arb, or sphere, which being fixed in the deferent of a planet, is carried along with it; and yet, by its own peculiar motion, carries the planet fastened to it round its proper centre. It was by means of epicycles, that Ptolemy and his followers solved the various phenomena of the planets, but more especially their stations and retrogradations. The great circle they called the excentric or deferent, and along its circumference the centre of the epicycle was conceived to move; carrying with it the planet fixed in its circumference, which in its motion downwards proceeded according to the order of the signs, but, in moving upwards, contrary to that order. The highest point of a planet's epicycle they called apogee, and the lowest perigee. EPICYCLOID, in geometry, a curve generated by the revolution of the periphery of a circle, ACE (Plate V. Miscel. fig. 4.) along the convex or concave side of the periphery of another circle, DG B. The length of any part of the curve, that any given point in the revolving circle has described, from the time it touched the circle it revolved upon, shall be to double the versed sine of half the arch, which all that time touched the circle at rest, as the sum of the diameters of the circles, to the semidiameter of the resting circle, if the revolving circle moves upon the convex side of the resting circle; but if upon the concave side, as the difference of the diameters to the semi-diameter of the resting circle. In the Philosoph. Transactions, No. 218, we have a general proposition for measuring the areas of all cycloids and epicycloids, viz. The area of any cycloid or epicycloid is to the area of the generating circle, as the sum of double the velocity of the centre and velocity of the circular motion to the velocity of the circular motion: and in the same proportion are the areas of segments of those curves to those of analogous segments of the generating circle. EPIDEMIC. A contagious disease is so termed that attacks many people at the same season, and in the same place; thus, putrid fever, plague, dysentery, &c. are often epidemic. Dr. James Sims observes, in the Memoirs of the Medical Society of London, that there are some grand classes of epidemics which prevail every year, and which are produced by the various changes of the seasons. Thus, spring is accompanied by inflammatory diseases; summer by complaints in the stomach and bowels; autumn by catarrhs; and winter by intermittents: these being obviously produced by the state of weather attendant upon them, other epidemics are supposed analogous to them, and obedient to the same rules, which, on examination, not being the case, all further scrutiny is laid aside, perhaps too hastily. The most natural and healthful seasons in this country are a moderately frosty winter, showery spring, dry summer, and rainy autumn; and whilst such prevail, the wet part of them is infested by vastly the greatest proportion of complaints, but those not of the most mortal kind. A long succession of wet seasons is accompanied by a prodigious number of diseases; but these being mild and tedious, the number of deaths are not in proportion to the coexistent ailments. On the other hand, a dry season, in the beginning, is attended with extremely few complaints, the body and mind both seeming invigorated by it; if, however, this kind of weather last very long, towards the close of it a number of dangerous complaints spring up, which, as they are very short in their duration, the mortality is much greater than one would readily suppose from the few persons that are ill at any one time and as soon as a wet season succeeds a long dry one, a prodigious sickness and mortality come on universally. So long as this wet weather continues, the sickness scarcely abates, but the mortality diminishes rapidly; so that in the last number of rainy years the number of deaths is at the minimum. The change of a long dry season, whether hot or cold, to a rainy one, appears to bring about the temperature of air favourable to the production of great epidemics. Some, however, seem more speedily to succeed the predisposing state of the air, others less so; or it may be that the state of air favourable to them exists at the very beginning of the change, whilst the state favourable to others progressively succeeds: of this last, however, Dr. Sims is very uncertain. Two infections diseases, it appears, are hardly ever prevalent together; therefore, although the same distemperature of air seems favourable to most epidemic disorders, yet some must appear sooner, others later. From observation and books, the Doctor describes the order in which these disorders have a tendency to succeed each other, to be plague, petechial fever, putrid sore throat, with or without scarlatina, dysentery, small-pox, measles, simple scar latina, hooping-cough, and catarrh: "I do not mean by this," says he, "that they always succeed each other as above; for often the individual infection is wanting, when another takes its place, until perhaps that infection is imported from a place, which has been so unfortunate as to have a coincidence of the two causes, without which it appears that no epidemic can take place; that is, a favourable disposition of the air, and that particular infection. Whenever it happens that one infectious disorder takes the place that should have been more properly occupied by another, it becomes much more virulent than it is naturally, whilst the former, if it afterwards succeeds, becomes milder in proportion: this, perhaps, is the reason why the same disorders, nay, the same appearance in a disorder, are at tended with much more fatality in one year than another." EPIDENDRUM, in botany, a genus of the Gynandria Diandria class and order. Natural order of Orchidea. Essential character: nectary turbinate, oblique, reflex; corolla spreading; spur none. There are 124 species. This numerous gems is obscure in its character, differences, and synonyms; for the flowers in dried specimens can hardly be unfolded; the plants are cultivated in gardens with difficulty; and the species have not been sufficiently described by authors; who have had an opportunity of seeing them in America, and the East Indies, their native places of growth. EPIDERMIS, in anatomy, the same with the cuticle. See CUTIS. EPIGÆA, in botany, a genus of the Decandria Monogynia class and order. Natural order of Bicornes. Erica, Jussieu. Essential character: calyx outer three-leaved; inner five-parted; corolla salver-form; capsule five-celled. There are but two species, viz. E. repens, creeping epigæa, or trailing arbutus, and E. cordifolia, heartleaved epigæa; the former is a native of Virginia and Canada, and the latter of Guadaloupe. EPIGLOTTIS, one of the cartilages of the larynx or wind-pipe. See ANATOMY. EPIGRAM, in poetry, a short poem or composition in verse, treating only of one thing, and ending with some lively, ingenious, and natural thought or point. EPILEPSY, in medicine, the same with what is otherwise called the falling-sickness, from the patient's falling suddenly to the ground. EPILOBIUM, in botany, a genus of the Octandria Monogynia class and order. Natural order of Calycanthema. Onagræ, Jussieu. Essential character: calyx fourcleft; petals four; capsule oblong, inferior; seeds downy. There are fourteen species. These plants are hardy perennials, not void of beauty; they are, however, commonly considered only as weeds, and are rarely cultivated in gardens. EPILOGUE, in dramatic poetry, a speech addressed to the audience after the play is over, by one of the principal actors therein, usually containing some reflections on certain incidents in the play, especially those in the part of the person that speaks it. EPIMEDIUM, in botany, English barrenwort, a genus of the Tetrandria Monogynia class and order. Natural order of Corydales. Berberides, Jussieu. Essential character: nectary four, cupform, leaning on the petals; corolla four-petalled; calyx very caducous; fruit a silique. There is but one species, viz. E. alpinum, alpine barrenwort. EPIPHANY, a christian festival, otherwise called the manifestation of Christ to the Gentiles, observed on the sixth of January, in honour of the appearance of our Saviour to the three magi, or wise men, who came to adore him, and bring him presents. The feast of epiphany was not originally a distinct festival, but made a part of that of the nativity of Christ, which being celebrated twelve days, the first and last of which were high or chief days of solemnity, either of these might properly be called epiphany, as that word signifies the appearance of Christ in the world. The kings of England and Spain offer gold, frankincense, and myrrh, on epiphany, or twelfth day, in memory of the offerings of the wise men to the infant Jesus. The festival of epiphany is called by the Greeks the feast of lights, because our Saviour is said to have been baptised on this day; and baptism is by them called illumination. EPISCOPALIANS, in the modern acceptation of the term, belong more especially to members of the Church of England, and derive this title from episcopus, the Latin word for bishop; or if it be referred to its Greek origin, implying the care and diligence with which bishops are expected to preside over those committed to their guidance and direction. They insist on the divine origin of their bishops, and other church officers, and on the alliance between church and state. Respecting these subjects, however, Warburton and Hoadley, together with others of the learned amongst them, have different opinions, as they have also on the thirtynine articles, which were established in the reign of Queen Elizabeth. These are to be found in most Common Prayer-Books; and the Episcopal Church in America has reduced their number to twenty. By some the articles are made to speak the language of Calvinism, and by others they have been interpreted in favour of Arminianism. The Church of England is governed by the King, who is the supreme head; by two archbishops, and by twenty-four bishops. The benefices of the bishops were converted by William the Conqueror into temporal baronies; so that every prelate has a seat and vote in the House of Peers. Dr. Benjamin Hoadley, however, in a sermon preached from this text, "My kingdom is not of this world," insisted that the clergy had no pretensions to temporal jurisdiction, which gave rise to various publications, termed by way of eminence the Bangorian Controversy, Hoadley being then bishop of Bangor. There is a bishop of Sodor and Man, who has no seat in the House of Peers. Since the death of the intolerant Arch. bishop Laud, men of moderate principles have been raised to the see of Canterbury, and this hath tended not a little to the tranquillity of church and state. The established Church of Ireland is the same as the Church of England, and is governed by four archbishops, and eighteen bishops. EPISODE, in poetry, a separate incident, story, or action, which a poet invents and connects with his principal action, that his work may abound with a greater diversity of events; though, in a more limited sense, all the particular incidents whereof the action or narration is compounded, are called episodes. EPITAPH, a monumental inscription in honour or memory of a person defunct, or an inscription engraven or cut on a tomb, to mark the time of a person's decease, his name, family; and, usually, some eulogium of his virtues, or good qualities. EPITHALAMIUM, in poetry, a nuptial song, or composition, in praise of the bride and bridegroom, praying for their prosperity, for a happy offspring, &c. EPITHET, in poetry and rhetoric, an adjective expressing some quality of a substantive to which it is joined; or such an adjective as is annexed to substantives by. way of ornament and illustration, not to make up an essential part of the descrip tion. "Nothing," says Aristotle, " tires the EPITOME, in literary history, an abridg- EPOCHA, in chronology, a term or fixed point of time, whence the succeeding years are numbered or accounted. See CHRO NOLOGY. EPODE, in lyric poetry, the third or last part of the ode, the antient ode being divided into strophe, antistrophe, and epode. EPOPOEIA, in poetry, the story, fable, or subject treated of, in an epic poem. The word is commonly used for the epic poem itself. See EPIC. EPSOM salt, another name for sulphate of magnesia. EQUABLE, an appellation given to such motions as always continue the same in degree of velocity, without being either accelerated or retarded. When two or more bodies are uniformly accelerated or retarded, with the same increase or dimunition of velocity in each, they are said to be equably accelerated or retarded. EQUAL, a term of relation between two or more things of the same magnitude, quantity, or quality. Mathematicians speak of equal lines, angles, figures, circles, ratios, solids, &c. EQUALITY, that agreement between two or more things whereby they are denominated equal. The equality of two quantities, in algebra, is denoted by two parallel lines placed between them: thus, 42 =6, that is, 4 added to 2 is equal to 6. EQUANIMITY, in ethics, denotes that even and calm frame of mind and temper under good or bad fortune, whereby a man appears to be neither puffed up or overjoyed with prosperity, nor dispirited, soured, or rendered uneasy by adversity. EQUATION, in algebra, the mutual comparing two equal things of different denominations, or the expression denoting this equality; which is done by setting the one in opposition to the other, with the sign of equality (=) between them; thus, 38= 36 d, or 3 feet = 1 yard. Hence, if we put a for a foot, and b for a yard, we shall have the equation 3 a=b, in algebraical characters. See ALGEBRA. EQUATIONS, construction of, in algebra, is the finding the roots or unknown quantitities of an equation, by geometrical con struction of right lines or curves, or the reducing given equations into geometrical figures. And this is effected by lines or curves, according to the order or rank of the equation. The roots of any equation may be determined, that is, the equation may be constructed, by the intersections of a straight line with another line or curve of the same dimensions as the equation to be constructed: for the roots of the equation are the ordinates of the curve at the points of intersection with the right line; and it is well known that a curve may be cut by a right line in as many points as its dimensions amount to. Thus, then, a simple equation will be constructed by the intersection of one right line with another; a quadratic equation, or an affected equation of the second rank, by the intersections of a right line with a circle, or any of the conic sections, which are all lines of the second order; and which may be cut by the right line in two points, thereby giving the two roots of the quadratic equation. A cubic equation may be constructed by the intersection of the right line with a line of the third order, and so on. But if, instead of the right line, some other line of a higher order be used, then the second line, whose intersections with the former are to determine the roots of the equation, may be taken as many dimensions lower as the former is taken higher. And, in general, an equation of any height will be constructed by the intersection of two lines, whose dimensions multiplied together produce the dimension of the given equation. Thus, the intersections of a circle with the conic sections, or of these with each other, will construct the biquadratic equations, or those of the fourth power, because 2 × 2 = 4; and the intersections of the circle, or conic sections, with a line of the third order, will construct the equations of the fifth and sixth power, and so on.-For example: To construct a simple equation. This is done by resolving the given simple equation into a proportion, or finding a third or fourth proportional, &c. Thus, 1. If the equation be a xbc; then a: b::c: x = he the fourth proportional to a, b, c. 2. If a fourth proportional to a, b+c, and b→ c. 4. If axb2+c; then construct the right-angled triangle A B C (Plate V. Miscel. fig. 5.) whose base is b, and perpendicular is c, so shall the square of the hypothenuse be b2+c2, which call h2; then the equation is a r = h, and r = = a third proportional to a and h. To construct a quadratic equation. 1. If it be a simple quadratic, it may be reduced to this form x2ab; and hence a:x:x:, or a = ya b, a mean proportional between u and b. Therefore upon a straight line take A B = a, and B C = 6 ; then upon the diameter AC describe a semicircle, and raise the perpendicular BD to meet it in D; so shall BD ber, the mean proportional sought between A B and B C, or between a and b. 2. If the quadratic be affected, let it first be t2 + 2 a r = b; then form the right-angled triangle whose base with the centre A and radius AC describe A B is a, and perpendicular BC is b; and the semicircle D CE; so shall D B and B E be the two roots of the given quadratic equation x2+2 ax = b2. 3. If the quadratic be x-2 axb2, then the construction will be the very same as of the preceding one x2+2 a x = b2. 4. But if the form be 2 ax-x2= b2, form a rightangled triangle (fig. .) whose hypothenuse FG is α, and perpendicular GH is b; then with the radius FG and centre F describe a semi-circle IGK: so shall I H and HK be the two roots of the given equation 2ax-x2=h2, or x2 — 2 ax — — b2. -- To construct cubic and biquadratic equasections of two conic sections; for the equations. These are constructed by the intertion will rise to four dimensions, by which are determined the ordinates from the four points in which these conic sections may cut one another; and the conic sections may be assumed in such a manner as to make this equation coincide with any proposed biquadratic; so that the ordinates from these four intersections will be equal to the roots of the proposed biquadratic. When one of the intersections of the conic section falls vanishes and the equation by which these orupon the axis, then one of the ordinates dinates are determined, will then be of three dimensions only, or a cubic to which any proposed cubic equation may be accommodated; so that the three remaining ordinates will be the roots of that proposed cubic. The conic sections for this purpose should he such as are most easily described; the ircle may be one, and the parabola is usually assumed for the other. See Simpson's and Maclaurin's algebra. reducible to 2- bx zc, &c. = 0, an equation one dimension lower, whose roots are b and c. Ex. One root of x' +1=0, or x+1= and the equation may be depressed to a quadratic in the following manner : x+1)x3+1(x2−x+1 x2+x2 EQUATIONS, nature of. Any equation involving the powers of one unknown quantity may be reduced to the form 2"-pz"-10, 9 z2, &c.=0: here the whole expression is equal to nothing, and the terms are arranged according to the dimensions of the unknown quantity, the coefficient of the highest dimension is unity, understood, and the coefficients p, q, r, and are affected with the proper signs. An equation, where the index 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 above form. Any quantity z” —p 2-1+ q za¬2, &c.+Pz-Q may be supposed to arise from the multiplication of x-u X -2 Hence the other two roots are the roots of Er. Two roots of the equation 26 — 1⇒ are +1 and — 1, or % -10, and +1=0; therefore it may be depressed to a biquadratic by dividing by z-1× ≈+1= x2-1. -bxz-c, &c. to n factors. For by actually multiplying the factors together, we obtain a quantity of n dimensions similar to the proposed quantity, 2"-pz"-+0, q-, &c.; and if a, b, c, &c. can be so assumed that the coefficients of the corre sponding terms in the two quantities become equal, the whole expressions coincide. And these coefficients may be made equal, because these will be n equations, to determine n quantities, a, b, c, &c. If then the quantities a, b, c, &c. be properly assumed, the equation_2′′ —p z′′ − 1 + 9 z”—2, &c. = = 0, is the same with z-axz-b × z —c, &c. =0. The quantities 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 equation. Every equation has as many roots as it has dimensions. If TM”—pz"-'+px"−1,&c. O, or z-ax z―b xx c, &c. to n factors =0, there are n quantities, a, b, c, &c. each of which when substituted for z makes the whole = 0, because in each case one of the factors becomes=0; but any given quantity different from these, as e when substituted for z, gives the product eux e-bxec, &c. which does not vanish, because none of the factors vanish, that is, e will not answer the condition which the equation requires. When one of the roots, a, is obtained, the equation za xz - b xz —c, &c. =0, z 2 — p z2 - 1 + q z " -2, &c. =0 is divisible by za without a remainder, and is thus Hence the equation +z2+1=0 contains the other four roots of the proposed equation. Conversely, if the equation be divisible by ra without a remainder, a is a root; if by xa xx-b, a and b are both roots. Let Q be the quotient arising from the division, then the equation is x-ax x b × Q=0, in which, if a or b be substituted for x the whole vanishes. EQUATIONS, cubic, solution of, by Cardan's rule. Let the equation be reduced to the form x3qxr0, where q and r may be positive or negative. - = Assume ra+b, then the equation becomes ab-q×u+b+r=0, or a3 +b+3ab xa+b-q× a+b+r=0; and since we have two unknown quantities, a and b, and have made only one supposi tion respecting them, viz. that a+b=4, we are at liberty to make another; let 3 ab |