VOLUME IX, SLICE VII

Equation to Ethics

ARTICLES IN THIS SLICE:

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

EQUATION (from Lat. _aequatio_, _aequare_, to equalize), an expression or statement of the equality of two quantities. Mathematical equivalence is denoted by the sign =, a symbol invented by Robert Recorde (1510-1558), who considered that nothing could be more equal than two equal and parallel straight lines. An equation states an equality existing between two classes of quantities, distinguished as known and unknown; these correspond to the data of a problem and the thing sought. It is the purpose of the mathematician to state the unknowns separately in terms of the knowns; this is called solving the equation, and the values of the unknowns so obtained are called the roots or solutions. The unknowns are usually denoted by the terminal letters, ... x, y, z, of the alphabet, and the knowns are either actual numbers or are represented by the literals a, b, c, &c..., i.e. the introductory letters of the alphabet. Any number or literal which expresses what multiple of term occurs in an equation is called the coefficient of that term; and the term which does not contain an unknown is called the absolute term. The degree of an equation is equal to the greatest index of an unknown in the equation, or to the greatest sum of the indices of products of unknowns. If each term has the sum of its indices the same, the equation is said to be homogeneous. These definitions are exemplified in the equations:--

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

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

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

_History._--There is little doubt that the earliest solutions of equations are given, in the Rhind papyrus, a hieratic document written some 2000 years before our era. The problems solved were of an arithmetical nature, assuming such forms as "a mass and its 1/7th makes 19." Calling the unknown mass x, we have given x + (1/7)x = 19, which is a simple equation. Arithmetical problems also gave origin to equations involving two unknowns; the early Greeks were familiar with and solved simultaneous linear equations, but indeterminate equations, such, for instance, as the system given in the "cattle problem" of Archimedes, were not seriously studied until Diophantus solved many particular problems. Quadratic equations arose in the Greek investigations in the doctrine of proportion, and although they were presented and solved in a geometrical form, the methods employed have no relation to the generalized conception of algebraic geometry which represents a curve by an equation and vice versa. The simplest quadratic arose in the construction of a mean proportional (x) between two lines (a, b), or in the construction of a square equal to a given rectangle; for we have the proportion a:x = x:b; i.e. x^2 = ab. A more general equation, viz. x^2 -ax + a^2 = 0, is the algebraic equivalent of the problem to divide a line in medial section; this is solved in _Euclid_, ii. 11. It is possible that Diophantus was in possession of an algebraic solution of quadratics; he recognized, however, only one root, the interpretation of both being first effected by the Hindu Bhaskara. A simple cubic equation was presented in the problem of finding two mean proportionals, x, y, between two lines, one double the other. We have a:x = x:y = y:2a, which gives x^2 = ay and xy = 2a^2; eliminating y we obtain x^3 = 2a^3, a simple cubic. The Greeks could not solve this equation, which also arose in the problems of duplicating a cube and trisecting an angle, by the ruler and compasses, but only by mechanical curves such as the cissoid, conchoid and quadratrix. Such solutions were much improved by the Arabs, who also solved both cubics and biquadratics by means of intersecting conics; at the same time, they developed methods, originated by Diophantus and improved by the Hindus, for finding approximate roots of numerical equations by algebraic processes. The algebraic solution of the general cubic and biquadratic was effected in the 16th century by S. Ferro, N. Tartaglia, H. Cardan and L. Ferrari (see ALGEBRA: _History_). Many fruitless attempts were made to solve algebraically the quintic equation until P. Ruffini and N.H. Abel proved the problem to be impossible; a solution involving elliptic functions has been given by C. Hermite and L. Kronecker, while F. Klein has given another solution.

In the geometric treatment of equations the Greeks and Arabs based their constructions upon certain empirically deduced properties of the curves and figures employed. Knowing various metrical relations, generally expressed as proportions, it was found possible to solve particular equations, but a general method was wanting. This lacuna was not filled until the 17th century, when Descartes discovered the general theory which explained the nature of such solutions, in particular those wherein conics were employed, and, in addition, established the most important facts that every equation represents a geometrical locus, and conversely. To represent equations containing two unknowns, x, y, he chose two axes of reference mutually perpendicular, and measured x along the horizontal axis and y along the vertical. Then by the methods described in the article GEOMETRY: _Analytical_, he showed that--(1) a linear equation represents a straight line, and (2) a quadratic represents a conic. If the equation be homogeneous or break up into factors, it represents a number of straight lines in the first case, and the loci corresponding to the factors in the second. The solution of simultaneous equations is easily seen to be the values of x, y corresponding to the intersections of the loci. It follows that there is only one value of x, y which satisfies two linear equations, since two lines intersect in one point only; two values which satisfy a linear and quadratic, since a line intersects a conic in two points; and four values which satisfy two quadratics, since two conics intersect in four points. It may happen that the curves do not actually intersect in the theoretical maximum number of points; the principle of continuity (see GEOMETRICAL CONTINUITY) shows us that in such cases some of the roots are imaginary. To represent equations involving three unknowns x, y, z, a third axis is introduced, the z-axis, perpendicular to the plane xy and passing through the intersection of the lines x, y. In this notation a linear equation represents a plane, and two linear simultaneous equations represent a line, i.e. the intersection of two planes; a quadratic equation represents a surface of the second degree. In order to graphically consider equations containing only one unknown, it is convenient to equate the terms to y; i.e. if the equation be [f](x) = 0, we take y = [f](x) and construct this curve on rectangular Cartesian co-ordinates by determining the values of y which correspond to chosen values of x, and describing a curve through the points so obtained. The intersections of the curve with the axis of x gives the real roots of the equation; imaginary roots are obviously not represented.

In this article we shall treat of: (1) Simultaneous equations, (2) indeterminate equations, (3) cubic equations, (4) biquadratic equations, (5) theory of equations. Simple, linear simultaneous and quadratic equations are treated in the article ALGEBRA; for differential equations see DIFFERENTIAL EQUATIONS.

I. _Simultaneous Equations._

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

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

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

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

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

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

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

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

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

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

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

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

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

II. _Indeterminate Equations._

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

III. _Cubic Equations_.

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

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

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

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

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

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

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

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

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

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

and

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

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

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

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

and put

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

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

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

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

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

which are therefore the three values of y.

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

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

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

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

IV. _Biquadratic Equations_.

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

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

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

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

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

so that x may have these four values,

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

two of which are real, and the others imaginary.

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

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

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

y^2 + qy + s = 0,

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

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

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

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

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

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

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

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

then, from the theory of equations we have

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

We square the assumed formula

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

and obtain

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

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

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

Let this equation be also squared, and we have

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

and since

ab + ac + bc = Q,

and

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

the same equation may be expressed thus:

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

Thus we have the biquadratic equation

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

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

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

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

and it immediately appears that

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

and from these equations we find

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

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

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

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

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

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

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

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

becomes, after reduction,

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

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

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

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

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

Then the roots of the proposed biquadratic equation are,

when q is negative, when q is positive,

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

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

V. _Theory of Equations_.

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

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

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

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

_Real Theory_.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

and then, if h be so small that the terms after the second may be neglected, [f]([gamma]) + h[f]'([gamma]) = 0, that is, h = {-[f]([gamma])/[f]'([gamma])}, or the new approximate value is x = [gamma] - {[f]([gamma])/[f]'([gamma])}; and so on, as often as we please. It will be observed that so far nothing has been assumed as to the separation of the roots, or even as to the existence of a real root; [gamma] has been taken as the approximate value of a root, but no precise meaning has been attached to this expression. The question arises, What are the conditions to be satisfied by [gamma] in order that the process may by successive repetitions actually lead to a certain real root of the equation; or that, [gamma] being an approximate value of a certain real root, the new value [gamma] - {[f]([gamma])/[f]'([gamma])} may be a more approximate value.

Referring to fig. 1, it is easy to see that if OC represent the assumed value [gamma], then, drawing the ordinate CP to meet the curve in P, and the tangent PC' to meet the axis in C', we shall have OC' as the new approximate value of the root. But observe that there is here a real root OX, and that the curve beyond X is convex to the axis; under these conditions the point C' is nearer to X than was C; and, starting with C' instead of C, and proceeding in like manner to draw a new ordinate and tangent, and so on as often as we please, we approximate continually, and that with great rapidity, to the true value OX. But if C had been taken on the other side of X, where the curve is concave to the axis, the new point C' might or might not be nearer to X than was the point C; and in this case the method, if it succeeds at all, does so by accident only, i.e. it may happen that C' or some subsequent point comes to be a point C, such that CO is a _proper_ approximate value of the root, and then the subsequent approximations proceed in the same manner as if this value had been assumed in the first instance, all the preceding work being wasted. It thus appears that for the proper application of the method we require _more_ than the mere separation of the roots. In order to be able to approximate to a certain root [alpha], =OX, we require to know that, between OX and some value ON, the curve is always convex to the axis (analytically, between the two values, [f](x) and [f]"(x) must have always the same sign). When this is so, the point C may be taken anywhere on the proper side of X, and within the portion XN of the axis; and the process is then the one already explained. The approximation is in general a very rapid one. If we know for the required root OX the two limits OM, ON such that from M to X the curve is always _concave_ to the axis, while from X to N it is always convex to the axis,--then, taking D anywhere in the portion MX and (as before) C in the portion XN, drawing the ordinates DQ, CP, and joining the points P, Q by a line which meets the axis in D', also constructing the point C' by means of the tangent at P as before, we have for the required root the new limits OD', OC'; and proceeding in like manner with the points D', C', and so on as often as we please, we obtain at each step two limits approximating more and more nearly to the required root OX. The process as to the point D', translated into analysis, is the ordinate process of interpolation. Suppose OD = [beta], OC = [alpha], we have approximately [f]([beta] + h) = [f]([beta]) + h{[f]([alpha]) - [f]([beta])} / ([alpha] - [beta]), whence if the root is [beta] + h then h = - ([alpha] - [beta])[f]([beta]) / {[f]([alpha]) - [f]([beta])}.

Returning for a moment to Horner's method, it may be remarked that the correction h, to an approximate value [alpha], is therein found as a quotient the same or such as the quotient [f]([alpha]) / [f]'([alpha]) which presents itself in Newton's method. The difference is that with Horner the integer part of this quotient is taken as the presumptive value of h, and the figure is verified at each step. With Newton the quotient itself, developed to the proper number of decimal places, is taken as the value of h; if too many decimals are taken, there would be a waste of work; but the error would correct itself at the next step. Of course the calculation should be conducted without any such waste of work.

_Imaginary Theory_.

7. It will be recollected that the expression _number_ and the correlative epithet _numerical_ were at the outset used in a wide sense, as extending to imaginaries. This extension arises out of the theory of equations by a process analogous to that by which number, in its original most restricted sense of positive integer number, was extended to have the meaning of a real positive or negative magnitude susceptible of continuous variation.

If for a moment number is understood in its most restricted sense as meaning positive integer number, the solution of a simple equation leads to an extension; ax - b = 0 gives x = (b/a), a positive fraction, and we can in this manner represent, not accurately, but as nearly as we please, any positive magnitude whatever; so an equation ax + b = 0 gives x = -(b/a), which (approximately as before) represents any negative magnitude. We thus arrive at the extended signification of number as a continuously varying positive or negative magnitude. Such numbers may be added or subtracted, multiplied or divided one by another, and the result is always a number. Now from a quadric equation we derive, in like manner, the notion of a complex or imaginary number such as is spoken of above. The equation x^2 + 1 = 0 is not (in the foregoing sense, number = real number) satisfied by any numerical value whatever of x; but we assume that there is a number which we call i, satisfying the equation i^2 + 1 = 0, and then taking a and b any real numbers, we form an expression such as a + bi, and use the expression number in this extended sense: any two such numbers may be added or subtracted, multiplied or divided one by the other, and the result is always a number. And if we consider first a quadric equation x^2 + px + q = 0 where p and q are real numbers, and next the like equation, where p and q are any numbers whatever, it can be shown that there exists for x a numerical value which satisfies the equation; or, in other words, it can be shown that the equation has a numerical root. The like theorem, in fact, holds good for an equation of any order whatever; but suppose for a moment that this was not the case; say that there was a cubic equation x^3 + px^2 + qx + r = 0, with numerical coefficients, not satisfied by any numerical value of x, we should have to establish a new imaginary j satisfying some such equation, and should then have to consider numbers of the form a + bj, or perhaps a + bj + cj^2 (a, b, c numbers [alpha] + [beta]i of the kind heretofore considered),--first we should be thrown back on the quadric equation x^2 + px + q = 0, p and q being now numbers of the last-mentioned extended form--_non constat_ that every such equation has a numerical root--and if not, we might be led to _other_ imaginaries k, l, &c., and so on _ad infinitum_ in inextricable confusion.

But in fact a numerical equation of any order whatever has always a numerical root, and thus numbers (in the foregoing sense, number = quantity of the form [alpha] + [beta]i) form (_what real numbers do not_) a universe complete in itself, such that starting in it we are never led out of it. There may very well be, and perhaps are, numbers in a more general sense of the term (quaternions are not a case in point, as the ordinary laws of combination are not adhered to), but in order to have to do with such numbers (if any) we must start with them.

8. The capital theorem as regards numerical equations thus is, every numerical equation has a numerical root; or for shortness (the meaning being as before), every equation has a root. Of course the theorem is the reverse of self-evident, and it requires proof; but provisionally assuming it as true, we derive from it the general theory of numerical equations. As the term root was introduced in the course of an explanation, it will be convenient to give here the formal definition.

A number a such that substituted for x it makes the function x1^n - p1x^(n - 1) ... [+-]p_n to be = 0, or say such that it satisfies the equation [f](x) = 0, is said to be a root of the equation; that is, a being a root, we have

a^n - p1a^(n - 1) ... [+-]p_n = 0, or say [f](a) = 0;

and it is then easily shown that x - a is a factor of the function [f](x), viz. that we have [f](x) = (x - a)[f]1(x), where [f]1(x) is a function x^(n - 1) - q1x^(n - 2) ... [+-]q_(n - 1) of the order n - 1, with numerical coefficients q1, q2 ... q_(n - 1).

In general a is not a root of the equation [f]1(x) = 0, but it may be so--i.e. [f]1(x) may contain the factor x - a; when this is so, [f](x) will contain the factor (x - a)^2; writing then [f](x) = (x - a)^2[f]2(x), and assuming that a is not a root of the equation [f]2(x) = 0, x = a is then said to be a double root of the equation [f](x) = 0; and similarly [f](x) may contain the factor (x - a)^3 and no higher power, and x = a is then a triple root; and so on.

Supposing in general that [f](x) = (x - a)^[alpha] F(x) ([alpha] being a positive integer which may be = 1, (x - a)^[alpha] the highest power of x - a which divides [f](x), and F(x) being of course of the order n - [alpha]), then the equation F(x) = 0 will have a root b which will be different from a; x - b will be a factor, in general a simple one, but it may be a multiple one, of F(x), and [f](x) will in this case be = (x - a)^[alpha] (x - b)^[beta] [Phi](x) ([beta] a positive integer which may be = 1, (x-b)^[beta] the highest power of x - b in F(x) or [f](x), and [Phi](x) being of course of the order n - [alpha] - [beta]). The original equation [f](x) = 0 is in this case said to have [alpha] roots each = a, [beta] roots each = b; and so on for any other factors (x - c)^[gamma], &c.

We have thus the _theorem_--A numerical equation of the order n has in every case n roots, viz. there exist n numbers, a, b, ... (in general all distinct, but which may arrange themselves in any sets of equal values), such that [f](x) = (x - a)(x - b)(x - c) ... identically.

If the equation has equal roots, these can in general be determined, and the case is at any rate a special one which may be in the first instance excluded from consideration. It is, therefore, in general assumed that the equation [f](x) = 0 has all its roots unequal.

If the coefficients p1, p2, ... are all or any one or more of them imaginary, then the equation [f](x) = 0, separating the real and imaginary parts thereof, may be written F(x) + i[Phi](x) = 0, where F(x), [Phi](x) are each of them a function with real coefficients; and it thus appears that the equation [f](x) = 0, with imaginary coefficients, has not in general any real root; supposing it to have a real root a, this must be at once a root of each of the equations F(x) = 0 and [Phi](x) = 0.

But an equation with real coefficients may have as well imaginary as real roots, and we have further the _theorem_ that for any such equation the imaginary roots enter in pairs, viz. [alpha] + [beta]i being a root, then [alpha] - [beta]i will be also a root. It follows that if the order be odd, there is always an odd number of real roots, and therefore at least one real root.

9. In the case of an equation with real coefficients, the question of the existence of real roots, and of their separation, has been already considered. In the general case of an equation with imaginary (it may be real) coefficients, the like question arises as to the situation of the (real or imaginary) roots; thus, if for facility of conception we regard the constituents [alpha], [beta] of a root [alpha] + [beta]i as the co-ordinates of a point _in plano_, and accordingly represent the root by such point, then drawing in the plane any closed curve or "contour," the question is how many roots lie within such contour.

This is solved theoretically by means of a theorem of A.L. Cauchy (1837), viz. writing in the original equation x + iy in place of x, the function [f](x + iy) becomes = P + iQ, where P and Q are each of them a rational and integral function (with real coefficients) of (x, y). Imagining the point (x, y) to travel along the contour, and considering the number of changes of sign from - to + and from + to - of the fraction corresponding to passages of the fraction through zero (that is, to values for which P becomes = 0, disregarding those for which Q becomes = 0), the difference of these numbers gives the number of roots within the contour.

It is important to remark that the demonstration does not presuppose the existence of any root; the contour may be the infinity of the plane (such infinity regarded as a contour, or closed curve), and in this case it can be shown (and that very easily) that the difference of the numbers of changes of sign is = n; that is, there are within the infinite contour, or (what is the same thing) there are in all n roots; thus Cauchy's theorem contains really the proof of the fundamental theorem that a numerical equation of the nth order (not only has a numerical root, but) has precisely n roots. It would appear that this proof of the fundamental theorem in its most complete form is in principle identical with the last proof of K.F. Gauss (1849) of the theorem, in the form--A numerical equation of the nth order has always a root.[3]

But in the case of a finite contour, the actual determination of the difference which gives the number of real roots can be effected only in the case of a rectangular contour, by applying to each of its sides separately a method such as that of Sturm's theorem; and thus the actual determination ultimately depends on a method such as that of Sturm's theorem.

Very little has been done in regard to the calculation of the imaginary roots of an equation by approximation; and the question is not here considered.

10. A class of numerical equations which needs to be considered is that of the binomial equations x^n - a = 0 (a = [alpha] + [beta]i, a complex number).

The foregoing conclusions apply, viz. there are always n roots, which, it may be shown, are all unequal. And these can be found numerically by the extraction of the square root, and of an nth root, of _real_ numbers, and by the aid of a table of natural sines and cosines.[4] For writing

/ [alpha] [beta] \ [alpha] + [beta]i = [root]([alpha]^2 + [beta]^2) ( ---------------------------- + ----------------------------i ), \[root]([alpha]^2 + [beta]^2) [root]([alpha]^2 + [beta]^2) /

there is always a real angle [lambda] (positive and less than 2[pi]), such that its cosine and sine are = [alpha] / [root]([alpha]^2 + [beta]^2) and [beta] / [root]([alpha]^2 + [beta]^2) respectively; that is, writing for shortness [root]([alpha]^2 + [beta]^2) = [rho], we have [alpha] + [beta]i = [rho](cos[lambda] + i sin[lambda]), or the equation is x^n = [rho](cos[lambda] + i sin [lambda]); hence observing that (cos [lambda]/n + i sin [lambda]/n )^n = cos[lambda] + i sin[lambda], a value of x is = [root n][rho] (cos [lambda]/n + i sin [lambda]/n). The formula really gives all the roots, for instead of [lambda] we may write [lambda] + 2s[pi], s a positive or negative integer, and then we have

/ [lambda] + 2s[pi] [lambda] + 2s[pi] \ x = [root n][rho] ( cos ----------------- + i sin ----------------- ), \ n n /

which has the n values obtained by giving to s the values 0, 1, 2 ... n - 1 in succession; the roots are, it is clear, represented by points lying at equal intervals on a circle. But it is more convenient to proceed somewhat differently; taking one of the roots to be [theta], so that [theta]^n = a, then assuming x = [theta]y, the equation becomes y^n - 1 = 0, which equation, like the original equation, has precisely n roots (one of them being of course = 1). And the original equation x^n - a = 0 is thus reduced to the more simple equation x^n - 1 = 0; and although the theory of this equation is included in the preceding one, yet it is proper to state it separately.

The equation x^n - 1 = 0 has its several roots expressed in the form 1, [omega], [omega]^2, ... [omega]^(n - 1), where [omega] may be taken = cos 2[pi]/n + i sin 2[pi]/n; in fact, [omega] having this value, any integer power [omega]^k is = cos 2[pi]k/n + i sin 2[pi]k/n, and we thence have ([omega]^k)^n = cos 2[pi]k + i sin 2[pi]k, = 1, that is, [omega]^k is a root of the equation. The theory will be resumed further on.

By what precedes, we are led to the notion (a numerical) of the radical a^(1/n) regarded as an n-valued function; any one of these being denoted by [root n]a, then the series of values is [root n]a, [omega][root n]a, ... [omega]^(n - 1)[root n]a; or we may, if we please, use [root n]a instead of a^(1/n) as a symbol to denote the n-valued function.

As the coefficients of an algebraical equation may be numerical, all which follows in regard to algebraical equations is (with, it may be, some few modifications) applicable to numerical equations; and hence, concluding for the present this subject, it will be convenient to pass on to algebraical equations.

_Algebraical Equations._

11. The equation is

x^n - p1x^(n-1) + ... [+-]p_n = 0,

and we here _assume_ the existence of roots, viz. we assume that there are n quantities a, b, c ... (in general all of them different, but which in particular cases may become equal in sets in any manner), such that

x^n - p1x^(n - 1) + ... [+-] p_n = 0;

or looking at the question in a different point of view, and starting with the roots a, b, c ... as given, we express the product of the n factors x - a, x - b, ... in the foregoing form, and thus arrive at an equation of the order n having the n roots a, b, c.... In either case we have

p1 = [Sigma]a, p2 = [Sigma]ab, ... p_n = abc ...;

i.e. regarding the coefficients p1, p2 ... p_n as given, then we assume the existence of roots a, b, c, ... such that p1 = [Sigma]a, &c.; or, regarding the roots as given, then we write p1, p2, &c., to denote the functions [Sigma]a, [Sigma]ab, &c.

As already explained, the epithet algebraical is not used in opposition to numerical; an algebraical equation is merely an equation wherein the coefficients are not restricted to denote, or are not explicitly considered as denoting, numbers. That the abstraction is legitimate, appears by the simplest example; in saying that the equation x^2 - px + q = 0 has a root x = 1/2{p + [root](p^2 - 4q)}, we mean that writing this value for x the equation becomes an identity, [1/2{p + [root](p^2 - 4q)}]^2 - p[1/2{p + [root](p^2 - 4q)}] + q = 0; and the verification of this identity in nowise depends upon p and q meaning numbers. But if it be asked what there is beyond numerical equations included in the term algebraical equation, or, again, what is the full extent of the meaning attributed to the term--the latter question at any rate it would be very difficult to answer; as to the former one, it may be said that the coefficients may, for instance, be symbols of operation. As regards such equations, there is certainly no proof that every equation has a root, or that an equation of the nth order has n roots; nor is it in any wise clear what the precise signification of the statement is. But it is found that the assumption of the existence of the n roots can be made without contradictory results; conclusions derived from it, if they involve the roots, rest on the same ground as the original assumption; but the conclusion may be independent of the roots altogether, and in this case it is undoubtedly valid; the reasoning, although actually conducted by aid of the assumption (and, it may be, most easily and elegantly in this manner), is really independent of the assumption. In illustration, we observe that it is allowable to express a function of p and q as follows,--that is, by means of a rational symmetrical function of a and b, this can, as a fact, be expressed as a rational function of a + b and ab; and if we prescribe that a + b and ab shall then be changed into p and q respectively, we have the required function of p, q. That is, we have F([alpha], [beta]) as a representation of [f](p, q), obtained as if we had p = a + b, q = ab, but without in any wise assuming the existence of the a, b of these equations.

12. Starting from the equation

x^n - p1x^(n - 1) + ... = x - a.x - b. &c.

or the equivalent equations p1 = [Sigma]a, &c., we find

a^n - p1a^(n - 1) + ... = 0, b^n - p1b^(n - 1) + ... = 0; . . . . . . . . .

(it is as satisfying these equations that a, b ... are said to be the roots of x^n - p1x^(n - 1) + ... = 0); and conversely from the last-mentioned equations, assuming that a, b ... are all different, we deduce

p1 = [Sigma]a, p2 = [Sigma]ab, &c.

and

x^n - p1x^(n - 1) + ... = x - a.x - b. &c.

Observe that if, for instance, a = b, then the equations a^n - p1a^(n - 1) + ... = 0, b^n - p1b^(n - 1) + ... = 0 would reduce themselves to a single relation, which would not of itself express that a was a double root,--that is, that (x - a)^2 was a factor of x^n - p1x^(n - 1) +, &c; but by considering b as the limit of a + h, h indefinitely small, we obtain a second equation

na^(n - 1) - (n - 1)p1a^(n - 2) + ... = 0,

which, with the first, expresses that a is a double root; and then the whole system of equations leads as before to the equations p1 = [Sigma]a, &c. But the existence of a double root implies a certain relation between the coefficients; the general case is when the roots are all unequal.

We have then the _theorem_ that every rational symmetrical function of the roots is a rational function of the coefficients. This is an easy consequence from the less general theorem, every rational and integral symmetrical function of the roots is a rational and integral function of the coefficients.

In particular, the sums of the powers [Sigma]a^2, [Sigma]a^3, &c., are rational and integral functions of the coefficients.

The process originally employed for the expression of other functions [Sigma]a^[alpha] b^[beta], &c., in terms of the coefficients is to make them depend upon the sums of powers: for instance, [Sigma]a^[alpha] b^[beta] = [Sigma]a^[alpha] [Sigma]a^[beta] - [Sigma]a^([alpha] + [beta]); but this is very objectionable; the true theory consists in showing that we have systems of equations

p1 = [Sigma]a,

p2 = [Sigma]ab, p1^2 = [Sigma]a^2 + 2[Sigma]ab,

p3 = [Sigma]abc, p1p2 = [Sigma]a^2 b + 3[Sigma]abc, p1^3 = [Sigma]a^3 + 3[Sigma]a^2 b + 6[Sigma]abc,

where in each system there are precisely as many equations as there are root-functions on the right-hand side--e.g. 3 equations and 3 functions [Sigma]abc, [Sigma]a^2 b, [Sigma]a^3. Hence in each system the root-functions can be determined linearly in terms of the powers and products of the coefficients:

[Sigma]ab = p2, [Sigma]a^2 = p1^2 - 2p2,

[Sigma]abc = p3, [Sigma]a^2 b = p1p2 - 3p3, [Sigma]a^3 = p1^3 - 3p1p2 + 3p3,

and so on. The other process, if applied consistently, would derive the originally assumed value [Sigma]ab = p2, from the two equations [Sigma]a = p, [Sigma]a^2 = p1^2 - 2p2; i.e. we have 2[Sigma]ab = [Sigma]a.[Sigma]a - [Sigma]a^2,= p1^2 - (p1^2 - 2p2), = 2p2.

13. It is convenient to mention here the theorem that, x being determined as above by an equation of the order n, any rational and integral function whatever of x, or more generally any rational function which does not become infinite in virtue of the equation itself, can be expressed as a rational and integral function of x, of the order n - 1, the coefficients being rational functions of the coefficients of the equation. Thus the equation gives x^n a function of the form in question; multiplying each side by x, and on the right-hand side writing for x^n its foregoing value, we have x^(n + 1), a function of the form in question; and the like for any higher power of x, and therefore also for any rational and integral function of x. The proof in the case of a rational non-integral function is somewhat more complicated. The final result is of the form [phi](x)/[psi](x) = I(x), or say [phi](x) -[psi](x)I(x) = 0, where [phi], [psi], I are rational and integral functions; in other words, this equation, being true if only [f](x) = 0, can only be so by reason that the left-hand side contains [f](x) as a factor, or we must have identically [phi](x) - [psi](x)I(x) = M(x)[f](x). And it is, moreover, clear that the equation [phi](x)/[psi](x) = I(x), being satisfied if only [f](x) = 0, must be satisfied by each root of the equation.

From the theorem that a rational symmetrical function of the roots is expressible in terms of the coefficients, it at once follows that it is possible to determine an equation (of an assignable order) having for its roots the several values of any given (unsymmetrical) function of the roots of the given equation. For example, in the case of a quartic equation, roots (a, b, c, d), it is possible to find an equation having the roots ab, ac, ad, bc, bd, cd (being therefore a sextic equation): viz. in the product

(y - ab)(y - ac)(y - ad)(y - bc)(y - bd)(y - cd)

the coefficients of the several powers of y will be symmetrical functions of a, b, c, d and therefore rational and integral functions of the coefficients of the quartic equation; hence, supposing the product so expressed, and equating it to zero, we have the required sextic equation. In the same manner can be found the sextic equation having the roots (a - b)^2, (a - c)^2, (a - d)^2, (b - c)^2, (b - d)^2, (c - d)^2, which is the equation of differences previously referred to; and similarly we obtain the equation of differences for a given equation of any order. Again, the equation sought for may be that having for its n roots the given rational functions [phi](a), [phi](b), ... of the several roots of the given equation. Any such rational function can (as was shown) be expressed as a rational and integral function of the order n - 1; and, retaining x in place of any one of the roots, the problem is to find y from the equations x^n - p1 x^(n - 1) ... = 0, and y = M0x^(n - 1) + M1x^(n - 2) + ..., or, what is the same thing, from these two equations to eliminate x. This is in fact E.W. Tschirnhausen's transformation (1683).

14. In connexion with what precedes, the question arises as to the number of values (obtained by permutations of the roots) of given unsymmetrical functions of the roots, or say of a given set of letters: for instance, with roots or letters (a, b, c, d) as before, how many values are there of the function ab + cd, or better, how many functions are there of this form? The answer is 3, viz. ab + cd, ac + bd, ad + bc; or again we may ask whether, in the case of a given number of letters, there exist functions with a given number of values, 3-valued, 4-valued functions, &c.

It is at once seen that for any given number of letters there exist 2-valued functions; the product of the differences of the letters is such a function; however the letters are interchanged, it alters only its sign; or say the two values are [Delta] and -[Delta]. And if P, Q are symmetrical functions of the letters, then the general form of such a function is P + Q[Delta]; this has only the two values P + Q[Delta], P - Q[Delta].

In the case of 4 letters there exist (as appears above) 3-valued functions: but in the case of 5 letters there does not exist any 3-valued or 4-valued function; and the only 5-valued functions are those which are symmetrical in regard to four of the letters, and can thus be expressed in terms of one letter and of symmetrical functions of all the letters. These last theorems present themselves in the demonstration of the non-existence of a solution of a quintic equation by radicals.

The theory is an extensive and important one, depending on the notions of _substitutions_ and of _groups_ (q.v.).

15. Returning to equations, we have the very important theorem that, given the value of any unsymmetrical function of the roots, e.g. in the case of a quartic equation, the function ab + cd, it is in general possible to determine rationally the value of any similar function, such as (a + b)^3 + (c + d)^3.

The _a priori_ ground of this theorem may be illustrated by means of a numerical equation. Suppose that the roots of a quartic equation are 1, 2, 3, 4, then if it is given that ab + cd = 14, this in effect determines a, b to be 1, 2 and c, d to be 3, 4 (viz. a = 1, b = 2 or a = 2, b = 1, and c = 3, d = 4 or c = 3, d = 4) or else a, b to be 3, 4 and c, d to be 1, 2; and it therefore in effect determines (a + b)^3 + (c + d)^3 to be = 370, and not any other value; that is, (a + b)^3 + (c + d)^3, as having a single value, must be determinable rationally. And we can in the same way account for cases of failure as regards particular equations; thus, the roots being 1, 2, 3, 4 as before, a^2 b = 2 determines a to be = 1 and b to be = 2, but if the roots had been 1, 2, 4, 16 then a^2 b = 16 does not uniquely determine a, b but only makes them to be 1, 16 or 2, 4 respectively.

As to the _a posteriori_ proof, assume, for instance,

t1 = ab + cd, y1 = (a + b)^3 + (c + d)^3, t2 = ac + bd, y2 = (a + c)^3 + (b + d)^3, t3 = ad + bc, y3 = (a + d)^3 + (b + c)^3:

then y1 + y2 + y3, t1y1 + t2y2 + t3y3, t1^2 y1 + t2^2y2 + t3^2y3 will be respectively symmetrical functions of the roots of the quartic, and therefore rational and integral functions of the coefficients; that is, they will be known.

Suppose for a moment that t1, t2, t3 are all known; then the equations being linear in y1, y2, y3 these can be expressed rationally in terms of the coefficients and of t1, t2, t3; that is, y1, y2, y3 will be known. But observe further that y1 is obtained as a function of t1, t2, t3 symmetrical as regards t2, t3; it can therefore be expressed as a rational function of t1 and of t2 + t3, t2t3, and thence as a rational function of t1 and of t1 + t2 + t3, t1t2 + t1t3 + t2t3, t1t2t3; but these last are symmetrical functions of the roots, and as such they are expressible rationally in terms of the coefficients; that is, y1 will be expressed as a rational function of t1 and of the coefficients; or t1 (alone, not t2 or t3) being known, y1 will be rationally determined.

16. We now consider the question of the algebraical solution of equations, or, more accurately, that of the _solution of equations by radicals_.

In the case of a quadric equation x^2 - px + q = 0, we can by the assistance of the sign [root]( ) or ( )^1/2 find an expression for x as a 2-valued function of the coefficients p, q such that substituting this value in the equation, the equation is thereby identically satisfied; it has been found that this expression is

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

and the equation is on this account said to be algebraically solvable, or more accurately solvable by radicals. Or we may by writing x = -1/2 p + z reduce the equation to z^2 = 1/4(p^2 - 4q), viz. to an equation of the form x^2 = a; and in virtue of its being thus reducible we say that the original equation is solvable by radicals. And the question for an equation of any higher order, say of the order n, is, can we by means of radicals (that is, by aid of the sign [root m]( ) or ( )^(1/m), using as many as we please of such signs and with any values of m) find an n-valued function (or any function) of the coefficients which substituted for x in the equation shall satisfy it identically?

It will be observed that the coefficients p, q ... are not explicitly considered as numbers, but even if they do denote numbers, the question whether a numerical equation admits of solution by radicals is wholly unconnected with the before-mentioned theorem of the existence of the n roots of such an equation. It does not even follow that in the case of a numerical equation solvable by radicals the algebraical solution gives the numerical solution, but this requires explanation. Consider first a numerical quadric equation with imaginary coefficients. In the formula x = 1/2{p [+-] [root](p^2 - 4q)}, substituting for p, q their given numerical values, we obtain for x an expression of the form x = [alpha] + [beta]i [+-] [root]([gamma] + [delta]i), where [alpha], [beta], [gamma], [delta] are real numbers. This expression substituted for x in the quadric equation would satisfy it identically, and it is thus an algebraical solution; but there is no obvious _a priori_ reason why [root]([gamma]+[delta]i) should have a value = c + di, where c and d are real numbers calculable by the extraction of a root or roots of real numbers; however the case is (what there was no _a priori_ right to expect) that [root]([gamma] + [delta]i) has such a value calculable by means of the radical expressions [root]{[root]([gamma]^2 + [delta]^2) [+-] [gamma]} : and hence the algebraical solution of a numerical quadric equation does in every case give the numerical solution. The case of a numerical cubic equation will be considered presently.

17. A cubic equation can be solved by radicals.

Taking for greater simplicity the cubic in the reduced form x^3 + qx - r = 0, and assuming x = a + b, this will be a solution if only 3ab = q and a^3 + b^3 = r, equations which give (a^3 - b^3)^2 = r^2 - (4/27)q^3, a quadric equation solvable by radicals, and giving a^3 - b^3 = [root](r^2 - (4/27)q^3), a 2-valued function of the coefficients: combining this with a^3 + b^3 = r, we have a^3 = 1/2{r + [root](r^2 - (4/27)q^3)}, a 2-valued function: we then have a by means of a cube root, viz.

a = [root 3][1/2{r + [root](r^2 - (4/27)q^3)}],

a 6-valued function of the coefficients; but then, writing q = b/3a, we have, as may be shown, a + b a 3-valued function of the coefficients; and x = a + b is the required solution by radicals. It would have been wrong to complete the solution by writing

b = [root 3][1/2{r - [root](r^2 - (4/27)q^3)}],

for then a + b would have been given as a 9-valued function having only 3 of its values roots, and the other 6 values being irrelevant. Observe that in this last process we make no use of the equation 3ab = q, in its original form, but use only the derived equation 27a^3 b^3 = q^3, implied in, but not implying, the original form.

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

3/2 4 3/2 4 a^3 = --- {r + [root](r^2 - --q^3)}, b^3 = --- {r - [root](r^2 - --q^3)}, q 27 q 27

i.e. here a^3, b^3 are each of them a 2-valued function, but as the only effect of altering the sign of the quadric radical is to interchange a^3, b^3, they may be regarded as each of them 1-valued; a and b are each of them 3-valued (for observe that here only a^3 b^3, not ab, is given); and ab(a + b) thus is in appearance a 9-valued function; but it can easily be shown that it is (as it ought to be) only 3-valued.

In the case of a numerical cubic, even when the coefficients are real, substituting their values in the expression

x = [root 3][1/2{r + [root](r^2 - (4/27)q^3)}] + (1/3)q / [root 3][1/2{r + [root](r^2 - (4/27)q^3)}],

this may depend on an expression of the form [root 3]([gamma] + [delta]i) where [gamma] and [delta] are real numbers (it will do so if r^2 - (4/27)q^3 is a negative number), and then we _cannot_ by the extraction of any root or roots of real positive numbers reduce [root 3]([gamma] + [delta]i) to the form c + di, c and d real numbers; hence here the algebraical solution does not give the numerical solution, and we have here the so-called "irreducible case" of a cubic equation. By what precedes there is nothing in this that might not have been expected; the algebraical solution makes the solution depend on the extraction of the cube root of a number, and there was no reason for expecting this to be a real number. It is well known that the case in question is that wherein the three roots of the numerical cubic equation are all real; if the roots are two imaginary, one real, then contrariwise the quantity under the cube root is real; and the algebraical solution gives the numerical one.

The irreducible case is solvable by a trigonometrical formula, but this is not a solution by radicals: it consists in effect in reducing the given numerical cubic (not to a cubic of the form z^3 = a, solvable by the extraction of a cube root, but) to a cubic of the form 4x^3 - 3x = a, corresponding to the equation 4 cos^3 [theta] - 3 cos[theta] = cos 3[theta] which serves to determine cos[theta] when cos 3[theta] is known. The theory is applicable to an algebraical cubic equation; say that such an equation, if it can be reduced to the form 4x^3 - 3x = a, is solvable by "trisection"--then the general cubic equation is solvable by trisection.

18. A quartic equation is solvable by radicals, and it is to be remarked that the existence of such a solution depends on the existence of 3-valued functions such as ab + cd of the four roots (a, b, c, d): by what precedes ab + cd is the root of a cubic equation, which equation is solvable by radicals: hence ab + cd can be found by radicals; and since abcd is a given function, ab and cd can then be found by radicals. But by what precedes, if ab be known then any similar function, say a + b, is obtainable rationally; and then from the values of a + b and ab we may by radicals obtain the value of a or b, that is, an expression for the root of the given quartic equation: the expression ultimately obtained is 4-valued, corresponding to the different values of the several radicals which enter therein, and we have thus the expression by radicals of each of the four roots of the quartic equation. But when the quartic is numerical the same thing happens as in the cubic, and the algebraical solution does not in every case give the numerical one.

It will be understood from the foregoing explanation as to the quartic how in the next following case, that of the quintic, the question of the solvability by radicals depends on the existence or non-existence of k-valued functions of the five roots (a, b, c, d, e); the fundamental theorem is the one already stated, a rational function of five letters, if it has less than 5, cannot have more than 2 values, that is, there are no 3-valued or 4-valued functions of 5 letters: and by reasoning depending in part upon this theorem, N.H. Abel (1824) showed that a general quintic equation is not solvable by radicals; and _a fortiori_ the general equation of any order higher than 5 is not solvable by radicals.

19. The general theory of the solvability of an equation by radicals depends fundamentally on A.T. Vandermonde's remark (1770) that, supposing an equation is solvable by radicals, and that we have therefore an algebraical expression of x in terms of the coefficients, then substituting for the coefficients their values in terms of the roots, the resulting expression must reduce itself to any one at pleasure of the roots a, b, c ...; thus in the case of the quadric equation, in the expression x = 1/2{p + [root](p^2 - 4q)}, substituting for p and q their values, and observing that (a + b)^2 - 4ab = (a - b)^2, this becomes x = 1/2{a + b + [root](a - b)^2}, the value being a or b according as the radical is taken to be +(a - b) or -(a - b).

So in the cubic equation x^3 - px^2 + qx - r = 0, if the roots are a, b, c, and if [omega] is used to denote an imaginary cube root of unity, [omega]^2 + [omega] + 1 = 0, then writing for shortness p = a + b + c, L = a + [omega]b + [omega]^2 c, M = a + [omega]^2 b + [omega]c, it is at once seen that LM, L^3 + M^3, and therefore also (L^3 - M^3)^2 are symmetrical functions of the roots, and consequently rational functions of the coefficients: hence

1/2{L^3 + M^3 + [root](L^3 - M^3)^2}

is a rational function of the coefficients, which when these are replaced by their values as functions of the roots becomes, according to the sign given to the quadric radical, = L^3 or M^3; taking it = L^3, the cube root of the expression has the three values L, [omega]L, [omega]^2 L; and LM divided by the same cube root has therefore the values M, [omega]^2M, [omega]M; whence finally the expression

(1/3)[p + [root 3]{1/2(L^3 + M^3 + [root](L^3 - M^3)^2)} + LM / [root 3]{1/2L^3 + M^3 + [root](L^3 - M^3)^2}]

has the three values

(1/3)(p + L + M), (1/3)(p + [omega]L + [omega]^2 M), (1/3)(p + [omega]^2 L + [omega]M);

that is, these are = a, b, c respectively. If the value M^3 had been taken instead of L^3, then the expression would have had the same three values a, b, c. Comparing the solution given for the cubic x^3 + qx - r = 0, it will readily be seen that the two solutions are identical, and that the function r^2 - (4/27)q^3 under the radical sign must (by aid of the relation p = 0 which subsists in this case) reduce itself to (L^3 - M^3)^2; it is only by each radical being equal to a rational function of the roots that the final expression _can_ become equal to the roots a, b, c respectively.

20. The formulae for the cubic were obtained by J.L. Lagrange (1770-1771) from a different point of view. Upon examining and comparing the principal known methods for the solution of algebraical equations, he found that they all ultimately depended upon finding a "resolvent" equation of which the root is a + [omega]b + [omega]^2 c + [omega]^3 d + ..., [omega] being an imaginary root of unity, of the same order as the equation; e.g. for the cubic the root is a + [omega]b + [omega]^2 c, [omega] an imaginary cube root of unity. Evidently the method gives for L^3 a quadric equation, which is the "resolvent" equation in this particular case.

For a quartic the formulae present themselves in a somewhat different form, by reason that 4 is not a prime number. Attempting to apply it to a quintic, we seek for the equation of which the root is (a + [omega]b + [omega]^2 c + [omega]^3 d + [omega]^4 e), [omega] an imaginary fifth root of unity, or rather the fifth power thereof (a + [omega]b + [omega]^2 c + [omega]^3d + [omega]^4 e)^5; this is a 24-valued function, but if we consider the four values corresponding to the roots of unity [omega], [omega]^2, [omega]^3, [omega]^4, viz. the values

(a + [omega]b + [omega]^2 c + [omega]^3 d + [omega]^4 e)^5, (a + [omega]^2 b + [omega]^4 c + [omega]d + [omega]^3e)^5, (a + [omega]^3 b + [omega]c + [omega]^4 d + [omega]^2e)^5, (a + [omega]^4 b + [omega]^3 c + [omega]^2 d + [omega]e)^5,

any symmetrical function of these, for instance their sum, is a 6-valued function of the roots, and may therefore be determined by means of a sextic equation, the coefficients whereof are rational functions of the coefficients of the original quintic equation; the conclusion being that the solution of an equation of the fifth order is made to depend upon that of an equation of the sixth order. This is, of course, useless for the solution of the quintic equation, which, as already mentioned, does not admit of solution by radicals; but the equation of the sixth order, Lagrange's resolvent sextic, is very important, and is intimately connected with all the later investigations in the theory.

21. It is to be remarked, in regard to the question of solvability by radicals, that not only the coefficients are taken to be arbitrary, but it is assumed that they are represented each by a single letter, or say rather that they are not so expressed in terms of other arbitrary quantities as to make a solution possible. If the coefficients are not all arbitrary, for instance, if some of them are zero, a sextic equation might be of the form x^6 + bx^4 + cx^2 + d = 0, and so be solvable as a cubic; or if the coefficients of the sextic are given functions of the six arbitrary quantities a, b, c, d, e, f, such that the sextic is really of the form (x^2 + ax + b)(x^4 + cx^3 + dx^2 + ex + f) = 0, then it breaks up into the equations x^2 + ax + b = 0, x^4 + cx^3 + dx^2 + ex + f = 0, and is consequently solvable by radicals; so also if the form is (x -a)(x - b)(x - c)(x - d)(x - e)(x - f) = 0, then the equation is solvable by radicals,--in this extreme case rationally. Such cases of solvability are self-evident; but they are enough to show that the general theorem of the non-solvability by radicals of an equation of the fifth or any higher order does not in any wise exclude for such orders the existence of particular equations solvable by radicals, and there are, in fact, extensive classes of equations which are thus solvable; the binomial equations x^n - 1 = 0 present an instance.

22. It has already been shown how the several roots of the equation x^n - 1 = 0 can be expressed in the form cos 2s[pi]/n + i sin 2s[pi]/n, but the question is now that of the algebraical solution (or solution by radicals) of this equation. There is always a root = 1; if [omega] be any other root, then obviously [omega], [omega]^2, ... [omega]^(n - 1) are all of them roots; x^n - 1 contains the factor x - 1, and it thus appears that [omega], [omega]^2, ... [omega]^(n - 1) are the n - 1 roots of the equation

x^(n - 1) + x^(n - 2) + ... x + 1 = 0;

we have, of course, [omega]^(n - 1) + [omega]^(n - 2) + ... + [omega] + 1 = 0.

It is proper to distinguish the cases n prime and n composite; and in the latter case there is a distinction according as the prime factors of n are simple or multiple. By way of illustration, suppose successively n = 15 and n = 9; in the former case, if [alpha] be an imaginary root of x^3 - 1 = 0 (or root of x^2 + x + 1 = 0), and [beta] an imaginary root of x^5 - 1 = 0 (or root of x^4 + x^3 + x^2 + x + 1 = 0), then [omega] may be taken = [alpha][beta]; the successive powers thereof, [alpha][beta], [alpha]^2 [beta]^2, [beta]^3, [alpha][beta]^4, [alpha]^2, [beta], [alpha][beta]^2, [alpha]^2[beta]^3, [beta]^4, [alpha], [alpha]^2 [beta], [beta]^2, [alpha][beta]^3, [alpha]^2 [beta]^4, are the roots of x^14 + x^13 + ... + x + 1 = 0; the solution thus depends on the solution of the equations x^3 - 1 = 0 and x^5 - 1 = 0. In the latter case, if [alpha] be an imaginary root of x^3 - 1 = 0 (or root of x^2 + x + 1 = 0), then the equation x^9 - 1 = 0 gives x^3 = 1, [alpha], or [alpha]^2; x^3 = 1 gives x = 1, [alpha], or [alpha]^2; and the solution thus depends on the solution of the equations x^3 - 1 = 0, x^3 - [alpha] = 0, x^3 - [alpha]^2 = 0. The first equation has the roots 1, [alpha], [alpha]^2; if [beta] be a root of either of the others, say if [beta]^3 = [alpha], then assuming [omega] = [beta], the successive powers are [beta], [beta]^2, [alpha], [alpha][beta], [alpha][beta]^2, [alpha]^2, [alpha]^2[beta], [alpha]^2 [beta]^2, which are the roots of the equation x^8 + x^7 + ... + x + 1 = 0.

It thus appears that the only case which need be considered is that of n a prime number, and writing (as is more usual) r in place of [omega], we have r, r^2, r^3, ... r^(n - 1) as the (n - 1) roots of the reduced equation

x^(n - 1) + x^(n - 2) + ... + x + 1 = 0;

then not only r^n - 1 = 0, but also r^(n - 1) + r^(n - 2) + ... + r + 1 = 0.

23. The process of solution due to Karl Friedrich Gauss (1801) depends essentially on the arrangement of the roots in a certain order, viz. not as above, with the indices of r in arithmetical progression, but with their indices in geometrical progression; the prime number n has a certain number of prime roots g, which are such that g^(n - 1) is the lowest power of g, which is [equivalent to] 1 to the modulus n; or, what is the same thing, that the series of powers 1, g, g^2, ... g^(n - 2), each divided by n, leave (in a different order) the remainders 1, 2, 3, ... n - 1; hence giving to r in succession the indices 1, g, g^2, ... g^(n - 2), we have, in a different order, the whole series of roots r, r^2, r^3, ... r^(n - 1).

In the most simple case, n = 5, the equation to be solved is x^4 + x^3 + x^2 + x + 1 = 0; here 2 is a prime root of 5, and the order of the roots is r, r^2, r^4, r^3. The Gaussian process consists in forming an equation for determining the periods P1, P2, = r + r^4 and r^2 + r^3 respectively;--these being such that the symmetrical functions P1 + P2, P1P2 are rationally determinable: in fact P1 + P2 = -1, P1P2 = (r + r^4)(r^2 + r^3), = r^3 + r^4 + r^6 + r^7, = r^3 + r^4 + r + r^2, = -1. P1, P2 are thus the roots of u^2 + u - 1 = 0; and taking them to be known, they are themselves broken up into subperiods, in the present case single terms, r and r^4 for P1, r^2 and r^3 for P2; the symmetrical functions of these are then rationally determined in terms of P1 and P2; thus r + r^4 = P1, r.r^4 = 1, or r, r^4 are the roots of u^2 - P1u + 1 = 0. The mode of division is more clearly seen for a larger value of n; thus, for n = 7 a prime root is = 3, and the arrangement of the roots is r, r^3, r^2, r^6, r^4, r^5. We may form either 3 periods each of 2 terms, P1, P2, P3 = r + r^6, r^3 + r^4, r^2 + r^5 respectively; or else 2 periods each of 3 terms, P1, P2 = r + r^2 + r^4, r^3 + r^6 + r^5 respectively; in each ease the symmetrical functions of the periods are rationally determinable: thus in the case of the two periods P1 + P2 = -1, P1P2 = 3 + r + r^2 + r^3 + r^4 + r^5 + r^6, = 2; and the periods being known the symmetrical functions of the several terms of each period are rationally determined in terms of the periods, thus r + r^2 + r^4 = P1, r.r^2 + r.r^4 + r^2.r^4 = P2, r.r^2.r^4 = 1.

The theory was further developed by Lagrange (1808), who, applying his general process to the equation in question, x^(n - 1) + x^(n - 2) + ... + x + 1 = 0 (the roots a, b, c ... being the several powers of r, the indices in geometrical progression as above), showed that the function (a + [omega]b + [omega]^2 c + ...)^(n - 1) was in this case a given function of [omega] with integer coefficients.

Reverting to the before-mentioned particular equation x^4 + x^3 + x^2 + x + 1 = 0, it is very interesting to compare the process of solution with that for the solution of the general quartic the roots whereof are a, b, c, d.

Take [omega], a root of the equation [omega]^4 - 1 = 0 (whence [omega] is = 1, -1, i, or -i, at pleasure), and consider the expression

(a + [omega]b + [omega]^2 c + [omega]^3 d)^4,

the developed value of this is

= a^4 + b^4 + c^4 + d^4 + 6(a^2 c^2 + b^2 d^2) + 12(a^2 bd + b^2 ca + c^2 db + d^2ac) +[omega] {4(a^3 b + b^3 c + c^3 + d^3 a) + 12(a^2 cd + b^2 da + c^2 ab + d^2 bc)} +[omega]^2{6(a^2 b^2 + b^2 c^2 + c^2 d^2 + d^2 a^2) + 4(a^3 c + b^3 d + c^3 a + d^3 b) + 24abcd} +[omega]^3{4(a^3 d + b^3 a + c^3 b + d^3 c) + 12(a^2 bc + b^2 cd + c^2 da + d^2 ab)}

that is, this is a 6-valued function of a, b, c, d, the root of a sextic (which is, in fact, solvable by radicals; but this is not here material).

If, however, a, b, c, d denote the roots r, r^2, r^4, r^3 of the special equation, then the expression becomes

r^4 + r^3 + r + r^2 + 6(1 + 1)+12(r^2 + r^4 + r^3 + r) + [omega] {4(1 + 1 + 1 + 1) + 12(r^4 + r^3 + r + r^2)} + [omega]^2{6(r + r^2 + r^4 + r^3) + 4(r^2 + r^4 + r^3 + r)} + [omega]^3{4(r + r^2 + r^4 + r^3) + 12(r^3 + r + r^2 + r^4)}

viz. this is

= -1 + 4[omega] + 14[omega]^2 - 16[omega]^3,

a completely determined value. That is, we have

(r + [omega]r^2 + [omega]^2 r^4 + [omega]^3 r^3) = -1 + 4[omega] + 14[omega]^2 - 16[omega]^3,

which result contains the solution of the equation. If [omega] = 1, we have (r + r^2 + r^4 + r^3)^4 = 1, which is right; if [omega] = -1, then (r + r^4 - r^2 - r^3)^4 = 25; if [omega] = i, then we have {r - r^4 + i(r^2 - r^3)}^4 = -15 + 20i; and if [omega] = -i, then {r - r^4 - i(r^2 - r^3)}^4 = -15 - 20i; the solution may be completed without difficulty.

The result is perfectly general, thus:--n being a prime number, r a root of the equation x^(n - 1) + x^(n - 2) + ... + x + 1 = 0, [omega] a root of [omega]^(n - 1) - 1 = 0, and g a prime root of g^(n - 1) [equivalent] 1 (mod. n), then

{r + [omega]r^g + ... + [omega]^(n - 2) r^g^(n - 2)}^(n - 1)

is a given function M0 + M1[omega] ... + M_(n - 2)[omega]^(n - 2) with integer coefficients, and by the extraction of (n - 1)th roots of this and similar expressions we ultimately obtain r in terms of [omega], which is taken to be known; the equation x^n - 1 = 0, n a prime number, is thus solvable by radicals. In particular, if n - 1 be a power of 2, the solution (by either process) requires the extraction of square roots only; and it was thus that Gauss discovered that it was possible to construct geometrically the regular polygons of 17 sides and 257 sides respectively. Some interesting developments in regard to the theory were obtained by C.G.J. Jacobi (1837); see the memoir "Ueber die Kreistheilung, u.s.w.," _Crelle_, t. xxx. (1846).

The equation x^(n - 1) + ... + x + 1 = 0 has been considered for its own sake, but it also serves as a specimen of a class of equations solvable by radicals, considered by N.H. Abel (1828), and since called Abelian equations, viz. for the Abelian equation of the order n, if x be any root, the roots are x, [theta]x, [theta]^2 x, ... [theta]^(n - 1)x ([theta]x being a rational function of x, and [theta]^nx = x); the theory is, in fact, very analogous to that of the above particular case.

A more general theorem obtained by Abel is as follows:--If the roots of an equation of any order are connected together in such wise that _all_ the roots can be expressed rationally in terms of any one of them, say x; if, moreover, [theta]x, [theta]1x being any two of the roots, we have [theta][theta]1x = [theta]1[theta]x, the equation will be solvable algebraically. It is proper to refer also to Abel's definition of an _irreducible_ equation:--an equation [phi]x = 0, the coefficients of which are rational functions of a certain number of known quantities a, b, c ..., is called irreducible when it is impossible to express its roots by an equation of an inferior degree, the coefficients of which are also rational functions of a, b, c ... (or, what is the same thing, when [phi]x does not break up into factors which are rational functions of a, b, c ...). Abel applied his theory to the equations which present themselves in the division of the elliptic functions, but not to the modular equations.

24. But the theory of the algebraical solution of equations in its most complete form was established by Evariste Galois (born October 1811, killed in a duel May 1832; see his collected works, _Liouville_, t. xl., 1846). The definition of an irreducible equation resembles Abel's,--an equation is reducible when it admits of a rational divisor, irreducible in the contrary case; only the word _rational_ is used in this extended sense that, in connexion with the coefficients of the given equation, or with the irrational quantities (if any) whereof these are composed, he considers any number of other irrational quantities called "adjoint radicals," and he terms rational any rational function of the coefficients (or the irrationals whereof they are composed) and of these adjoint radicals; the epithet irreducible is thus taken either absolutely or in a relative sense, according to the system of adjoint radicals which are taken into account. For instance, the equation x^4 + x^3 + x^2 + x + 1 = 0; the left hand side has here no rational divisor, and the equation is irreducible; but this function is = (x^2 + 1/2 x + 1)^2 -(5/4)x^2, and it has thus the irrational divisors x^2 + 1/2(1 + [root]5)x + 1, x^2 + 1/2(1 - [root]5)x + 1; and these, if we _adjoin_ the radical [root]5, are rational, and the equation is no longer irreducible. In the case of a given equation, assumed to be irreducible, the problem to solve the equation is, in fact, that of finding radicals by the adjunction of which the equation becomes reducible; for instance, the general quadric equation x^2 + px + q = 0 is irreducible, but it becomes reducible, breaking up into rational linear factors, when we adjoin the radical [root](1/4 p^2 - q).

The fundamental theorem is the Proposition I. of the "Memoire sur les conditions de resolubilite des equations par radicaux"; viz. given an equation of which a, b, c ... are the m roots, there is always a group of permutations of the letters a, b, c ... possessed of the following properties:--

1. Every function of the roots invariable by the substitutions of the group is rationally known.

2. Reciprocally every rationally determinable function of the roots is invariable by the substitutions of the group.

Here by an invariable function is meant not only a function of which the form is invariable by the substitutions of the group, but further, one of which the value is invariable by these substitutions: for instance, if the equation be [phi](x) = 0, then [phi](x) is a function of the roots invariable by any substitution whatever. And in saying that a function is rationally known, it is meant that its value is expressible rationally in terms of the coefficients and of the adjoint quantities.

For instance in the case of a general equation, the group is simply the system of the 1.2.3 ... n permutations of all the roots, since, in this case, the only rationally determinable functions are the symmetric functions of the roots.

In the case of the equation x^(n - 1) ... + x + 1 = 0, n a prime number, a, b, c ... k = r, r^g, r^g^2 ... r^g^(n - 2), where g is a prime root of n, then the group is the cyclical group abc ... k, bc ... ka, ... kab ... j, that is, in this particular case the number of the permutations of the group is equal to the order of the equation.

This notion of the group of the original equation, or of the group of the equation as varied by the adjunction of a series of radicals, seems to be the fundamental one in Galois's theory. But the problem of solution by radicals, instead of being the sole object of the theory, appears as the first link of a long chain of questions relating to the transformation and classification of irrationals.

Returning to the question of solution by radicals, it will be readily understood that by the adjunction of a radical the group may be diminished; for instance, in the case of the general cubic, where the group is that of the six permutations, by the adjunction of the square root which enters into the solution, the group is reduced to abc, bca, cab; that is, it becomes possible to express rationally, in terms of the coefficients and of the adjoint square root, any function such as a^2 b + b^2 c + c^2 a which is not altered by the cyclical substitution a into b, b into c, c into a. And hence, to determine whether an equation of a given form is solvable by radicals, the course of investigation is to inquire whether, by the successive adjunction of radicals, it is possible to reduce the original group of the equation so as to make it ultimately consist of a single permutation.

The condition in order that an equation of a given prime order n may be solvable by radicals was in this way obtained--in the first instance in the form (scarcely intelligible without further explanation) that every function of the roots x1, x2 ... x_n, invariable by the substitutions x_(ak + b) for x_k, must be rationally known; and then in the equivalent form that the resolvent equation of the order 1.2 ... (n - 2) must have a rational root. In particular, the condition in order that a quintic equation may be solvable is that Lagrange's resolvent of the order 6 may have a rational factor, a result obtained from a direct investigation in a valuable memoir by E. Luther, _Crelle_, t. xxxiv. (1847).

Among other results demonstrated or announced by Galois may be mentioned those relating to the modular equations in the theory of elliptic functions; for the transformations of the orders 5, 7, 11, the modular equations of the orders 6, 8, 12 are depressible to the orders 5, 7, 11 respectively; but for the transformation, n a prime number greater than 11, the depression is impossible.

The general theory of Galois in regard to the solution of equations was completed, and some of the demonstrations supplied by E. Betti (1852). See also J.A. Serret's _Cours d'algebre superieure_, 2nd ed. (1854); 4th ed. (1877-1878).

25. Returning to quintic equations, George Birch Jerrard (1835) established the theorem that the general quintic equation is by the extraction of only square and cubic roots reducible to the form x^5 + ax + b = 0, or what is the same thing, to x^5 + x + b = 0. The actual reduction by means of Tschirnhausen's theorem was effected by Charles Hermite in connexion with his elliptic-function solution of the quintic equation (1858) in a very elegant manner. It was shown by Sir James Cockle and Robert Harley (1858-1859) in connexion with the Jerrardian form, and by Arthur Cayley (1861), that Lagrange's resolvent equation of the sixth order can be replaced by a more simple sextic equation occupying a like place in the theory.

The theory of the modular equations, more particularly for the case n = 5, has been studied by C. Hermite, L. Kronecker and F. Brioschi. In the case n = 5, the modular equation of the order 6 depends, as already mentioned, on an equation of the order 5; and conversely the general quintic equation may be made to depend upon this modular equation of the order 6; that is, assuming the solution of this modular equation, we can solve (not by radicals) the general quintic equation; this is Hermite's solution of the general quintic equation by elliptic functions (1858); it is analogous to the before-mentioned trigonometrical solution of the cubic equation. The theory is reproduced and developed in Brioschi's memoir, "Uber die Auflosung der Gleichungen vom funften Grade," _Math. Annalen_, t. xiii. (1877-1878).

26. The modern work, reproducing the theories of Galois, and exhibiting the theory of algebraic equations as a whole, is C. Jordan's _Traite des substitutions et des equations algebriques_ (Paris, 1870). The work is divided into four books--book i., preliminary, relating to the theory of congruences; book ii. is in two chapters, the first relating to substitutions in general, the second to substitutions defined analytically, and chiefly to linear substitutions; book iii. has four chapters, the first discussing the principles of the general theory, the other three containing applications to algebra, geometry, and the theory of transcendents; lastly, book iv., divided into seven chapters, contains a determination of the general types of equations solvable by radicals, and a complete system of classification of these types. A glance through the index will show the vast extent which the theory has assumed, and the form of general conclusions arrived at; thus, in book iii., the algebraical applications comprise Abelian equations, equations of Galois; the geometrical ones comprise Q. Hesse's equation, R.F.A. Clebsch's equations, lines on a quartic surface having a nodal line, singular points of E.E. Kummer's surface, lines on a cubic surface, problems of contact; the applications to the theory of transcendents comprise circular functions, elliptic functions (including division and the modular equation), hyperelliptic functions, solution of equations by transcendents. And on this last subject, solution of equations by transcendents, we may quote the result--"the solution of the general equation of an order superior to five cannot be made to depend upon that of the equations for the division of the circular or elliptic functions"; and again (but with a reference to a possible case of exception), "the general equation cannot be solved by aid of the equations which give the division of the hyperelliptic functions into an odd number of parts." (See also GROUPS, THEORY OF.) (A. Ca.)

BIBLIOGRAPHY.--For the general theory see W.S. Burnside and A.W. Panton, _The Theory of Equations_ (4th ed., 1899-1901); the Galoisian theory is treated in G.B. Matthews, _Algebraic Equations_ (1907). See also the _Ency. d. math. Wiss._ vol. ii.

FOOTNOTES:

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

[2] The third edition (1826) is a reproduction of that of 1808; the first edition has the date 1798, but a large part of the contents is taken from memoirs of 1767-1768 and 1770-1771.

[3] The earlier demonstrations by Euler, Lagrange, &c, relate to the case of a numerical equation with real coefficients; and they consist in showing that such equation has always a real quadratic divisor, furnishing two roots, which are either real or else conjugate imaginaries [alpha] + [beta]i (see Lagrange's _Equations numeriques_).

[4] The square root of [alpha] + [beta]i can be determined by the extraction of square roots of positive real numbers, without the trigonometrical tables.

EQUATION OF THE CENTRE, in astronomy, the angular distance, measured around the centre of motion, by which a planet moving in an ellipse deviates from the mean position which it would occupy if it moved uniformly. Its amount is the correction which must be applied positively or negatively to the mean anomaly in order to obtain the true anomaly. It arises from the ellipticity of the orbit, is zero at pericentre and apocentre, and reaches its greatest amount nearly midway between these points. (See ANOMALY and ORBIT.)

EQUATION OF TIME, the difference between apparent time, determined by the meridian passage of the real sun, and mean time, determined by the passage of the mean sun. It goes through a double period in the course of a year. Its amount varies a fraction of a minute for the same date, from year to year and from one longitude to another, on the same day. The following table shows an average value for any date and for the Greenwich meridian for a number of years, from which the actual value will seldom deviate more than 20 seconds until after 1950. The + sign indicates that the real sun reaches the meridian _after_ mean noon; the - sign _before_ mean noon.

_Table of the Equation of Time._

m. s. m. s. m. s. Jan. 1 +3 26 Mar. 1 +12 39 May 1 -2 55 6 5 45 6 11 35 6 -3 27 11 7 51 11 10 20 11 -3 46 16 9 43 16 8 58 16 -3 51 21 11 19 21 7 30 21 -3 40 26 12 36 26 5 59 26 -3 16

Feb. 1 +13 42 Apr. 1 +4 9 June 1 -2 32 6 14 14 6 2 40 6 -1 44 11 14 25 11 +1 15 11 -0 48 16 14 17 16 -0 3 16 +0 14 21 13 52 21 -1 12 21 1 19 26 13 11 26 -2 10 26 2 24

July 1 +3 26 Sept. 1 +0 9 Nov. 1 -16 18 6 4 21 6 -1 28 6 -16 19 11 5 8 11 -3 10 11 -15 58 16 5 44 16 -4 55 16 -15 15 21 6 8 21 -6 41 21 -14 12 26 6 18 26 -8 25 26 -12 49

Aug. 1 +6 10 Oct. 1 -10 5 Dec. 1 -11 7 6 5 47 6 -11 38 6 - 9 9 11 5 9 11 -13 2 11 - 6 57 16 4 17 16 -14 14 16 - 4 35 21 3 12 21 -15 11 21 - 2 7 26 1 55 26 -15 52 26 + 0 23

EQUATOR (Late Lat. _aequator_, from _aequare_, to make equal), in geography, that great circle of the earth, equidistant from the two poles, which divides the northern from the southern hemisphere and lies in a plane perpendicular to the axis of the earth; this is termed the "geographical" or "terrestrial equator." In astronomy, the "celestial equator" is the name given to the great circle in which the plane of the terrestrial equator intersects the celestial sphere; it is consequently equidistant from the celestial poles. The "magnetic equator" is an imaginary line encircling the earth, along which the vertical component of the earth's magnetic force is zero; it nearly coincides with the terrestrial equator.

EQUERRY (from the Fr. _ecurie_, a stable, through its older form _escurie_, from the Med. Lat. _scuria_, a word of Teutonic origin for a stable or shed, cf. Ger. _Scheuer_; the modern spelling has confused the word with the Lat. _equus_, a horse), a contracted form of "gentleman of the equerry," an officer in charge of the stables of a royal household. At the British court, equerries are officers attached to the department of the master of the horse, the first of whom is called chief equerry (see HOUSEHOLD, ROYAL).

EQUIDAE, the family of perissodactyle ungulate mammals typified by the horse (_Equus caballus_); see HORSE. According to the older classification this family was taken to include only the forms with tall-crowned teeth, more or less closely allied to the typical genus _Equus_. There is, however, such an almost complete graduation from the former to earlier and more primitive mammals with short-crowned cheek-teeth, at one time included in the family _Lophiodontidae_ (see PERISSODACTYLA), that it has now become a very general practice to include the whole "phylum" in the family _Equidae_. The _Equidae_, in this extended sense, together with the extinct _Palaeotheriidae_, are indeed now regarded as forming one of four main groups into which the Perissodactyla are divided, the other groups being the Tapiroidea, Rhinocerotoidea and Titanotheriide. For the horse-group the name Hippoidea is employed. All four groups were closely connected in the Lower Eocene, so that exact definition is almost impossible.

In the Hippoidea there is generally the full series of 44 teeth, but the first premolar is often deciduous or wanting in the lower or in both jaws. The incisors are chisel-shaped, and the canines tend to become isolated so as in the now specialized forms to occupy nearly the middle of a longer or shorter gap between the incisors and premolars. In the upper molars the two outer columns of the primitive tubercular molar coalesce to form an outer wall, from which proceed two crescentic transverse crests; the connexion between the crests and the wall being imperfect or slight, and the crests themselves sometimes tubercular. Each of the lower molars carries two crescentic ridges. The number of toes ranges from four to one in the fore-foot, and from three to one in the hind-foot. The paroccipital, postglenoid and post-tympanic processes of the skull are large, and the latter always distinct. Normally there are no traces of horn-cores. The calcaneum lacks the facet for the fibula found in the Titanotheroidea.

In the earlier _Equidae_ the teeth were short-crowned, with the premolars simpler than the molars; but there is a gradual tendency to an increase in the height of the crowns of the teeth, accompanied by increasing complexity of structure and the filling up of the hollows with cement. Similarly the gap on each side of the canine tooth in each jaw continues to increase in length; while in all the later forms the orbit is surrounded by a ring of bone. A third modification is the increasing length of limb (as well as in general bodily size), accompanied by a gradual reduction in the number of toes from three or four to one.

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_.

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.

Be this as it may, the next North American representatives of the family constitute the genera _Protohippus_ and _Merychippus_ of the Miocene, in both of which the lateral digits are fully developed and terminate in small though perfect hoofs. In both the cheek-teeth have moderately tall crowns, and in the first named of the two those of the milk-series are nearly similar to their permanent successors. In _Merychippus_, on the other hand, the milk-molars have short crowns, without any cement in the hollows, thus resembling the permanent molars of the under-mentioned genus _Anchitherium_. From the well-known _Hipparion_, or _Hippotherium_, typically from the Lower Pliocene of Europe, but also occurring in the corresponding formation in North Africa, Persia, India and China, and represented in the Upper Miocene Loup Fork beds of the United States by species which it has been proposed to separate generically as _Neohipparion_, we reach small horses which are now generally regarded as a lateral offshoot from the _Merychippus_ type. The cheek-teeth, which have crowns of moderate height, differ from those of all the foregoing in that the postero-internal pillar (the projection on the right-hand top corner of c in fig. 2) is isolated in place of being attached by a narrow neck to the adjacent crescent. The skull, which is relatively short, has a large depression in front of the orbit, commonly supposed to have contained a gland, but this may be doubtful. In the typical, and also in the North American forms these were complete, although small, lateral toes in both feet (fig. 3, d), but it is possible that in _H. antilopinum_ of India the lateral toes had disappeared. If this be so, we have the development of a monodactyle foot in this genus independently of _Equus_.

The foregoing genera constitute the subfamily _Equinae_, or the _Equidae_ as restricted by the older writers. In all the dentition is of the hypsodont type, with the hollows of the cheek-teeth filled by cement, the premolars molariform, and the first small and generally deciduous. The orbit is surrounded by a bony ring; the ulna and radius in the fore, and the tibia and fibula in the hind-limb are united, and the feet are of the types described above. Between this subfamily and the second subfamily, _Hyracotheriinae_, a partial connexion is formed by the North American Upper Miocene genera _Desmatippus_ and _Anchippus_ or _Parahippus_. The characteristics of the group will be gathered from the remarks on the leading genera; but it may be mentioned that the orbit is open behind, the cheek-teeth are short-crowned and without cement (fig. 1, a), the gap between the canine and the outermost incisor is short, the bones of the middle part of the leg are separate, and there are at least three toes to each foot.

The longest-known genus and the one containing the largest species is _Anchitherium_, typically from the Middle Miocene of Europe, but also represented by one species from the Upper Miocene of North America. The European _A. aurelianense_ was of the size of an ordinary donkey. The cheek-teeth are of the type shown in a of figs. 1 and 2; the premolars, with the exception of the small first one, being molar-like; and the lateral toes (fig. 3, c) were to some extent functional. The summits of the incisors were infolded to a small extent. Nearly allied is the American _Mesohippus_, ranging from the Lower Miocene to the Lower Oligocene of the United States, of which the earliest species stood only about 18 in. at the shoulder. The incisors were scarcely, if at all, infolded, and there is a rudiment of the fifth metacarpal (fig. 3, b). By some writers all the species of _Mesohippus_ are included in the genus _Miohippus_, but others consider that the two genera are distinct.

_Mesohippus_ and _Miohippus_ are connected with the earliest and most primitive mammal which it is possible to include in the family _Equidae_ by means of _Epihippus_ of the Uinta or Upper Eocene of North America, and _Pachynolophus_, or _Orohippus_, of the Middle and Lower Eocene of both halves of the northern hemisphere. The final stage, or rather the initial stage, in the series is presented by _Hyracotherium_ (_Protorohippus_), a mammal no larger than a fox, common to the Lower Eocene of Europe and North America. The general characteristics of this progenitor of the horses are those given above as distinctive of the group. The cheek-teeth are, however, much simpler than those of _Anchitherium_; the transverse crests of the upper molars not being fully connected with the outer wall, while the premolars in the upper jaw are triangular, and thus unlike the molars. The incisors are small and the canines scarcely enlarged; the latter having a gap on each side in the lower, but only one on their hinder aspect in the upper jaw. The fore-feet have four complete toes (fig. 3, a), but there are only three hind-toes, with a rudiment of the fifth metatarsal. The vertebrae are simpler in structure than in _Equus_. From _Hyracotherium_, which is closely related to the Eocene representatives of the ancestral stocks of the other three branches of the Perissodactyla, the transition is easy to _Phenacodus_, the representative of the common ancestor of all the Ungulata.

See also H.F. Osborn, "New Oligocene Horses," _Bull. Amer. Mus._ vol. xx. p. 167 (1904); J.W. Gidley, _Proper Generic Names of Miocene Horses_, p. 191; and the article PALAEONTOLOGY. (R. L.*)

EQUILIBRIUM (from the Lat. _aequus_, equal, and _libra_, a balance), a condition of equal balance between opposite or counteracting forces. By the "sense of equilibrium" is meant the sense, or sensations, by which we have a feeling of security in standing, walking, and indeed in all the movements by which the body is carried through space. Such a feeling of security is necessary both for maintaining any posture, such as standing, or for performing any movement. If this feeling is absent or uncertain, or if there are contradictory sensations, then definite muscular movements are inefficiently or irregularly performed, and the body may stagger or fall. When we stand erect on a firm surface, like a floor, there is a feeling of resistance, due to nervous impulses reaching the brain from the soles of the feet and from the muscles of the limbs and trunk. In walking or running, these feelings of resistance seem to precede and guide the muscular movements necessary for the next step. If these are absent or perverted or deficient, as is the case in the disease known as locomotor ataxia, then, although there is no loss of the power of voluntary movement, the patient staggers in walking, especially if he is not allowed to look at his feet, or if he is blind-folded. He misses the guiding sensations that come from the limbs; and with a feeling that he is walking on a soft substance, offering little or no resistance, he staggers, and his muscular movements become irregular. Such a condition maybe artificially brought about by washing the soles of the feet with chloroform or ether. And it has been observed to exist partially after extensive destruction of the skin of the soles of the feet by burns or scalds. This shows that tactile impulses from the skin take a share in generating the guiding sensation. In the disease above mentioned, however, tactile impressions may be nearly normal, but the guiding sensation is weak and inefficient, owing to the absence of impulses from the muscles. The disease is known to depend on morbid changes in the posterior columns of the spinal cord, by which impulses are not freely transmitted upwards to the brain. These facts point to the existence of impulses coming from the muscles and tendons. It is now known that there exist peculiar spindles, in muscle, and rosettes or coils or loops of nerve fibres in close proximity to tendons. These are the end organs of the sense. The transmission of impulses gives rise to the _muscular sense_, and the guiding sensation which precedes co-ordinated muscular movements depends on these impulses. Thus from the limbs streams of nervous impulses pass to the sensorium from the skin and from muscles and tendons; these may or may not arouse consciousness, but they guide or evoke muscular movements of a co-ordinated character, more especially of the limbs.

In animals whose limbs are not adapted for delicate touch nor for the performance of complicated movements, such as some mammals and birds and fishes, the guiding sensations depend largely on the sense of vision. This sense in man, instead of assisting, sometimes disturbs the guiding sensation. It is true that in locomotor ataxia visual sensations may take the place of the tactile and muscular sensations that are inefficient, and the man can walk without staggering if he is allowed to look at the floor, and especially if he is guided by transverse straight lines. On the other hand, the acrobat on the wire-rope dare not trust his visual sensations in the maintenance of his equilibrium. He keeps his eyes fixed on one point instead of allowing them to wander to objects below him, and his muscular movements are regulated by the impulses that come from the skin and muscles of his limbs. The feeling of insecurity probably arises from a conception of height, and also from the knowledge that by no muscular movements can a man avoid a catastrophe if he should fall. A bird, on the other hand, depends largely on visual impressions, and it knows by experience that if launched into the air from a height it can fly. Here, probably, is an explanation of the large size of the eyes of birds. Cover the head, as in hooding a falcon, and the bird seems to be deprived of the power of voluntary movement. Little effect will be produced if we attempt to restrain the movements of a cat by covering its eyes. A fish also is deprived of the power of motion if its eyes are covered. But both in the bird and in the fish tactile and muscular impressions, especially the latter, come into play in the mechanism of equilibrium. In flight the large-winged birds, especially in soaring, can feel the most delicate wind-pressures, both as regards direction and force, and they adapt the position of their body so as to catch the pressure at the most efficient angle. The same is true of the fish, especially of the flat-fishes. In mammals the sense of equilibrium depends, then, on streams of tactile, muscular and visual impressions pouring in on the sensorium, and calling forth appropriate muscular movements. It has also been suggested that impulses coming from the abdominal viscera may take part in the mechanism. The presence in the mesentery of felines (cats, &c.) of large numbers of Pacinian corpuscles, which are believed to be modified tactile bodies, favours this supposition. Such animals are remarkable for the delicacy of such muscular movements, as balancing and leaping.

There is another channel by which nervous impulses reach the sensorium and play their part in the sense of equilibrium, namely, from the semicircular canals, a portion of the internal ear. It is pointed out in the article HEARING that the appreciation of sound is in reality an appreciation of variations of pressure. The labyrinth consists of the vestibule, the cochlea and the semicircular canals. The cochlea receives the sound-waves (variations of pressure) that constitute musical tones. This it accomplishes by the structures in the ductus cochlearis. In the vestibule we find two sacs, the saccule next to and communicating with the ductus cochlearis, and the utricle communicating with the semicircular canals. The base of the stapes communicates pressures to the utricle. The membranous portion of the semicircular canals consists of a tube, dilated at one end into a swelling or pouch, termed the ampulla, and each end communicates freely with the utricle. On the posterior wall of both the saccule and of the utricle there is a ridge, termed in each case the macula acustica, bearing a highly specialized epithelium. A similar structure exists in each ampulla. This would suggest that all three structures have to do with hearing; but, on the other hand, there is experimental evidence that the utricle and the canals may transmit impressions that have to do with equilibrium. Pressure of the base of the stapes is exerted on the utricle. This will compress the fluid in that cavity, and tend to drive the fluid into the semicircular canals that communicate with that cavity by five openings. Each canal is surrounded by a thin layer of perilymph, so that it may yield a little to this pressure, and exert a pull or pressure on the nerve-endings in each ampulla. Thus impulses may be generated in the nerves of the ampullae.

The three semicircular canals lie in the three directions in space, and it has been suggested that they have to do with our appreciation of the direction of sound. But our appreciation of sound is very inaccurate: we look with the eyes for the source of a sound, and instinctively direct the ears or the head, or both, in the direction from which the sound appears to proceed. But the relationship of the canals on the two sides must have a physiological significance. Thus (1) the six canals are parallel, two and two; or (2) the two horizontal canals are in the same plane, while the superior canal on one side is nearly parallel with the posterior canal of the other. These facts point to the two sets of canals and ampullae acting as one organ, in a manner analogous to the action of two retinae for single vision.

We have next to consider how the canals may possibly act in connexion with the sense of equilibrium. In 1820 J. Purkinje studied the vertigo that follows rapid rotation of the body in the erect position on a vertical axis. On stopping the rotation there is a sense of rotation in the opposite direction, and this may occur even when the eyes are closed. Purkinje noticed that the position of the imaginary axis of rotation depends on the axis around which the head revolves. In 1828 M.J.P. Flourens discovered that injury to the canals causes disturbance to the equilibrium and loss of co-ordination, and that sections of the canals produce a rotatory movement of a kind corresponding to the canal that had been divided. Thus division of a membranous canal causes rotatory movements round an axis at right angles to the plane of the divided canal. The body of the animal always moves in the direction of the cut canal. Many other observers have corroborated these experiments. F. Goltz was the first who formulated the conditions necessary for equilibration. He put the matter thus:--(1) A central co-ordinating organ--in the brain; (2) centripetal fibres, with their peripheral terminations--in the ampullae; and (3) centrifugal fibres, with their terminal organs--in the muscular mechanisms. A lesion of any one of these portions of the mechanism causes loss or impairment of balancing. Cyon also investigated the subject, and concluded:--(1) To maintain equilibrium, we must have an accurate notion of the position of the head in space; (2) the function of the semicircular canals is to communicate impressions that give a representation of this position--each canal having a relation to one of the dimensions of space; (3) disturbance of equilibrium follows section; (4) involuntary movements following section are due to abnormal excitations; (5) abnormal movements occurring a few days after the operation are caused by irritation of the cerebellum.

On theoretical considerations of a physical character, E. Mach, Crum-Brown and Breuer have advanced theories based on the idea of the canals being organs for sensations of acceleration of movement, or for the sense of rotation. Mach first pointed out that Purkinje's phenomena, already alluded to, were in all probability related to the semicircular canals. "He showed that when the body is moved in space, in a straight line, we are not conscious of the velocity of motion, but of variations in this velocity. Similarly, if a body is rotated round a vertical axis, we perceive only angular acceleration and not angular velocity. The sensations produced by angular acceleration last longer than the acceleration itself, and the position of the head during the movements enables us to determine direction." Both Mach and Goltz state that varying pressures of the fluid in the canals produced by angular rotation produce sensations of movement (always in a direction opposite to the rotation of the body), and that these, in turn, cause the vertigo of Purkinje and the phenomena of Flourens. Mach, Crum-Brown and Breuer advance hydrodynamical theories in which they assume that the fluids move in the canals. Goltz, on the other hand, supports a hydrostatical theory in which he assumes that the phenomena can be accounted for by varying pressures. Crum-Brown differs from Mach and Breuer as follows:--(1) In attributing movement or variation of pressure not merely to the endolymph, but also to the walls of the membranous canals and to the surrounding perilymph; and (2) in regarding the two labyrinths as one organ, all the six canals being required to form a true conception of the rotating motion of the head. He sums up the matter thus: "We have two ways in which a relative motion can occur between the endolymph and the walls of the cavity containing it--(1) When the head begins to move, here the walls leave the fluid behind; (2) when the head stops, here the fluid flows on. In both cases the sensation of rotation is felt. In the first this sensation corresponds to a real rotation, in the second it does not, but in both it corresponds to a real acceleration (positive or negative) of rotation, using the word acceleration in its technical kinematical sense."

Cyon states that the semicircular canals only indirectly assist in giving a notion of spatial relations. "He holds that knowledge of the position of bodies in space depends on nervous impulses coming from the contracting ocular muscles; that the oculomotor centres are in intimate physiological relationship with the centres receiving impulses from the nerves of the semicircular canals; and that the oculomotor centres, thus excited, produce the movements of the eyeballs, which then determine our notions of spatial relations." These views are supported by experiments of Lee on dog-fish. When the fish is rotated round different axes there are compensating movements of the eyes and fins. "It was observed that if the fish were rotated in the plane of one of the canals, exactly the same movements of the eyes and fins occurred as were produced by experimental operation and stimulation of the ampulla of that canal." Sewall, in 1883, carried out experiments on young sharks and skates with negative results. Lee returned to the subject in 1894, and, after numerous experiments on dog-fish, in which the canals or the auditory nerves were divided, obtained evidence that the ampullae contain sense-organs connected with the sense of equilibrium.

It has been found by physicians and aurists that disease or injury of the canals, occurring rapidly, produces giddiness, staggering, nystagmus (a peculiar twitching movement of the muscles of the eyeballs), vomiting, noises in the ear and more or less deafness. It is said, however, that if pathological changes come on slowly, so that the canals and vestibule are converted into a solid mass, none of these symptoms may occur. On the whole, the evidence is in favour of the view that from the semicircular canals nervous impulses are transmitted, which, co-ordinated with impulses coming from the visual organs, from the muscles and from the skin, form the bases of these guiding sensations on which the sense of equilibrium depends. These impulses may not reach the level of consciousness, but they call into action co-ordinated mechanisms by which complicated muscular movements are effected.

Full bibliographical references are given in the article on "The Ear" by J.G. McKendrick, in Schafer's _Textbook of Physiology_, vol. ii. p. 1194. (J. G. M.)

EQUINOX (from the Lat. _aequus_, equal, and _nox_, night), a term used to express either the moment at which, or the point at which, the sun apparently crosses the celestial equator. Since the sun moves in the ecliptic, it is in the last-named sense the point of intersection of the ecliptic and the celestial equator. This is the usual meaning of the term in astronomy. There are two such points, opposite each other, at one of which the sun crosses the equator toward the north and at the other toward the south. They are called vernal and autumnal respectively, from the relation of the corresponding times to the seasons of the northern hemisphere. The line of the equinoxes is the imaginary diameter of the celestial sphere which joins them.

The vernal equinox is the initial point from which the right ascensions and the longitudes of the heavenly bodies are measured (see ASTRONOMY: _Spherical_). It is affected by the motions of Precession and Nutation, of which the former has been known since the time of Hipparchus. The actual equinox is defined by first taking the conception of a fictitious point called the Mean Equinox, which moves at a nearly uniform rate, slow varying, however, from century to century. The true equinox then moves around the mean equinox in a period equal to that of the moon's nodes. These two motions are defined with greater detail in the articles PRECESSION OF THE EQUINOXES and NUTATION.

_Equinoctial Gales._--At the time of the equinox it is commonly believed that strong gales may be expected. This popular idea has no foundation in fact, for continued observations have failed to show any unusual prevalence of gales at this season. In one case observations taken for fifty years show that during the five days from the 21st to the 25th of March and September, there were fewer gales and storms than during the preceding and succeeding five days.

EQUITES ("horsemen" or "knights," from _equus_, "horse"), in Roman history, originally a division of the army, but subsequently a distinct political order, which under the empire resumed its military character. According to the traditional account, Romulus instituted a cavalry corps, consisting of three _centuriae_ ("hundreds"), called after the three tribes from which they were taken (Ramnes, Tities, Luceres), divided into ten _turmae_ ("squadrons") of thirty men each. The collective name for the corps was _celeres_ ("the swift," or possibly from [Greek: keles], "a riding horse"); Livy, however, restricts the term to a special body-guard of Romulus. The statements in ancient authorities as to the changes in the number of the equites during the regal period are very confusing; but it is regarded as certain that Servius Tuillus found six centuries in existence, to which he added twelve, making eighteen in all, a number which remained unchanged throughout the republican period. A proposal by M. Porcius Cato the elder to supplement the deficiency in the cavalry by the creation of four additional centuries was not adopted. The earlier centuries were called _sex suffragia_ ("the six votes"), and at first consisted exclusively of patricians, while those of Servius Tullius were entirely or for the most part plebeian. Until the reform of the comitia centuriata (probably during the censorship of Gaius Flaminius in 220 B.C.; see COMITIA), the equites had voted first, but after that time this privilege was transferred to one century selected by lot from the centuries of the equites and the first class. The equites then voted with the first class, the distinction between the _sex suffragia_ and the other centuries being abolished.

Although the equites were selected from the wealthiest citizens, service in the cavalry was so expensive that the state gave financial assistance. A sum of money (_aes equestre_) was given to each eques for the purchase of two horses (one for himself and one for his groom), and a further sum for their keep (_aes hordearium_); hence the name _equites equo publico_. In later times, pay was substituted for the _aes hordearium_, three times as much as that of the infantry. If competent, an eques could retain his horse and vote after the expiration of his ten years' service, and (till 129 B.C.) even after entry into the senate.

As the demands upon the services of the cavalry increased, it was decided to supplement the regulars by the enrolment of wealthy citizens who kept horses of their own. The origin of these _equites equo privato_ dates back, according to Livy (v. 7), to the siege of Veii, when a number of young men came forward and offered their services. According to Mommsen, although the institution was not intended to be permanent, in later times vacancies in the ranks were filled in this manner, with the result that service in the cavalry, with either a public or a private horse, became obligatory upon all Roman citizens possessed of a certain income. These _equites equo privato_ had no vote in the centuries, received pay in place of the _aes equestre_, and did not form a distinct corps.

Thus, at a comparatively early period, three classes of equites may be distinguished: (a) The patrician equites _equo publico_ of the _sex suffragia_; (b) the plebeian equites in the twelve remaining centuries; (c) the equites _equo privato_, both patrician and plebeian.

The equites were originally chosen by the curiae, then in succession by the kings, the consuls, and (after 443 B.C.) by the censors, by whom they were reviewed every five years in the Forum. Each eques, as his name was called out, passed before the censors, leading his horse. Those whose physique and character were satisfactory, and who had taken care of their horses and equipments, were bidden to lead their horse on (_traducere equum_), those who failed to pass the scrutiny were ordered to sell it, in token of their expulsion from the corps. This inspection (_recognitio_) must not be confounded with the full-dress procession (_transvectio_) on the 15th of July from the temple of Mars or Honos to the Capitol, instituted in 304 B.C. by the censor Q. Fabius Maximus Rullianus to commemorate the miraculous intervention of Castor and Pollux at the battle of Lake Regillus. Both inspection and procession were discontinued before the end of the republic, but revived and in a manner combined by Augustus.

In theory, the twelve plebeian centuries were open to all freeborn youths of the age of seventeen, although in practice preference was given to the members of the older families. Other requirements were sound health, high moral character and an honourable calling. At the beginning of the republican period, senators were included in the equestrian centuries. The only definite information as to the amount of fortune necessary refers to later republican and early imperial times, when it is known to have been 400,000 sesterces (about L3500 to L4000). The insignia of the equites were, at first, distinctly military--such as the purple-edged, short military cloak (_trabea_) and decorations for service in the field.

With the extension of the Roman dominions, the equites lost their military character. Prolonged service abroad possessed little attraction for the pick of the Roman youth, and recruiting for the cavalry from the equestrian centuries was discontinued. The equites remained at home, or only went out as members of the general's staff, their places being taken by the _equites equo privato_, the cavalry of the allies and the most skilled horsemen of the subject populations. The first gradually disappeared, and Roman citizens were rarely found in the ranks of the effective cavalry. In these circumstances there grew up in Rome a class of wealthy men, whose sole occupation it was to amass large fortunes by speculation, and who found a most lucrative field of enterprise in state contracts and the farming of the public revenues. These tax-farmers (see PUBLICANI) were already in existence at the time of the Second Punic War; and their numbers and influence increased as the various provinces were added to the Roman dominions. The change of the equites into a body of financiers was further materially promoted (a) by the lex Claudia (218 B.C.), which prohibited senators from engaging in commercial pursuits, especially if (as seems probable) it included public contracts (cf. FLAMINIUS, GAIUS); (b) by the enactment in the time of Gaius Gracchus excluding members of the senate from the equestrian centuries. These two measures definitely marked off the aristocracy of birth from the aristocracy of wealth--the landed proprietor from the capitalist. The term equites, originally confined to the purely military equestrian centuries of Servius Tullius, now came to be applied to all who possessed the property qualification of 400,000 sesterces.

As the equites practically monopolized the farming of the taxes, they came to be regarded as identical with the _publicani_, not, as Pliny remarks, because any particular rank was necessary to obtain the farming of the taxes, but because such occupation was beyond the reach of all except those who were possessed of considerable means. Thus, at the time of the Gracchi, these _equites-publicani_ formed a close financial corporation of about 30,000 members, holding an intermediate position between the nobility and the lower classes, keenly alive to their own interests, and ready to stand by one another when attacked. Although to some extent looked down upon by the senate as following a dishonourable occupation, they had as a rule sided with the latter, as being at least less hostile to them than the democratic party. To obtain the support of the capitalists, Gaius Gracchus conceived the plan of creating friction between them and the senate, which he carried out by handing over to them the control (a) of the jury-courts, and (b) of the revenues of Asia.

(a) Hitherto, the list of jurymen for service in the majority of processes, both civil and criminal, had been composed exclusively of senators. The result was that charges of corruption and extortion failed, when brought against members of that order, even in cases where there was little doubt of their guilt. The popular indignation at such scandalous miscarriages of justice rendered a change in the composition of the courts imperative. Apparently Gracchus at first proposed to create new senators from the equites and to select the jurymen from this mixed body, but this moderate proposal was rejected in favour of one more radical (see W.W. Fowler in _Classical Review_, July 1896). By the lex Sempronia (123 B.C.) the list was to be drawn from persons of free birth over thirty years of age, who must possess the equestrian census, and must not be senators. Although this measure was bound to set senators and equites at variance, it in no way improved the lot of those chiefly concerned. In fact, it increased the burden of the luckless provincials, whose only appeal lay to a body of men whose interests were identical with those of the _publicani_. Provided he left the tax-gatherer alone, the governor might squeeze what he could out of the people, while on the other hand, if he were humanely disposed, it was dangerous for him to remonstrate.

(b) The taxes of Asia had formerly been paid by the inhabitants themselves in the shape of a fixed sum. Gracchus ordered that the taxes, direct and indirect, should be increased, and that the farming of them should be put up to auction at Rome. By this arrangement the provincials were ignored, and everything was left in the hands of the capitalists.

From this time dates the existence of the equestrian order as an officially recognized political instrument. When the control of the courts passed into the hands of the property equites, all who were summoned to undertake the duties of judices were called equites; the _ordo judicum_ (the official title) and the _ordo equester_ were regarded as identical. It is probable that certain privileges of the equites were due to Gracchus; that of wearing the gold ring, hitherto reserved for senators; that of special seats in the theatre, subsequently withdrawn (probably by Sulla) and restored by the lex Othonis (67 B.C.); the narrow band of purple on the tunic as distinguished from the broad band worn by the senators.

Various attempts were made by the senate to regain control of the courts, but without success. The lex Livia of M. Livius Drusus (q.v.), passed with that object, but irregularly and by the aid of violence, was annulled by the senate itself. In 82 Sulla restored the right of serving as judices to the senate, to which he elevated 300 of the most influential equites, whose support he thus hoped to secure; at the same time he indirectly dealt a blow at the order generally, by abolishing the office of the censor (immediately revived), in whom was vested the right of bestowing the public horse. To this period Mommsen assigns the regulation, generally attributed to Augustus, that the sons of senators should be knights by right of birth. By the lex Aurelia (70 B.C.) the judices were to be chosen in equal numbers from senators, equites and tribuni aerarii (see AERARIUM), (the last-named being closely connected with the equites), who thus practically commanded a majority. About this time the influence of the equestrian order reached its height, and Cicero's great object was to reconcile it with the senate. In this he was successful at the time of the Catilinarian conspiracy, in the suppression of which he was materially aided by the equites. But the union did not last long; shortly afterwards the majority ranged themselves on the side of Julius Caesar, who did away with the tribuni aerarii as judices, and replaced them by equites.

Augustus undertook the thorough reorganization of the equestrian order on a military basis. The _equites equo privato_ were abolished (according to Herzog, not till the reign of Tiberius) and the term equites was officially limited to the _equites equo publico_, although all who possessed the property qualification were still considered to belong to the "equestrian order." For the _equites equo publico_ high moral character, good health and the equestrian fortune were necessary. Although free birth was considered indispensable, the right of wearing the gold ring (_jus anuli aurei_) was frequently bestowed by the emperor upon freedmen, who thereby became _ingenui_ and eligible as equites. Tiberius, however, insisted upon free birth on the father's side to the third generation. Extreme youth was no bar; the emperor Marcus Aurelius had been an eques at the age of six. The sons of senators were eligible by right of birth, and appear to have been known as _equites illustres_. The right of bestowing the _equus publicus_ was vested in the emperor; once given, it was for life, and was only forfeitable through degradation for some offence or the loss of the equestrian fortune.

Augustus divided the equites into six _turmae_ (regarded by Hirschfeld as a continuation of the _sex suffragia_). Each was under the command of a _sevir_ ([Greek: hilarchos]), who was appointed by the emperor and changed every year. During their term of command the _seviri_ had to exhibit games (_ludi sevirales_). Under these officers the equites formed a kind of corporation, which, although not officially recognized, had the right of passing resolutions, chiefly such as embodied acts of homage to the imperial house. It is not known whether the _turmae_ contained a fixed number of equites; there is no doubt that, in assigning the public horse, Augustus went far beyond the earlier figure of 1800. Thus, Dionysius of Halicarnassus mentions 5000 equites as taking part in a review at which he himself was present.

As before, the equites wore the narrow, purple-striped tunic, and the gold ring, the latter now being considered the distinctive badge of knighthood. The fourteen rows in the theatre were extended by Augustus to seats in the circus.

The old _recognitio_ was replaced by the _probatio_, conducted by the emperor in his censorial capacity, assisted by an advisory board of specially selected senators. The ceremony was combined with a procession, which, like the earlier _transvectio_, took place on the 15th of July, and at such other times as the emperor pleased. As in earlier times, offenders were punished by expulsion.

In order to provide a supply of competent officers, each eques was required to fill certain subordinate posts, called _militiae equestres_. These were (1) the command of an auxiliary cohort; (2) the tribunate of a legion; (3) the command of an auxiliary cavalry squadron, this order being as a rule strictly adhered to. To these Septimius Severus added the centurionship. Nomination to the _militiae equestres_ was in the hands of the emperor. After the completion of their preliminary military service, the equites were eligible for a number of civil posts, chiefly those with which the emperor himself was closely concerned. Such were various procuratorships; the prefectures of the corn supply, of the fleet, of the watch, of the praetorian guards; the governorships of recently acquired provinces (Egypt, Noricum), the others being reserved for senators. At the same time, the abolition of the indirect method of collecting the taxes in the provinces greatly reduced the political influence of the equites. Certain religious functions of minor importance were also reserved for them. In the jury courts, the equites, thanks to Julius Caesar, already formed two-thirds of the judices; Augustus, by excluding the senators altogether, virtually gave them the sole control of the tribunals. One of the chief objects of the emperors being to weaken the influence of the senate by the opposition of the equestrian order, the practice was adopted of elevating those equites who had reached a certain stage in their career to the rank of senator by _adlectio_. Certain official posts, of which it would have been inadvisable to deprive senators, could thus be bestowed upon the promoted equites.

The control of the imperial correspondence and purse was at first in the hands of freedmen and slaves. The emperor Claudius tentatively entrusted certain posts connected with these to the equites; in the time of Hadrian this became the regular custom. Thus a civil career was open to the equites without the obligation of preliminary military service, and the emperor was freed from the pernicious influence of freedmen. After the reign of Marcus Aurelius (according to Mommsen) the equites were divided into: (a) _viri eminentissimi_, the prefects of the praetorian guard; (b) _viri perfectissimi_, the other prefects and the heads of the financial and secretarial departments; (c) _viri egregii_, first mentioned in the reign of Antoninus Pius, a title by right of the procurators generally.

Under the empire the power of the equites was at its highest in the time of Diocletian; in consequence of the transference of the capital to Constantinople, they sank to the position of a mere city guard, under the control of the prefect of the watch. Their history may be said to end with the reign of Constantine the Great.

Mention may also be made of the _equites singulares Augusti_. The body-guard of Augustus, consisting of foreign soldiers (chiefly Germans and Batavians), abolished by Galba, was revived from the time of Trajan or Hadrian under the above title. It was chiefly recruited from the pick of the provincial cavalry, but contained some Roman citizens. It formed the imperial "Swiss guard," and never left the city except to accompany the emperor. In the time of Severus, these equites were divided into two corps, each of which had its separate quarters, and was commanded by a tribune under the orders of the prefect of the praetorian guard. They were subsequently replaced by the _protectores Augusti_.

See further article ROME: _History_; also T. Mommsen, _Romisches Staatsrecht_, iii.; J.N. Madvig, _Die Verfassung des romischen Staates_, i.; R. Cagnat in Daremberg and Saglio's _Dictionnaire des antiquites_, where full references to ancient authorities are given in the footnotes; A.S. Wilkins in Smith's _Dictionary of Greek and Roman Antiquities_ (3rd ed., 1891); E. Belot, _Histoire des chevaliers romains_ (1866-1873); H.O. Hirschfeld, _Untersuchungen auf dem Gebiete der romischen Verwaltungsgeschichte_ (Berlin, 1877); E. Herzog, _Geschichte und System der romischen Staatsverfassung_ (Leipzig, 1884-1891); A.H. Friedlander, _Sittengeschichte Roms_, i. (1901); A.H.J. Greenidge, _History of Rome_, i. (1904); J.B. Bury, _The Student's Roman Empire_ (1893); T.M. Taylor, _Political and Constitutional History of Rome_ (1899). For a concise summary of different views of the _sex suffragia_ see A. Bouche-Leclercq's _Manuel des antiquites romaines_, quoted in Daremberg and Saglio; and on the _equites singulares_, T. Mommsen in _Hermes_, xvi. (1881), p. 458. (J. H. F.)

EQUITY (Lat. _aequitas_), a term which in its most general sense means equality or justice; in its most technical sense it means a system of law or a body of connected legal principles, which have superseded or supplemented the common law on the ground of their intrinsic superiority. Aristotle (_Ethics_, bk. v. c. 10) defines equity as a better sort of justice, which corrects legal justice where the latter errs through being expressed in a universal form and not taking account of particular cases. When the law speaks universally, and something happens which is not according to the common course of events, it is right that the law should be modified in its application to that particular case, as the lawgiver himself would have done, if the case had been present to his mind. Accordingly the equitable man ([Greek: epieikes]) is he who does not push the law to its extreme, but, having legal justice on his side, is disposed to make allowances. Equity as thus described would correspond rather to the judicial discretion which modifies the administration of the law than to the antagonistic system which claims to supersede the law.

The part played by equity in the development of law is admirably illustrated in the well-known work of Sir Henry Maine on _Ancient Law_. Positive law, at least in progressive societies, is constantly tending to fall behind public opinion, and the expedients adopted for bringing it into harmony therewith are three, viz. legal fictions, equity and statutory legislation. Equity here is defined to mean "any body of rules existing by the side of the original civil law, founded on distinct principles, and claiming incidentally to supersede the civil law in virtue of a superior sanctity inherent in those principles." It is thus different from legal fiction, by which a new rule is introduced surreptitiously, and under the pretence that no change has been made in the law, and from statutory legislation, in which the obligatory force of the rule is not supposed to depend upon its intrinsic fitness. The source of Roman equity was the fertile theory of natural law, or the law common to all nations. Even in the Institutes of Justinian the distinction is carefully drawn in the laws of a country between those which are peculiar to itself and those which natural reason appoints for all mankind. The connexion in Roman law between the ideas of equity, nature, natural law and the law common to all nations, and the influence of the Stoical philosophy on their development, are fully discussed in the third chapter of the work we have referred to. The agency by which these principles were introduced was the edicts of the praetor, an annual proclamation setting forth the manner in which the magistrate intended to administer the law during his year of office. Each successive praetor adopted the edict of his predecessor, and added new equitable rules of his own, until the further growth of the irregular code was stopped by the praetor Salvius Julianus in the reign of Hadrian.

The place of the praetor was occupied in English jurisprudence by the lord high chancellor. The real beginning of English equity is to be found in the custom of handing over to that officer, for adjudication, the complaints which were addressed to the king, praying for remedies beyond the reach of the common law. Over and above the authority delegated to the ordinary councils or courts, a reserve of judicial power was believed to reside in the king, which was invoked as of grace by the suitors who could not obtain relief from any inferior tribunal. To the chancellor, as already the head of the judicial system, these petitions were referred, although he was not at first the only officer through whom the prerogative of grace was administered. In the reign of Edward III. the equitable jurisdiction of the court appears to have been established. Its constitutional origin was analogous to that of the star chamber and the court of requests. The latter, in fact, was a minor court of equity attached to the lord privy seal as the court of chancery was to the chancellor. The successful assumption of extraordinary or equitable jurisdiction by the chancellor caused similar pretensions to be made by other officers and courts. "Not only the court of exchequer, whose functions were in a peculiar manner connected with royal authority, but the counties palatine of Chester, Lancaster and Durham, the court of great session in Wales, the universities, the city of London, the Cinque Ports and other places silently assumed extraordinary jurisdiction similar to that exercised in the court of chancery." Even private persons, lords and ladies, affected to establish in their honours courts of equity.

English equity has one marked historical peculiarity, viz. that it established itself in a set of independent tribunals which remained in standing contrast to the ordinary courts for many hundred years. In Roman law the judge gave the preference to the equitable rule; in English law the equitable rule was enforced by a distinct set of judges. One cause of this separation was the rigid adherence to precedent on the part of the common law courts. Another was the jealousy prevailing in England against the principles of the Roman law on which English equity to a large extent was founded.

When a case of prerogative was referred to the chancellor in the reign of Edward III., he was required to grant such remedy as should be consonant to honesty (_honestas_). And honesty, conscience and equity were said to be the fundamental principles of the court. The early chancellors were ecclesiastics, and under their influence not only moral principles, where these were not regarded by the common law, but also the equitable principles of the Roman law were introduced into English jurisprudence. Between this point and the time when equity became settled as a portion of the legal system, having fixed principles of its own, various views of its nature seem to have prevailed. For a long time it was thought that precedents could have no place in equity, inasmuch as it professed in each case to do that which was just; and we find this view maintained by common lawyers after it had been abandoned by the professors of equity themselves. G. Spence, in his book on the _Equitable Jurisdiction of the Court of Chancery_, quotes a case in the reign of Charles II., in which chief justice Vaughan said:

"I wonder to hear of citing of precedents in matter of equity, for if there be equity in a case, that equity is an universal truth, and there can be no precedent in it; so that in any precedent that can be produced, if it be the same with this case, the reason and equity is the same in itself; and if the precedent be not the same case with this it is not to be cited."

But the lord keeper Bridgeman answered:

"Certainly precedents are very necessary and useful to us, for in them we may find the reasons of the equity to guide us, and besides the authority of those who made them is much to be regarded. We shall suppose they did it upon great consideration and weighing of the matter, and it would be very strange and very ill if we should disturb and set aside what has been the course for a long series of times and ages."

Selden's description is well known: "Equity is a roguish thing. 'Tis all one as if they should make the standard for measure the chancellor's foot." Lord Nottingham in 1676 reconciled the ancient theory and the established practice by saying that the conscience which guided the court was not the natural conscience of the man, but the civil and political conscience of the judge. The same tendency of equity to settle into a system of law is seen in the recognition of its limits--in the fact that it did not attempt in all cases to give a remedy when the rule of the common law was contrary to justice. Cases of hardship, which the early chancellors would certainly have relieved, were passed over by later judges, simply because no precedent could be found for their interference. The point at which the introduction of new principles of equity finally stopped is fixed by Sir Henry Maine in the chancellorship of Lord Eldon, who held that the doctrines of the court ought to be as well settled and made as uniform almost as those of the common law. From that time certainly equity, like common law, has professed to take its principles wholly from recorded decisions and statute law. The view (traceable no doubt to the Aristotelian definition) that equity mitigates the hardships of the law where the law errs through being framed in universals, is to be found in some of the earlier writings. Thus in the _Doctor and Student_ it is said:

"Law makers take heed to such things as may often come, and not to every particular case, for they could not though they would; therefore, in some cases it is necessary to leave the words of the law and follow that reason and justice requireth, and to that intent equity is ordained, that is to say, to temper and mitigate the rigour of the law."

And Lord Ellesmere said:

"The cause why there is a chancery is for that men's actions are so divers and infinite that it is impossible to make any general law which shall aptly meet with every particular act and not fail in some circumstances."

Modern equity, it need hardly be said, does not profess to soften the rigour of the law, or to correct the errors into which it falls by reason of its generality.

To give any account, even in outline, of the subject matter of equity within the necessary limits of this article would be impossible. It will be sufficient to say here that the classification generally adopted by text-writers is based upon the relations of equity to the common law, of which some explanation is given above. Thus equitable jurisdiction is said to be exclusive, concurrent or auxiliary. Equity has _exclusive_ jurisdiction where it recognizes rights which are unknown to the common law. The most important example is trusts. Equity has _concurrent_ jurisdiction in cases where the law recognized the right but did not give adequate relief, or did not give relief without circuity of action or some similar inconvenience. And equity has _auxiliary_ jurisdiction when the machinery of the courts of law was unable to procure the necessary evidence.

"The evils of this double system of judicature," says the report of the judicature commission (1863-1867), "and the confusion and conflict of jurisdiction to which it has led, have been long known and acknowledged." A partial attempt to meet the difficulty was made by several acts of parliament (passed after the reports of commissions appointed in 1850 and 1851), which enabled courts of law and equity both to exercise certain powers formerly peculiar to one or other of them. A more complete remedy was introduced by the Judicature Act 1873, which consolidated the courts of law and equity, and ordered that law and equity should be administered concurrently according to the rules contained in the 26th section of the act. At the same time many matters of equitable jurisdiction are still left to the chancery division of the High Court in the first instance. (See CHANCERY.)

AUTHORITIES.--The principles of equity as set out by the following writers may be consulted: J. Story, J.W. Smith, H.A. Smith and W. Ashburner; and for the history see G. Spence, _The Equitable Jurisdiction of the Court of Chancery_ (2 vols., 1846-1849); D.M. Kerly, _Historical Sketch of the Equitable Jurisdiction of the Court of Chancery_ (1890).

EQUIVALENT, in chemistry, the proportion of an element which will combine with or replace unit weight of hydrogen. When multiplied by the valency it gives the atomic weight. The determination of equivalent weights is treated in the article STOICHIOMETRY. (See also CHEMISTRY.) In a more general sense the term "equivalent" is used to denote quantities of substances which neutralize one another, as for example NaOH, HCl, 1/2 H2SO4, 1/2 Ba(OH)2.

ERARD, SEBASTIEN (1752-1831), French manufacturer of musical instruments, distinguished especially for the improvements he made upon the harp and the pianoforte, was born at Strassburg on the 5th of April 1752. While a boy he showed great aptitude for practical geometry and architectural drawing, and in the workshop of his father, who was an upholsterer, he found opportunity for the early exercise of his mechanical ingenuity. When he was sixteen his father died, and he removed to Paris where he obtained employment with a harpsichord maker. Here his remarkable constructive skill, though it speedily excited the jealousy of his master and procured his dismissal, almost equally soon attracted the notice of musicians and musical instrument makers of eminence. Before he was twenty-five he set up in business for himself, his first workshop being a room in the hotel of the duchesse de Villeroi, who gave him warm encouragement. Here he constructed in 1780 his first pianoforte, which was also one of the first manufactured in France. It quickly secured for its maker such a reputation that he was soon overwhelmed with commissions, and finding assistance necessary, he sent for his brother, Jean Baptiste, in conjunction with whom he established in the rue de Bourbon, in the Faubourg St Germain, a piano manufactory, which in a few years became one of the most celebrated in Europe. On the outbreak of the Revolution he went to London where he established a factory. Returning to Paris in 1796, he soon afterwards introduced grand pianofortes, made in the English fashion, with improvements of his own. In 1808 he again visited London, where, two years later, he produced his first double-movement harp. He had previously made various improvements in the manufacture of harps, but the new instrument was an immense advance upon anything he had before produced, and obtained such a reputation that for some time he devoted himself exclusively to its manufacture. It has been said that in the year following his invention he made harps to the value of L25,000. In 1812 he returned to Paris, and continued to devote himself to the further perfecting of the two instruments with which his name is associated. In 1823 he crowned his work by producing his model grand pianoforte with the double escapement. Erard died at Passy, on the 5th of August 1831. (See also HARP and PIANOFORTE.)

ERASMUS, DESIDERIUS (1466-1536), Dutch scholar and theologian, was born on the night of the 27/28th of October, probably in 1466; but his statements about his age are conflicting, and in view of his own uncertainty (_Ep._ x. 29: 466) and the weakness of his memory for dates, the year of his birth cannot be definitely fixed. His father's name seems to have been Rogerius Gerardus. He himself was christened Herasmus; but in 1503, when becoming familiar with Greek, he assimilated the name to a fancied Greek original, which he had a few years before Latinized into Desyderius. A contemporary authority states that he was born at Gouda, his father's native town; but he adopted the style _Rotterdammensis_ or _Roterodamus_, in accordance with a story to which he himself gave credence. His first schooling was at Gouda under Peter Winckel, who was afterwards vice-pastor of the church. In the dull round of instruction in "grammar" he did not distinguish himself, and was surpassed by his early friend and companion, William Herman, who was Winckel's favourite pupil. From Gouda the two boys went to the school attached to St Lebuin's church at Deventer, which was one of the first in northern Europe to feel the influence of the Renaissance. Erasmus was at Deventer from 1475 to 1484, and when he left, had learnt from Johannes Sinthius (Syntheim) and Alexander Hegius, who had come as headmaster in 1483, the love of letters which was the ruling passion of his life. At some period, perhaps in an interval of his time at Deventer, he was a chorister at Utrecht under the famous organist of the cathedral, Jacob Obrecht.

About 1484 Erasmus' father died, leaving him and an elder brother Peter, both born out of wedlock, to the care of guardians, their mother having died shortly before. Erasmus was eager to go to a university, but the guardians, acting under a perhaps genuine enthusiasm for the religious life, sent the boys to another school at Hertogenbosch; and when they returned after two or three years, prevailed on them to enter monasteries. Peter went to Sion, near Delft; Erasmus after prolonged reluctance became an Augustinian canon in St Gregory's at Steyn, a house of the same Chapter near Gouda. There he found little religion and less refinement; but no serious difficulty seems to have been made about his reading the classics and the Fathers with his friends to his heart's content. The monastery once entered, there was no drawing back; and Erasmus passed through the various stages which culminated in his ordination as priest on the 25th of April 1492.

But his ardent spirit could not long be content with monastic life. He brought his attainments somehow to the notice of Henry of Bergen, bishop of Cambrai, the leading prelate at the court of Brussels; and about 1494 permission was obtained for him to leave Steyn and become Latin secretary to the bishop, who was then preparing for a visit to Rome. But the journey was abandoned, and after some months Erasmus found that even with occasional chances to read at Groenendael, the life of a court was hardly more favourable to study than that of Steyn. At the suggestion of a friend, James Batt, he applied to his patron for leave to go to Paris University. The bishop consented and promised a small pension; and in August 1495 Erasmus entered the "domus pauperum" of the college of Montaigu, which was then under the somewhat rigid rule of the reformer Jan Standonck. He at once introduced himself to the distinguished French historian and diplomatist Robert Gaguin (1425-1502) and published a small volume of poems; and he became intimate with Johann Mauburnus (Mombaer), the leader of a mission summoned from Windesheim in 1496 to reform the abbey of Chateau-Landon. But the life at Montaigu was too hard for him. Every Lent he fell ill and had to return to Holland to recover. He continued to read nevertheless for a degree in theology, and at some time completed the requirements for the B.D. After a year or two he left Montaigu and eked out his money from the bishop by taking pupils. One of these, a young Englishman, William Blount, 4th Baron Mountjoy (d. 1534), persuaded him to visit England in the spring of 1499.

Being without a benefice, he had no settled income to look to, and apart from the precarious profits of teaching and writing books, could only wait on the generosity of patrons to supply him with the leisure he craved. The faithful Batt had sought a pension for him from his own patroness, Anne of Borsselen, the Lady of Veere, who resided at the castle of Tournehem near Calais, and whose son Batt was now teaching. But as nothing promised at once, Erasmus accepted Mountjoy's offer, and thus a tie was formed which led Mountjoy then or a few years later to grant him a pension of L20 for life. Otherwise the visit to England gave no hope of preferment; and in the summer Erasmus prepared to leave. He was delayed, and used the interval to spend two or three months at Oxford, where he found John Colet lecturing on the Epistle to the Romans. Discussions between them on theological questions soon convinced Colet of Erasmus' worth, and he sought to persuade him to stay and teach at Oxford. But Erasmus could not be content with the Bible in Latin. Oxford could teach him no Greek, so away he must go.

In January 1500 he returned to Paris, which though it could offer no Greek teacher better than George Hermonymus, was at least a better centre for buying and for printing books. The next few years were spent still in preparation, supported by pupils' fees and the dedications of books; the _Collectanea adagiorum_ in June 1500 to Mountjoy, and some devotional and moral compositions to Batt's patroness and her son. When the plague drove him from Paris, he went to Orleans or Tournehem or St Omer, as the way opened. From 1502 to 1504 he was at Louvain, still declining to teach publicly; among his friends being the future Pope Adrian VI. In January 1504 the archduke Philip gave him fifty livres for the Panegyric which "_ung religieux de l'ordre de St Augustin_" had composed on his Spanish journey; and in October, ten more, for the maintenance of his studies.

He had been working hard at Greek, of which he now felt himself master, at the Fathers (above all at Jerome), and at the Epistles of St Paul, fulfilling the promise made to Colet in Oxford, to give himself to sacred learning. But the bent of his reading is shown by the manuscript with which he returned to Paris at the close of 1504--Valla's _Annotations on the New Testament_, which Badius printed for him in 1505.

Shortly afterwards Lord Mountjoy invited him again to England, and this visit was more successful. He found in London a circle of learned friends through whom he was introduced to William Warham, archbishop of Canterbury, Richard Foxe, bishop of Winchester and other dignitaries. John Fisher (bishop of Rochester), who was then superintending the foundation of Christ's College for the Lady Margaret, took him down to Cambridge for the king's visit; and at length the opportunity came to fulfil his dream of seeing Italy. Baptista Boerio, the king's physician, engaged him to accompany his two sons thither as supervisor of their studies. In September 1506 he set foot on that sacred soil, and took his D.D. at Turin. For a year he remained with his pupils at Bologna, and then, his engagement completed, negotiated with Aldus Manutius for a new edition of his _Adagia_ upon a very different scale. The volume of 1500 had been jejune, written when he knew nothing of Greek; 800 adages put together with scanty elucidations. In 1508 he had conceived a work on lines more to the taste of the learned world, full of apt and recondite learning, and now and again relieved by telling comments or lively anecdotes. Three thousand and more collected justified a new title--_Chiliades adagiorum_; and the author's reputation was now established. So secure in public favour did the book in time become, that the council of Trent, unable to suppress it and not daring to overlook it, ordered the preparation of a castrated edition.

To print the _Adagia_ he had gone to Venice, where he lived with Andrea Torresano of Asola (Asulanus) and did the work of two men, writing and correcting proof at the same time. When it was finished, with an ample re-dedication to Mountjoy, a new pupil presented himself, Alexander Stewart, natural son of James IV. of Scotland--perhaps through a connexion formed in early days at Paris. They went together to Siena and Rome and then on to Campania, thirsty under the summer sun. When they returned to Rome, his pupil departed to Scotland, to fall a few years later by his father's side at Flodden; Erasmus also found a summons to call him northwards.

On the death of Henry VII. Lord Mountjoy, who had been companion to Prince Henry in his studies, had become a person of influence. He wrote to Erasmus of a land flowing with milk and honey under the "divine" young king, and with Warham sent him L10 for journey money. At first Erasmus hesitated. He had been disappointed in Italy, to find that he had not much to learn from its famed scholarship; but he had made many friends in Aldus's circle--Marcus Musurus, John Lascaris, Baptista Egnatius, Paul Bombasius, Scipio Carteromachus; and his reception had been flattering, especially in Rome, where cardinals had delighted to honour him. But to remain in Rome was to sell himself. He might have the leisure which was so indispensable, but at price of the freedom to read, think, write what he liked. He decided, therefore, to go, though with regrets; which returned upon him sometimes in after years, when the English hopes had not borne fruit.

In the autumn he reached London, and in Thomas More's house in Bucklersbury wrote the witty satire which Milton found "in every one's hands" at Cambridge in 1628, and which is read to this day. The _Moriae encomium_ was a sign of his decision. In it kings and princes, bishops and popes alike are shown to be in bondage to Folly; and no class of men is spared. Its author was willing to be beholden to any one for leisure; but he would be no man's slave. For the next eighteen months he is entirely lost to view; when he reappears in April 1511, he is leaving More's house and taking the _Moria_ to be printed privily in Paris. Wherever they were spent, these must have been months of hard work, as were the years that followed. His time was now come. The long preparation and training, bought by privation and uncongenial toil, was over, and he was ready to apply himself to the scientific study of sacred letters. His English patrons were liberal. Fisher sent him in August 1511 to teach in Cambridge; Warham gave him a benefice, Aldington in Kent, worth _L_33, 6_s._ 8_d._ a year, and in violation of his own rule commuted it for a pension of L20 charged on the living; and the dedications of his books were fruitful. In Cambridge he completed his work on the New Testament, the Letters of Jerome, and Seneca; and then in 1514, when there seemed no prospect of ampler preferment, he determined to transfer himself to Basel and give the results of his labours to the world.

The origin of Erasmus's connexion with Johann Froben is not clear. In 1511 he was preparing to reprint his _Adagia_ with Jodocus Badius, who in the following year was to have also Seneca and Jerome. But in 1513 Froben, who had just reprinted the Aldine _Adagia_, acquired through a bookseller-agent Erasmus' amended copy which had been destined for Badius. That the agent was acting entirely on his own responsibility may be doubted; for within a few months Erasmus had decided to betake himself to Basel, bearing with him Seneca and Jerome, the latter to be incorporated in the great edition which Johannes Amerbach and Froben had had in hand since 1510. In Germany he was widely welcomed. The Strassburg Literary Society feted him, and Johannes Sapidus, headmaster of the Latin school at Schlettstadt, rode with him into Basel. Froben received him with open arms, and the presses were soon busy with his books. Through the winter of 1514-1515 Erasmus worked with the strength of ten; and after a brief visit to England in the spring, the New Testament was set up. Around him was a circle of students, some young, some already distinguished--the three sons of Froben's partner, Johannes Amerbach, who was now dead, Beatus Rhenanus, Wilhelm Nesen, Ludwig Ber, Heinrich Glareanus, Nikolaus Gerbell, Johannes Oecolampadius--who looked to him as their head and were proud to do him service.

Though from this time forward Basel became the centre of occupation and interest for Erasmus, yet for the next few years he was mainly in the Netherlands. On the completion of the New Testament in 1516 he returned to his friends in England; but his appointment, then recent, as councillor to the young king Charles, brought him back to Brussels in the autumn. In the spring of 1517 he went for the last time to England, about a dispensation from wearing his canonical dress, obtained originally from Julius II. and recently confirmed by Leo X., and in May 1518 he journeyed to Basel for three months to set the second edition of the New Testament in progress. But with these exceptions he remained in proximity to the court, living much at Louvain, where he took great interest in the foundation of Hieronymus Busleiden's Collegium Trilingue. His circumstances had improved so much, by pensions, the presents 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-Comte. 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 ete 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._