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eircle may be one, and the parabola is usually assumed for the other. See Simpson's and Maclaurin's algebra.

reducible to z

bxz-c, &c. 0, an equation one dimension lower, whose roots

are b and c.

Ex. One root of x3+1=0, or x+1= and the equation may be depressed to a quadratic in the following manner : x+1)x3+1(x2 −x + 1 x3+x2

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Hence the other two roots are the roots of

the quadratic x2-x+1=0. If two roots, a and b, be obtained, the equation is divisible by x-a × x-b, and may be reduced in the same manner two dimensions lower.

EQUATIONS, nature of. Any equation involving the powers of one unknown quantity may be reduced to the form 2"--p2"-10, 92-2, &c.—0: here the whole expression is equal to nothing, and the terms are arranged according to the dimensions of the unknown quantity, the coefficient of the highest dimension is unity, understood, and the coefficients p, q, r, and are affected with the proper signs. An equation, where the index is of the highest power of the unknown quantity is n, is said to be of n dimensions, and in speaking simply of an equation of n dimensions, we understand one reduced to the above form. Any quantity 2" - pz" ~ 1 + 424-2, &c.+Pz-Q may be supposed to arise from the multiplication of xax z-bxz-c, &c. to n factors. For by actually multiplying the factors together, we obtain a quantity of n dimensions simi- Er. Two roots of the equation 26 lar to the proposed quantity, z"-pz-10, are +1 and 1, or 2-10, and -2, &c.; and if a, b, c, &c. can be so x+1=0; therefore it may be depressed assumed that the coefficients of the correto a biquadratic by dividing by z=1x sponding terms in the two quantities become 1=2— 1. equal, the whole expressions coincide. And these coefficients may be made equal, because these will be n equations, to determine a quantities, a, b, c, &c. If then the quantities a, b, c, &c. be properly assumed, the equation z-p z" − 1 + q z′′ ~ 2, &c. =0, is the same with z-ax z —b × ≈ —c, &c. =0. The quantities a, b, c, d, &c. are called roots of the equation, or values of z; because, if any one of them be substituted for z, the whole expression becomes nothing, which is the condition proposed by the equation.

Every equation has as many roots as it has dimensions. If "—pz"-'+pz"−2, &c. o, or z-ax z-bxz-c, &c. to n factors =0, there are n quantities, a, b, c, &c. each of which when substituted for z makes the whole = 0, because in each case one of the factors becomes=0; but any given quantity different from these, ase when substituted for z, gives the product e-ax e-bxe-c, &c. which does not vanish, because none of the factors vanish, that is, e will not answer the condition which the equation requires.

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When one of the roots, a, is obtained, the equation za × z − b × z —c, &c. =0, *~pz" - 1+qx2-2, &c.—0 is divisible by za without a remainder, and is thus

1

2

-

1(2+2+1

2 — 1):6. 4
36

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24

-1=

Hence the equation *+*2+1=0 con tains the other four roots of the proposed equation.

Conversely, if the equation be divisible by x-a without a remainder, a is a root; if by x axxb, a and b are both roots. Let Q be the quotient arising from the division, then the equation is x-axx-b × Q=0, in which, if a or b be substituted for x the whole vanishes.

rule. Let the equation be reduced to the EQUATIONS, cubic, solution of, by Cardan's form x3-qxr0, where q and r may be positive or negative.

Assume x=a+b, then the equation becomes a +-6)3 — q × u+b+r=0, or a3 +b+3ab xa+b¬q×a+b+r=0; and since we have two unknown quantities, a and b, and have made only one supposi tion respecting them, viz. that a+b=x, we are at liberty to make another; let Sab

-q=0, then the equation becomes a3+ and the values of the cube root of b3, are b, b'+r=0, also, since 3 a b — q = 0, b =

3

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3

b,

1

2

3

b.

Hence, it appears, that there are nine values of a+b, three only of which can answer the conditions of the equation, the others having been introduced by involu tion. These nine values are,

1. a +h.

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q3 the biquadratic to be a+b= 1;
√=1; −a+c√ —1;

4 27

6 m2n+n', and == -nx Sm2. -n, a quantity manifestly impossible, unless n be negative, that is, unless two roots of the proposed cubic be impossible.

the values of e are 2 a, b+c. √ — 1, ・c. / — 1, − b —c. √ −1, −b+c.

1 and 2a; and the three values of y are 2 ul2, — b + c2, — b c2, which are all possible, as in the preceding case. But if the roots of the biquadratic be a+b √ −1, a b√1, -a+c,

-

the values of y are 2 u2, c + b √√ — 1}3, cb1', two of which are impossible; therefore the cubic may be solved by Cardan's rule.

EQUATIONS, biquadratic, solution of, by Des Carte's method. Any biquadratic may be reduced to the form x+qx2+rx+ =0, by taking away the second term. Suppose this to be made up of the two quadratics, ex+ƒ= 0, and x2- ex+ g=0, where +e and -e are made the coefficients of the second terms, because the second term of the biquadratic is wanting, that is, the sum of its roots is 0. By multiplying these quadratics together we have xˆ+g+ƒ—e.x2+eg—ej.x+tion depends upon the excentricity of the fg=0, which equation is made to coincide with the former by equating their coeffi cients, or making g +ƒ— e2 = q, eg -ef =r, and fg = s; hence, g +ƒ=q+e,

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also g —ƒ=2, and by taking the sum and difference of these equations, 2g=4+e2 +, and 2ƒ = 2 + e2 therefore 4fg

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are known; the biquadratic is thus resolved into two quadratics, whose roots may be found.

It may be observed, that which ever value of y is used, the same values of x are obtained.

This solution can only be applied to those cases, in which two roots of the biquadratic are possible and two impossible.

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Let the roots be a, b, c, a+b+c; then since e, the coefficient of the second term of one of the reducing quadratics, is the sum of two roots, its different values are a + b, a + c, b + c, - a + b, a+c, -bc, and the values of e2, or y, are a + b2, a + c2, b+c2; all of which being possible, the cubic cannot be solved by any direct method. Suppose the roots of

EQUATION, annual, of the mean motion of the sun and moon's apogee and nodes. The annual equation of the sun's mean mo

earth's orbit round him, and is 16 such the sun and the earth is 1000; whence some parts, of which the mean distance between have called it the equation of the centre, which, when greatest, is 1° 56′ 20′′.

is 11′ 40′′; of the apogee, 20'; and of its The equation of the moon's mean motion node, 9′ 30′′.

These four annual equations are always mutually proportionable to each other; so that when any of them is at the greatest, the when one diminishes, the rest diminish in three others will also be greatest; and the same ratio. Wherefore the annual equation of the centre of the sun being given, the other three corresponding equations will be given, so that one table of the central equations will serve for all.

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EQUATION of a curve, is an equation shewing the nature of a curve by expressing the relation between any absciss and its corresponding ordinate, or else the relation of their fluxions, &c. Thus, the equation to the circle is a x — r2y2, where a is its diameter, any absciss or part of that diameter, and y the ordinate at that point of the diameter; the meaning being that whatever absciss is denoted by x, thenthe square of its corresponding ordinate will be - x2. In like manner the equation

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EQUATION of time, in astronomy and chronology, the reduction of the apparent time or motion of the sun, to equable, mean, or true time. The difference between true and apparent time arises from two causes, the excentricity of the earth's orbit, and the obliquity of the ecliptic. See TIME, equation of.

EQUATOR, in geography, a great circle of the terrestrial globe, equidistant from its poles, and dividing it into two equal hemispheres; one north and the other south. It passes through the east and west points of the horizon, and at the meridian is raised as much above the horizon as is the complement of the latitude of the place. From this circle the latitude of places, whether north or south, begin to be reckoned in degrees of the meridian. All people living on this circle, called by geographers and navigators the line, have their days and nights constantly equal. It is in degrees of the equator that the longitude of places are reckoned; and as the natural day is mea. sured by one revolution of the equator, it follows that one hour answers to 36 15 degrees: hence one degree of the equator will contain four minutes of time; 15 minutes of a degree will make a minute of an hour; and consequently, four seconds answer to one minute of a degree.

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EQUATIONAL. See OBSERVATORY. EQUERRY, in the British customs, an officer of state, under the master of the horse. There are five equerries who ride abroad with his Majesty; for which purpose they give their attendance monthly, one at a time, and are allowed a table.

EQUISETUM, in botany, English horsetail, a genus of the Cryptogamia Filices class and order. Natural order of Filices or Ferns. There are seven species. They are natives of most parts of Europe in woods and shady places.

EQUIANGULAR, in geometry, an epithet given to figures, whose angles are all equal: such are a square, an equilateral triangle, &c.

EQUICRURAL, in geometry, the same with isosceles. See ISOSCELES TRIANGLE. EQUIDIFFERENT numbers, in arithmetic, are of two kinds. 1. Continually

equidifferent is when, in a series of three numbers, there is the same difference be tween the first and second, as there is between the second and third; as 3, 6, 9. And 2. Discretely equidifferent, is when, in a series of four numbers or quantities, there is the same difference between the first and second as there is between the third and fourth: such are 3, 6, 7, 10.

EQUIDISTANT, an appellation given to things placed at equal distance from some fixed point, or place, to which they are referred.

EQUILATERAL, in general, something that hath equal sides, as an equilateral angle.

EQUILATERAL hyperbola, one whose transverse diameter is equal to its parameter; and so all the other diameters equal to their parameters: in such an hyperbola, the asymptotes always cut one another at right angles in the centre. Its most simple equation, with regard to the transverse axis, is y2= x -- a2; and with regard to the conjugate, y2 = x2+a2, when a is the semitransverse, or semiconjugate. The length of the curve cannot be found by means of the quadrature of any space, of which a conic section is any part of the perimeter.

EQUILIBRIUM, in mechanics, is when the two ends of a lever or balance hang so exactly even and level, that neither doth ascend or descend, but keep in a position parallel to the horizon, which is occasioned by their being both charged with an equal weight.

EQUIMULTIPLES, in arithmetic and geometry, are numbers and quantities multiplied by one and the same number or quantity. Hence, equimultiples are always in the same ratio to each other, as the simple quantities before multiplication: thus, if 6 and 8 are multiplied by 4, the equimultiples 24 and 32 will be to each other, as 6

to 8.

EQUINOCTIAL, in astronomy, a great circle of the celestial globe, whose poles are the poles of the world. It is so called, because whenever the sun comes to this circle, the days and nights are equal all over the globe; being the same with that which the sun seems to describe, at the time of the two equinoxes of spring and autumn. All stars directly under this circle, have no declination, and always rise due east, and set full west. The hour circles are drawn at right angles to it, passing through every fifteenth degree; and the parallels to it are called parallels of declination.

EQUINOX, the time when the sun enters either of the equinoctial points, where the ecliptic intersects the equinoctial. It was evidently an important problem in practical astronomy, to determine the exact moment of the sun's occupying these stations; for it was natural to compute the course of the year from that moment. Accordingly this has been the leading problem in the astronomy of all nations. It is Susceptible of considerable precision, without any apparatus of instruments. It is only necessary to observe the sun's declination on the noon of two or three days before and after the equinoctial day. On two consecutive days of this number, his declination must have changed from north to south, or from south to north. If his declination on one day was observed to be 21′ north, and on the next 5′ south, it follows that his declination was nothing, or that he was in the equinoctial point about 23 minutes after 7 in the morning of the second day. Knowing the precise moments, and knowing the rate of the sun's motion in the ecliptic, it is easy to ascertain the precise point of the ecliptic in which the equator intersected it. By a series of such observations made at Alexandria, between the years 161 and 127 before Christ, Hipparchus, the father of our astronomy, found that the point of the autumnal equinox was about six degrees to the eastward of the star called spica virginis. Eager to determine every thing by multiplied observations, he ransacked all the Chaldean, Egyptian, and other records, to which his travels could procure him access, for observations of the same kind; but he does not mention his having found any. He found, however, some observations of Aristillus and Timochares, made about 150 years before. From these it appeared evident that the point of the autumnal equinox was then about eight degrees east of the same star. He discusses these observations with great sagacity and rigour: and on their authority, he asserts that the equinoctial points are not fixed in the heavens, but move to the westward about a degree in 75 years, or somewhat less.

This motion is called the precession of the equinoxes, because by it the time and place of the sun's equinoctial station precedes the usual calculations: it is fully confirmed by all subsequent observations. In 1750, the autumnal equinox was observed to be 20° 21′ westward of spica virginis. Supposing the motion to have been uniform

during this period of ages, it follows that the annual precession is about 50'; that is, if the celestial equator cuts the ecliptic in a particular point on any day of this year, it will on the same day of the following year, cut it in a point 50' to the west of it, and the sun will come to the equinox 20′ 23′′ before he has completed his round of the heavens. Thus the equinoctial, or tropical year, or true year of seasons, is so much shorter than the revolution of the sun or the sidereal year. It is this discovery that has chiefly immortalized the name of Hipparchus, though it must be acknowledged that all his astronomical researches have been conducted with the same sagacity and intelligence. It was natural, therefore, for him to value himself highly for the discovery. It must be acknowledged to be one of the most singular that has been made, that the revolution of the whole heavens should not be stable, but its axis continually changing. For it must be observed, that since the equator changes its position, and the equator is only an imagi. nary circle, equidistant from the two poles, or extremities of the axis, these poles, and this axis must equally change their positions. The equinoctial points make a complete revolution in about 25,745 years, the equator being all the while inclined to the ecliptic in nearly the saine angle. Therefore the poles of this diurnal revolution must describe a circle round the poles of the ecliptic, at the distance of about 23 degrees in 25,745 years; and in the time of Timochares, the north pole of the heavens must have been 30 degrees eastward of where it now is.

EQUITY, quasi æqualitus, is generally understood in law, a liberal correction, or qualification of the law, where it is too strict, too confined, or severe, and is sometimes applied, where, by the words of a statute, a case does not fall within it, yet being within the mischief, the judges, by an equitable construction, have extended its application to that case. Equity is understood as a correction of the law: the difference between courts of equity and law is known only in this country, and arises principally, if not entirely, from the different modes of trial which must ever render them essentially distinct. For it is obvious, that where men form contracts in the ordinary course of law, the legal consequence, and the enforcement of them, must be, according to general rules, applicable to general cases; and the nature of our

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