Krb, 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. Miseel, fig. 4.) along the convex or concave side of the periphery of another circle, DGB. 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. 318, 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 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 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 scarlatina, hooping-cough, and catarrh: " I do not mean by EPIDENDRUM, in botany, a genus of the Gynandria Diandria class and order. Natural order of Orchidcac Essential character: nectary turbinate, obliqne, reflex; corolla spreading; spur none. There are 124 species. This numerous genus is obscure in its character, differences, and synonyms; for the flowers in dried specimens can hardly be unfolded; the plants EPIDERMIS, in anatomy, the same with the cuticle. See Cutis. EPIG.EA, in botany, a genus of the Decandria Monogynia class and order. Natural order of Bicornes. Erica?, Jussicu. Essential character: calyx outer three-leaved; inner five-parted ; corolla salver-form ; capsule five-celled. There are but two species, 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 EPILOBIUM, in botany, a genus of the Octandria Monogynia class and order. Natural order of Calycanthemi. Onager, Jussicu. 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 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 j fruit a siliuue. There is but one species, viz. E. alpinum, alpine barreuwort. 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 Since the death of the intolerant Archbishop 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 reader more than too great a redundancy of epithets, or epithets placed improperly; and yet nothing is so essential in poetry as a proper use of them." EPITOME, in literary history, an abridgment or summary of any book, particularly of a history. EPOCHA, in chronology, a term or fixed point of time, whence the succeeding years are numbered or accounted. See ChroNology. EPODE, in lyric poetry, the third or last part of the ode, the antieiit ode being divided into strophe, antistrophc, 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 sinie in degree of velocity, without being either accelerated or retarded. When two or more bodies are uniformly accelerated or retarded, with the same increasc-or diminution 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 or" two quantities, in algebra, is denoted by two parallel lines placed between them: thus, 4 -j- 2 = 6, that is, i added to 'i is equal h> 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, 3« = 36 d, or 3 feet = 1 yard. Hence, if we put u for a foot, and b for a yard, we shall have the equation ." a = b, in algebraical characters. Sec Ai.gebha. EQUATIONS,construction of, in algebra, is the finding the roots or unknown quantities of an equation, by geometrical constrnction of right lines or curves, or the reducing given equations into geometrical figures. And this is effected by lines or To construct a simple equation. This 2-^, the fourth proportional to a, b, c. 2. If h' ax = 6'; then a:6::o:x=—,a third proportional to a and 6. 3. If a x = b1 — tJ; then, since b1 — c2 = b-\-c x b — c, it a fourth proportional to a, b-\-c, and b— c. 4. If a x xs 6' -J- <?; then construct the right-angled triangle ABC (Plate V. Miscel. fig. A.) whose base is b, and perpendicnlar is c, so is ais A', and x =— 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 x2 = a 6; and hence a : x c = 4 ; then upon the diameter AC describe a semicircle, and raise the perpendicular B D to meet it in D; so shall B D be = .T, the mean proportional sought between A B and B C, or between a and 6. 2. If the quadratic be affected, let it first be x' -f- 2 a x = b*; then form the right-angled triangle whose base A B is a, and perpendicular B C is 6; and with the centre A and radius A C describe the semicircle DCE;so shall D B and It E be the two roots of the given quadratic equation x2 -}- 2 a x = o2. 3. If the quadratic be x2 — 2ax = 42, then the construction will be the very same as of the preceding one x2 -f- 2 a x = b>. 4. But if the form be 2 ax — x! = 42, form a rightangled triangle (fig. .) whose hypothenuse F G is a, and perpendicular G H is 4; then with the radius F G and centre F describe a semi-circle IG K: so shall I H and H K be the two roots of the given equation 2ax—x'xzli', orx2— 2ax= — b1.
R 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 upon the axis, then one of the ordinates vanishes and the equation by which these ordinates 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 be such as are most easily described; the
circle may be one, and the parabola is usually assumed for the other. See Simpson's and Maelaurin's algebra.Equations, nature of. Any equation involving Every equation has as many roots as it has dimensions. If 2"—px 0,or z — a X z — b X z = 0, there are n quantities, a, b, c, ice. each of which when substituted for z makes the whole =r 0, because in each case oue of the factors becomes = 0; but any given quantity different from these, as e when substituted for z, gives the product <-«x e — iX< — c, &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 z— a X * — » X z — c, &c. =0, z"—pz"-'-\-qz"-2,Stc.zz:0 is divisible by z — a without a remainder, and is thus Hence the equation z4-j-z]-j-l Conversely, if the equation be divisible by Equations, cubic, solution of, by Cardan's rule. Let the equation be reduced to Assume x = a -)- |