to L and g, are known; '2 ''2 the biquadratic is thus resolved into two quadratics, whose roors may be found. It may be observed, that which ever This solution can only be applied to those cases, in which two roots of the biquadratic are possible and two impossible. Let the roots be a, b, c, — a -|- the biquadratic to be a -\- b </ — 1; a — h */ — !> —a-\-C\/ — 1; —a — c */ — 1; the values of c are 2 a, b + c . ^/ — l, o — c.\/ — 1, —6 — c.^/ — 1, — 4-(-c. tj— l and — 2a; and the three values ofyareiT?2, — b~+c]\ — b^c]', which are all possible, as in the preceding case. But if the roots of the biquadratic be a + 6 \/ — 1, a — b*/ ■— 1, —a-\-c, —a —c, the values of y are i~a)2, c -\-b^/'—l)2, c — b *S — l r, two of which are impossible; therefore the cubic may be solved by Cardan's rule. Equation, annual, of the mean motion of the sun and moon's apogee and nodes. The annual equation of the sun's mean motion depends upon the excentricily of the earth's orbit round him, and is 16JJ such parts, of which the mean distance between the sun and the earth is 10.00; whence some have called it the equation of the centre, which, when greatest, is 1° 56' 20". The equation of the moon's mean motion is 11' 40"; of the apogee, 20'; and of its 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 three 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 ax — -r2 = if, where a is its diameter, x 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 u i — x2. In like manner the equation of the ellipse U £. ax — .t' = f a p of the hyperbola is - . a x -(- iu = if, of the parabola is px = y2. Where This method of expressing the nature of curves by algebraical equations, was first introduced by Des Cartes, who, by thug connecting together the two sciences of algebra and geometry, made them mutually assisting to each other, and so laid the foundation of the greatest improvements that have been made in every branch of them since that time. 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 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, Sec. EQUICRURAL, in geometry, the same with isosceles. See Isosceles Triangle. EQUIDIFFERENT numbers, in arithmetic, are of two kinds. 1. Continually cqiiidifferent is when, in a series of three numbers, there is the same difference between the first and second, as EQUIDISTANT, 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 y1 = x — oa; and with regard to the conjugate, if =ii + °2> wncn o 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 cast, 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.' EQVIN'OX, the time when the sun enters either of the equinoctial poir; <, where the ecliptic intersects the 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 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 imaginary circle, equidistant from the two poles, or extremities ot 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 same 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 SO degrees eastward of where it now is.EQUITY, quasi aqualitat, 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, Iu equity, however, the plaintiff by filing his bill, waves the objection, and submits to take the answer of each defendant, though he cannot be admitted to give evidence himself. This is the process by what is called English bill in equity, and the form of proceeding, though somewhat tardy, gives the parties the fullest opportunity of obtaining a final decision according to good conscience. It is this difference in the proceeding, which has rendered the best judges in courts of law, averse to introducing equitable distinctions and principles applicable to courts of equity in courts of law, because they have not the same means of informing their consciences upon all the circumstances necessary, to induce them to alter the strict law according to the peculiar facts, or conscientious circumstances of the case. Formerly, it is supposed, the King, upon petition, referred the case upon a harsh decision at law to a committee, together with the Chancellor; but in the time of Edward III, when uses, or trusts of lands, which were discountenanced at common law, were considered as binding in conscience by the clergy, John Waltham, i Immvllor to Richard II, introduced the writ cessors, has established a practical system of equity, which is as definite and well understood as the law itself; and taking into consideration the leading circumstances above mentioned, is nothing more thaiwlie' law administered according to. the justice of the case. There are some cases which belong more peculiarly to a court of chancery, as the care of infants, and appointing guardians to them, so of lunatics and charities, in which the Chancellor acts for the Ming as keeper of his conscience. In other cases, as in cases of trust, matters of fraud,' account, suits for a discovery, matters of accident, and the like, courts of equity act, in aid of the courts of law, and give relief, where, from the nature of the case, a court of law cannot relieve. Thus, where an agreement is to be performed, courts of law can only give damages for the breach, but a court of equity, taking all the circumstances into consideration, directs and enjoins a specific performance of it according ,> to good conscience. So where it apprehends an injury likely to be done, it will interfere to prevent it. We have thought this explanation of the general principles, which distinguish courts of law and equity, better suited to a work like the present, than an attempt to abridge any more particular account of the practice and principles of courts of equity, which will be found to proceed upon the ordinary rules of good conscience, as far as they can be reduced to practice. An appeal lies from the Chancellor to the House of Lords. The Court of Exchequer has a court of equity, and so have most courts of peculiar jurisdiction. Equity of redemption. Upon a mortgage, although the estate upon non-payment of the money becomes vested in the mortagee, yet equity considers it only a pledge for the money, and gives the party a right to redeem, which is called his equity of redemption. If the mortgagee is desirous to bar the equity of redemption, he may oblige the mortgager either to pay the money, or be foreclosed of his equity, which is done by proceedings in the Court of Chancery by bill of foreclosure. EQUUS, the horse, in natural history, a genus of mammalia of the order Belhur. Generic character: upper fore-teeth parallel, and six in number ; in the lower jaw six, rather more projecting ; tasks on each side, in both jaws, remote from the rest; feet with undivided hoofs. There are six species, and very many varieties. |