from the sun's longitude, there remains the value of his right ascension. 191. It is evident that by spherical trigonometry, or, at once, from the formulæ in art. 181., there may be obtained the latitude and longitude of the moon, a planet, or a star, when the right ascension and declination have been obtained from observation, and the obliquity of the ecliptic is known. Let (fig. to art. 187.) be part of the equator, Y C part of the ecliptic, and let the angle YQ, the obliquity of the ecliptic, be represented by 0. Let's be the celestial body, TR its right ascension (a) and RS its declination (= d); also let L, the longitude, be represented by L, and LS, the latitude, by; then by substitution in (A) and (B) (art. 181.), we have the values of sin. λ and tan. L. Conversely, the longitude and latitude being taken from tables, there may be found from the formulæ for sin. d and tan. a, the values of the declination and right ascension. CHAP. VIII. THE ORBIT OF THE EARTH. ITS FIGURE SHOWN TO BE ELLIPTICAL. SITUATION AND MOVEMENT OF THE PERIHELION POINT.-THE MEAN, TRUE, AND EXCENTRIC ANOMALIES.-EQUATION OF THE CENTRE. 192. FOR the sake of a more ready comprehension of the manner in which the figure of the earth's orbit may be determined, it will be convenient, for a moment, to imagine that the earth is at rest, and that the sun describes about it the periphery of a curve similar to that which the earth describes about the sun; it is evident that, as the means employed to determine the figure involve only the mutual distances of the earth and sun, and the angular movement of either, the form of the required curve will be the same whether the former or the latter be supposed to move about the other. 193. From the observed declinations and right ascensions of the sun obtained daily during a whole year, let the longitudes of that luminary be computed (art. 189.), and let the daily differences of the longitudes be found by subtraction. These daily differences, which may be considered as the velocities in longitude, are not equal to one another; and a comparison of them will show that, in the present age, they are the greatest soon after mid-winter, and the least soon after mid-summer; on the first day of January they are about 61' 11.5, and on the first day of July, about 57' 12.5, the mean of which is 59′ 12′′. The angular distance between the places of the sun on the days of the greatest and least velocity is 180 degrees or half the circuit of the celestial sphere, so that the two places appear to be at the extremities of a line drawn through the earth and produced each way to the heavens. Again, if the angle subtended by the diameter of the sun be accurately measured by means of a micrometer daily, or at intervals of a few days, it will be found that this element is variable; on the first day of July it is the least, being then equal to 31′ 30′′, and on the first of January it is the greatest, being then equal to 32′ 35′′. Now, by the laws of optics, the distance from the observer, of any object which subtends a small angle, is inversely proportional to its apparent magnitude; it must be inferred therefore that the sun is at a greater distance from the earth in summer than in winter; the ratio of the two distances being as 32′ 35′′ to 31' 30", or as 1.0169 to 0.9831. The variations of the longitudes and of the angular magnitudes follow the same law on both sides of the line of greatest and least distances; and it follows that the curve apparently described by the sun about the earth in one year, or that which is described by the earth about the sun in the same time, is symmetrical on the two sides of that line. E BA X Y P S 194. An approximation to the figure of the earth's orbit may be conceived to be obtained from a graphic construction in the following manner. From any point s representing the sun draw lines making the angles ASB, ASD, ASE, &c., equal to the sun's increase of longitude for one day, two days, three days, &c., and make the lengths of SA, SB, SD, &c. inversely proportional to the apparent angular measure of the sun's diameter; then if a line be drawn through the points A, B, D, &c., it will represent the figure of the orbit, and will be found to be nearly the periphery of an ellipse of which the sun occupies one of the foci. Or if, with the given values of the angles ASB, ASD, &c., assuming AC or the half of AP to be unity, the lengths of the lines SB, SD, &c. be computed by the formula for the distances of B, D, &c. from the focus of an ellipse, the values of those lines will be found to agree very nearly with the values obtained by supposing the same lines to be inversely proportional to the apparent angular measures of the sun's diameter; and thus the ellipticity of the orbit may with more certainty be proved. On computing the areas comprehended between the radii vectores SA and SB, SB and SD, &c., they will be found to be equal to one another when the times of describing the arcs AB, BD, &c. are equal to one another, that is, the sectoral areas imagined to be described by the radii vectores about s vary with the times of the description; or, if t be the time in which the earth moves from в to D for example, and T be the time in which it describes the periphery of the ellipse, we have area BSD area of the ellipse::t: T. The ellipticity of the earth's orbit was first discovered by Kepler, and the above relation between the areas is designated one of Kepler's laws. 195. Though it is now known that the orbit is not, strictly speaking, an ellipse, yet the latter being the regular curve which is next in simplicity to a circle, astronomers for convenience consider it as the figure of the orbit described by the sun about the earth, or of the earth about the sun. The point A or P at which the sun or the earth is when the two bodies are at the greatest or the least distance from one another, is called an apsis, and a line joining the points of greatest and least distance is called the line of the apsides. If it be assumed that the earth revolves about the sun the same points are respectively called the aphelion and the perihelion points, the sun being supposed to be in one of the two foci of the ellipse. 196. If BSD be one of the triangles described by a radius vector r in a unit of time (one second, one minute, or one hour), and if v represent the angular velocity of the sun or earth (a circular arc intercepted between SB and SD, and supposed to be described about as a centre with a radius equal to the unit of length); then (by similarity of sectors), rv may be considered as equal to the line Dm (a perpendicular let fall from D on SB), and by mensuration, r2v may be considered as equal to the area BSD, or v x nearly; but the area BSD is constant when the times are equal; therefore the angular velocity is inversely proportional to the square of the radius vector (nearly). area BSD 2 197. The earth's orbit being symmetrical on each side of the line of apsides, it will follow that the instants when the earth is in the aphelion and perihelion points successively, must differ in time by half the period of a complete revolution of the earth about the sun, as well as that the longitudes of the points must differ from one another by 180 degrees; for it is evident that if any other line as XY be imagined to be drawn through the sun it will cut the orbit in two points which will differ in longitude by 180 degrees, while the times in which the earth moves from one to the other on opposite sides of the line will be unequal, because the movement is more rapid about the perihelion than about the aphelion point. It should be observed that, among the registered observations which may have been made during a year, there may not be two which give by computation longitudes differing by exactly 180 degrees; but if there be found two longitudes which differ by nearly that quantity, then from these, with the known velocity of the sun, the times when the longitudes so differ may be computed by proportions simply. 198. If the longitudes of the sun be computed for two times distant by many years, the daily differences of longitude being at both times the least or the greatest, or being equal to one another and nearly equal to the least or greatest, the longitudes will be those of the sun at, or nearly at, the instants when the earth is in the aphelion or perihelion point, or at equal distances from either; these longitudes will be found to differ from one another, and the difference will evidently express the quantity by which the perihelion point has moved in longitude in the interval. This movement which takes place in the "order of the signs," is called the progression of the perigee, or of the perihelion point, and its mean value for one year, if reckoned by the different angles which, at given times, SP or SA makes with a line drawn through the sun and the first point of Aries, is found to be 61.9. 199. The time in which the earth revolves once about the sun from the perihelion point to the same is called the anomalistic year it is evidently equal to the length of the tropical year, together with the time in which the sun moves through 61′′.9 in longitude. But the equinoctial point (the first point of Aries) retrograding 50′′.2 (art. 174.) annually by the general precession, the progression of the perigee, if measured by the different angles which at given times SP or SA makes with a line drawn through s and a fixed point (supposed to be the place of a star) in the heavens is only equal to 11".7 annually. The anomalistic year, therefore, exceeds the length of a sidereal year by the time in which the sun moves through 11".7 in longitude. 200. If from a register of the computed longitudes of the sun there be selected two which differ from one another by exactly 180 degrees, and which correspond to times when (the daily differences of longitude being nearly the least and the greatest) it may be considered that the earth was nearly in the aphelion and perihelion points; the instant at which the earth was in the latter point, and also the longitude of that point, may be found in the following manner. Let AP (fig. to art. 194.) be the line of the apsides, s the sun, X and Y the two places of the earth when near A and P, and when in the direction of a right line through s; also let the required time in which the sun will move from Y to P be represented by t: then, by Kepler's law (art. 194.) sector YSP sector ASX:: t: t' (t' denoting the time of moving from x to A). But the sectors being supposed to be similar to one another, SP2 SA2 :: sector YSP: sector ASX, or as t: t'; and the angular velocities at P and A, being inversely as the squares of the distances from s, if those velocities be represented respectively by v and v' (which are known, being equal to the increments of longitude in equal times, at or near the perihelion and aphelion points), we have |