frenchman, knew Greek well; Erasmus, though a Dutchman, very imperfectly. But the Frenchman Budaeus wrote an execrable Latin style, unreadable then as now, while the Teuton Erasmus charmed the reading world with a style which, though far from good Latin, is the most delightful which the Renaissance has left us.

The style of Erasmus is, considered as Latin, incorrect, sometimes even barbarous, and far removed from any classical model. But it has qualities far above purity. The best Italian Latin is but an echo and an imitation; like the painted glass which we put in our churches, it is an anachronism. Bembo, Sadoleto and the rest write purely in a dead language. Erasmus’s Latin was a living and spoken tongue. Though Erasmus had passed nearly all his life in England, France and Germany, his conversation was Latin; and the language in which he talked about common things he wrote. Hence the spontaneity and naturalness of his page, its flavour of life and not of books. He writes from himself, and not out of Cicero. Hence, too, he spoiled nothing by anxious revision in terror lest some phrase not of the golden age should escape from his pen. He confesses apologetically to Christopher Longolius (Ep. iii. 63; 402) that it was his habit to extemporize all he wrote, and that this habit was incorrigible; “effundo verius quam scribo omnia.” He complains that much reading of the works of St Jerome had spoiled his Latin; but, as Scaliger says (Scaliga 2a), “Erasmus’s language is better than St Jerome’s.” The same critic, however, thought Erasmus would have done better “if he had kept more closely to the classical models.”

In the annals of classical learning Erasmus may be regarded as constituting an intermediate stage between the humanists of the Latin Renaissance and the learned men of the age of Greek scholarship, between Angelo Poliziano and Joseph Scaliger. Erasmus, though justly styled by Muretus (Varr. Lectt. 7, 15) “eruditus sane vir, ac multae lectionis,” was not a “learned” man in the special sense of the word—not an “érudit.” He was more than this; he was the “man of letters”—the first who had appeared in Europe since the fall of the Roman empire. His acquirements were vast, and they were all brought to bear upon the life of his day. He did not make a study apart of antiquity for its own sake, but used it as an instrument of culture. He did not worship, imitate and reproduce the classics, like the Latin humanists who preceded him; he did not master them and reduce them to a special science, as did the French Hellenists who succeeded him. He edited many authors, it is true, but he had neither the means of forming a text, nor did he attempt to do so. In editing a father, or a classic, he had in view the practical utility of the general reader, not the accuracy required by the gild of scholars. “His Jerome,” says J. Scaliger, “is full of sad blunders” (Scaliga 2a). Even Julien Garnier could discover that Erasmus “falls in his haste into grievous error in his Latin version of St Basil, though his Latinity is superior to that of the other translators” (Pref. in Opp. St. Bas., 1721). It must be remembered that the commercial interests of Froben’s press led to the introduction of Erasmus’s name on many a title page when he had little to do with the book, e.g. the Latin Josephus of 1524 to which Erasmus only contributed one translation of 14 pages; or the Aristotle of 1531, of which Simon Grynaeus was the real editor. Where Erasmus excelled was in prefaces—not philological introductions to each author, but spirited appeals to the interest of the general reader, showing how an ancient book might be made to minister to modern spiritual demands.

Of Erasmus’s works the Greek Testament is the most memorable. It has no title to be considered as a work of learning or scholarship, yet its influence upon opinion was profound and durable. It contributed more to the liberation of the human mind from the thraldom of the clergy than all the uproar and rage of Luther’s many pamphlets. As an edition of the Greek Testament it has no critical value. But it was the first, and it revealed the fact that the Vulgate, the Bible of the church, was not only a second-hand document, but in places an erroneous document. A shock was thus given to the credit of the clergy in the province of literature, equal to that which was given in the province of science by the astronomical discoveries of the 17th century. Even if Erasmus had had at his disposal the MSS. subsidia for forming a text, he had not the critical skill required to use them. He had at hand a few late Basel MSS., one of which he sent straight to press, correcting them in places by collations of others which had been sent to him by Colet in England. In four reprints, 1519, 1522, 1527, 1535, Erasmus gradually weeded out many of the typographical errors of his first edition, but the text remained essentially such as he had first printed it. The Greek text indeed was only a part of his scheme. An important feature of the volume was the new Latin version, the original being placed alongside as a guarantee of the translator’s good faith. This translation, with the justificatory notes which accompanied it, though not itself a work of critical scholarship, became the starting-point of modern exegetical science. Erasmus did nothing to solve the problem, but to him belongs the honour of having first propounded it.

Besides translating and editing the New Testament, Erasmus paraphrased the whole, except the Apocalypse, between 1517 and 1524. The paraphrases were received with great applause, even by those who had little appreciation for Erasmus. In England a translation of them made in 1548 was ordered to be placed in all parish churches beside the Bible. His correspondence is perhaps the part of his works which has the most permanent value; it comprises about 3000 letters, which form an important source for the history of that period. For the same purpose his Colloquia may be consulted. They are a series of dialogues, written first for pupils in the early Paris days as formulae of polite address, but afterwards expanded into lively conversations, in which many of the topics of the day are discussed. Later in the century they were read in schools, and some of Shakespeare’s lines are direct reminiscences of Erasmus.

His complete works have been printed twice; by the Froben firm under the direction of his literary executors (9 vols., Basel, 1540); and by Leclerc at Leiden (11 vols., 1703-1706). For his life the chief contemporary sources are a Compendium vitae written by himself in 1524, and a sketch prefixed by Beatus Rhenanus to the Basel edition of 1540. Of his writings he gives an account in his Catalogus lucubrationum, composed first in January 1523 and enlarged in September 1524; and also in a letter to Hector Boece of Aberdeen, written in 1530. An elaborate bibliography, entitled Bibliotheca Erasmiana, was undertaken by the officials of the Ghent University Library; it is divided into three sections, for Erasmus’s writings, the books he edited, and the literature about him. Listes sommaires were issued in 1893; and since 1897 the completed volumes have been appearing at intervals. There is an excellent sketch of Erasmus’s life down to 1519 in F. Seebohm’s Oxford Reformers (3rd ed., 1887); and of the many biographies those by S. Knight (1726), J. Jortin (2 vols., 1758-1760) and R.B. Drummond (2 vols., 1873) may be mentioned. There are also two volumes (1901-1904) of translations by F.M. Nichols from Erasmus’s letters down to 1517, with an ample commentary which amounts almost to a biography; and an edition of the letters, in Latin, was begun by the Oxford University Press in 1906 (vol. ii., 1910).

(M. P.; P. S. A.)

ERASTUS, THOMAS (1524-1583), German-Swiss theologian, whose surname was Lüber, Lieber, or Liebler, was born of poor parents on the 7th of September 1524, probably at Baden, canton of Aargau, Switzerland. In 1540 he was studying theology at Basel. The plague of 1544 drove him to Bologna and thence to Padua as student of philosophy and medicine. In 1553 he became physician to the count of Henneberg, Saxe-Meiningen, and in 1558 held the same post with the elector-palatine, Otto Heinrich, being at the same time professor of medicine at Heidelberg. His patron’s successor, Frederick III., made him (1559) a privy councillor and member of the church consistory. In theology he followed Zwingli, and at the sacramentarian conferences of Heidelberg (1560) and Maulbronn (1564) he advocated by voice and pen the Zwinglian doctrine of the Lord’s Supper, replying (1565) to the counter arguments of the Lutheran Johann Marbach, of Strassburg. He ineffectually resisted the efforts of the Calvinists, led by Caspar Olevianus, to introduce the Presbyterian polity and discipline, which were established at Heidelberg in 1570, on the Genevan model. One of the first acts of the new church system was to excommunicate Erastus on a charge of Socinianism, founded on his correspondence with Transylvania. The ban was not removed till 1575, Erastus declaring his firm adhesion to the doctrine of the Trinity. His position, however, was uncomfortable, and in 1580 he returned to Basel, where in 1583 he was made professor of ethics. He died on the 31st of December 1583. He published several pieces bearing on medicine, astrology and alchemy, and attacking the system of Paracelsus. His name is permanently associated with a posthumous publication, written in 1568. Its immediate occasion was the disputation at Heidelberg (1568) for the doctorate of theology by George Wither or Withers, an English Puritan (subsequently archdeacon of Colchester), silenced (1565) at Bury St Edmunds by Archbishop Parker. Withers had proposed a disputation against vestments, which the university would not allow; his thesis affirming the excommunicating power of the presbytery was sustained. Hence the treatise of Erastus. It was published (1589) by Giacomo Castelvetri, who had married his widow, with the title Explicatio gravissimae quaestionis utrum excommunicatio, quatenus religionem intelligentes et amplexantes, a sacramentorum usu, propter admissum facinus arcet, mandato nitatur divino, an excogitata sit ab hominibus. The work bears the imprint Pesclavii (i.e. Poschiavo in the Grisons) but was printed by John Wolfe in London, where Castelvetri was staying; the name of the alleged printer is an anagram of Jacobum Castelvetrum. In the Stationers’ Register (June 20, 1589) the printing is said to have been “alowed” by Archbishop Whitgift. It consists of seventy-five Theses, followed by a Confirmatio in six books, and an appendix of letters to Erastus by Bullinger and Gualther, showing that his Theses, written in 1568, had been circulated in manuscript. An English translation of the Theses, with brief life of Erastus (based on Melchior Adam’s account), was issued in 1659, entitled The Nullity of Church Censures; it was reprinted as A Treatise of Excommunication (1682), and, as revised by Robert Lee, D.D., in 1844. The aim of the work is to show, on Scriptural grounds, that sins of professing Christians are to be punished by civil authority, and not by withholding of sacraments on the part of the clergy. In the Westminster Assembly a party holding this view included Selden, Lightfoot, Coleman and Whitelocke, whose speech (1645) is appended to Lee’s version of the Theses; but the opposite view, after much controversy, was carried, Lightfoot alone dissenting. The consequent chapter of the Westminster Confession (“Of Church Censures”) was, however, not ratified by the English parliament. “Erastianism,” as a by-word, is used to denote the doctrine of the supremacy of the state in ecclesiastical causes; but the problem of the relations between church and state is one on which Erastus nowhere enters. What is known as “Erastianism” would be better connected with the name of Grotius. The only direct reply made to the Explicatio was the Tractatus de vera excommunicatione (1590) by Theodore Beza, who found himself rather savagely attacked in the Confirmatio thesium; e.g. “Apostolum et Mosen adeoque Deum ipsum audes corrigere.”

See A. Bonnard, Thomas Éraste et la discipline ecclésiastique (1894); Gass, in Allgemeine deutsche Biog. (1877); G.V. Lechler and R. Stähelin, in A. Hauck’s Realencyklop. für prot. Theol. u. Kirche (1898).

(A. Go.*)

ERATOSTHENES OF ALEXANDRIA (c. 276-c. 194 B.C.), Greek scientific writer, was born at Cyrene. He studied grammar under Callimachus at Alexandria, and philosophy under the Stoic Ariston and the Academic Arcesilaus at Athens. He returned to Alexandria at the summons of Ptolemy III. Euergetes, by whom he was appointed chief librarian in place of Callimachus. He is said to have died of voluntary starvation, being threatened with total blindness. Eratosthenes was one of the most learned men of antiquity, and wrote on a great number of subjects. He was the first to call himself Philologos (in the sense of the “friend of learning”), and the name Pentathlos was bestowed upon him in honour of his varied accomplishments. He was also called Beta as being second in all branches of learning, though not actually first in any. In mathematics he wrote two books On means (Περὶ μεσοτήτων) which are lost, but appear, from a remark of Pappus, to have dealt with “loci with reference to means.” He devised a mechanical construction for two mean proportionals, reproduced by Pappus and Eutocius (Comm. on Archimedes). His κόσκινον or sieve (cribrum Eratosthenis) was a device for discovering all prime numbers. He laid the foundation of mathematical geography in his Geographica, in three books. His greatest achievement was his measurement of the earth. Being informed that at Syene (Assuan), on the day of the summer solstice at noon, a well was lit up through all its depth, so that Syene lay on the tropic, he measured, at the same hour, the zenith distance of the sun at Alexandria. He thus found the distance between Syene and Alexandria (known to be 5000 stadia) to correspond to 1⁄50th of a great circle, and so arrived at 250,000 stadia (which he seems subsequently to have corrected to 252,000) as the circumference of the earth. He is credited by Ptolemy and his commentator Theon with having found the distance between the tropics to be 11⁄83 rds. of the meridian circle, which gives 23° 51’ 20″ for the obliquity of the ecliptic. His astronomical poem Hermes began apparently with the birth and exploits of Hermes, then passed to the legend of his having ordered the heavens, the zones and the stars, and gave a history of the latter. His Erigone, of which a few fragments are also preserved, is sometimes spoken of as a separate poem, but it may have belonged to the Hermes, which appears also to have been known by other names such as Catalogi. The still extant Catasterismi, containing the story of certain stars in prose, is probably not by Eratosthenes.

Eratosthenes was the founder of scientific chronology in his χρονογραφία in which he endeavoured to fix the dates of the chief literary and political events from the conquest of Troy. An important work was his treatise on the old comedy, dealing with theatres and theatrical apparatus generally, and discussing the works of the principal comic poets themselves. Works on moral philosophy, history, and a number of letters were also attributed to him.

There is a complete edition of the fragments of Eratosthenes by Bernhardy (1822); poetical fragments, Hillier (1872); geographical, Seidel (1799) and Berger (1880); καταστερισμοι, Schaubach (1795) and Robert (1878). See Sandys, Hist. Class. Schol. i. (1906).

(T. L. H.)

ERBACH, a town of Germany, in the grand-duchy of Hesse-Darmstadt, on the Mümling, 22 m. S.E. of Darmstadt. It has cloth mills and ivory-turning, for which last branch it possesses a technical school. Wool and cattle fairs are held twice a year. Pop. 2800. The castle contains an interesting collection of weapons and pictures, and in the chapel are the coffins of Einhard, the friend and biographer of Charlemagne, and his wife, Emma.

Erbach has long been the residence of the counts of Erbach, who trace their descent back to the 12th century, and who held the office of cupbearer to the electors palatine of the Rhine until 1806. In 1532 the emperor Charles V. made the county a direct fief of the Empire, on account of the services rendered by Count Eberhard during the Peasants’ War. Since 1717 the family has been divided into the three lines of Erbach-Fürstenau, Erbach-Erbach and Erbach-Schönberg, who rank for precedence, not according to the age of their descent, but according to the age of the chief of their line. In 1818 the counts of Erbach-Erbach inherited the county of Wartenberg-Roth, and in 1903 the count of Erbach-Schönberg was granted the title of prince. The county was mediatized in 1806, and is now incorporated with the duchy of Hesse-Darmstadt.

See Simon, Die Geschichte der Dynasten und Grafen zu Erbach (Frankfort, 1858).

ERBIUM (symbol, Er; atomic weight, 165-166), one of the metals of the rare earths. The first of the rare earth minerals was discovered in 1794 by J. Gadolin and was named gadolinite from its discoverer. In 1797 Ekeberg showed that gadolinite contained another rare earth, which was given the name yttria. Yttria is an exceedingly complex mixture, which has been decomposed, yielding as an intermediate product terbia. This latter substance in its turn has been split by J.L. Soret, P.T. Cleve, Lecoq de Boisbaudran and others into erbia, holmia, thulia and dysprosia, but it is still doubtful whether any one of these four splitting products is a single substance. The rare earth metals are found in the minerals gadolinite, samarskite, fergusonite, euxenite and cerite. They are separated from the minerals by converting them into oxalates, which by ignition give the corresponding oxides. The oxides are then converted into double sulphates which are separated from each other by repeated fractional crystallization or by fractional precipitation with ammonia or some other base. Erbium forms rose-coloured salts and a rose-coloured oxide. The oxide dissolves slowly in acids; it is not reduced by hydrogen and is infusible. The salts show a characteristic absorption spectrum.

See J.F. Bahr and R. Bunsen (Ann., 1866, 137, p. 1); A. v. Welsbach (Monats., 1883, 4, p. 641; 1884, 5, p. 508; 1885, 6, p. 477); P.T. Cleve (Comptes rendus, 1879, 89, p. 478; 1880, 91, pp. 328, 381; 1882, 95, p. 1225; Bull. de la soc. chim., 1874, 21, p. 196; 1883, 39, p. 287); C. Marignac (Ann. Chim. phys., 1849 27, p. 226); B. Brauner (Monats., 1882, 3, p. 13); W. Crookes (Proc. Roy. Soc., 1886, 40, p. 502); Lecoq de Boisbaudran (Comptes rendus, 1886, 102, p. 1005); A. Bettendorf (Ann., 1892, 270, p. 376); M. Muthmann (Ber., 1898, 31, p. 1718; 1900, 33, p. 42); G. Krüss (Zeit. f. anorg. Chem., 1893, 3, p. 108).

ERCILLA Y ZÚNIGA, ALONSO DE (1533-1595), Spanish soldier and poet, was born in Madrid on the 7th of August 1533. In 1548 he was appointed page to the heir-apparent, afterwards Philip II. In this capacity Ercilla visited Italy, Germany and the Netherlands, and was present in 1554 at the marriage of his master to Mary of England. Hearing that an expedition was preparing to subdue the Araucanians of Chile, he joined the adventurers. He distinguished himself in the ensuing campaign; but, having quarrelled with a comrade, he was condemned to death in 1558 by his general, Garcia Hurtado de Mendoza. The sentence was commuted to imprisonment, but Ercilla was speedily released and fought at the battle of Quipeo (14th of December 1558). He returned to Spain in 1562, visited Italy, France, Germany, Bohemia, and in 1570 married Maria de Bazán, a lady distantly connected with the Santa Cruz family; in 1571 he was made knight of the order of Santiago, and in 1578 he was employed by Philip II. on a mission to Saragossa. He complained of living in poverty but left a modest fortune, and was obviously disappointed at not being offered the post of secretary of state. His principal work is La Araucana, a poem based on the events of the wars in which he had been engaged. It consists of three parts, of which the first, composed in Chile and published in 1569, is a versified narrative adhering strictly to historic fact; the second, published in 1578, is encumbered with visions and other romantic machinery; and the third, which appeared in 1589-1590, contains, in addition to the subject proper, a variety of episodes mostly irrelevant. This so-called epic lacks symmetry, and has been over-praised by Cervantes and Voltaire; but it is written in excellent Spanish, and is full of vivid rhetorical passages. An analysis of the poem was given by Hayley in his Essay on Epic Poetry (1782).

A good biography precedes the Morceaux choisis (Paris, 1900) by Jean Ducamin.

ERCKMANN-CHATRIAN, the joint names of two French writers whose collaboration made their work that of, so to speak, one personality. Émile Erckmann (1822-1899) was born on the 20th of May 1822 at Phalsbourg, and Louis Gratien Charles Alexandre Chatrian (1826-1890) on the 18th of December 1826 at Soldatenthal, Lorraine. In 1847 they began to write together, and continued doing so till 1889. Chatrian died in 1890 at Villemomble near Paris, and Erckmann at Lunéville in 1899. The list of their publications is a long one, ranging from the Histoires et contes fantastiques (1849; reprinted from the Démocrate du Rhin), L’Illustre Docteur Mathéus (1859), Madame Thérèse (1863), L’Ami Fritz (1864), Histoire d’un conscrit de 1813 (1864), Waterloo (1865), Le Blocus (1867), Histoire d’un paysan (4 vols., 1868-1870), L’Histoire du plébiscite (1872), to Le Grand-père Lebigue (1880); besides dramas like Le Juif polonais (1869) and Les Rantzau (1882). Without any special literary claim, their stories are distinguished by simplicity and genuine descriptive power, particularly in the battle scenes and in connexion with Alsatian peasant life. They are marked by a genuine democratic spirit, and by real patriotism, which developed after 1870 into hatred of the Germans. The authors attacked militarism by depicting the horrors of war in the plainest terms.

See also J. Claretie, Erckmann-Chatrian (1883), in the series of “Célébrités contemporaines.”

ERDÉLYI, JÁNOS (1814-1868), Hungarian poet and author, was born in 1814 at Kapos, in the county of Ungvár, and educated at the Protestant college of Sárospatak. In 1833 he removed to Pest, where he was, in 1839, elected member of the Hungarian Academy of Sciences. His literary fame was made by his collection of Hungarian national poems and folk-tales, Magyar Népköltési Gyüjtemény, Népdalok és Mondák (Pest, 1846-1847). This work, published by the Kisfaludy Society, was supplemented by a dissertation upon Hungarian national poetry, afterwards partially translated into German by Stier (Berlin, 1851). Erdélyi also compiled for the Kisfaludy Society an extensive collection of Hungarian proverbs—Magyar Közmondások könyve (Pest, 1851),—and was for some time editor of the Szépirodalmi Szemle (Review of Polite Literature). In 1848 he was appointed director of the national theatre at Pest; but after 1849 he resided at his native town. He died on the 23rd of January 1868. A collection of folklore was published the year after his death, entitled A Nép Koltészete népdalok, népmesék és közmondások (Pest, 1869). This work contains 300 national songs, 19 folk-tales and 7362 Hungarian proverbs.

ERDMANN, JOHANN EDUARD (1805-1892), German philosophical writer, was born at Wolmar in Livonia on the 13th of June 1805. He studied theology at Dorpat and afterwards at Berlin, where he fell under the influence of Hegel. From 1829 to 1832 he was a minister of religion in his native town. Afterwards he devoted himself to philosophy, and qualified in that subject at Berlin in 1834. In 1836 he was professor-extraordinary at Halle, became full professor in 1839, and died there on the 12th of June 1892. He published many philosophical text-books and treatises, and a number of sermons; but his chief claim to remembrance rests on his elaborate Grundriss der Geschichte der Philosophie (2 vols., 1866), the 3rd edition of which has been translated into English. Erdmann’s special merit is that he does not rest content with being a mere summarizer of opinions, but tries to exhibit the history of human thought as a continuous and ever-developing effort to solve the great speculative problems with which man has been confronted in all ages. His chief other works were: Leib und Seele (1837), Grundriss der Psychologie (1840), Grundriss der Logik und Metaphysik (1841), and Psychologische Briefe (1851).

ERDMANN, OTTO LINNÉ (1804-1869), German chemist, son of Karl Gottfried Erdmann (1774-1835), the physician who introduced vaccination into Saxony, was born at Dresden on the 11th of April 1804. In 1820 he began to attend the medico-chirurgical academy of his native place, and in 1822 he entered the university of Leipzig where in 1827 he became extraordinary professor, and in 1830 ordinary professor of chemistry. This office he held until his death, which happened at Leipzig on the 9th of October 1869. He was particularly successful as a teacher, and the laboratory established at Leipzig under his direction in 1843 was long regarded as a model institution. As an investigator he is best known for his work on nickel and indigo and other dye-stuffs. With R.F. Marchand (1813-1850) he also carried out a number of determinations of atomic weights. In 1828, in conjunction with A.F.G. Werther (1815-1869), he founded the Journal für technische und ökonomische Chemie, which became in 1834 the Journal für praktische Chemie. He was also the author of Über das Nickel (1827), Lehrbuch der Chemie (1828), Grundriss der Waarenkunde (1833), and Über das Studium der Chemie (1861).

EREBUS, in Greek mythology, son (according to Hesiod, Theog. 123) of Chaos, and father of Aether (upper air) and Hemera (day) by his sister Nyx (night). The word, which signifies darkness, is in Homer the gloomy subterranean region through which the departed shades pass into Hades. The entrance to it was in the extreme west, on the borders of Ocean, in the mythical land of the Cimmerians. It is to be distinguished from Tartarus, the place of punishment for the wicked.

ERECH (Uruk in the Babylonian inscriptions; Gr. Orchoë), the Biblical name of an ancient city of Babylonia, situated E. of the present bed of the Euphrates, on the line of the ancient Nil canal, in a region of marshes, about 140 m. S.S.E. from Bagdad. It was one of the oldest and most important cities of Babylonia, and the site of a famous temple, called E-Anna, dedicated to the worship of Nana, or Ishtar. Erech played a very important part in the political history of the country from an early time, exercising hegemony in Babylonia at a period before the time of Sargon. Later it was prominent in the national struggles of the Babylonians against Elam (2000 B.C. and earlier), in which it suffered severely; recollections of these conflicts are embodied in the Gilgamesh epic, as it has come down to us through the library of Assur-bani-pal. Erech enjoyed much distinction in the later times, as a seat of learning and of the worship of Ishtar, and Assur-bani-pal drew largely on its literary stores for his library at Nineveh, from which we derive our principal information concerning ancient Babylonian literature. The inscriptions found here show that it continued in existence through the Persian and Seleucid periods. The ruins of the ancient site, known as Warka, which are among the largest in all Babylonia, forming an irregular circle nearly 6 m. in circumference, bounded by a wall, still standing in some places to the height of 40 ft., were explored and partially excavated by W.K. Loftus in 1850 and 1854. The most conspicuous ruin, now called Abu-Berdi, “Father of Marsh Grass,” or Buwariye, “reed matting,” because of the layers of reeds between each twelve courses of unbaked brick, is the ziggurat (tower) of the ancient temple of E-Anna. It is about 100 ft. in height, and strikingly resembles in general appearance the ruins of the ziggurat of the temple of Enlil at Nippur. Second to this in size was the ruin called Wuswas, a walled quadrangle, including an area of more than seven and a half acres, within which was an edifice 246 ft. long and 174 ft. wide, elevated on an artificial platform 50 ft. in height. The south-west façade, still standing in some places to the height of 23 ft., exhibited an interesting use of half columns, and stepped recesses for purposes of decoration. In another ruin Loftus found a wall, 30 ft. long, composed entirely of small yellow terra-cotta nail-headed cones, such as have been discovered in great numbers, inscribed and uninscribed, used for votive purposes in connexion with walls at Tello and elsewhere in Babylonia. His excavations being superficial, the Babylonian inscriptions found by him, about one hundred in all, exclusive of the ancient Ur-Gur bricks from the temple, belong in general to the neo-Babylonian, Persian and Seleucid periods. The older remains are buried deep beneath the huge mass of later debris. Loftus also discovered at Erech, almost everywhere within and without the walls, great numbers of clay coffins, piled one above another, to the height of over 30 ft., forming a vast and, on the whole, well-ordered cemetery belonging to the Persian, Parthian and later occupations of Babylonia, during which period Erech, like other cities of the south, evidently became a necropolis for a large extent of country. After Loftus’s time the mounds were visited by various travellers, but no further excavations have been conducted. Work on this important part of the site is attended with very great difficulties, owing to the inaccessible position of the ruins, the unsettled character of the country, the frequent sand-storms, and above all, the immense mass of material of later periods which must be removed before a systematic excavation of the more ancient and interesting ruins could be undertaken. A curious feature of the Warka neighbourhood is the existence of conical sand-hills, rising to a considerable height, so compact as to be almost like stone. These hills extend from Warka northward as far as Tel Ede.

See W.K. Loftus, Chaldaea and Susiana (1857); J.P. Peters, Nippur (1897); E. Sachau, Am Euphrat und Tigris (1900). Cf. also Nippur and authorities there quoted.

(J. P. Pe.)

ERECHTHEUM, a temple (commonly called after Erechtheus, to whom a portion of it was dedicated) on the acropolis at Athens, unique in plan, and in its execution the most refined example of the Ionic order. There is no clear evidence as to when the building was begun, some placing it among the temples projected by Pericles, others assigning it to the time after the peace of Nicias in 421 B.C. The work was interrupted by the stress of the Peloponnesian War, but in 409 B.C. a commission was appointed to make a report on the state of the building and to undertake its completion, which was carried out in the following year.

The peculiar plan of the Erechtheum has given rise to much speculation. It may be due partly to the natural conformation of the rock and the differences of level, partly to the necessity of enclosing within a single building several objects of ancient sanctity, such as the mark of Poseidon’s trident and the spring that arose from it, the sacred olive tree of Athena, and the tomb of Cecrops. But there are some features which cannot be so explained, and which have led Professor W. Dörpfeld and others to believe that the plan, as we now have it, is a modification or abridgment of the original design, due to the same conservative influences as led to the curtailment of the plan of the Propylaea (q.v.).

The building as completed consisted of a temple of the ordinary type, opening by a door and two windows to the east front, before which stood a portico of six Ionic columns. This part was the temple of Athena Polias. Adjoining it on the west was the central chamber, on a lower level; this chamber was separated by a partition, originally of wood and later of marble, from the western compartment of the temple, which was of peculiar construction. The west end was formed by a wall, on which stood four columns between antae; but the main entrance to this western compartment was through a large and very ornate doorway on the north; and a large Ionic portico, consisting of four columns in the front, and one in the return on each side, was placed in front of this door. At the south end of the western compartment was a smaller door, with steps leading up to the higher level, within a projecting space enclosed by a low wall and covered with a projecting porch carried by six “maidens” or caryatides. The construction of the building at this south-western corner shows that there was some sacred object that had to be bridged over by a huge block of marble; this we know from inscriptions to have been the Cecropeum or tomb of Cecrops. In the north portico a square hole in the floor, with a corresponding hole in the roof above it, must have given access to another sacred object, the mark of Poseidon’s trident in the rock. The sacred olive tree probably stood just outside the temple to the west in the Pandroseion. The Ionic order, as used in this temple, is of the most ornate Attic type. The bases of the columns are either reeded or decorated with a plait-pattern; the capital has the broad channel between the volutes subdivided by a carefully-profiled incision; and the top of the shafts is ornamented by a broad band of palmette or honeysuckle pattern. A similar band of ornament runs round the top of the walls outside, and at their base is a reeded torus. The frieze consisted of white marble figures in relief, affixed to a background of black Eleusinian stone.

The contents of the Erechtheum are described by Pausanias. It contained the ancient image of Athena Polias, and three altars, one to Poseidon and Erechtheus, one to Butes and one to Hephaestus; there were portraits of the family of the Butadae on the walls. Within it was also the gold lamp of Callimachus, which burnt for a year without refilling, and had a chimney in the form of a palm-tree.

The Erechtheum was damaged by a fire, soon after its completion, in 406 B.C., but was repaired early in the following century. The west end appears to have been damaged in Roman times and to have been replaced by the attached columns with windows between them which appear in old drawings and are still partially extant. It was used as a church in Christian times, and under Turkish rule as the harem of the governor of Athens. Lord Elgin carried off to London, about 1801-1803, one of the columns of the east portico and one of the caryatides; these were replaced later by terra-cotta casts. During the siege of the Acropolis in 1827, the roof of the north portico was thrown down and the building was otherwise much damaged. It was partially rebuilt between 1838 and 1846; the west front was blown down in a storm in 1852. Since 1900 the project of rebuilding the Erechtheum as far as possible with the original blocks has again been undertaken.

See Stuart, Antiquities of Athens; Inwood, The Erechtheum; H. Forster in Papers of American School at Athens, i. (1882-1883); J.H. Middleton, Plans and Drawings of Athenian Buildings (1900), pls. xiv.-xxii.; E.A. Gardner, Ancient Athens, chap. viii.; W. Dörpfeld, “Der ursprungliche Plan des Erechtheion” in Mitteil. Athen., 1904, p. 101, taf. 6; G.P. Stevens, “The East Wall of the Erechtheum,” in American Journ. Arch., 1906, pls. vi.-ix.

(E. Gr.)

ERECHTHEUS, in Greek legend, a mythical king of Athens, originally identified with Erichthonius, but in later times distinguished from him. According to Homer, who knows nothing of Erichthonius, he was the son of Aroura (Earth), brought up by Athena, with whom his story is closely connected. In the later story, Erichthonius (son of Hephaestus and Atthis or Athena herself) was handed over by Athena to the three daughters of Cecrops—Aglauros (or Agraulos), Herse and Pandrosos—in a chest, which they were forbidden to open. Aglauros and Herse disobeyed the injunction, and when they saw the child (which had the form of a snake, or round which a snake was coiled) they went mad with fright, and threw themselves from the rock of the Acropolis (or were killed by the snake). Athena herself then undertook the care of Erichthonius, who, when he grew up, drove out Amphictyon and took possession of the kingdom of Athens. Here he established the worship of Athena, instituted the Panathenaea, and built an Erechtheum. The Erechtheus of later times was supposed to be the grandson of Erechtheus-Erichthonius, and was also king of Athens. When Athens was attacked by the Thracian Eumolpus (or by the Eleusinians assisted by Eumolpus) victory was promised Erechtheus if he sacrificed one of his daughters. Eumolpus was slain and Erechtheus was victorious, but was himself killed by Poseidon, the father of Eumolpus, or by a thunderbolt from Zeus. The contest between Erechtheus and Eumolpus formed the subject of a lost tragedy by Euripides; Swinburne has utilized the legend in his Erechtheus. The scene of the opening of the chest is represented on a Greek vase in the British Museum. The name Erichthonius is connected with χθών (“earth”) and the representation of him as half-snake, like Cecrops, indicates that he was regarded as one of the autochthones, the ancestors of the Athenians who sprung from the soil.

See Apollodorus iii. 14. 15; Euripides, Ion; Ovid, Metam. ii. 553; Hyginus, Poët. astron. ii. 13; Pausanias i. 2. 5. 8; E. Ermatinger, Die attische Autochthonensage (1897); article by J.A. Hild in Daremberg and Saglio’s Dictionnaire des antiquités; B. Powell in Cornell Studies, xvii. (1906), who identifies Erechtheus, Erichthonius, Poseidon and Cecrops, all denoting the sacred serpent of Athena, whose cult she first contested, but then amalgamated with her own. The birth of Erichthonius (as a corn-spirit) is interpreted by Mannhardt as a mythical way of describing the growth of the corn, and by J.E. Harrison (Myths and Monuments of Ancient Athens, xxvii.-xxxvi.) as a fiction to explain the ceremony performed by the two maidens called Arrephori. See also Farnell, Cults of the Greek States, i. 270; and Frazer’s Pausanias, ii. 169.

ERESHKIGAL, also known as Allatu, the name of the chief Babylonian goddess of the nether-world where the dead are gathered. Her name signifies “lady of the nether-world.” She is known to us chiefly through two myths, both symbolizing the change of seasons, but intended also to illustrate certain doctrines developed in the temple-schools of Babylonia. One of these myths is the famous story of Ishtar’s descent to Irkalla or Arālu, as the lower world was called, and her reception by her sister who presides over it; the other is the story of Nergal’s offence against Ereshkigal, his banishment to the kingdom controlled by the goddess and the reconciliation between Nergal and Ereshkigal through the latter’s offer to have Nergal share the honours of the rule over Irkalla. The story of Ishtar’s descent is told to illustrate the possibility of an escape from Irkalla, while the other myth is intended to reconcile the existence of two rulers of Irkalla—a goddess and a god.

It is evident that it was originally a goddess who was supposed to be in control of Irkalla, corresponding to Ishtar in control of fertility and vegetation on earth. Ereshkigal is therefore the sister of Ishtar and from one point of view her counterpart, the symbol of nature during the non-productive season of the year. As the doctrine of two kingdoms, one of this world and one of the world of the dead, becomes crystallized, the dominions of the two sisters are sharply differentiated from one another. The addition of Nergal represents the harmonizing tendency to unite with Ereshkigal as the queen of the nether-world the god who, in his character as god of war and of pestilence, conveys the living to Irkalla and thus becomes the one who presides over the dead.

(M. Ja.)

ERETRIA (mod. Aletria), an ancient coast town of Euboea about 15 m. S.E. of Chalcis, opposite to Oropus. Eretria, like its neighbour Chalcis (q.v.), early entered upon a commercial and colonizing career. Besides founding townships in the west and north of Greece, it acquired dependencies among the Cyclades and joined the great mercantile alliance of Miletus and Aegina. Since the so-called Lelantine War (7th century B.C.) against the coming league of Chalcis, it began to be overshadowed by its rivals. The interference of Eretria in the Ionian revolt (498) brought upon it the vengeance of the Persians, who captured and destroyed it shortly before the battle of Marathon (490). The city was soon rebuilt, and as a member of both the Delian Leagues attached itself by numerous treaties to the Athenians. The latter, through their general Phocion, rescued it from the tyrants suborned by Philip of Macedon (354 and 341). Under Macedonian and Roman rule Eretria fell into insignificance; for a short period under Mark Antony, the triumvir, it became a possession of Athens. Eretria was the birthplace of the tragedian Achaeus and of the “Megarian” philosopher Menedemus.

The modern village, which is sometimes called Nea Psará because the inhabitants of Psará were transferred there in 1821, is on unhealthy low-lying ground near the sea. The excavation of the site was carried out by the American School of Athens (1890-1895). At the foot of the Acropolis Hill, where the ground begins to rise, the theatre lies; and though the material of which this was built is rough, and only seven imperfect rows of seats remain, a good part of the scena and of the chambers behind it is preserved, and beneath these there runs a tunnel, which, together with other peculiar features, has raised interesting questions in connexion with the arrangement of the Greek theatre, the orchestra being at present on a level about 12 ft. below that of the rooms in the scena. Near by are the substructions of a temple of Dionysus and a large altar, and also a gymnasium with arrangements for bathing. Besides these, in 1900 the substructions of a temple of Apollo Daphnephoros were unearthed. Both the northern and the southern side of the hill are flanked by walls, which seem to have reached the sea, where there was a mole and a harbour; and the wall of the acropolis itself remains in one part to the height of eight courses.

Authorities.—Strabo x. 447 f.; Herodotus v. 99, vi. 101; Corpus Inscr. Atticarum, i. 339, iv. (2), pp. 5, 10, 22; H. Heinze, De rebus Eretriensium (Göttingen, 1869); W.M. Leake, Travels in Northern Greece (London, 1835), ii. 266, 443; B.V. Head, Historia numorum (Oxford, 1887), pp. 305-308; Papers of the American School at Athens, vol. vi.

(E. Gr.)

ERETRIAN SCHOOL OF PHILOSOPHY. This Greek school was the continuation of the Elian school, which was transferred to Eretria by Menedemus. It was of small importance, and in the absence of certain knowledge must be supposed to have adhered to the doctrines of Socrates. (See Menedemus.)

ERFURT, a city of Germany, in Prussian Saxony, on the Gera, and the railway Halle-Bebra, about midway between Gotha and Weimar, which are 14 m. distant. Pop. (1875) 48,025; (1905) 100,065. The city, which is dominated on the west by the two citadels of Petersberg and Cyriaxburg, is irregularly built, the only feature in its plan, or want of plan, being the Friedrich Wilhelmsplatz, a broad open space of irregular shape abutting on the Petersberg. On the south-western side of this square, which contains a monument to the elector Frederick Charles Joseph of Mainz (1719-1802), is the Domberg, an eminence on which stand, side by side, the cathedral and the great church of St Severus with its three spires (14th century). The churches are approached by a flight of forty-eight stone steps, the grouping of the whole mass of buildings being exceedingly impressive. The cathedral (Beatae Mariae Virginis) is one of the finest churches in Germany. It was begun in the 12th century, but the nave was rebuilt in the 13th in the Gothic style. The magnificent chancel (1349-1372), with the 14th-century crypt below, rests on massive substructures, known as the Cavate. The twin towers are set between the chancel and nave. The cathedral contains, besides fine 15th-century glass, some very rich portal sculptures and bronze castings, among others the coronation of the Virgin by Peter Vischer. In one of its towers is the famous bell, called Maria Gloriosa, which bears the date 1497, and weighs 270 cwt. Besides the cathedral and St Severus, which are Roman Catholic, Erfurt possesses several very interesting medieval churches, now Evangelical. Among these may be mentioned the Predigerkirche, dating from the latter half of the 12th century; the Reglerkirche, a Romanesque building (restored in 1859) with a 12th-century tower; and the Barfüsserkirche, a Gothic building containing fine 14th-century monuments. All these were originally monastic churches. Of the former religious houses there survive a Franciscan convent, with a girls’ school attached, and an Ursuline convent. The Augustinian monastery, in which Luther lived as a friar, is now used as an orphanage, under the name of the Martinsstift. The cell of Luther was destroyed by fire in 1872. A bronze statue of the reformer was erected in the Anger, the chief street of the town, in 1890. At one time Erfurt had a university, of which the charter dated from 1392; but it was suppressed in 1816, and its funds devoted to other purposes, among these being the endowment of an institution founded in 1758 and now called the royal academy of sciences, and the support of the royal library, which now contains 60,000 volumes and over 1000 manuscripts. On the W. and S.W. extensive new quarters have grown up within recent years, e.g. Hirschbrühl. The interior of the town hall (1869-1875) is adorned with legendary and historical frescoes by Kämpfer and Peter Janssen. Erfurt possesses also a picture gallery and an antiquarian collection.

The educational establishments of the town include a gymnasium, a realgymnasium, a realschule, technical schools for building and handicrafts, a high-class commercial school, a school of agriculture, and an academy of music. The most notable industry of Erfurt is the culture of flowers and of vegetables, which is very extensively carried on. This industry had its origin in the large gardens attached to the monasteries. It has also important and growing manufactures of ladies’ mantles, boots and shoes, machines, furniture, woollen goods, musical instruments, agricultural machinery and implements, leather, tobacco, chemicals, &c. Brewing, bleaching and dyeing are also carried on on a large scale, and there are extensive railway works and a government rifle factory.

Erfurt (Med. Erpesfurt, Erphorde, Lat. Erfordia) is a town of great antiquity. Its origin is obscure, but in 741 it was sufficiently important for St Boniface to found a bishopric here, which was, however, after the martyrdom of the first bishop, Adolar, in 755, reabsorbed in that of Mainz. In 805 the place received certain market rights from the emperor Charlemagne. Later the overlordship was claimed by the archbishops of Mainz, on the strength of charters granted by the emperor Otto I., and their authority in Erfurt was maintained by a burgrave and an advocatus, the office of the latter becoming in the 12th century hereditary in the family of the counts of Gleichen. In spite of many vicissitudes (from 1109 to 1137, for instance, the town was subject to the landgraves of Thuringia), and of a charter granted in 1242 by the emperor Frederick II., the archbishops succeeded in upholding their claims. In 1255, however, Archbishop Gerhard I. had to grant the city municipal rights, the burgraviate disappeared, and Erfurt became practically a free town. Its power was at its height early in the 15th century, when it joined the Hanseatic League. It had acquired by force or purchase various countships and other fiefs in the neighbourhood, and ruled a considerable territory; and its wealth was so great that in 1378 it established a university, the first in Europe that embraced the four faculties. By the end of the century, however, its prosperity had sunk owing to the perpetual feud with Mainz, the internecine war in Saxony, and the consequent dwindling of trade. By the convention of Amorbach in 1483 the overlordship of Erfurt was ultimately transferred by the electors of Mainz to Saxony. The political and religious quarrels of the 16th century still further depressed the city, in which the reformed religion was established in 1521. Then came the Thirty Years’ War, during which Erfurt was for a while occupied by the Swedes. After the peace of Westphalia (1648) the city was assigned by the emperor to the elector of Mainz, and, on its refusal to submit, it was placed under the ban of the Empire (1660). In 1664 it was captured by the troops of the archbishop of Mainz, and remained in the possession of the electorate till 1802, when it came into the possession of Prussia. In 1808 it was the scene of the memorable interview between Napoleon and the emperor Alexander I. of Russia, at which the kings of Bavaria, Saxony, Westphalia and Württemberg also assisted, which is known as the congress of Erfurt. Here in 1850 the parliament of the short-lived Prussian Northern Union (known as the Erfurt parliament) held its sittings. In 1902 the 100th anniversary of the city’s incorporation with Prussia was celebrated.

See W.J.A. von Tettau, Erfurt in seiner Vergangenheit und Gegenwart (Erfurt, 1880); C. Beyer, Geschichte der Stadt Erfurt (Erfurt, 1900); and F.W. Kampschulte, Die Universität Erfurt in ihrem Verhältnisse zu dem Humanismus und der Reformation (1856-1858). For a detailed bibliography see U. Chevalier, Répertoire des sources. Topo-bibliographie (Montebéliard, 1894-1899), s.v.

ERGOT, or Spurred Rye, the drug ergota or Secale cornutum (Ger. Mutterkorn; Fr. seigle ergoté), consisting of the sclerotium (or hard resting condition) of a fungus, Claviceps purpurea, parasitic on the pistils of many members of the Grass family, but obtained almost exclusively from rye, Secale cereale. In the ear of rye that is infected with ergot a species of fermentation takes place, and there exudes from it a sweet yellowish mucus, which after a time disappears. The ear loses its starch, and ceases to grow, and its ovaries become penetrated with the white spongy tissue of the mycelium of the fungus which towards the end of the season forms the sclerotium, in which state the fungus lies dormant through the winter.

The drug consists of grains, usually curved (hence the name, from the O. Fr. argot, a cock’s spur), which are violet-black or dark-purple externally, and whitish with a tinge of pink within, are between 1⁄3 and 1½ in. long, and from 1 to 4 lines broad, and have two lateral furrows, a close fracture, a disagreeable rancid taste, and a faint, fishy odour, which last becomes more perceptible when the powder of the drug is mixed with potash solution. Ergot should be kept in stoppered bottles in order to preserve it from the attacks of a species of mite, and to prevent the oxidation of its fatty oil.

The extremely complex composition of this drug has been studied in great detail, and with such important results that instead of giving ergot itself by the mouth in doses of 20 to 60 grains, it is now possible to obtain much more rapid and certain results by giving one three-hundredth of a grain of one of its constituents hypodermically. This constituent is the alkaloid cornutine, which is the valuable ingredient of the drug. Other ingredients are a fixed oil, present to the extent of 30%, ergotinic acid, a glucoside, trimethylamine, which gives the drug its unpleasant odour, and sphacelinic acid, a non-nitrogenous resinoid body. Of the numerous preparations only two need be mentioned—the liquid extract (dose 10 minims to 2 drachms or more), and the hypodermic injection. The latter does not keep well, and the best way of using ergot is to dissolve tablets obtained from a reputable maker, and containing some of the active principles, in pure water, the solution being injected subcutaneously.

Ergot has no external action. Given internally it stimulates the intestinal muscles and may cause diarrhoea. After absorption it slows the pulse by stimulation of the vagus nerves. It has indeed been asserted that the slow pulse characteristic of the puerperal period is really due to the common administration of ergot at that time. This is probably an exaggeration. The important actions of ergot are on the blood-vessels and the uterus. The drug greatly raises the blood-pressure by causing extreme contraction of the arteries. This is mainly due to a direct action on the muscular coats of the vessels, but is also partly of central origin, since the drug also stimulates the vaso-motor centre in the medulla oblongata. This action on the vessels is so marked as to constitute the drug a haemostatic, not only locally but also remotely. It may arrest bleeding from the nose, for instance, when injected hypodermically. Nearly all the constituents share in causing this action, but the sphacelinic acid is probably the most potent. Ergot is the most powerful known stimulant of the pregnant uterus. The action is a double one. At least four of its constituents act directly on the muscular fibre of the uterus, whilst the cornutine acts through the nerves. Of great practical importance is the fact that the cornutine causes rhythmic contractions such as naturally occur, whilst the sphacelinic acid produces a tonic contraction of the uterus, which is unnatural and highly inimical to the life of the foetus. Ergot is used in therapeutics as a haemostatic, and is very valuable in haemoptysis and sometimes in haematemesis. But its great use is in obstetrics. The drug should regularly be given hypodermically, and it is important to note that if the injection be made immediately under the skin, an abscess, or considerable discomfort, may ensue. The injection should be intra-muscular, the needle being boldly plunged into a muscular mass, such as that of the deltoid or the gluteal region. The indications for the use of ergot in obstetrics are highly complex and demand detailed treatment. It can only be said here that the drug should only in the rarest possible cases be given whilst the child is still in utero. This rule is necessitated by the sphacelinic acid, which causes an unnatural state of the organ. When it is possible to obtain pure cornutine, which is unfortunately very expensive, the precautions necessary in other cases may be abrogated.

Chronic poisoning, or ergotism, used frequently to occur amongst the poor fed on rye infected with the Claviceps. As it is practically impossible to reproduce the symptoms of ergotism nowadays, whether experimentally in the lower animals, or when the drug is being administered to a human being for some therapeutic purpose, it is believed that the symptoms of ergotism were rendered possible only by the semi-starvation which must have ensued from the use of such rye-bread; for the grain disappears as the fungus develops. There were two types of ergotism. In the gangrenous form various parts of the body underwent gangrene as a consequence of the arrest of blood-supply produced by the action of sphacelinic acid on the arteries. In the spasmodic form the symptoms were of a nervous character. The initial indications of the disease were cutaneous itching, tingling and formication, which gave place to actual loss of cutaneous sensation, first observed in the extremities. Amblyopia and some loss of hearing also occurred, as well as mental failure. With weakness of the voluntary muscles went intermittent spasms which weakened the patient and ultimately led to death by implication of the respiratory muscles. The last-known “epidemic” of ergotism occurred in Lorraine and Burgundy in the year 1816.

ERIC XIV. (1533-1577), king of Sweden, was the only son of Gustavus Vasa and Catherine of Saxe-Lauenburg. The news of his father’s death reached Eric as he was on the point of embarking for England to press in person his suit for the hand of Queen Elizabeth. He hastened back to Stockholm, after burying his father, summoned a Riksdag, which met at Arboga on the 15th of April 1561, and adopted the royal propositions known as the Arboga articles, considerably curtailing the authority of the royal dukes, John and Charles, in their respective provinces. Two months later Eric was crowned at Upsala, on which occasion he first introduced the titles of baron and count into Sweden, by way of attaching to the crown the higher nobility, these new counts and barons receiving lucrative fiefs adequate to the maintenance of their new dignities.

From the very beginning of his reign Eric’s morbid fear of the upper classes drove him to give his absolute confidence to a man of base origin and bad character, though, it must be admitted, of superior ability. This was Göran Persson, born about 1530, who had been educated abroad in Lutheran principles, and after narrowly escaping hanging at the hands of Gustavus Vasa for some vile action entered the service of his son. This powerful upstart was the natural enemy of the nobility, who suffered much at his hands, though it is very difficult to determine whether the initiative in these prosecutions proceeded from him or his master. Göran was also a determined opponent of Duke John, with whom Eric in 1563 openly quarrelled, because John, contrary to the royal orders, had married (Oct. 4, 1562) Catherine, daughter of Sigismund I. of Poland, engaging at the same time to assist the Polish king to conquer Livonia. This act was a flagrant breach of that paragraph of the Arboga articles which forbade the royal dukes to contract any political treaty without the royal assent. An army of 10,000 men was immediately sent by Eric to John’s duchy of Finland, and John and his consort were seized, brought over to Sweden and detained as prisoners of state in Gripsholm Castle. But Eric did not stop here. His suspicion suggested to him that, if his own brother failed him, the loyalty of the great nobles, especially the members of the ancient Sture family, who had been notable in Sweden when the Vasas were unknown, could not be depended upon. The head of the Sture family at this time was Count Svante, who had married a sister of Gustavus Vasa’s second wife, and had by her a numerous family, of whom two sons, Nils and Eric, still survived. The dark tragedy, known as the Sture murders, began with Eric XIV.’s strange treatment of young Count Nils. In 1566 he was summoned before a newly erected tribunal and condemned to death for gross neglect of duty, though not one of the frivolous charges brought against him could be substantiated. The death penalty was commuted into a punishment worse because more shameful than death. On the 15th of June 1566 the unfortunate youth, bruised and bleeding from shocking ill-treatment, was placed upon a wretched hack, with a crown of straw on his head, and led in derision through the streets of Stockholm. The following night he was sent a prisoner to the fortress of Örbyhus. A few days later he was appointed ambassador extraordinary, and despatched to Lorraine to resume the negotiations for Eric’s marriage with the princess Renata. Before he returned, however, Eric had resolved to marry Karin, or Kitty Månsdatter, the daughter of a common soldier, who had been his mistress since 1565. In January 1567 Eric extorted a declaration from two of his senators that they would assist him to punish all who should try to prevent his projected marriage; and, in the middle of May, a Riksdag was summoned to Upsala to judge between the king and those of the aristocracy whom he regarded as his personal enemies. Eric himself arrived at Upsala on the 16th in a condition of incipient insanity. On the 19th he opened parliament in a speech which, as he explained, he had to deliver extempore owing to “the treachery” of his secretary. Two days later Nils Sture arrived at Upsala fresh from his embassy to Lorraine, and was at once thrown into prison, where other members of the nobility were already detained. On the following day Eric murdered Nils in his cell with his own hand, and by his order the other prisoners were despatched by the royal provost marshal forthwith. These murders were committed so promptly and secretly that it is doubtful whether the estates, actually in session at the same place, knew what had been done when, on the 26th of May, under violent pressure from Göran Persson, they signed a document declaring that all the accused gentlemen under detention had acted like traitors, and confirming all sentences already passed or that might be passed upon them.

During the greater part of 1567 Eric was so deranged that a committee of senators was appointed to govern the kingdom. One of his illusions was that not he was king but his brother John, whom he now set at liberty. When, at the beginning of 1568, Eric recovered his reason, a reconciliation was effected between the king and the duke, on condition that John recognized the legality of his brother’s marriage with Karin Månsdatter, and her children as the successors to the throne. A month later, on the 4th of July, he was solemnly married to Karin at Stockholm by the primate. The next day Karin was crowned queen of Sweden and her infant son Gustavus proclaimed prince-royal. Shortly after his marriage Eric issued a circular ordering a general thanksgiving for his delivery from the assaults of the devil. This document, in every line of which madness is legible, convinced most thinking people that Eric was unfit to reign. The royal dukes, John and Charles, had already taken measures to depose him; and in July the rebellion broke out in Östergötland. Eric at first offered a stout resistance and won two victories; but on the 17th of September the dukes stood before Stockholm, and Eric, after surrendering Göran Persson to the horrible vengeance of his enemies, himself submitted, and resigned the crown. On the 30th of September 1568 John III. was proclaimed king by the army and the nobility; and a Riksdag, summoned to Stockholm, confirmed the choice and formally deposed Eric on the 25th of January 1569. For the next seven years the ex-king was a source of the utmost anxiety to the new government. No fewer than three rebellions, with the object of releasing and reinstating him, had to be suppressed, and his prison was changed half a dozen times. On the 10th of March 1575, an assembly of notables, lay and clerical, at John’s request, pronounced a formal sentence of death upon him. Two years later, on the 24th of February 1577, he died suddenly in his new prison at Örbyhus, poisoned, it is said, by his governor, Johan Henriksen.

See Sveriges Historia, vol. iii. (Stockholm, 1880); Robert Nisbet Bain, Scandinavia, cap. 4-6 (Cambridge, 1905); Eric Tegel, Konung Eriks den XIV. historia (Stockholm, 1751).

(R. N. B.)

Fig. 1.—Vaccinium vitis-idaea, with leaf and flower, nat. size. 1, Flower of V. myrtillus, cut lengthwise. 2, Fruit of same.

ERICACEAE, in botany, a natural order of plants belonging to the higher or gamopetalous division of Dicotyledons. They are woody plants, sometimes with a slender creeping stem as in bilberry, Vaccinium (fig. 1), or Andromeda (fig. 2), or forming low bushes as in the heaths, or larger, sometimes becoming tree-like, as in species of Rhododendron. The leaves are alternate, opposite or whorled in arrangement, and in their form and structure show well-marked adaptation for life in dry or exposed situations. Thus in the true heaths they are needle-like, with the margins often rolled back to form a groove or an almost closed chamber on the under side. In others such as Rhododendron or Arbutus they are often leathery and evergreen, the strongly cuticularized upper surface protecting a water-storing tissue situated above the green layers of the leaf. The flowers are sometimes solitary and axillary or terminal as in Andromeda, but are generally arranged in racemose inflorescences at the end of the branches as in Arbutus and Rhododendron, or on small lateral shoots as in Erica. They are hermaphrodite and generally regular with parts in 4 or 5, thus: sepals 4 or 5, petals 4 or 5, stamens 8 or 10 in two series, the outer of which is opposite the petals, and carpels 4 or 5. The corolla is usually more or less bell-shaped, and in the heaths persists in a dry state in the fruit. The petals with the stamens are situated on the outer edge of a honey-secreting disk. The anthers show a very great variety in shape, the halves are often more or less free and often appendaged; they open to allow the escape of the pollen by a terminal pore or slit. The carpels are united to form a 4- to 5-chambered ovary, which bears a simple elongated style ending in a capitate stigma; each ovary-chamber contains one to many ovules attached to a central placenta. The brightly coloured corolla, the presence of nectar and the scent render the flowers attractive to insects, and the projection of the stigma beyond the anthers favours crossing. The fruit is generally a capsule containing many seeds, as in Erica (fig. 3) or Rhododendron; sometimes a berry as in Arbutus.

1, Flowering shoot of Erica cinerea, about 1½ nat. size.

2, Flower cut lengthwise.

3, Stamen showing appendages and porous dehiscence of anther.

4, Capsule showing the loculicidal dehiscence; a few seeds remain attached to the central axis.

5, Diagram of the flower having four sepals, four divisions of the corolla, eight stamens in two rows, and four divisions of the pistil.

The order falls into four distinct tribes, which are characterized by the relative position of the ovary and by the fruit and seed. They are as follows:—

1. Rhododendron tribe, characterized by capsular fruit, seed with a loose coat, deciduous petals and anthers without appendages. It consists mainly of the great genus Rhododendron (in which Azalea is included by recent botanists), which is chiefly developed in the mountains of eastern Asia, many species occurring on the Himalayas. Dabeocia, St Dabeoc’s heath, occurs in Ireland.

2. Arbutus Tribe.—Fruit a berry or capsule, petals deciduous and anthers with bristle-like appendages, chiefly north temperate to arctic in distribution. Arbutus Unedo, the strawberry-tree, so called from its large scarlet berry, is a southern European species which extends into south Ireland. Arctostaphylos (bearberry) and Andromeda are arctic and alpine genera occurring in Britain. Epigaea repens is the trailing arbutus or mayflower of Atlantic America.

3. Vaccinium Tribe.—Ovary inferior, fruit a berry. Extends from the north temperate zone to the mountains of the tropics. Vaccinium, the largest genus, has four British species: V. Myrtillus is the bilberry(q.v.), blaeberry or whortleberry, V. Vitis-Idaea the cowberry, and V. Oxycoccos the cranberry (q.v.). This tribe is sometimes regarded as a separate order Vacciniaceae, distinguished by its inferior ovary.

4. Erica Tribe.—Fruit usually a capsule, seeds round, not winged; corolla persisting round the ripe fruit; anthers often appendaged. The largest genus is Erica, the true heath (q.v.), with over 400 species, the great majority of which are confined to the Cape; others occur on the mountains of tropical Africa and in Europe and North Africa, especially the Mediterranean region. E. cinerea (purple heather) and E. Tetralix (cross-leaved heath) are common British heaths. Calluna is the ling or Scotch heather.

ERICHSEN, SIR JOHN ERIC, Bart. (1818-1896), British surgeon, born on the 19th of July 1818 at Copenhagen, was the son of Eric Erichsen, a member of a well-known Danish family. He studied medicine at University College, London, and at Paris, devoting himself in the early years of his career to physiology, and lecturing on general anatomy and physiology at University College hospital. In 1844 he was secretary to the physiological section of the British Association, and in 1845 he was awarded the Fothergillian gold medal of the Royal Humane Society for his essay on asphyxia. In 1848 he was appointed assistant surgeon at University College hospital, and in 1850 became full surgeon and professor of surgery, his lectures and clinical teaching being much admired; and in 1875 he joined the consulting staff. His Science and Art of Surgery (1853) went through many editions. He rose to be president of the College of Surgeons in 1880. From 1879 to 1881 he was president of the Royal Medical and Chirurgical Society. He was created a baronet in 1895, having been for some years surgeon-extraordinary to Queen Victoria. As a surgeon his reputation was world-wide, and he counts (says Sir W. MacCormac in his volume on the Centenary of the Royal College of Surgeons) “among the makers of modern surgery.” He was a recognized authority on concussion of the spine, and was often called to give evidence in court on obscure cases caused by railway accidents, &c. He died at Folkestone on the 23rd of September 1896.

ERICHT, LOCH, a lake partly in Inverness-shire and partly in Perthshire, Scotland, lying between the districts of Badenoch on the N. and Rannoch on the S. The boundary line is drawn from a point opposite to the mouth of the Alder, and follows the centre of the longitudinal axis north-eastwards to 56° 50′ N., where it strikes eastwards to the shore. All of the lake to the S. and E. of this line belongs to Perthshire, the rest, forming the major portion, to Inverness-shire. It is a lonely lake, situated in extremely wild surroundings at a height of 1153 ft. above the sea, being thus the loftiest lake of large size in the United Kingdom. It is over 14½ m. long, with a mean breadth of half a mile and over 1 m. at its maximum. Its area amounts to some 7¼ sq. m., and it receives the drainage of an area of nearly 50½ sq. m. The mean depth is 189 ft., and the maximum 512 ft. It has a general trend from N.E. to S.W., the head lying 1 m. from Dalwhinnie station on the Highland railway. It receives many streams, and discharges at the south-western extremity by the Ericht. Salmon and trout afford good fishing. The surrounding mountains are lofty and rugged. Ben Alder (3757 ft.) on the west shore is the chief feature of the great Corrour deer forest. The only point of interest on the banks is the cavern, near the mouth of the Alder, in which Prince Charles Edward concealed himself for a time after the battle of Culloden.

ERICSSON, JOHN (1803-1889), Swedish-American naval engineer, was born at Langbanshyttan, Wermland, Sweden, on the 31st of July 1803. He was the second son of Olaf Ericsson, an inspector of mines, who died in 1818. Showing from his earliest years a strong mechanical bent, young Ericsson, at the age of twelve, was employed as a draughtsman by the Swedish Canal Company. From 1820 to 1827 he served in the army, where his drawing and military maps attracted the attention of the king, and he soon attained the rank of captain. In 1826 he went to London, at first on leave of absence from his regiment, and in partnership with John Braithwaite constructed the “Novelty,” a locomotive engine for the Liverpool & Manchester railway competition at Rainhill in 1829, when the prize, however, was won by Stephenson’s “Rocket.” The number of Ericsson’s inventions at this period was very great. Among other things he worked out a plan for marine engines placed entirely below the water-line. Such engines were made for the “Victory,” for Captain (afterwards Sir) John Ross’s voyage to the Arctic regions in 1829, but they did not prove satisfactory. In 1833 his caloric engine was made public. In 1836 he took out a patent for a screw-propeller, and though the priority of his invention could not be maintained, he was afterwards awarded a one-fifth share of the £20,000 given by the Admiralty for it. At this time Captain Stockton, of the United States navy, gave an order for a small iron vessel to be built by Laird of Birkenhead, and to be fitted by Ericsson with engines and screw. This vessel reached New York in May 1839. A few months later Ericsson followed his steamer to New York, and there he resided for the rest of his life, establishing himself as an engineer and a builder of iron ships. In 1848 he was naturalized as a citizen of the United States. He had many difficulties to contend with, and it was only by slow degrees that he established his fame and won his way to competence. At his death he seems to have been worth about £50,000. The provision of defensive armour for ships of war had long occupied his attention, and he had constructed plans and a model of a vessel lying low in the water, carrying one heavy gun in a circular turret mounted on a turntable. In 1854 he sent his plans to the emperor of the French. Louis Napoleon, however, acting probably on the advice of Dupuy de Lôme, declined to use them. The American Civil War, and the report that the Confederates were converting the “Merrimac” into an ironclad, caused the navy department to invite proposals for the construction of armoured ships. Among others, Ericsson replied, and as it was thought that his design might be serviceable in inland waters, the first armoured turret ship, the “Monitor,” was ordered; she was launched on the 30th of January 1862, and on the 9th of March she fought the celebrated action with the Confederate ram “Merrimac.” The peculiar circumstances in which she was built, the great importance of the battle, and the decisive nature of the result gave the “Monitor” an exaggerated reputation, which further experience did not confirm. In later years Ericsson devoted himself to the study of torpedoes and sun motors. He published Solar Investigations (New York, 1875) and Contributions to the Centennial Exhibition (New York, 1877). He died in New York on the 8th of March 1889, and in the following year, on the request of the Swedish government, his body was sent to Stockholm and thence into Wermland, where, at Filipstad, it was buried on the 15th of September.

A Life of Ericsson by William Conant Church was published in New York in 1890 and in London in 1893.

ERIDANUS, or Fluvius (“the river”), in astronomy, a constellation of the southern hemisphere, mentioned by Eudoxus (4th century B.C.) and Aratus (3rd century B.C.); Ptolemy catalogued 34 stars in it. θ Eridani, a fine double star of magnitudes 3.5 and 5.5, is now of the third magnitude. It is supposed to be identical with the Achernar of Al-Sufi, who described it as of the first magnitude; this star has therefore decreased in brilliancy in historic times. The star ο2 Eridani (numbered 40 by Flamsteed) was discovered to be a ternary star group by Herschel in 1783; it consists of a close pair, of magnitudes 9.2 and 10.9, revolving in a period of 180 years, associated with a star of magnitude 4.5, which is distant from the pair by 82″; these stars have an exceptionally swift proper motion, about 4″ per annum. Eridanus was the ancient name of the river Po.

ERIDU, one of the oldest religious centres of the Sumerians, described in the ancient Babylonian records as the “city of the deep.” The special god of this city was Ea (q.v.), god of the sea and of wisdom, and the prominence given to this god in the incantation literature of Babylonia and Assyria suggests not only that many of our magical texts are to be traced ultimately to the temple of Ea at Eridu, but that this side of the Babylonian religion had its origin in that place. Certain of the most ancient Babylonian myths, especially that of Adapa, may also be traced back to the shrine of Ea at Eridu. But while of the first importance in matters of religion, there is no evidence in Babylonian literature of any special political importance attaching to Eridu, and certainly at no time within our knowledge did it exercise hegemony in Babylonia. The site of Eridu was discovered by J.E. Taylor in 1854, in a ruin then called by the natives Abu-Shahrein, a few miles south-south-west of Moghair, ancient Ur, nearly in the centre of the dry bed of an inland sea, a deep valley, 15 m. at its broadest, covered for the most part with a nitrous incrustation, separated from the alluvial plain about Moghair by a low, pebbly, sandstone range, called the Hazem, but open toward the north to the Euphrates and stretching southward to the Khanega wadi below Suk-esh-Sheiukh. In the rainy season this valley becomes a sea, flooded by the discharge of the Khanega; in summer the Arabs dig holes here which supply them with brackish water. The ruins, in which Taylor conducted brief excavations, consist of a platform of fine sand enclosed by a sandstone wall, 20 ft. high, the corners toward the cardinal points, on the N.W. part of which was a pyramidal tower of two stages, constructed of sun-dried brick, cased with a wall of kiln-burned brick, the whole still standing to a height of about 70 ft. above the platform. The summit of the first stage was reached by a staircase on the S.E. side, 15 ft. wide and 70 ft. long, constructed of polished marble slabs, fastened with copper bolts, flanked at the foot by two curious columns. An inclined road led up to the second stage on the N.W. side. Pieces of polished alabaster and marble, with small pieces of pure gold and gold-headed copper nails, found on and about the top of the second stage, indicated that a small but richly adorned sacred chamber, apparently plated within or without in gold, formerly crowned the top of this structure. Around the whole tower was a pavement of inscribed baked bricks, resting on a layer of clay 2 ft. thick. On the S.E. part of the terrace were the remains of several edifices, containing suites of rooms. Inscriptions on the bricks identified the site as that of Eridu. Since Taylor’s time the place has not been visited by any explorer, owing to the unsafe condition of the neighbourhood; but T.K. Loftus (1854) and J.P. Peters (1890) both report having seen it from the summit of Moghair. The latter states that the Arabs at that time called the ruin Nowawis, and apparently no longer knew the name Abu-Shahrein. Through an error, in many recent maps and Assyriological publications Eridu is described as located in the alluvial plain, between the Tigris and the Euphrates. It was, in fact, an island city in an estuary of the Persian Gulf, stretching up into the Arabian plateau. Originally “on the shore of the sea,” as the old records aver, it is now about 120 m. from the head of the Persian Gulf. Calculating from the present rate of deposit of alluvium at the head of that gulf, Eridu should have been founded as early as the seventh millennium B.C. It is mentioned in historical inscriptions from the earliest times onward, as late as the 6th century B.C. From the evidence of Taylor’s excavations, it would seem that the site was abandoned about the close of the Babylonian period.

See J.E. Taylor, Journal of the Royal Asiatic Society, vol. xv. (1855); F. Delitzsch, Wo lag das Paradies? (1881); J.P. Peters, Nippur (1897); M. Jastrow, The Religion of Babylonia and Assyria (1898); H.V. Hilprecht, Excavations in Assyria and Babylonia (1904); L.W. King, A History of Sumer and Akkad (1910).

(J. P. Pe.)