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6m n, andere
products of every two with their signs the biquadratic is thus resolved into two changed) = - 3 ma
quadratics, whose roots may be found. 3
It may be observed, that whichever +n; also, " (the product of all the roots
value of y is used, the same values of 2 with their signs changed)
= 2 m3 - are obtained.
This solution can only be applied to and - 3 mn; and by invo- those cases, in which two roots of the lution,
biquadratic are possible, and two impossi
ble. = mb - 6 m4 n + 9 ma no Let the roots be a, b, c, - a + b + c;
then since e, the coefficient of the second q
= m +3m+ +3mon' tnen3 term of one of the reducing quadratics, is goz q?
the sum of two roots, its different values Hence,
9m+n+ 6mon"-n3 = are a + b, a + c, b + c, – a + b 27
a + 93
6 + c, and the values of e', ory, - 1 x 9m+-6m’ntno, and
are a +61, a +0,6 + 0); all of which
being possible, the cubic cannot be solved =v-n x 3m-n, a quantity manifest
by any direct method. Suppose the roots ly impossible, unless n be negative, that is, unless two roots of the proposed cubic
of the biquadratic to be a + bv - 1; be impossible.
-bv-1; - atv - 1;
-1; the values of e are 2 a, b + c. EQUATIONS, biquadratic, solution of, by
- 1,6-c. -1,-6-c. ✓-1, Des Carter's method. Any biquadratic may be reduced to the form x++ x +rx+
b+c.v - 1 and 2 a; and the 0, by taking away the second term. three values of y are 2 al, -5 +0,Suppose this to be made up of the two 5 - c', which are all possible, as in the quadratics, x'textf0, and x?
preceding case. But if the roots of the biex+5
0, where te and -e are made the coefficients of the second terms, be quadratic be a tov - 1,2-6v-1,
a+c, - a - c, the values of y are cause the second term of the biquadratic is wanting, that is, the sum of its roots 2 a), c+6 -1',c-by two is 0. By multiplying these quadratics of which are impossible; therefore the together we have 2+ +5+5-22.x + cubic may be solved by Cardan's rule.
EQUATION, annual, of the mean moegref.x +39= 0, which equation is tion of the sun and moon's apogee and made to coincide with the former by nodes. The annual equation of the sun's equating their coefficients, or making
mean motion depends upon the excen$+/-e =q,egref=r, and fg=8; tricity of the earth's orbit round him, and hence, s +f=9+e', also g -f=, is 161; such parts, of which the mean
distance between the sun and the earth is and by taking the sum and difference of 1000 ; whence some have called it the these equations, 2 grates +
and equation of the centre, which, when great
est, is 1° 56' 20". 2f =q+et-therefore 4 f = = q*
The equation of the moon's mean motion is 11° 40'; of the apogee, 20'; and
of its node, 9'300'. +2getet = 4 s, and multiply. These four annual equations are always ing by e®, and arranging the terms accord- mutually proportionable to each other; ing to the dimensions of e, em + 2q et
so that when any of them is at the great+(^-4 8 X eo -r = : 0;
or making and when one diminishes, the rest di
est, the three others will also be greatest; y=e", y3 + 2q yd + q 48 y-p=0. minish in the same ratio. Wherefore the
annual equation of the centre of the sun By the solution of this cubic, a value of being given, the other three correspondy, and therefore of y, or e, is obtained; ing equations will be given, so that one also f and g, which are respectively equal table of the central equations will serve
for all. atest
EQUATION of a curve, is an equation and
are known; shewing the nature of a curve by express2. 2
ing the relation between any absciss and
its corresponding ordinate, or else the re. officer of state, under the master of the lation of their Auxions, &c. Thus, the horse. There are five equerries who ride equation to the circle is a r - r: = y, abroad with his Majesty; for which purwhere a is its diameter, * any absciss or pose they give their attendance monthly, part of that diameter, and y the ordinate one at a time, and are allowed a table. at that point of the diameter: the mean. EQUISETUM, in botany, English horse. ing being, that whatever absciss is de tail, a genus of the Cryptogamia Filices noted by x, then the square of its cor class and order. Natural order of Filices responding ordinate will be a x - x? In or Ferns. There are seven species. They like manner the equation
are natives of most parts of Europe, in
woods and shady places. of the ellipse is bas
x-x=y, EQUIANGULAR, in geometry, an epi.
thet given to figures, whose angles are of the hyperbola is 2. a xtr: = y*, all equal; such are, a square, an equilate. of the parobola is.......
ral triangle, &c.
..pX = y Where a is an axis, and p the parameter. with isosceles. See ISOSCELES TRIAN.
EQUICRURAL, in geometry, the same And in like manner for any other curves. This method of expressing the nature
EQUIDIFFERENT numbers, in arith. of curves by algebraical equations was first introduced by Des @artes, who, by equidifferent, is when, in a series of three
metic, are of two kinds. 1. Continually thus connecting together the two sciences numbers, there is the same difference beof algebra and geometry, made them mu.
tween the first and second, as there is tually assisting to each other, and so laid
between the second and third ; as 3, 6, 9. the foundation of the greatest improve. And 2. Discretely equidifferent, is when, ments that have been made in every in a series of four numbers or quantities, branch of them since that time.
there is the same difference between the EQUATION of time, in astronomy and
first and second as there is between the chronology, the reduction of the appa- third and fourth : such are, 3, 6, 7, 10. rent time or motion of the sun to equa
EQUIDISTANT, an appellation given ble, mean, or true time. The difference between true and apparent time arises
to things placed at equal distance from from two causes, the excentricity of the
some fixed point, or place, to which they
are referred. earth's orbit, and the obliquity of the
EQUILATERAL, in general, someecliptic. See Time, equation of.
thing that hath equal sides, as an equilaEQUATOR, in geography, a great cir- teral angle. cle of the terrestrial globe, equidistant EQUILATERAL hyperbola, one whose from its poles, and dividing it into two transverse diameter is equal to its paraequal hemispheres; one north and the
meter; and so all the other diameters other south. It passes through the east equal to their parameters: in such an hy. and west points of the horizon, and at the perbola, the asymptotes always cut one meridian is raised as much above the another at right angles in the centre. Its horizon as is the complement of the lati
most simple equation, with regard to the tude of the place. From this circle the latitude of places, whether north or south, regard to the conjugate, yo = ** t av,
transverse axis, is yo =X-a; and with begin to be reckoned in degrees of the when a is the semitransverse, or semiconmeridian. All people living on this cir- jugate. The length of the curve cannot cle, called by geographers and navigators be found by means of the quadrature of the line, have their days and nights con
any space, of which a conic section is any stantly equal. It is in degrees of the part of the perimeter. equator that the longitude of places are
EQUILIBRIUM, in mechanics, is when reckoned; and as the natural day is mea the two ends of a lever or balance hang sured by one revolution of the equator, it so exactly even and level, that neither follows that one hour answers to 3629= 15 doth ascend or descend, but keep in a degrees: hence one degree of the equator position parallel to the horizon, which is will contain four minutes of time ; 15 mi- occasioned by their being both charged nutes of a degree will make a minute of with an equal weight. an hour ; and, consequently, four seconds EQUIMULTIPLES, in arithmetic and answer to one minute of a degree. geometry, are numbers and quantities
EQUATIONAL. See OBSERVATORY. multiplied by one and the same number EQUERRY, in the British customs, an or quantity. Hence, equimultiples arc
always in the same ratio to each other, as He found, however, some observations of the simple quantities before multiplica. Aristilics and Timochares, made about tion: thus, if 6 and 8 are multiplied by 4, 150 years before. From these it appeared the equimultiples 24 and 32 will be tó evident that the point of the autumnal each other as 6 to 8.
equinox was then about eight degrees EQUINOCTIAL, in astronomy, a great east of the same star. He discusses circle of the celestial globe, whose poles tiese observations with great sagacity are the poles of the world. It is so call. and rigour; and, on their authority, he ed, because, whenever the sun comes to asserts that the equinoctial points are not this circle, the days and nights are equal fixed in the heavens, but move to the all over the globe'; being the same with westward about a degree in 75 years, or that which the sun seems to describe at somewhat less. the time of the two equinoxes of spring This motion is called the procession of and autumn. All stars directly under this the equinoxes, because by it the time and circle have no declination, and always place of the sun's equinoctial station prerise due east, and set full west. The cedes the usual calculation : it is fully hour circles are drawn at right angles to confirmed by all subsequent observations. it, passing through every fifteenih de- In 1750, the autumnal equinox was obgree ; and the parallels to it are called served to be 20° 21' westward of spica parallels of declination.
virginis. Supposing the motion to have EQUINOX, the time when the sun en. been uniform during this period of ages, ters either of the equinoctial points, it follows that the annual precession is where the ecliptic intersects the equi- about 50' and one-third ; that is, if the noctial. It was evidently an important celestial equator cuts the ecliptic in a problem in practical astronomy, 10 de particular point on any day of this year, termine the exact moment of ihe sun's it will on the same day of the following occupying these stations ; for it was na. year cut it in a point 50' and one.third to tural to compute the course of the year the west of it, and the sun will come to from that moment. Accordingly, this the equinox 2023" before he has comhas been the leading problem in the as- pleted his round of the heavens. Thus tronomy of all nations. It it susceptible the equinoctial, or tropical year, or true of considerable precision, without any year of seasons, is so much shorter than apparatus of instruments. It is only ne the revolution of the sun, or the sidereal cessary to observe the sun's declination year. It is this discovery that has chiefly on the noon of two or three days before immortalized the name of Hipparchus, and after the equinoctial day. On two though it must be acknowledged that all consecutive days of this number, his de his astronomical researches have been clination must have changed from north conducted with the same sagacity and into south, or from south 10 north. If his telligence. It was natural, therefore, declination on one day was observed to for him to value bimself highly for the be 21' north, and on the next 5' south, it discovery. It must be acknowledged to follows that his declination was nothing, be one of the most singular that has been or that he was in the equinoctial point made, that the revolution of the whole about 23 minutes after 7 in the morning heavens should not be stable, but its axis of the second day. Knowing the pre- continually changing: For it must be cise moments, and knowing the rate of observed, that since the equator changes the sun's motion in the ecliptic, it is easy its position, and the equator is only an to ascertain the precise point of the imaginary circle, equidistant from the ecliptic in which the equator intersected two poles, or extremities of the axis, it. By a series of such observations these poles, and this axis, must equally made at Alexandria, between the years change their positions. The equinoctial 161 and 127 before Christ, Hipparchus, points make a complete revolution in the father of our astronomy, found that about 25,745 years, the equator being all the point of the autumnal equinox was the while inclined to the ecliptic in nearabout six degrees to the eastward of the ly the same angle. Therefore the poles star called spica virginis. Eager to de- of this diurnal revolution must describe termine every thing by multiplied obser. a circle round the poles of the ecliptic, Fations, he ransacked all the Chaldean, at the distance of about 23 degrees in Egyptian, and other records to which 25,745 years; and in the time of Timohis travels could procure him access, for chares, the north pole of the heavens observations of the same kind; but he must have been 30 degrees eastward of does not mention bis having found any. where it now is.
EQUITY, quasi æqualitas, is generally or conscientious circumstances of the understood, in law, a liberal correction, or case. Formerly, it is supposed, the qualification of the law, where it is too King, upon petition, referred the case strict, too confined, or severe, and is upon a harsh decision at law to a comsometimes applied, where, by the words mittee, together with the Chancellor ; of a statute, a case does not fall within it, but in the time of Edward IU. when yet being within the mischief, the judges, uses, or trusts of lands, which were disby an equitable construction, have ex countenanced at common law, were contended its application to that case. Equi. sidered as binding in conscience by the ty is understood as a currection of the clergy, John Waltham, Chancellor to law: the difference between courts of Richard II. introduced the writ of subequity and law is known only in this pæna, returnable in the Court of Chance. country, and arises principally, if not en ry only, to make the tenant, or feoffee to tirely, from the different modes of trial, uses, answerable for the confidence rewhich must ever render them essentially posed in him, and this writ is the com. distinct. For it is obvious, that where mencement of a suit in equity, which men form contracts in the ordinary has been chiefly modelled by Lord Elles course of law, the legal consequence, and mere, the great Lord Bacon, and Sir the enforcement of them, must be, ac. Heneage Finch, in the time of Charles I cording to general rules, applicable to ge Lord Hardwicke followed, at some disneral cases; and the nature of our mode tance, after these great men, and by his of trial by jury is so strict in the evidence decisions, together with those of his sucwhich it requires, that a strict legal deci- cessors, has established a practical sys, sion alone can justly be founded upon it. tem of equity, which is as definite and There are, however, many cases, in which well understood as the law itself; and there are particular circumstances be. taking into consideration the leading cirtween the different parties peculiar to cumstances above mentioned, is nothing their case, which give rise to exceptions more than the law administered accordand equitable decisions wholly different ing to the justice of the case. There from the general rule. These cases of are some cases which belong more peexception are such, that unless the judge culiarly to a court of chancery, can inquire into all the circumstances af care of infants, and appointing guardians fecting the conscience of the several par. to them ; so of lunatics and charities, in ties, a perfectly equitable decision cannot which the Chancellor acts for the King be given. For this purpose the court of as keeper of his conscience. In other equity is empowered to examine all the cases, as in cases of trust, matters of Pitigant parties upon their oaths, and to fraud, account, suits for a discovery, make every one answer to the full, as to matters of accident, and the like, courts all the circumstances affecting the case, of equity act, in aid of the courts of law, which is not done in a court of law, and give relief, where, from the nature where no person can be a witness in his of the case, a court of law cannot relieve.
Thus, where an agreement is to be per: In equity, however, the plaintiff by formed, courts of law can only give da
: filing his bill waves the objections, and mages for the breach ; but a court of submits to take the answer of each de equity, taking all the circumstances into fendant, though he cannot be admitted consideration, directs and enjoins a speci: to give evidence himself. This is the fic performance of it according to good process by what is called English bill conscience. So, where it apprehends an in equity; and the form of proceeding, injury likely to be done, it will interfere though somewhat tardy, gives the par to prevent it. ties the fullest opportunity of obtaining We have thought this explanation of a final decision according to good con the general principles, which distinguish science. It is this difference in the pro courts of law and equity, better suited to ceeding, which has rendered the best a work like the present, than an attempt judges in courts of law averse to intro to abridge any more particular account ducing equitable distinctions and princi- of the practice and principles of courts ples, applicable to courts of equity, in of equiiy, which will be found to pro; courts of law, because they have not the ceed upon the ordinary rules of gnod same means of informing their con conscience, as far as they can be reduced sciences upon all the circumstances ne to practice. An appeal lies from the cessary to induce them to alter the Chancellor to the House of Lords. The strict law according to the peculiar facts, Court of Exchequer has a court of equi
\, and so have most courts of peculiar ju. cle of commerce. These American horrisdiction.
ses are the descendants of those which Equity, of redemption. Upon a mort were introduced by the Spaniards on gage, although the estate upon non-pay
their discovery of America, as none prement of the money becomes vested in viously existed on that continent. They the mortgagee, yet equity considers it on. are, in general, small and clumsily form. }y a pledge for the money, and gives the ed, and their height is rarely above fourparty a right to redeem, which is called teen hands. In the deserts of Arabia, it his equity of redemption. If the mort has been stated by several writers, wild gagee is desirous to bar the equity of re horses are extremely abundant; but demption, he may oblige the mortgager Shaw and Sonnini, with greater probabieither to pay the money, or be foreclosed lity, confine their appearance in that of his equity, which is done by proceed. country to the borders of the desert, the ings in the Court of Chancery by bill of latter not supplying materials for their foreclosure.
subsistence. Mr. Bruce mentions the EQUUS, the horse, in natural history, a horses of Nubia as unequalled in beauty, genus of mammalia of the order of Belluæ. and far superior to those of Arabia. Of Generic character : upper fore-teeth pa
the former little notice has been taken rallel, and six in number; in the lower but from that observant traveller; of the jaw six, rather more projecting; tusks on latter the fame has long been distinguisheach side, in both jaws, remote from the ed ; and the Arabian horse, celebrated rest; feet with undivided hoofs. There for his beauty and swiftness, has been are six species, and very many varieties. long exported to the most remote coun
E. caballus, or the common horse. tries of Europe, to correct and improve The elegance, grace, and usefulness of the native breeds. In Arabia, almost evethe horse entitle him to particular atten. ry man possesses his horse, which lives tion, and certainly confer upon him a in the same apartment, or tent, with his pre-eminence above all other quadru. family, and is considered as constituting peds. There are few parts of the world by no means the lest important part of in which horses are not to be found; and it. Harsh and violent applications, such in various parts of Africa they maintain as the whip or spur, are rarely inflicted their original independence, and range on it. It is fed with the most regular at. at pleasure in herds of several hundreds, tention, and cleaned with incessant assihaving always one or more as an advanc- duity. The Arab occasionally appears to ed guard, to alarm against approaching carry on a conversational intercourse danger. These alarms are expressed by with his horse, and bis external attacha sudden snorting, at which the main body ment to this animal excites in return a gallop off with the most surprising corresponding affection. The horse beswiftness. In the south of Siberia also, ing purified under his management from and at the north-west of China, wild every vicious propensity, and guarded ahorses are to be found in considerable gainst casual injury with the utmost soli. abundance; and it is stated, that differ. citude, suffering the infant children to ent berds will carry on hostilities, and climb its legs without the slightest atone party frequently surround an enemy tempt to kick or shake them off. The inferior in number, and conduct them to Arabs never cross the breeds of horses, the hostile territory, manæuvring perpe and preserve the genealogies of these anitually to bafle all their attempts to
mals for a considerable number of geescape. On each bank of the river Don, nerations. The horses of Barbary are in towards the Palus Mæotis, horses are high reputation, also, for speed and elefound wild, but are supposed to be the gance, as are likewise those of Spain. In descendants of domesticated horses, be. various parts of the East, as in India and longing to the Russian army occupied in in some parts of China, there exists a the siege of Asoph, at the close of the race of these animals, scarcely exceeding seventeenth century. In America, like. the height of a large mastiff, and with wise, horses are found wild in vast abun. their diminutive size are generally condance, sweeping the extensive plains of nected not a little intractability and misBuenos Ayres, and the Brazils particu. chievousness. In no country of the larly
, in immense herds. They are ta. globe has the breeding of the horse been ken by the inhabitants, by throwing, with attended to on more enlarged and philogreat dexterity, a noosed cord over their sophic principles than in Great Britain heads, at full speed; and are often de. and with such success have the efforts stroved merely for their hides, as an arti. of the English on this subject been at