arb, or sphere, which being fixed in the complaints in the stomach and bowels; audeferent of a planet, is carried along with tumn by catarrhs; and winter by interit; and yet, by its own peculiar motion, mittents: these being obviously produced carries the planet fastened to it round its by the state of weather attendant upon proper centre. them, other epidemics are supposed analoIt was by means of epicycles, that Pto- gous to them, and obedient to the same lemy and his followers solved the various rules, which, on examination, not being phenomena of the planets, but more espe- the case, all further scrutiny is laid aside, cially their stations and retrogradations. perhaps too hastily. The great circle they called the excentric The most patural and healthful seasons or deferent, and along its circumference in this country are a moderately frosty win. the centre of the epicycle was conceived to ter, showery spring, dry summer, and rainy move; carrying with it the planet fixed in autumn; and whilst such prevail, the wet its circumference, which in its motion down- part of them is infested by vastly the greatwards proceeded according to the order of est proportion of complaints, but those not the signs, but, in moving upwards, con- of the most mortal kind. A long succestrary to that order. The highest point of sion of wet seasons is accompanied by a a planet's epicycle they called apogee, and prodigious number of diseases ; but these the lowest perigee. being mild and tedious, the number of EPICYCLOID, in geometry, a curve deaths are not in proportion to the cogenerated by the revolution of the peri- existent ailments. On the other hand, a phery of a circle, ACE (Plate V. Mis- dry season, in the beginning, is attended cel. fig. 4.) along the convex or concave with extremely few complaints, the body side of the periphery of another circle, and mind both seeming invigorated by it; DGB. if, however, this kind of weather last very The length of any part of the curve, long, towards the close of it a number of that any given point in the revolving circle dangerons complaints spring up, which, as has described, from the time it touched the they are very short in their duration, the circle it revolved upon, shall be to double mortality is much greater than one would the versed sine of half the arch, which all readily suppose from the few persons that that time touched the circle at rest, as the are ill at any one time : and as soon as a sum of the diameters of the circles, to the wet season sacceeds a long dry one, a prosemidiameter of the resting circle, if the digious sickness and mortality come on unirevolving circle moves upon the convex versally. So long as this wet weather conside of the resting circle ; but if upon the tinyes, the sickness scarcely abates, but concave side, as the difference of the dia- the mortality diminishes rapidly; so that in meters to the semi-diameter of the resting the last number of rainy years the number circle. of deaths is at the minimum. The change In the Philosoph. Transactions, No. 218, of a long dry season, whether hot or cold, we have a general proposition for measuring to a rainy one, appears to bring about the the areas of all cycloids and epicycloids, temperature of air favourable to the proviz. The area of any cycloid or epicycloid duction of great epidemics. Some, howis to the area of the generating circle, as the ever, seem more speedily to succeed the sum of double the velocity of the centre predisposing state of the air, others less so; and velocity of the circular motion to the or it may be that the state of air favourable velocity of the circular motion : and in the to them exists at the very beginning of the same proportion are the areas of segments change, whilst the state favourable to others of those curves to those of analogous seg- progressively succeeds : of this last, howments of the generating circle. ever, Dr. Sims is very uncertain. EPIDEMIC. A contagious disease is so Two infections diseases, it appears, are termed that attacks many people at the hardly ever prevalent together; therefore, same season, and in the same place ; thus, although the same distemperature of air putrid fever, plague, dysentery, &c. are seems favourable to most epidemic disoroften epidemic. Dr. James Sims observes, ders, yet some must appear sooner, others in the Memoirs of the Medical Society of later. From observation and books, the London, that there are some grand classes Doctor describes the order in which these of epidemics which prevail every year, and disorders have a tendency to succeed each which are produced by the various changes other, to be plague, petechial fever, puof the seasons. Thus, spring is accompa- trid sore throat, with or without scarlatina, nied by inflammatory diseases; summer by dysentery, small-pox, measles, simple scar. Jatina, hooping-cough, and catarrh : “I do Natural order of Calycanthemæ. Onagræ, not mean by this,” says he, “ that they al- Jussiou. Essential character: calyx fourways succeed each other as above; for cleft; petals four; capsule oblong, inferior; often the individual infection is wanting, seeds downy. There are fourteen species. when another takes its place, until perhaps These plants are hardy perennials, not void that infection is imported from a place, of beauty ; they are, however, commonly which has been so unfortunate as to have a considered only as weeds, and are rarely coincidence of the two causes, without which cultivated in gardens. it appears that no epidemic can take place; EPILOGUE, in dramatic poetry, a that is, a favourable disposition of the air, and speech addressed to the audience after the that particular infection. Whenever it hap- play is over, by one of the principal actors pens that one infectious disorder takes the therein, usually containing some reflections place that should have been more properly on certain incidents in the play, especially occupied by another, it becomes much more those in the part of the person that speaks virulent than it is naturally, whilst the for- it. mer, if it afterwards succeeds, becomes EPIMEDIUM, in botany, English bar. milder in proportion : this, perhaps, is the renwort, a gerius of the Tetrandria Monoreason why the same disorders, nay, the gynja class and order. Natural order of same appearance in a disorder, are at. Corydales. Berberides, Jussieu. Essential tended with much more fatality in one year character: nectary four, cupform, leaning than another.” on the petals ; corolla four-petalled; calyx EPIDENDRUM, in botany, a genus of very caducous ; fruit a silique. There is the Gynandria Diandria class and order. but one species, viz. E. alpinum, alpine barNatural order of Orchideæ. Essential cha- renwort. racter : nectary turbinate, oblique, reflex ; EPIPHANY, a christiau festival, othercorolla spreading; spur none. There are wise called the manifestation of Christ to 124 species. This numerous gems is ob- the Gentiles, observed on the sixth of Jascure in its character, differences, and syno- nuary, in honour of the appearance of our nyms; for the flowers in dried specimens Saviour to the three magi, or wise men, can hardly be unfolded; the plants are cul- wbo came to adore him, and bring him pretivated in gardens with difficulty; and the sents. The feast of epiphany was not orispecies have not been sufficiently described ginally a distinct festival, but made a part by authors; who have had an opportunity of that of the nativity of Christ, which be. of seeing them in America, and the Easting celebrated twelve days, the first and last Indies, their native places of growth of which were high or chief days of solem EPIDERMIS, in anatomy, the same nity, either of these might properly be with the cuticle. See Cutis. called epiphany, as that word signifies the EPIGÆA, in botany, a genus of the appearance of Christ in the world. Decandria Monogynia class and order. Na- The kings of England and Spain offer tural order of Bicornes. Ericæ, Jussieu. Es- gold, frankincense, and myrrh, on epiphany, sential character: calyx outer three-leaved; or twelfth day, in memory of the offerings inner five-parted; corolla salver form ; cap- of the wise men to the infant Jesus. sule five-celled. There are but two spe- The festival of epiphany is called by the cies, viz. E. repens, creeping epigæa, or Greeks the feast of lights, because our Satrailing arbutus, and E. cordifolia, heart, viour is said to have been baptised on this leaved epigæa; the former is a native of day; and baptism is by them called illumiVirginia and Canada, and the latter of nation. Guadaloupe. EPISCOPALIANS, in the modern acEPIGLOTTIS, one of the cartilages of ceptation of the term, belong more especially the larynx or wind-pipe. See ANATOMY. to members of the Church of England, and de EPIGRAM, in poetry, a short poem or' rive this title from episcopus, the Latin word composition in verse, treating only of one for bishop; or if it be referred to its Greek thing, and ending with some lively, inge- origin, implying the care and diligence with nious, and natural thought or point. which bishops are expected to preside over EPILEPSY, in medicine, the same with those committed to their guidance and diwhat is otherwise called the falling-sickness, rection. They insist on the divine origin of from the patient's falling suddenly to the their bishops, and other church officers, and ground. on the alliance between church and state. EPILOBIUM, in botany, a genus of Respecting these subjects, however, Warthe Octandria Monogynia class and order. burton and Hoadley, together with others NOLOGY. of the learned amongst them, have different tion. “ Nothing," says Aristotle, “ tires the opinions, as they have also on the thirty- reader more than too great a redundancy nine articles, which were established in the of epithets, or epithets placed improperly ; reign of Queen Elizabeth. These are to be and yet nothing is so essential in poetry as found in most Common Prayer-Books; and a proper nise of them." the Episcopal Church in America has re- EPITOME, in literary history, an abridge duced their number to twenty. By some ment or summary of any book, particularly the articles are made to speak the language of a history. of Calvinism, and by others they have EPOCHA, in chronology, a term or fixed been interpreted in favour of Arminianism. point of time, whence the succeeding years The Church of England is governed by are numbered or accounted. See Curothe King, who is the supreme head; by two archbishops, and by twenty-four bi. EPODE, in lyric poetry, the third or shops. The benefices of the bishops were last part of the ode, the antient ode being converted by William the Conqueror into divided into strophe, antistrophe, and epode. temporal baronies; so that every prelate EPOPOEIA, in poetry, the story, fable, has a seat and vote in the House of Peers. or subject treated of, in an epic poem. Dr. Benjamin Hoadley, however, in a ser- The word is commonly used for the epic mon preached from this text, “ My king- poem itself. See Epic. dom is not of this world,” insisted that the EPSOM salt, another name for sulpliate clergy had no pretensions to temporal juris- of magnesia. diction, which gave rise to various publica- EQUABLE, an appellation given to such tions, termed by way of eminence the motions as always continue the same in Bangorian Controversy, Hoadley being degree of velocity, without being either acthen bishop of Bangor. There is a bishop .celerated or retarded. When two or more of Sodor and Man, who has no seat in the bodies are uniformly accelerated or retardHouse of Peers. ed, with the same increase or dimunition of Since the death of the intolerant Arch- velocity in each, they are said to be eqnably bishop Laud, men of moderate principles accelerated or retarded. have been raised to the see of Canterbury, EQUAL, a term of relation between two and this bath tended not a little to the or more things of the same magnitule, tranquillity of church and state. The esta- quantity, or quality. Mathematicians speak blished Church of Ireland is the same as the of equal lines, angles, figures, circles, ratios, Church of England, and is governed by solids, &c. four archbishops, aud eighteen bishops. EQUALITY, that agreement between EPISODE, in poetry, a separate inci- two or more things whereby they are denodent, story, or action, which a poet invents minated equal. The equality or two quanand connects with his principal action, that tities, in algebra, is denoted by two parallel his work may abound with a greater diver- lines placed between them: thus, 4 to 2 sity of events; though, in a more limited =6, that is, 4 added to 2 is equal to 6. sense, all the particular incidents whereof EQUANIMITY, in ethics, denotes that the action or narration is compounded, are even and calm frame of mind and temper called episodes. under good or bad fortune, whereby a man EPITAPH, a monumental inscription in appears to be neither puffed up or overhonour or memory of a person defunct, joyed with prosperity, nor dispirited, souror an inscription engraven or cut on a ed, or rendered easy by adversity. tomb, to mark the time of a person's de. EQUATION, in algebra, the nuutual cease, his name, family; and, usually, some comparing two equal things of difierent deeulogium of his virtues, or good qualities. nominations, or the expression denoting this EPITHALAMIUM, in poetry, a nup- equality; which is done by setting the one tial song, or composition, in praise of the in opposition to the other, with the sign of bride and bridegroom, praying for their equality (=) between them: thus, 33 = prosperity, for a happy oft spring, &c. 36d, or 3 feet == 1 yard. Hence, if we put EPITHET, in poetry and rhetoric, an u for a foot, and b for a yard, we shall have adjective expressing some quality of a sub- the equation 3a =), in algebraical characstantive to which it is joined; or such an ters. See ALGEBRA. adjective as is annexed to substantives hy EQUATIONS, construction of, in algebra, way of ornament and illustration, not to is the finding the roots or unknown quantimake up an essential part of the discrip. tities of an equation, by geometrical con straction of right lines or curves, or the re- a fourth proportional to , btc, and b -- . ducing given equations into geometrical 4. If ax=b' +6; then construct the figures. And this is effected by lines or right-angled triangle A B C (Plate V. Miscel. curves, according to the order or rank of fig. 5.) whose base is b, and perpendicular the equation. The roots of any equation is c, so shall the square of the hypothenuse may be determined, that is, the equation be b? +c, which call h>; then the equation may be constructed, by the intersections of a straight line with another line or curve of is 4x=h', and s= a third proportional the same dimensions as the equation to be to a and h. constructed: for the roots of the equation To construct a quadratic equation. 1. If it are the ordinates of the curye at the points be a simple quadratic, it may be reduced to of intersection with the right line; and it is this form x=ab; and hence a : x ; x : 1, well known that a carve inay be cut by a or a=vab, a mean proportional between right line in as many points as its dimen- u and 6. Therefore upon a straight line sions amount to. Thus, then, a simple take A B=a, and BC=b; then upon the equation will be constructed by the inter diameter AC describe a semicircle, and section of one right line with another; a raise the perpendicular B D to meet it in quadratic equation, or an affected equation D; so shall B D be=x, the mean proporof the second rank, by the intersections of a tional sought between A B and BC, or be. right line with a circle, or any of the conic tween a and b. 2. If the quadratic be af. sections, which are all lines of the second fected, let it first be ..? + 2ax=b2; then order; and which may be cut by the right form the right-angled triangle whose base line in two points, thereby giving the two AB is a, and perpendicular B C is b; and roots of the quadratic equation. A cubic with the centre A and radius AC describe equation may be constructed by the inter- the semicircle D CE; so shall D B and B E section of the right line with a line of the be the two roots of the given quadratic third order, and so on. But if, instead of equation *x + 2 ax=b. 3. If the quathe right line, some other line of a higher dratic be až — 2 ax=6", then the conorder be used, then the second line, whose struction will be the very same as of the intersections with the former are to deter- preceding one x2 + 2 a xzb4. But if mine the roots of the equation, may be the form be 2 u x - x?=b, form a righttaken as many dimensions lower as the for- angled triangle (fig. .) whose hypothenuse mer is taken higher. And, in general, an F G is a, and perpendicular G H is b; then equation of any height will be constructed with the radius FG and centre F describe by the intersection of two lines, whose di- a semi-circle I GK: so shall I H and HK mensions multiplied together prodnce the be the two roots of the given equation dimension of the given equation. Thus, the 2a x– x=b, or x? — 2ax=-62. intersections of a circle with the conic sec. To construct cubic and biquadratic equations, or of these with each other, will con- tions. These are constructed by the interstruct the bịquadratic equations, or those of sections of two conic sections; for the equathe fourth power, because 2 x 2 =4; and tion will rise to four dimensions, by which the intersections of the circle, or conic sec- are determined the ordinates from the four tions, with a line of the third order, will points in which these conic sections may cut construct the equations of the fifth and sixth one another; and the conic sections may be power, and so on. For example: assumed in such a manner as to make this To construct a simple equation. This is equation coincide with any proposed biquadone by resolving the given simple equa- dratie; so that the ordinates from these four tion into a proportion, or finding a third or intersections will be equal to the roots of fourth proportional, &c. Thus, 1. If the the proposed biquadratic. When one of equation be ax=be; then a :b ::c: x the intersections of the conic section falls bc upon the axis, then one of the ordinates the fourth proportional to u, b, c. 2. If vanishes and the equation by which these or dinates are determined, will then be of three ax=b?; then a : 6:30: x=-, a third dimensions only, or a cubic to which any proproportional to a and b. 3. If ax=b? – posed cubic equation may be accommodat ed; so that the three remaining ordinates ; then, since 6-=otoxo--c, it will be the roots of that proposed cubic. btoxb The conic sections for this purpose should will be a :b +0::6---6:3= he such as are most easily described; the an circle may be one, and the parabola is usu- reducible to : - bxz-, &c. = 0, ally assumed for the other. See Simpson's equation one dimension lower, whose roots and Maclaurin’s algebra. are b and c. EQUATIONS, nature of. Any equation involving the powers of one unknown quantity Ex. One root of x' ti=n, or x+1= may be reduced to the form zu — pza-+ O, and the equation may be depressed to a 9 zh-, &c.=0: here the whole expression quadratic in the following manner : is equal to nothing, and the terms are ar x+1)x' +1(x? —- +1 ranged according to the dimensions of the r'22 unknown quantity, the coefficient of the highest dimension is unity, understood, and gora the coefficients p, q, r, and are affected with the proper signs. An equation, where the in +1+1 dex 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 Hence the other two roots are the roots of above form. Any quantity — p*-+ the quadratic x? — +1 = 0. If two 937–, &c. + Pz-Q may be supposed roots, a and b, be obtained, the equation is to arise from the multiplication of 2 - ux divisible by I-axx-6, and may be re duced in the same manner two dimensions z-Oxz-, &c. to n factors. For by lower. actually multiplying the factors together, we obtain a quantity of n dimensions simi- Er. Two roots of the equation z6 -13 lar to the proposed quantity, 2" -- pza-it 0, are + 1 and — 1, or % -1=0, and q?**?, &c.; and if a, b, c, &c. can be so %+1=0; therefore it may be depressed assumed that the coefficients of the corre- to a biquadratic by dividing by 7 mix sponding terms in the two quantities become zti=z_ 1. equal, the whole expressions coincide. And these coefficients may be made equal, be z? — 1)26 — 1(* + x2 + 1 cause these will be nequations, to determine 26. n quantities, a, b, c, &c. If then the quanti + ties a, b, c, &c. be properly assumed, the equation zu—p 24-' +9z*—?, &c.= +22 - 1 the same with -axz-6X2-6, &c. +32 - 1 = 0. The qnantities a, b, c, d, &c. are called roots of the equation, or values of z; because, if any one of them be substituted Hence the equation z' +x+1=0 confor z, the whole expression becomes nothing, tains the other four roots of the proposed which is the condition proposed by the equation. equation. Conversely, if the equation be divisible Every eqnation has as many roots as it has by - a without a remainder, a is a root; dimensions. If xa — pz-'tpx?-?,&c.= if by r — axx - b, a and b are both roots. 0, or z-ax z-bxz— c,&c. to n factors Let Q be the quotient arising from the di=0, there are n quantities, a, b, c, &c. vision, then the equation is maxT- -6 each of which when substituted for z makes XQ=0, in which, if a or b be substituted the whole = 0, because in each case one of for x the whole vanishes. the factors becomes=0; but any given quantity different from these, as e when EQUATIONS, cubic, solution of, by Cardan's rule. Let the equation be reduced to the substituted for 2, gives the product eux form x-9x+"=0, where q and r may e-oxe-i, &c. which does not vanish, be positive or negative. because none of the factors vanish, that is, Assume x= satb, then the equation bee will not answer the condition which the comes a ta'-9 xütbtr=0, or a' equation requires. +6+3ab xato-q xatotr=0; When one of the roots, a, is obtained, the and since we have two unknown quantities, equation z - X7-6xz-e, &c. = 0, a and b, and have made only one supposi. z'-p:-:+929-2, &c.=0 is divisible tion respecting them, riz. that a +6=4, by z - a without a remainder, and is thus we are at liberty to make another; let 3 ab = 0, is |