made use of by Ptolemy, Cenforinus, and feveral other authors, began 747 years before the incarnation, and in the 3967 of the julian period, Julian EPOCHA. The firit year of Julius Cæfar's correcting the calendar ftands 45 years before our Saviour's birth, and coincides with the 4669 of the julian period. EPOCHA of Chrift. The chriftian world generally reckoned from the epocha of the creation, the building of Rome, the confuls register, or the emperor's reign, till about sco years after Christ, when the epocha of the nativity of our blessed Lord was introduced by Dionyfius Exiguus. He began his account from the conception or incarnation properly called Lady-day. Most countries in Europe, however, at prefent reckon from the first of January next following, except the court of Rome, where the epocha of the incarnation ftill obtains for the date of their bulls and briefs. But here we are to obferve, that there are different opinions touching the year of our Saviour's birth. Capellus and Kepler fix it at about the 748th year from the building of Rome. Deckar and Petavius place the incarnation in the 749th of Rome. Scaliger and Voffius make it fall on the 751ft of Rome, Dionyfius Exiguus, Bede, &c. fix the birth of our Saviour to the year 751 of Rome; the diverfity of thefe opinions proceeding from the difficulty of fixing Herod the great's death, who, as is evident from the evangelists, was living at our Saviour's birth, the taxation of Cyrenius, and the time of our Saviour's beginning his miniftry. But let this be as it will, it is generally agreed, that as to computation and ufe, the common epocha is to be followed, which places the birth of Chrift in the 4713th of the julian period, although the true birth rather correfponds with the 4711th of the fame period. Dioclefian EPOCHA, or EPOCHA of martyrs, year To reduce the years of one epocha to those of another, obferve the following rule: add the given year of an epocha to the year of the julian period correfpond. ing with its rife, and that will give the year of the period. For example, if to 1754, the present year of the chriftian epocha, we add 4713, the year of the julian period correspond. ing with its rife, the fum, 6467, will be the prefent year of the julian period: now if we fubtract from the year thus found, the year of the julian period corresponding with the rife of any epocha, the remainder fhews the true method of mak ing a just connexion betwixt that epocha and the known year of Chrift. Again, if we want to find the year of Spanish EPOCHA. See the article ÆRA. EPOMIS, in anatomy, a muscle, other The word is commonly used for the epic poem itself. See the article EPIC. EPOTIDES, in the naval architecture of the antients, two thick blocks of wood, one on each fide the prow of a galley, for warding off the blows of the roftra of the enemy's veffels. See the articles GALLEY and ROSTRUM. EPPINGEN, a town of Germany, fituated about ten miles north of Hailbron. EPSOM, a town of Surry, about fifteen miles miles fouth-weft of London; much reforted to on account of its medicinal wa ters; from which the bitter purging falt being firft extracted, got the name of eplom falt. At prefent, however, the bitter purging falt is procured from the bittern, remaining after the cryftallization of common falt; and this is found to answer all the purposes of that firft obtained from Epfom-waters, and goes by its name. Epfom-falt is esteemed good in colics, the fcurvy, diabetes, lofs of appetite, the rheumatifm, jaundice, hypochondriac affection, and other chronic complaints. The best way of taking it is with any chalybeate waters, as thofe of Tunbridge; for inftance, a dram, or a dram and an half, diffolved in the three or four first draughts. EPULIDES, or PARULIDES, in furgery. See the article PARULIDES. EPULONES, in roman antiquity, ministers who affifted at the facrifices, and had the care of the facred banquet committed to them. At first they were only three in number, but afterwards increased to seven. Their office was, to give notice when feafts were to be held in honour of the gods; and, to take care that nothing was wanting towards the celebration. See the article EPULUM. EPULOTICS, Exa, the fame with cicatrizants. See CICATRIZANTS. EPULUM, banquet, in antiquity, a holy feaft prepared for the gods. The statues of the gods were commonly laid upon a bed, and served in the epula, as if they had been very hungry; to perform which was the function of the minifters of facrifice, hence called epulones. EQUABLE, an appellation given to fuch motions as always continue the fame in degree of velocity, without being either accelerated or retarded. When two or more bodies are uniformly accelerated or retarded, with the fame increase or diminution of velocity in each, they are faid to be equably accelerated or retarded. EQUAL; a term of relation between two or more things of the fame magnitude, quantity, or quality. Mathematicians speak of equal lines, angles, figures, circles, ratios, folids, &c. See the articles LINE, ANGLE, &C. EQUALITY, that agreement between two or more things, whereby they are denominated equal. The equality of two quantities, in alge. bra, is denoted by two parallel lines placed between them: thus, 4+ 2 = 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 profperity; nor difpirited, foured, or rendered uneafy by adverfity. EQUANT, in the old aftronomy, a circle defcribed on the center of the deferent, for accounting for the excentricity of the planets. See EXCENTRICITY, EQUATION, in algebra, the mutual comparing two equal things of different denominations, or the expreffion denoting this equality; which is done by fetting the one in oppofition to the other, with the fign of equality (=) between them: thus 3 s 36 d, or 3 feet1 yard. Hence, if we put a for a foot, and b for a yard, we will have the equation 3 ab, in algebraical characters. When a problem is propofed to be refolved by means of equations, the first thing to be done is to form a clear conception of the conditions and nature of it; taking care to fubftitute the first letters of the alphabet for known quantities, and the laft letters of the alphabet for unknown ones. Then by due reafoning from the conditions of the quef tion, let the quantities concerned therein be justly stated, and carefully compared; fo that their relation to one another may appear, and the difference, which renders them unequal, be difcovered; and, confequently, the fame thing found expreffible two ways, or brought into an equation, or feveral equations independent on each other. And here it is to be obferved, 1. That if there are as many equations given, as there are quantities fought, then the question has a determinate number of folutions, or is truly li mited, viz. each quantity fought hath but one fingle value. Thus, fuppofe a question propofed concerning the age of three perfons, was conditioned as follows, viz. the fecond is feven years older than the first, the age of the third is triple that of the first and fecond, and the fum of all their ages is 68. Required the age of each. In order to bring this question to an equation, put ≈ for the age of the fift; then will the age of the fecond be +7, and the age of the third 6 z +213 the fum of all their ages +*+7+ 6x+21=68, So that here is but one 7 C 2 equation equation given, and one quantity required, viz. the age of the fift. 2. When the number of the quantities fought exceed the number of the given equations, the question is capable of an indeterminate number of anfwers; and, therefore, can be but impe: featly determined. Reduction of EQUATIONS. If the quef tion, when stated, is found to have a determinable number of folutions, then the equation, directly drawn from the conditions of the question, must be reduced into another, by equal augmenta. tion and diminution; fo that the known quantities may ftand on one fide, and one of the unknown quantities, or some power of it, on the other fide of the equation. This is called reduction of equations, and depends upon a right application of the five following axioms: i. If equal quantities be added to equal quantities, the fum of thote quantities will be equal. 2. If equal quantities be fubtracted or taken from equal quantities, the quantities remaining will be equal. 3. If equal quantities be multiplied by equal quantities, their products will be equal. 4. If equal quantities be divided by equal quantities, their quotients will be equal. 5. Quantities that are equal to one and the fame thing, are alfo equal to one another. If thele axioms be well understood, the reduction of equations will appear very plain, and the operations be easily performed. 1. Reduction by tranfpofition, is performed by transferring a quantity to the other fide of the equation with a contrary fign; or by equal addition, if the quantity be negative; and by equal fub. traction, if affirmative. Thus the equation to 40, is reduced by adding +10 10 each fide, and the refult will be the fame as if 10 had been tranfpofed to the oppofire fide with the contrary fign; for x 10 + 10 40+ 10, is the fame with x 40 +10, the 10 and + 10 deftroying each other. In the fame manner x 10 40. is reduced to x = 49 10, by tranfpofing the + 10 with a contrary fign. 2. Reduction is performed by equal multiplication, in cafe there are fractional quantities; for by multiplying every term in the equation by the deno minators of the fractions, it will be cleared of fractions: thus by multiplying every term of the equation by the deno ༤ in the equation az + excb, by divid ing each fide by a +e, we get the equa св tion z 4. Equations are clear. ate ed of furd quantities by involution: thus, if the equation hea= 6; then by involution or quariag each fide of the equa tion, we have the equation a 36. If both fides be fimilar furds, or of the fame power, all that we have to do is to reject the radical fign; thus, for √ a=√46 we write ad+c, rejeƐling the radical fign of both. 5. When any fingle pow. er of the unknown quantity is on one fide of the equation, evolve or extract the root of both fides, according as the index of that power denotes, and their roots will be equal. Thus if ≈≈25, by ex. tracting the root of each fide we have 5. In the fame manner, if aaa= 27, their cube roots will be equal, viz. a=3. Or, if any compound power of the unknown quantity be on one fide of an equation, that hath a true root of its kind; then, by evolving both fides of the equation, it will be expreffed in lower terms: for example, a 2+ zba+b2= d2, by evolving both fides, comes out a+b=d. 6. A proportion may be converted into an equation, afferting the pro duct of the extremes to be equal to that of the means; or, any one of the ex tremes may be made equal to the product of the means divided by the other extreme: thus, if 12 -::: 411, then x 2 2. If there are two unknown quantities, then there must be two equations arifing from the conditions of the question; fuppofe x and y. The rule is, to find a value of xory from each of the equations, and then by putting thefe two values equal to each other, there will arife a new equation involving only one unknown quantity, which must be reduced by the same rules as formerly. Example: let the fums of two quantities bes, and their difference d; let s and d be given, and let it be required to find the 3. When in one of the given equations, the unknown quantity is of one dimenfion, and in the other of a higher dimenfion; you must find a value of the unknown quantity from that equation where it is of one dimenfion, and then raife that value to the power of the unknown quantity in the other equation; and by comparing it, fo involved, with the value you deduce from that other equation, you will obtain an equation that will have only one unknown quantity and its powers that is, when you have two equations of different dimenfions, if you cannot reduce the higher to the fame dimenfion with the lower, you must raise the lower to the fame dimenfion with the higher. Example: the fum of two quantities, and the difference of their fquares, being given, to find the quantities themselves. Suppofe them to be x and y, their fum s, and the difference of their fquares d. Then, J(822)=4 This method is general, and will extend to all equations that involve three unknown quantities; but there are often eafier and fhorter methods, to deduce an equation involving only one unknown quantity, which is best learned from practice. Solution of quadratic EQUATIONS. 1. If, after the equation is reduced as directed above, and the unknown quantity brought to stand on one fide, it is found to be a fimple fquare power, all that you have to do is to evolve both fides of the equation, by which means you will find the value of the fimple unknown quantity. Thus, if xx=36; then, by evolution or extraction, x6. See the article EXTRACTION. Add the fquare of a to both fides, 2. In the folution of any question, whet you have got an equation that involve only one unknown quantity, but involve at the fame time the fquare of that qua tity, and the product of it multiplied by fome known quantity; then you hư: what is called an adfected quadratic equa tion, which may be refolved by the fo lowing rules: 1. Tranfpofe all the terms that involve the unknown quantity a one fide, and the known terms to the other fide of the equation. 2. If the fquare of the unknown quantity is me tiplied by any coefficient, you are to d vide all the terms by that coefficient, the the coefficient of the fquare of the known quantity may be unit. 3. Að to both fides the square of half the coef ficient prefixed to the unknown quar. tity itself, and the fide of the equation that involves the unknown quantity then be a complete fquare. 4. Extrat the fquare root from both fides of the equation, which you will find, on one fide, always to be the unknown quanti tity with half the forefaid coefficient fubjoined to it; fo that by tranfpofing this half, you may obtain the value of the unknown quantity expreffed in known terms. Thus, fuppofe the quadratic equation to be, Extract the root, Transpose, and 2 Here it is to be observed, that the square root of any quantity, as+a2, may be +a, ora; and hence all quadratic equations admit of two solutions. Al |