=1X 7854 D2-4e where D is the greatest diameter ELLIPTICITY of the Terrestrial Spheroid, is the difference between the major and the minor semiaxes; it is generally expressed in terms of the former, that is, of the radius of the equator. ELONGATION, in Astronomy, the angle under which a planet would appear from the sun, when reduced to the ecliptic; or it is the angle formed by two lines drawn from the earth to the sun and planet, when reduced as above. The greatest elongation is the greatest distance to which the planets recede from the sun; which, however, only relates to the inferior planets Mercury and Venus. The greatest elongation of Venus, is from 44° 571 to 47° 48', and that of Mercury from 17° 36 to 28° 20. EPACTS, in Chronology, the excesses of the solar month above the lunar synodical month, and of the solar year above the lunar year of twelve synodical months. EPACTS, Annual, are the excesses of the solar year above lúnar. As the new moons are the same, that is, as they fall on the same day every 19 years, so the difference between the lunar and solar years is the same every 19 years. And because the said difference is always to be added to the lunar year in order to adjust or make it equal to the solar year; therefore, the said difference respectively belonging to each year of the moon's cycle is called the epact of the said year, that is, the number to be added to the same year, to make it equal to the solar year. Rule for finding the Gregorian Epact.-Subtract 1 from the golden number, multiply the remainder by 11, and reject the 30's. This rule will serve till the year 1900; but after that year the Gregorian Epact will be found by this rule: divide the centuries of the given year by 4, multiply the remainder by 17; then to this pro duct add 43 times the quotient, and also the number 86, and divide the whole sum by 25, reserving the quotient: next multiply the golden number by 11, and from the product subtract the reserved quotient, so shall the remainder, after rejecting all the 30's contain ed in it, be the epact sought. EPHEMERIDES, in Astronomy, tables calculated by astronomers, showing the present state of the heavens for every day at noon; that is, the places wherein all the planets are found at that time. It is from these tables that the eclipses, conjunctions, and aspects of the planets are determined: horoscopes, or celestial schemes constructed, &c. EPICYCLOID, a curve gene rated by a point in one circle, which revolves about another cir cle, either on the convacity or con. vexity of its circumference. EPICYCLOIDS, are distinguished into exterior and interior. EPICYCLOIDS, Exterior, are those which are formed by the revolu tion of the generating circle, about the convex circumference, of the quiescent circle. EPICYCLOIDS, Interior, are those that are formed when the generating circle revolves on concave eircumference. To these curves belong several. curious properties, of which we shall only mention a few of the most remarkable. 1. If the generating and equies. cent circle have to each other any commensurable ratio, then is the epicycloid both rectifiable and quadrable, although the area of the common cycloid, which is so much more simple in appearance, can never be completely obtained. 2. If the generating and quiescent circle are incommensurable with each other, then the area of the epicycloid cannot be found, but it is still in this case also rectifiable. 3. If in the interior epicycloid the diameter of the generant is equal to the radius of the quiescent circle, the curve becomes a right line, equal and coincident with the diameter of the latter. To find the Length of any exterior revolve upon another equal to it, its focus will describe a right line Epicycloid. As the semi-diameter of the qui-perpendicular to the axis of the escent circle, Is to the sum of the diameters of the two circles; So is double the versed sine of the arc of the generant, which has passed over any part of the quescent circle, To the length of epicycloidal arc generated by the point which touched the quiescent circle at the beginning of motion. When the whole arc is required, the versed sine becomes the diameter of the generant. The length of any arc of an inte. rior cycloid is found by the same proportion, only using the difference of the diameters, in the second term of the proportion, in stead of the sum. To find the Areas of Epicycloids. As the radius of the quiescent circle, Is to three times that radius, So is the circular segment quiescent parabola; the vertex of the revolving parabola will also describe the cissoid of Diocles, and any other point of it will describe some of Newton's defective hyper bolas, having a double point in the like point of the quiescent parabola. In like manner, if an ellipse revolve upon another el lipse equal and similar to it, its focus will describe a circle, whose centre is in the other focus, and consequently the radius is equal to the axis of the ellipse; and any other point in the plane of the ellipse will describe a line of the fourth order. The same also may be said of an hyperbola revolving upon another equal and similar to it; for one of the foci will des cribe a circle, having its centre in the other focus, and the radius will be the principal axis of the hyberbola & and any other point of the hyperbola will describe a line of the fourth order. EPOCH, or EPOCHA, a term or fixed point of time, from whence the succeeding years are number Or, so is the whole area of the ed or reckoned. The word is from generant, Tox", from TEXECY, to sustain, To the whole area of the epicy-stop; because epochs define or cloid, Which proportion holds good both for the exterior and interior epicycloids. limit a certain space or time. Different nations make use of different epochs. The Christians Dr. Halley gives us a general chiefly use the epoch of the natiproposition for the measurement vity or incarnation of Jesus Christ; of all cycloids and epicycloids: the Mahometans, that of the Hethus, the area of a cycloid or epi-gira; the Jews, that of the creacycloid, either primary, or con- tion of the world, or of the detracted, or prolate, is to the area luge; the ancient Greeks used of its generating circle; and also that of the Olympiads; the Rothe areas of the parts generated in mans, that of the building of their those curves, to the areas of ana-city; the ancient Persians and Aslogous segments of the circle; as the sum of double the velocity of the centre and velocity of the circular motion, to the velocity of the circular motion. EPICYCLOIDS, Spherical, are formed by a point of the revolv ing cirele when its plane makes an invariable angle with the plane of the circle on which it revolves. EPICYCLOIDS, Parabolic, Ellip tic, &c. If a parabola be made to syrians, that of Nabonnassar, &c. EQUABLY accelerated or retarded Motion, &c. is when the motion or change is increased or decreased by equal quantities or degrees, in equal times. EQUAL, a term of relation be. tween two or more things of the same magnitude, quantity, or qua lity. It is an axiom in geometry, that two things which are equal to the same thing are also equal to each EQUALITY, Triple, is when there are three formulæ of the same kind as above to be made perfect powers. EQUALITY, Ratio, is the ratio of two equal quantities. EQUATION, in A.gebra, is any expression in which two quanti ties differently represented are put equal to each other, by the sign placed between them. EQUALITY, Double, in the Dio other. And again, if to or from phantine Analysis, is when we equals you add or subtract equals,have two formulæ containing the the sum or remainder will be equal. same unknown quantity or quanEQUAL Altitudes, in Practical tities, each of which is to be made Astronomy, one of the most prac equal to a perfect power. ticable and certain methods of determining the time, and thus ascertaining the error of a clock or chronometer, is by observing equal altitudes of the sun or of a fixed star. For this purpose, all that is necessary is to observe the instant the sun or star is at any altitude towards the east, before the meridian passage; and the instant must likewise be marked when the same object attains exactly the same altitude towards the west, after the meridian passage; the mean between the above quantities will be the instant marked by the clock at the moment the sun or star was on the meridian. The preceding operation, however, supposes that the A Simple EQUATION, is that in declination of the object has not which the unknown quantity en varied daring the elapsed inter-ters only in the first degree; or, val, but this with the sun seldom which contains only one power of happens. The observation must, therefore, be corrected by a table, or by a direct calculation. A Literal EQUATION, is that in which all the quantities, both known and unknown, are expressed by letters. A Numeral EQUATION, is that in which the co-efficients of the unknown quantity and absolute terms are given numbers. the unknown quantity. An Adfected EQUATION is that which contains two or more powers of the unknown quantity It is called a Quadratic, a Cubic, a Biquadratic, &c., according to the highest power of the unknown quantity, Binomial EQUATIONS, are such as have only two terms. are Determinute EQUATIONS, those equations in which only one unknown quantity enters; or if more than one enter, there are always given as many independent equations as there are unknown quantities. Indeterminate EQUATIONS, are those equations in which there are more unknown quantities than there are independent equations. Reciprocal EQUATIONS, are those in which the co-efficients of each pair of terms, equally distant from the extremes, are equal to each other. are Exponential EQUATIONS, those in which the exponent or index of the power is unknown. Roots of an EQUATION, are those numbers or quantities which, when substituted for the unknown quan. tity, will make the whole expres- | ratic, Cubic, and Biquadratic Equasion become equal to 0. The roots of an equation are Positive, when they have a positive sign prefixed or understood; Negative, when they have a negative sign prefixed to them; Real, when they are expressed by any real or possible quantity; and Imaginary, when any imaginary quantity enters, as a√b; which is imaginary, or impossible, because a negative quantity has no square root. Every equation has as many roots, real and imaginary, as there are units in the exponent of the highest power of the unknown quantity. tions; and the several methods of procedure, in each of these, is given under the respective articles. Equations that exceed the fourth degree cannot be solved by any direct rule except in a few partial cases, in which there are certain relations either between the roots or co-efficients. We have, there fore, no means of obtaining the roots in those cases, but by approxi mation. EQUATIONS in the Differential and Integral Calculus, are of dif ferent denominations; as Differential Equations, Equation of Finite Differences, Equations of Partial Differences, &c. If one of the imaginary roots of Differential EQUATION, is that an equation be a +-b, another which contains in it certain differof its roots will be ab.ential quantities; and it is said to Hence it follows, that the number be of the first, second, third, &c. of imaginary roots in any equation order, according as it involves the is always even; or, which is the first, second, third, &c. differen same, they always enter in pairs; tial." therefore in an equation of odd dimensions, that is, when the highest power of the unknown quantity is an odd number, there being an odd number of roots, one of them must necessarily be real; whereas in an equation of even dimensions, all its roots may be imaginary. The co-efficient of the second term is equal to the sum of all the roots, having their signs changed; the co-efficient of the third term is equal to the sum of the product of every two roots; the co-efficient of the fourth term is equal to the product of every three, with their signs changed; and so on. Reduction of EQUATIONS, is of two kinds; viz. first, the reduction of them from a higher to a lower dimension; and second, the reduction of them to some particular form, to prepare them for solution. The former of these cases is more commonly called the Depression of Equations; and the latter usually consists in exterminating the second term of the equation, this being the most eligible form for solution. Solution of EQUATIONS, is the method of finding their roots; which, however, can only be done in a direct manner, for the first four degrees, viz. in Simple, Quad EQUATION of Finite Differences, in the Theory of Finite Differences or Increments, is that into which the finite differences of the variable quantities of any function enter.' EQUATION of Partial Differences, is that which has place between the differential of any function and the differentials of the vari ables on which it depends, com. bined with the variables them selves, and with or without con stant quantities. EQUATION of a Curve, in Analysis, is an equation showing the na ture of a curve, by expressing the relation between any absciss and its corresponding ordinate, or the relation of their fluxions, &c. EQUATION of Payments, in Arith metic, is the finding the time to pay at once several debts due at different times, and bearing no interest till after the time of payment, so that no loss shall be sustained by either party. The rule commonly given for this purpose is as follows: Multiply each sum by the time at which it is due; then divide the sum of the products by the sum of the payments, and the quotient will be the time required. The true equated time for two payments is expressed by the following formula: Let p and pl be sums due, at the end of the time n, n', respectively; let also r represent the rate of interest, and a the required time of payment of the whole. Make and. then. pr (n+n')+pl+p.. true equated time. =r, the sun's mean motion for the same interval of time; then half the dif ference between the true and mean motions will show the greatest equation of the centre. When the mean anomaly and eccentricity of an orbit are given, the equation of the centre may be readily obtained by the following rule, as radius to the cosine of the bgiven anomaly, so is five-fourths of the eccentricity of the orbit to a fourth number; which number add to half the greater axis, if the anomaly be less than 90° or more than 270°, otherwise subtract from the same. Then say, as the sum or remainder is to double the eccentricity, so is the logarithmic sine of the giveu anomaly to the sine of a first arch, from three times which sine subtract double radius, the remainder will be the sine of a second arch, whose oneleaves the equation sought. third part, taken from the former, This formula is founded on the principle of allowing only simple interest. If compound interest be allowed it is more simple, being as follows: Let a represent the sum of the several debts, and 6 the sum of their present values, computed as in the case of compound interest; also r the interest on 17. for a year, and a the equated time required; then log. b x= log. a log. (1+r) EQUATION of Time, in Astronomy, denotes the difference between mean and apparent time, or the reduction of the apparent unequal time, or motion of the sun or a planet, to equable and mean time, or motion. EQUATION, in Astronomy, is a term used to express the correction or quantity to be added to, or subtracted from, the mean position of The equation of time is calculaa heavenly body, to obtain the ted by tracing out the effects of true position; it also, in a more three combined causes; the obligeneral sense, implies the correc-quity of the ecliptic, the sun's untion arising from any erroneous equal apparent motion therein, supposition whatever. and the precession of the equinocEQUATION to Corresponding Alti-tial points. In consequence of the tudes, is a correction which must be applied to the apparent time of noon (found by means of the time elapsed between the instants when the sun had equal altitudes, both before and after noon,) in order to ascertain the true time. EQUATION of the Centre, is the difference between the true and mean place of a planet, or the angles made by the true and mean place; or, which amounts to the same, between the mean and equated anomaly. The greatest equation of the sun's centre may be obtained by finding the sun's longitude, at the times when he is near his mean distances, for then the difference will give the true motion for that interval of time; next find the first of these, in the first and third quadrants of the ecliptic from Aries; that is, between Aries and Cancer, and between Libra and Capricorn, the right ascension being less than the mean longitude, the point of right ascension is to the west, and therefore the apparent noon precedes the mean noon; but in the second and fourth quadrants, namely, between Cancer and Libra, or Capricorn and Aries, the right ascension being greater than the longitude, or the mean motion taken in the equator, the mean noon is westward, and there. fore precedes the apparent noon. The mean and apparent solar days are never equal, except when the sun's daily motion in right as cension is 59' 8"; this is nearly the |