of certain letters representing the unknown quantities is called an equation. These particular values are called the roots of the equation, and the determination of these roots is called the solution of the equation. Thus 2x + 3 = 9 is an equation, because the equality is true only for a particular value of the unknown quantity x, viz., for = 3. The expression 2+5=7 expresses an equality, but it is not an equation as the word is technically used in mathematics. Expressions like (ax) a2 + 2ax + x2 are true for all values of the letters and are called identities to distinguish them from equations. If an algebraic function f(x) equals zero, and is arranged according to the descending, integral, positive powers of x, and in its relation to 0 expressed as an equation, it has the form f(x) Иах + а, х + · + αn 1 ∞ + a11 = 0. Such an equation is called a complete equation of the nth degree with one unknown quantity; e.g. ɑx2 + å ̧x + α ̧=0 is a complete equation, while ax2 + a2 = 0 is an incomplete equation, both of the second degree. The letters a,, a1, ap-1, a stand for known quantities, and in the theory of equations, so called, they stand for real quantities. They are all coefficients of powers of x, except the absolute term, a, which might, however, be considered the coefficient of €o. In case ao α a are all expressed as numbers, the equation is said to be numerical; otherwise it is known as literal.

n-l

If two or more equations are satisfied by the same value of the unknown quantities they are said to be simultaneous, as in the case of a2+ y = 7, x + y2 = 11, where x = 2, y = 3; but x2 + y = 7 and 3x2+3y=9 are not simultaneous; they are inconsistent, there being no values of x and y that will satisfy both; and x2+y=7 and 3x2 + 3y = 21 are said to be identical, each being derivable from the other. In case sufficient relations are not given to enable the roots of an equation to be determined, exactly or approximately, the equation is said to be indeterminate; e.g. in the equation + 2y = 10, any of the following pairs of values satisfies the equation: (0,5), (1, 4.5), (2, 4), (3, 3.5), (10, 0), (11, -0.5), In general, n linear equations, each containing n+1 or more unknown quantities, are indeterminate. Thus 2x+3y+≈ 10, 3x + 2y + z = 8, give rise to the simple equation + y2, which is indeterminate. Equations may also be classified as rational, irrational, integral, or fractional, according as the two members, when like terms are united, are composed of expressions

which are rational, irrational (or partly so), integral, or fractional (or partly so), respectively, with respect to the unknown quantities; e.g.: 3x + 50 is a rational integral equation, 6+3x1 O is an irrational integral equation, 2 +√14 O is a rational fractional equation, (x+3)= 5 is an irrational fractional equation. Algebra is chiefly concerned with the solution of equations, and definite methods have been devised for determining the roots of algebraic equations of the first, second, third, and fourth deby applying the common axioms: grees. Equations of the first degree are solved If equals are added to equals, the results are equal; if equals are subtracted from equals, the results are equal; ; and the corresponding ones of multiplication and division. Equations of the second degree may be solved by reducing the quadratic function to the product of two linear factors, thus making the solution of the quadratic equation depend upon that of two linear equations. Thus x2+ px+q=0 reduces to

Р

x + ¦ ± } √ p2 + q + 0.

2

whence Similarly, the solution of the cubic equation is made to depend upon that of the quadratic equation, and that of the biquadratic equation upon that of the cubic equation. These formulas, however, when applied to numerical equations often involve operations upon complex numbers not readily performed, and hence are of little value in such cases; e.g. in applying the general formula for the roots of the cubic equation, the cube root of a complex number is often required, in which case the methods of trigonometry are employed. The real roots of numerical equations of any degree may be calculated_approximately by the methods of Newton, Lagrange, and Horner, the last being the most recent and generally preferred of the three.

Equations of the first degree were familiar to the Egyptians in the time of Ahmes (q.v.), since a papyrus transcribed by him contains an equation in the following form: Heap (hau), its 23, its, its, its whole, gives 37; that is,

z x + z x + 4 x + x = 37. The ancient Greeks knew little of linear equations except through proportion, but they treated in geometric form many quadratic and cubic equations. (See CUBE.) Diophantus (c.300 A.D.), however, distinguished the coefficients (#Xos ), of the unknown quantity, gave the equation a symbolic form, classified equations, and gave definite rules for reducing them to their simplest forms. His work was chiefly concerned with indeterminate systems of equations, and his method of treatment is known as Diophantine analysis (q.v.).

The Chinese likewise solved quadratic equations geometrically, and Sun Tse (third century), like Diophantus, developed a method of solving linear indeterminate equations. The Hindus advanced the knowledge of the Greeks. Bhaskara (twelfth century) used only one type of quadratic equation, ar2 + bx + c =0. considered both signs of the square root, and distinguished the surd values, while the Greeks accepted only positive integers. The Arabs improved the methods of their predecessors. They developed quite an

elaborate system of symbolism. The equations of Al Kalsadi (fifteenth century) are models of brevity, and this plan for solving linear equations, a modified Hindu method, was what was later known as the regula falsi. See FALSE POSITION.

The Europeans of the Middle Ages made little advance in the solving of equations until the discovery by Ferro, Tartaglia, and Cardan (sixteenth century) of the general solution of the cubic equation. The solution of the biquadratic equation soon followed, and the general quintic was attacked. But, although much was done to advance the general theory of the equation by Vandermonde, Euler, Lagrange, Bézout, Waring, Malfatti, and others, it was not until the beginning of the nineteenth century that equations of a degree higher than the fourth received satisfactory treatment. Ruffini and Abel were the first to demonstrate that the solution, by algebraic methods, of a general equation of a degree higher than the fourth is impossible, and to direct investigation into new channels. Mathematicians now sought to classify equations which could be solved algebraically, and to discover higher methods for those which could not. Gauss solved the cyclotomic group, Abel the group known as the Abelian equations, and Galois stated the necessary and sufficient condition for the algebraic solubility of any equation as follows: If the degree of an irreducible equation is a prime number, the equation is soluble by radicals alone, provided the roots of this equation can be expressed rationally in terms of any two of them. As to higher methods, Tschirnhausen, Bring, and Hermite have shown that the general equation of the fifth degree can be put in the form tt A= 0; Hermite and Kronecker solved the equation of the fifth degree by elliptic functions; and Klein has given the simplest solution by transcendental functions.

A few of the more important properties of equations are: (1) If r is a root of the equation f(x) = 0, then r is a factor of f(x); e.g. 2 being a root of x2+2x-8= 0, then a 2 is a factor of 2 + 2x

8.

(2) If f(x) is divisible by x-r, r is a root of f(x) = 0; e.g. in (x 2) (x2 + x + 1) 0, r2 is a factor, hence -- 20 satisfies the equation, and x = 2.

(3) Every equation of the nth degree has n roots and no more (the fundamental theorem of equations due to Harriot. or, in its complete form, to D'Alembert): e.g. x-10 has four roots, r = 1, — 1, i, — i, and no more.

(4) The coefficients of an equation are functions of its roots. Thus, in

0-1

n-1

....

D-2

2

+.. . a =0 if r1, r,. . . r are the roots, then" · (r1 + r2+.. +-r1), a = r1 r2 + r1rs + +r. == (r1 r2 r3 + r1 r2r1 +. + T-2 r) an = ±r1r2 (5) The number of positive roots of f(x) = 0 does not exceed the number of changes of signs in f(x). (Descartes's rule of signs.) E.g. in x-3x3- - 2x2 + x −1 = 0 there are 3 changes of signs, hence there can be no more than 3 positive roots.

(6) The special functions associated with the roots of an equation which serve to distinguish the nature of these roots are called discriminants; e.g. the general form of the roots of the

from which it is required to find the relation between y and x. The theory of the solution of such equations is an extension of the integral calculus, and is a branch of study of the highest importance.

For the general theory of equations, consult: Burnside and Panton, Theory of Equations (4th ed., London, 1899-1901), the appendix to which Théorie des équations, algébriques (by Laurent, Paris, 1897); Salmon, Lessons Introductory to Modern Higher Algebra (Dublin, 1859, and subsequent editions); Serret, Cours d'algèbre supérieure (3d ed., Paris, 1866); Jordan, Traité des substitutions et des equations algébriques (Paris, 1870). An extensive work, covering both history and method, is Matthiessen, Grundzüge der antiken und modernen Algebra der literalen Gleichungen (Leipzig, 1896).

contains valuable historical material; Peterson,

It

EQUATION, ANNUAL. One of the most conspicuous of the subordinate fluctuations in the moon's motion, due to the action of the sun. consists in an alternate increase and decrease in the moon's longitude, corresponding with the earth's situation in its annual orbit, i.e. to its angular distance from the perihelion, and therefore it has a year instead of a month, or aliquot part of a month, for its period.

EQUATION, CHEMICAL. See CHEMISTRY.

EQUATION, PERSONAL. A very important factor in astronomical observations. Two observers, each of admitted skill, often differ in their record of the same event-as the passage of a star before the wires of a transit instrument-by a quantity nearly the same for all observations by those persons. This quantity is their relative personal equation. Each observer habitually notes the time too early or too late, by a small and nearly uniform portion of a second. This quantity is his absolute personal equation. Machines have been invented for determining the amount of personal equation by reproducing artificially the kind of observation. usually affected with this form of error in actual work on the sky. The so-called Repsold apparatus is a mechanical device which so changes the condition of observation with a transit instrument or meridian circle that the personal equation is removed altogether, and its quantitative evaluation is rendered unnecessary.

EQUATION OF EQUINOXES. The difference between the actual position of the equinoxes (q.v.) and the position calculated on the assumption that their motion is uniform. See PRECESSION.

EQUATION OF LIGHT. In astronomical observations, the ray of light by which we see any celestial body is not that which it emits at the moment we look at it, but which it did emit some time before, viz., the time occupied by light in traversing the space which separates us from the celestial body. The quantity of time so required for the passage of light from the sun to the earth is the so-called light-equation. It amounts to about eight minutes twenty seconds. EQUATION OF PAYMENTS. A method of

finding the time when, if a sum of money be paid by a debtor, instead of several debts payable by him at different times, no loss will be sustained by either the debtor or creditor. The common rule is Multiply each debt by its term of credit, and divide the sum of the products by the sum of the debts. The quotient will be the average term of credit. This added to the date from which the credits were reckoned will give the average time of payments; e.g. to find the average time of paying $200 due April 1st, $200 due May 11th, and $400 due June 30th: $200 + 40 $200+ 90 $400 $44,000; $44,000 $800 55. April 1st +55 days May 26th the equated time. This method is incorrect, except for equal debts, because it takes no account of the balance of interest and discount. It is, however, sufficiently accurate for ordinary use.

=

EQUATION OF TIME. The earth's motion in the ecliptic is not uniform. This want of uniformity would of itself be sufficient to cause an irregularity in the intervals of time between successive returns of the sun to the meridian, day after day; but besides this want of uniformity in the sun's apparent motion in the ecliptic, there is another cause of inequality in the time of his coming to the meridian, the obliquity of the ecliptic to the equinoctial. These two independent causes conjointly produce the inequality in the time of his appearance on the meridian, the correction for which is the 'equation of time.' But from the causes above explained, mean and apparent noon differ, the latter taking place sometimes as much as 164 minutes before the former, and at others as much as 14 minutes after. The difference for any day, called the equation of time, is to be found inserted in ephemerides for every day of the year. (See EPHEMERIS.) It is nothing, or zero, at four different times in the year, at which the mean and unequal motions would exactly agree-viz., about the 15th of April. the 15th of June, the 31st of August, the 24th of December. At all other times the sun is either too fast or too slow for clock-time.

EQUATOR, CELESTIAL (ML. æquator, equalizer, from Lat. æquare, to equalize). The great circle which would be cut out on the sky by extending the plane of the earth's equator.

EQUATOR, TERRESTRIAL. The great circle on the earth's surface, half way between the poles, which divides the earth into the northern and southern hemispheres.

E'QUATORIAL (from ML. æquator, equalizer). A term applied in astronomy to a method of mounting astronomical telescopes, by which a celestial body may be observed at any point of its diurnal course. It consists of a telescope fastened to a graduated circle, called the declination circle, whose axis is attached at right angles to that of another graduated circle called the hour-circle, and is wholly supported by it. The hour-circle axis, which is called the principal axis of the instrument, turns on fixed supports; it is pointed to the pole of the heavens, and the hour-circle is of course parallel to the equinoctial. This combination of axes gives us a universal joint, thus enabling us to point the tube at any star in the sky; and with the pair of circles we can measure and record the exact position in the sky of the star under observation. On account of one axis being pointed at the pole, about which all the stars revolve in their diurnal course, it becomes possible to follow their motions by rotating the telescope about this one axis only, and this rotation can be effected easily and conveniently with clockwork.

EQUESTRIAN ORDER (Lat. ordo equester), or EQUITES (Lat., knights, from equus, horse). Originally the cavalry of the Roman Romulus, who selected from the three principal It is said to have been instituted by Army. Roman tribes a body-guard of 300, called Celeres. The number was afterwards gradually increased to 3600, who were partly of patrician and partly of plebeian rank, and required to possess a certain amount of property (400,000 sestertii-about $17,000). Each of these equites received a horse from the State; but about B.C. 403 a new body of equites began to make their appearance, who were obliged to furnish a horse at their own expense. Until B.C. 123, the equites were exclusively a military body; but in that year Caius Gracchus carried a measure, by which all the judices (jurors) had to be selected from them. Now, for the first time, they became a distinct order or class in the State, and were called ordo equester. Sulla deprived them of this privilege; but their power did not then decrease. as the farming of the public revenues appears to have fallen into their hands. After the conspiracy of Catiline, the ordo equester began to be looked upon as a third estate in the Republic; and to the title of senatus populusque Romanus was added et ordo equester.

EQUESTRIAN STATUE. The representation in sculpture of a person on horseback. Equestrian statues were not commonly erected in Greece, but in Rome they were often awarded as a high honor to military commanders and persons of distinetion, and latterly were, for the most part, restricted to the emperors, the most famous in existence being that of the Emperor Marcus Aurelius, which now stands in the piazza of the Capitol at Rome. It is the only ancient equestrian statue in bronze that has been preserved. The most famous equestrian statues of the Renais