2 circle may be one, and the parabola is usu- reducible to z 6 XZ-C, &c. = 0, an 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 in. volving the powers of one unknown quantity Er. One root of x' +1=0, or x+1= may be reduced to the form z" - pza-it 0, and the equation may be depressed to a 978-, &c. =0: here the whole expression quadratic in the following manner : is equal to nothing, and the terms are ar x +-1).r' +1(x? —- x + 1 ranged according to the dimensions of the r? + unknown quantity, the coefficient of the highest dimension is unity, understood, and the coefficients p, q,', and are affected with the proper siyns. An equation, where the in trt1 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 thie Hence the other two roots are the roots of above form. Any quantity zo - pza-'t the quadratic x - x+1= 0. If two © 2*-?, &c.+PzPz-Q may be supposed roots, a and b, be obtained, the equation is to arise from the multiplication of z-ax divisible by x — axr- - b, and may be re duced in the same manner two dimensions 2-6x7c, &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 26 —1= lar to the proposed quantity, 2" — pz"-'t 0, are + 1 and -- 1, or % -1=0, and 93-2, &c.; and if a, b, c, &c. can be so z+1=0; therefore it may be depressed assumed that the coefficients of the corre- to a biquadratic by dividing by 2sponing terms in the two quantities become 271=z? — 1. Equal, the whole expressions coincide. And these coefficients may be made equal, be - 1):6-1(++1 cause these will be n equations, to determine # quantities, a, b, c, &c. If then the quanti +3* ties a, b, c, &c. be properly assumed, the equation 2 -p 2*-1 +924-?, &c. = +z? - 1 the same with z-a X2-0X3-6, &c. +22 - 1 =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 Hence the equation 24 +2+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 equation has as many roots as it has by r- a without a remainder, a is a root; dimensions. If z" — pzm-'tpx"–,&c. = if by r -axx -- b, a and bare both roots. 0,or z-axz-bxx-c,&c. to n factors -b Let Q be the quotient arising from the di=0, there are n quantities, a, b, c, &c. vision, then the equation is x— axb each of which when substituted for z makes XQ=0, in which, if a or 6 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, ae when EQUATIONS, cubic, solution of, ly Cardan's rule. Let the equation be reduced to the substituted for z, gives the product e-ax form x' - 9x tor= 0, where g and r may Proxec, &c. which does not vanish, be positive or negative. because none of the factors vanish, that is, Assume x=a+b, then the equation bee will not answer the condition which the comes u t-01-q xütbtr=0, or a' equation requires. +63 +3ab xa +0-9xa+b+r=0; When one of the roots, a, is obtained, the and since we have two unknown quantities, equation z-axz- X2 – , &c. 0, a and b, and have made only one supposi. **--px*-' +979—?, &c.=0 is divisible tion respecting them, viz. that a +b=x, by i-- a without a remainder, and is thus we are at liberty to make another; let Sab + = 0, is 十 1 3 a 3 1 3 2 or as 27 -9=0, then the equation becomes a't and the values of the cube root of b, are b, bitr=0, also, since 3 ab -9=0,b= -1tr ✓ 9 gy - b. 2 and by substitution, a'+owtr= -1=0, 3 a' q? Hence, it appears, that there are nine tra't = 0, an equation of a values of a +b, three only of which can answer the conditions of the equation, the quadratic form; and by completing the others having been introduced by involu s2 q square, as tra't and a' + tion. These nine values are, 27' 1. a +1. ; therefore a' = 2. at b q3 Ev and a = b. 2 4 1 1+ v 3. 27 2 1 4 27 3. at 2 3 6. -1+v b. at 3 8. b. 2 2 7. atb. ap 1+v at V b. In the operation we assume 3 ab = 9, is positive, it is negative in the other, that that is, the product of the corresponding 92 is r = + values of a and b is supposed to be possible. This consideration excludes the qdo gd• Athe therefore the three roots of the equation are ☺ 1+v Since b = the value of x is also a+b; at itv b. 2 2 9 Cor. 3. This solution only extends to those q cases in which the cubic has two impossi ble roots. Er. Let x + 6 x For if the roots be mtv3n, m -- 20 = 0; here q= V 3n, and -- 2 m, then q (the sum of the pro- 6, r=- – 20, x= 10 +v 108 + ducts of every two with their signs changed) 10 – 108 = 2.732 — 732 = 2. - 3 ma ? 3 n, and = m +n; also, r Cor. 1. Having obtained one value of r, '(the product of all the roots with their signs the equation may be depressed to a quadra- changed) = 2 m’ — 6 mn, and a = m'. tic, and the other roots found, 3 mn; and by involution, Cor. 2. The possible values of a and b be go2 ing discovered, the other roots are known = m' — 6 m*n + 9 m2 12 without the solution of a quadratic. gi The values of the cube roots of a' are a, =m6 + 3 in*n+3 m?n? + n-n' -1+v. ✓ ge2 o =-9min + 6 m2?n? --n'- . C. C. also g g' the biquadratic to be a+b 1; n-1 -* X 9 !2* — o m'n + n-, and 27 V-1; -aton - 1; a-N = - nx 3 m2 — n, a quantity manifest- the values of e are 2 a, 5 tov - 1, hy impossible, unless n be negative, that is, 6--0.1=1, --5-c.v - 1,--.to. unless two roots of the proposed cubic be 1 and - 2 a; and the three values is possible. of y are 2 u/?, -5+c, -b-cl?, which EQUATIONS, biquadratic, solution of, by are all possible, as in the preceding case. Des Curtis method. Any biquadratic may But if the roots of the biquadratic be a to be reduced to the form rt+qro + rx - rx+ ✓ s=0, by taking away the second term. 1, a-bv-1, -atac, Suppose this to be made up of the two qua- the values of y are 2ul, ctbv11, dratics, I textf= 0, and x? - ext COD11, two of which are impossix=0, wliere te and -e are made the ble; therefore the cubic may be solved by coefficients of the second terms, because Cardan's rule. the second term of the biquadratic is want EQUATION, annual, of the mean motion ing, that is, the sum of its roots is 0. By of the sun and moon's apogee and nodes. multiplying these quadratics together we The annual equation of the sun's mean mo. have x' + tt-.x2 +eg - ef:x+ tion depends upon the excentricity of the fz=0, which equation is made to coincide earth's orbit round him, and is 16h such with the former by equating their coeffi- parts, of which the mean distance between cients, or making +f-=9, eg - ef the sun and the earth is 1000; whence some =r, and fg=s; hence, g +f=ite, have called it the equation of the centre, -f=, and by taking the sum and which, when greatest, is 1° 56" 20". difference of these equations, 2 g=ite is 11' 40"; of the apogee, 20'; and of its The equation of the moon's mean motion +, and Of=9té-; therefore 45g node, 930". + , . These four annual equations are always m2 =g taget et = 48, and multi- mutually proportionable to each other; so tiat when any of them is at the greatest, the plying by e, and arranging the terms ac- three others will also be greatest; and cording to the dimensions of e, co + 2qe* when onc diminishes, the rest diminish in 7 - 48 xé— po? = 0; or, making the same ratio. Wherefore the annual equay=e, y' + 20y + y - 48.y--=0. tion of the centre of the sun being given, By the solution of this cubic, a value of y, the other three corresponding equations will and therefore of vy, or e, is obtained; be given, so that one table of the central a'so f and g, which are respectively equal equations will serve for all. EQUATION of a curre, is an equation ate 9+e+ shewing the nature of a curve by expressing are known; the relation between any absciss and its the biquadratic is thus resolved into two corresponding ordinate, or else the relation of their fluxions, &c. Thus, the equation to quadratics, whose roots may be found. the circle is ax - 72 = y’, where a is its It may be observed, that which ever diameter, x any absciss or part of that value of y is used, the sanie values of x are diameter, and y the ordinate at that point obtained. of the diameter; the meaning being that This solution can only be applied to those whatever absciss is denoted by x, thenthe cases, in which two roots of the biquadratic square of its corresponding ordinate will be are possible and two impossible. ur - x. In like manner the equation Let the roots be a, b, c, a toto; then since e, the coefficient of the second of the ellipse is...... P x = y term of one of the reducing quadratics, is P the sum of two roots, its different values are of the hyperbola is • ax + x = y, e+b, a tc, btc, - atb, atc, of the parabola is......... px = y. - ht e, and the values of e’, or y, are Where a is an axis, and p the parameter. ato, a tol, 6+c; all of which be- And in like manner for any other curves. ing possible, the cubic cannot be solved by This method of expressing the nature of any direct metiiod. Suppose the roots of curves by algebraical equations, was first to e and 2 2 ar introduced by Des Cartes, who, by thus equidifferent is when, in a series of three connecting together the two sciences of numbers, there is the same difference be. algebra and geometry, made them mutually tween the first and second, as there is beassisting to each other, and so laid the tween the second and third; as 3, 6, 9. foundation of the greatest improvements And 2. Discretely equidifferent, is when, in that have been made in every branch of a series of four numbers or quantities, there them since that time. is the same ditference between the first EQUATION of time, in astronomy and and second as there is between the third chronology, the reduction of the apparent and fourth: such are 3, 6, 7, 10. time or motion of the sun, to equable, EQUIDISTANT, an appellation given mean, or true time. The difference be- to things placed at equal distance from tween true and apparent time arises from some fixed point, or place, to which they two causes, the excentricity of the earth's are referred. orbit, and the obliquity of the ecliptic. See EQUILATERAL, in general, something TIME, equation of that bath equal sides, as an equilateral angle. EQUATOR, in geography, a great cir- EQUILATERAL hyperbola, one whose transcle of the terrestrial globe, equidistant verse diameter is equal to its parameter ; from its poles, and dividing it into two equal and so all the other diameters equal to their hemispheres; one north and the other south parameters: in such an hyperbola, the It passes through the east and west points asymptotes always cut one another at right of the horizon, and at the meridian is raised angles in the centre. Its most simple equaas much above the horizon as is the com- tion, with regard to the transversc axis, is plement of the latitude of the place. From y = = r --u?; and with regard to the conthis circle the latitude of places, whether jugate, y? = x2 + a', when a is the seminorth or south, begin to be reckoned in transverse, or semiconjugate. The length degrees of the meridian. All people liv- of the curve cannot be found by means of ing on this circle, called by geographers the quadrature of any space, of which a coand navigators the line, have their days and nic section is any part of the perimeter. nights constantly equal. It is in degrees of EQUILIBRIUM, in mechanics, is when the equator that the longitude of places are the two ends of a lever or balance hang so reckoned; and as the natural day is mea. exactly even and level, that neither doth sured by one revolution of the equator, it ascend or descend, but keep in a position follows that one hour answers to a = 15 parallel to the horizon, which is occasioned degrecs : hence one degree of the equator by their being both charged with an equal will contain four minutes of time; 15 mi- weight. putes of a degree will make a minute of an EQUIMULTIPLES, in arithmetic and and consequently, four seconds an- geometry, are numbers and quantities mulswer to one minute of a degree. tiplied by one and the same number or EQUATIONAL. See ObservATORY. quantity. Hence, equimultiples are always EQUERRY, in the British customs, an in the same ratio to each other, as the simofficer of state, under the master of the ple quantities before multiplication : thus, horse. There are five equerries who ride if 6 and 8 are multiplied by 4, the equimulabroad with his Majesty; for which purpose tiples 24 and 32 will be to each other, as 6 they give their attendance monthly, one at to 8. a time, and are allowed a table. EQUINOCTIAL, in astronomy, a great EQUISETUM, in botany, English horse- circle of the celestial globe, whose poles tail, a genus of the Cryptogamia Filices are the poles of the world. It is so called, class and order. Natural order of Filices because whenever the sun comes to this • or Ferns. There are seven species. They circle, the days and nights are equal all are natives of most parts of Europe in over the globe ; being the same with that woods and shady places. which the sun seems to describe, at the EQUIANGULAR, in geometry, an epi. time of the two equinoxes of spring and thet given to figures, whose angles are all autumn. All stars directly under this cir. equal: such are a square, an equilateral tri- cle, have no declination, and always rise angle, &c. due east, and set full west. The hour cir. EQUICRURAL, in geometry, the same cles are drawn at right angles to it, passing with isosceles. See IsoscELES TRIANGLE. through every fifteenth degree; and the EQUIDIFFERENT numbers, in arith. parallels to it are called parallels of declimetic, are of two kinds. 1. Continually nation, 360 hour; EQUINOX, the time when the sun en. during this period of ages, it follows that ters either of the equinoctial points, where the annual precession is about 50" }; that the ecliptic intersects the eqninoctial. It is, if the celestial equator cuts the ecliptic was evidently an important problem in in a particular point on any day of this practical astronomy, to determine the cxact year, it will on the same day of the followmoment of the sun's occupying these sta- ing year, cut it in a point 50'| to the west tions; for it was natural to compute the of it, and the sun will come to the equinox course of the year from that moment. 20' 23" before he has completed his round Accordingly this has been the leading pro- of the heavens. Thus the equinoctial, or blem in the astronomy of all nations. It is tropical year, or true year of seasons, is susceptible of considerable precision, with- so much shorter than the revolution of the out any apparatus of instruments. It is sun or the sidereal year. It is this discoonly necessary to observe the sun's decli- very that has chiefly immortalized the name nation on the noon of two or three days of Hipparchus, though it must be acknowbefore and after the equinoctial day. On ledged that all his astronomical researches two consecutive days of this number, his have been conducted with the same sagadeclination must have changed from north city and intelligence. It was natural, thereto south, or from south to north. If his fore, for him to value himself highly for the declination on one day was observed to be discovery. It must be acknowledged to 21 north, and on the next 5' south, it fol- be one of the most singular that has been lows that his declination was nothing, or made, that the revolution of the whole that he was in the equinoctial point about heavens should not be stable, but its axis 23 minutes after 7 in the morning of the continually changing. For it must be obsecond day. Knowing the precise mo served, that since the equator changes its ments, and knowing the rate of the sun's position, and the equator is only an imagi. motion in the ecliptic, it is easy to ascertain nary circle, equidistant from the two poles, the precise point of the ecliptic in which or extremities of the axis, these poles, and the equator intersected it. By a series of this axis must equally change their positions. sich observations made at Alexandria, be- The equinoctial points make a complete tween the years 161 and 127 before Christ, revolution in about 25,745 years, the equaHipparchus, the father of our astronomy, tor being all the while inclined to the eclip. found that the point of the autumnal equi- tic in nearly the saine angle. Therefore nox was about six degrees to the eastward the poles of this diurnal revolution must of the star called spica virginis. Eager to describe a circle round the poles of the determine every thing by multiplied obser- ecliptic, at the distance of about 234 de. Fations, he ransacked all the Chaldean, grees in 25,745 years; and in the time of Egyptian, and other records, to which his Timochares, the north pole of the heavens travels could procure him access, for obser must have been so degrees eastward of vations of the same kind; but he does not where it now is. mention his having found any. He found, EQUITY, quasi æqualitus, is generally however, some observations of Aristillus understood in law, a liberal correction, or and Timochares, made about 150 years qualification of the law, where it is too before. From these it appeared evident strict, too contiped, or severe, and is :somethat the point of the autumnal equinox was times applied, where, by the words of a then about eight degrees east of the same statute, a case does not fall within it, yet star. He discusses these observations with being within the mischief, the judges, by great sagacity and rigour: and on their au- an equitable construction, have extended thority, he asserts that the equinoctial its application to that case. Equity is unpoints are not fixed in the heavens, but derstood as a correction of the law : the move to the westward about a degree in difference between courts of equity and 75 years, or somewhat less. law is known only in this country, and This motion is called the precession of arises principally, if not entirely, from the the equinoxes, because by it the time and different modes of trial which must ever place of the sun's equinoctial station pre- render thiem essentially distinct. For it is cedes the usual calcnlations : it is fully con- obvious, that where men form contracts in firmed by all subsequent observations. In the ordinary course of law, the legal conse1750, the autumnal equinox was observed quence, and the enforcement of them, must to be 20° 21' westward of spica virginis. be, according to general rules, applicable Supposing the motion to haye been uniform to general caşes; and the nature of our |