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have had an opportunity of seeing them men, who came to adore him, and bring in America and the East Indies, their na. him presents. The feast of Epiphany tive places of growth.

was not originally a distinct festival, but EPIDERMIS, in anatomy, the same made a part of that of the nativity of with the cuticle. See Cutis.

Christ, which being celebrated twelve EPIGÆA, in botany, a genus of the days, the first and last of which were Decandria Monogynia class and order. high or chief days of solemnity, either of Natural order of Bicornes. Ericæ, Jus. these might properly be called Epiphany, sieu. Essential character: calyx outer as that word signifies the appearance of three-leaved ; inner five-parted; corolla Cbrist in the world. salver-form ; capsule five-celled. There The kings of England and Spain offer are but two species, riz. E. repens, creep. gold, frankincense, and myrrh, on Epiphing epigæa, or trailing arbulus, and E. any, or twelfth day, in memory of the cordifolia, heart-leaved epigæa : the form- offerings of the wise men to the infant er, remarkable for its fine odour, is a na Jesus. tive of Virginia and Canada, and the latter The festival of Epiphany is called by of Guadaloupe.

the Greeks the feast of lights, because EPIGLOTTIS, one of the cartilages our Saviour is said to have been baptised of the larynx or wind-pipe. See Ana on this day ; and baptism is by them call

ed illumination. EPIGRAM, in poetry, in short poem or EPISCOPALIANS, in the modern accomposition in verse, treating only of one ceptation of the term, belong more espething, and ending with some lively, inge. cially to members of the Church of Eng. nious, and natural thought or point. land, and derive this title from episcopus,

EPILEPSY, in medicine, the same the Latin word for bishop; or, if it be rewith what is otherwise called the falling ferred to its Greek origin, implying the sickness, from the patient's falling sud care and diligence with which bishops denly to the ground.

are expected to preside over those comEPILOBIUM, in botany, a genus of mitted to their guidance and direction. the Octandria Monogynia class and or- They insist on the divine origin of their der. Natural order of Calycanthemæ. bishops, and other church officers, and Onagræ, Jussieu. Essential character: on the alliance between church and calyx four-cleft; petals four; capsule state. Respecting these subjects, how. oblong, inferior ; seeds downy. There ever, Warburton and Hoadley, together are fourteen species. These plants are with others of the learned amongst them, hardy perennials, not void of beauty; have different opinions, as they have also they are, however, commonly considered on the thirty-nine articles, which were only as weeds, and are rarely cultivated established in the reign of Queen Eliza. in gardens. The American species are, beth. These are to be found in most 1. E. estoratum ; 2. E. spicatum ; 3. E. Common Prayer-Books; and the Episco. strictum ; 4. E. linerate.

pal Church in America bas reduced their EPILOGUE, in dramatic poetry, a number to twenty. By some the articles speech addressed to the audience after are made to speak the language of Calvi. the play is over, by one of the principal nism, and by others they have been in. actors therein, usually containing some terpreted in favour of Arminianism. reflections on certain incidents in the The Church of England is governed by play, especially those in the part of the the King, who is the supreme head : by person that speaks it.

two archbishops, and twenty-four bishops. EPIMEDIUM, in botany, English bar. The benefices of the bishops were conrenwort, a genus of the Tetrandria Mono- verted by William the Conqueror into gynia class and order. Natural order of temporal baronies; so that every prelate Corydales. Berberides, Jussieu. Essen. has a seat and vote in the House of Peers. tial character : nectary four, cup-form, Dr. Benjamin Hoadley, however, in a serleaning on the petals ; corolla four petal- mon preached from this text, “ My kingled ; calys very caducous; fruit a silique. dom is not of this world,” insisted that There is but one species, viz. E. alpinum, the clergy had no pretensions to tempo. alpine barrenwort.

ral jurisdiction, which gave rise to vari. 'EPIPHANY, a christian festival, other. ous publications, termed, by way of emiwise called the manifestation of Christ to nence, the Bangorian Controversy, Hoad. the Gentiles, observed on the sixth of Ja- ley being then bishop of Bangor. There nuary, in honour of the appearance of is a bishop of Sodor and Man, who has our Saviour to the three magi, or wise no seat in the House of Peers.

Since the death of the intolerant Arch- crease or diminution of velocity in each, bishop Laud, men of moderate princi- they are said to be equally accelerated, ples have been raised to the see of or retarded. Canterbury, and this hath tended not a EQUAL, a term of relation between little to the tranquillity of church and two or more things of the same magnistate. The established Church of Ire- tude, quantity, or quality. Mathematiland is the same as the Church of Eng- cians speak of equal lines, angles, figures, land, and is governed by four archbi. circles, ratios, solids, &c. shops, and eighteen bishops.

EQUALITY, that agreement between EPISODE, in poetry, a separate inci- two or more things whereby they are de. dent, story or action, which a poet invents nominated equal. The equality of two and connects with his principal action, quantities, in algebra, is denoted by two that his work may abound with a greater parallel lines placed between them; thus, diversity of events; though, in a more 4+2 = 6, that is, 4 added to 2 is equal limited sense, all the particular incidents

to 6. whereof the action or narration is com EQUANIMITY, in ethics, denotes that pounded are called episodes.

even and calm frame of mind and temEPITAPH, a monumental inscription per, under good or bad fortune, whereby in honour or memory of a person defunct, a man appears to be neither puffed up or an inscription engraven or cut on a or overjoyed with prosperity, nor dispitomb, to mark the time of a person's de- rited, soured, or rendered uneasy, by adcease, his name, family, and, usually, versity. some eulogium of his virtues, or good EQUATION, in algebra, the mutual qualities.

comparing two equal things of differEPITHALAMIUM, in poetry, a nup ent denominations, or the expression tial song, or composition, in praise of the denoting this equality; which is done by bride and bridegroom, praying for their setting the one in opposition to the prosperity, for a happy offspring, &c. other, with the sign of equality ( =)

EPITHET, in poetry and rhetoric, an between them: thus, 38. = 36d. or 3 adjective expressing some quality of a feet = 1 yard. Hence, if we put a for a substantive to which it is joined; or such foot, and 6 for a yard, we shall have the an adjective as is annexed to substantives equation 3 a = 6, in algebraical charac. by way of ornament and illustration, not

See ALGEBRA. to make up an essential part of the de EQUATIONS, construction of, in algescription." “ Nothing,” says Aristotle, bra, is the finding the roots or unknown * tires the reader more than too great a quantities of an equation, by geometrical redundancy of epithets, or epithets plac. construction of right lines or curves, or ed improperly; and yet nothing is so es the reducing given equations into geosential in poetry as a proper use of them.” metrical figures. And this is effected

EPITOME, 'in literary history, an by lines or curves, according to the order abridgment or summary of any book, par or rank of the equation. The roots of ticularly of a history.

any equation may be determined, that is, EPOCHA, in chronology, a term or the equation may be constructed, by the fixed point of time, whence the succeed intersections of a straight line with ano. ing years are numbered or accounted. ther line or curve of the same dimensions See CHRONOLOGY,

as the equation to be constructed : for EPODE, in lyric poetry, the third or the roots of the equation are the ordinates last part of the ode, the ancient ode be. of the curve at the points of intersection ing divided into strophe, antistrophe, and with the right line ; and it is well known epode.

that a curve may be cut by a right line in EPOPOEIA, in poetry, the story, fable, as many points as its dimensions amount or subject, treated of in an epic poem. to. Thus, then, a simple equation will be The word is commonly used for the epic constructed by the intersection of one poem itself. See Epic.

right line with another; a quadratic EPSOM salt, another name for sulphate equation, or an affected equation of the of magnesia.

second rank, by the intersections of a EQUABLE, an appellation given to right line with a circle, or any of the cosuch motions as always continue the nic sections, which are all lines of the sesame in degree of velocity, without be cond order; and which may be cut by the ing either accelerated or retarded. When right line in two points, thereby giving two or more bodies are uniformly acce the two roots of the quadratic equation. lerated or retarded, with the same in. A cubic equation may be constructed by


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the intersection of the right line with a and radius A C describe the semicircle D line of the third order, and so on. But CE; so shall D B and BE be the two roots if, instead of the right line, some other of the given quadratic equation x +2 ar line of a higher order be used, then the = 6*. 3. If the quadratic be x - 303 second line, whose intersections with the =b, then the construction will be the very former are to determine the roots of the same as of the preceding one rs + 2 at equation, may be taken as many dimen = b. 4. But if the form be 2 arsions lower as the former is taken high , form a right-angled triangle (fig. er. And, in general, an equation of any 1.) whose hypothenuse F G is a, and perheight will be constructed by the inter pendicular G H is b; then with the radius section of two lines, whose dimensions F G and centre F describe a semicircle I multiplied together produce the dimen GK; so shall I H and H K be the two sion of the given equation. Thus, the in roots of the given equation 2 ax -- ** = tersections of a circle with the conic sec b, or 2 ax=-ba. tions, or of these with each other, will construct the biquadratic equations, or To construct cubic and biquadratic equathose of the fourth power, because 2 X 2 tions. These are constructed by the inter= 4; and the intersections of the circle, sections of two conic sections; for the or conic sections, with a line of the third equation will rise to four dimensions, by order, will construct the equations of the which are determined the ordinates from fifth and sixth power, and so on. — For the four points in which these conic sec. example:

tions may cut one another; and the

conic sections may be assumed in such To construct a simple equation. This is a manner as to make this equation codone by resolving the given simple equa: incide with any proposed biquadratic ; tion into a proportion, or finding a third so that the ordinates from these four or fourth proportional, &c. Thus, 1. If intersections will be equal to the roots the equation be a x=bc; then a:b::C:x of the proposed biquadratic. When one be

of the intersections of the conic section the fourth proportional to a,b,c. 2.

falls upon the axis, then one of the orb',

dinates vanishes, and the equation by If a x=b; then a:0::::x = a third which these ordinates are determined proportional to a and b. 3. If a x = b.

will then be of three dimensions only, cə; then, since b: -c=b+cxb-c,

or a cubic to which any proposed cubic

it b+cxb

equation may be accommodated ; so that will be a:b+c::6c:x=

the three remaining ordinates will be

the roots of that proposed cubic. The a fourth proportional to a, btc, and bc. conic sections for this purpose should 4. If a x = b +ca; then construct the be such as are most easily described ; right-angled triangle ABC (Plate V. Mis the circle may be one, and the paracel fig. 5.) whose base is b, and perpen. bola is usually assumed for the other. dicular is c, so shall the square of the hy. See Simpson's and Maclaurin's Algebra. pothenuse be be+c, which call h; then the equation is ax=h', and

a third

EQUATIONS, nature of. Any equation

involving the powers of one unknown proportional to a and h.

quantity may be reduced to the form zł

pza-1 +9.39–3, &c. = 0, here the whole To construct a quadratic equation. 1. If it expression is equal to nothing, and the be a simple quadratic, it may be reduced terms are arranged according to the dito this form, xà = ab; and hence a :x:: 3 mensions of the unknown quantity, the :b, or x = ✓ab, a mean proportional coefficient of the highest dimension is between a and b. Therefore upon a

unity, understood, and the coefficients straight line take A B = a, and BC= b;

p, q, r, and are effected with the proper then upon the diameter A C describe a signs. An equation, where the index is semicircle, and raise the perpendicular of the highest power of the unknown BD to meet it in D; so shall B D be = quantity is n, is said to be of n dimenx, the mean proportional sought between sions, and in speaking simply of an equaA B and B C, or between a and b. 2. If tion of n dimensions, we understand one the quadratic be affected, let it first be xa

reduced to the above form. Any quan. + 2 ax = = 6; then form the right-angled tity mp zn–1 + 9 217—2, &c. + P :-Q triangle,

whose base A B is a, and perpen- may be supposed to arise from the multi. dicular B C is b; and with the centre Aplication of : - - ax -Ex


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f, &c.

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&c. =

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to n factors. For by actually multiply. Hence the other two roots are the roots ing the factors together, we obtain a of the quadratic x - x +1 = 0. If quantity of n dimensions similar to the two roots, a and b, be obtained, the equaproposed quantity Znap z' +92n, tion is divisible by x - a xx

b, and &c.; and if a, b, c, &c. can be so assum

may be reduced in the same manner two ed, that the coefficients of the corres.

dimensions lower. ponding terms in the two quantities become equal, the whole expressions coin Ex. Two roots of the equation 2 - 1 cide. And these coefficients may be made

• 0, are + 1 and -1, orz-1=0, and equal, because these will be n equations, 2+1= 0'; therefore it may be depressto determine n quantities, a, b, c, &c. If then the quantities, a, b, c, &c. be pro

ed to a biquadratic by dividing by 7 perly assumed, the equation zoopz+

X:+1=q - 1. q 27.-, &c. = 0, is the same with z - a

Xz-bx? — C, &c. 0. The quanti-

ties 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 equa-


Every equation has as many roots as it
has dimensions. If on — pzms +pzn,

: 0, or :- axz-6X7-C, &c.
to n factors = 0, there are n quantities,
a, b, c, &c. each of which when substi-
tuted for z makes the whole = 0, because

Hence the equation z4 + zd+1=0 con. in each case one of the factors becomes

tains the other four roots of the proposed = 0; but any given quantity different

equation. from these, as e when substituted for 2, Conversely, if the equation be divisigives the product e-axe-o xe-C, ble by x – a without a remainder, a is &c, which does not vanish, because none of the factors vanish, that is, e will not

a root; if by xa x x - b, a and b are

both roots. Let Q be the quotient arisanswer the condition which the equation

ing from the division, then the equation requires. When one of the roots, a, is obtained,

is x -a X x-bXQ= 0, in which, if

a or 6 be substituted for x, the whole the equation :-ax:-6 XI-C,


0,27-2 +92n–, &c. = 0 is di.
visible by : - a without a remainder,

EQUATIONS, cubic solution of, by Car. and is thus reducible to z — b X2 – C, dan's rule. Let the equation be reduced &c. - 0, an equation one dimension low to the form x3—9 xtr=0, where q and er, whose roots are b and c.

r may be positive or negative.

Assume x = a + b, then the equation Er. One root of x3 +1= 0, or x + becomes a +63-9 xa+b+r=1, or 1= 0, and the equation may be depressed to a quadratic in the following

a3 + b3 + 3 a b xa+b-q xa+b+ p=0; and since we have two unknown

quantities, a and b, and have made only *+1).x3+1(x-x+1

one supposition respecting them, viz. that ritx


= x, we are at liberty to make another; let 3 a 6 - 9 = 0, then the equation becomes a3+63 for=0; also, since 3 ab-q=0, b:

3 a'

and by sub+x+1


0, or a + x+1

q} ra3 +

= 0, an equation of a qua.

27 dratic form; and by completing the

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stitution, a3 + 27 az

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Ex. Let x3 + 6 x

20: = 0; here

+/10 - 108= 2.732-732 = 2.

=-6,r=–20,x= xy 10+ V 108 a+b;=1+x=



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Cor. 1. Having obtained ône value of 3, the equation may be depressed to a quadratic, and the other roots found.

Cor. 3. This solution only extends to those cases in which the cubic has two impossible roots.

Cor. 2. The possible values of a and b being discovered, the other roots are known without the solution of a quadratic.

For if the roots be m + /3nm-311

, and - 2 m, then 9 (the sum of the

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