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— n × 9 m2 — 6 m2 un2, a = √ = n x 3m2 n, a quantity manifestly impossible, unless n be negative, that is, unless two roots of the proposed cubic be impossible.
the biquadratic to be a + b√1;
EQUATION, annual, of the mean motion of the sun and moon's apogee and nodes. The annual equation of the sun's mean mo
EQUATIONS, biquadratic, solution of, by Des Carte's method. Any biquadratic may be reduced to the form a+q2+rx+ $0, by taking away the second term. Suppose this to be made up of the two quadratics, x2+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 x+g+ƒ—2. x2 + eg− eƒ.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 = 9, e g—ef =r, and fg = s; hence, g +f=4+e2, also g —ƒ=2, and by taking the sum and difference of these equations, 2g = qe +2, and 2ƒ=9+e therefore 4fg
= q2 + 29 e2 + e2. =48, and multi-
4 8 × e2 — r2 = 0; or, making y = e2, y3 + 2 qy2 + q2 — 4 8. y — r2 = 0. By the solution of this cubic, a value of y, and therefore of, or e, is obtained; also ƒ and g, which are respectively equal 9+ e2
and 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.
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 -b+c, and the values of e2, or y, are a + b2, a + c 2, b+c)2; all of which being possible, the cubic cannot be solved by any direct method. Suppose the roots of
earth's orbit round him, and is 16 such parts, of which the mean distance between the sun and the earth is 1000; 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 others will also be greatest; and when one diminishes, the rest diminish in 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.
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- ry, 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 r, thenthe square of its corresponding ordinate will be uxx'. In like manner the equation - x2 = y2
introduced by Des Cartes, who, by thus 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. Sce 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 liv. ing 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 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.
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 between 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, au 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 y=x -- a2; and with regard to the conjugate, y2 = x2+a', 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 decli̟nation.
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 decli nation 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 equiBox 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 précedes 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 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 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, minst be, according to general rules, applicable to general cases; and the nature of our
mode of trial by jury, is so strict in the evidence which it requires, that a strict legal decision, alone can justly be founded upon it. There are, however, many cases in which there are particular circumstances between the different parties peculiar to their case, which give rise to exceptions and equitable decisions wholly different from the general rule. These cases of exception are such, that unless the judge can inquire into all the circumstances affecting the conscience of the several parties, a perfectly equitable decision cannot be given, For this purpose the court of equity is empowered to examine all the litigant parties upon their oaths, and to make every one answer to the full, as to all the circumstances affecting the case, which is not done in a court of law, where no person can be a witness in his own cause.
In 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, Chancellor to Richard II. introduced the writ of subpoena, returnable in the Court of Chancery only, to make the tenant, or feoffee to uses, answerable for the confidence reposed in him, and this writ is the commencement of a suit in equity, which has been chiefly modelled by Lord Ellesmere, the great Lord Bacon, and Sir Heneage Finch, in the time of Charles I. Lord Hardwicke followed, at some distance, after these great men, and by his decisions, together with those of his suc
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 than the 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 King 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 Belluæ. Generic character: upper fore-teeth parallel, and six in number; in the lower jaw six, rather more projecting; tusks on each side, in both jaws, remote from the rest; feet with undivided hoofs. There are six species, and very many varieties,