whence Cos. A cos. L COS. NSX' COS. NSY' + sin. λ=0. In like manner, for another place P' of the moon, we should have cos. ' cos. L' COS. NSX' +cos. λ' sin. L' COS. N SY' from these two equations there may be obtained the values of COS. NSX' COS. N SY' COS. NSZ" and also of cos. NSZ" cos.2 NSX' + cos.2 NSY' COS.2 NSZ ; the last fraction expresses (art. 185.) the square of the tangent of the angle z's N, and this is equal to the required inclination of the orbit to the ecliptic. The position of the line of nodes is found as in art. 186., a and ẞ being the co-ordinates of any point in that line, which, as above observed, is to pass through s: for we have whence a a cos. NSX' + ß cos. NSY' =0: COS. NSY' = tan. X's M', and Bsm, the comβ COS. NSX' plement of X's M', is the angle which the line of nodes makes with sx', that is, with the line of the equinoxes. 211. The longitude of the moon's nodes as well as the obliquity of her orbit to the plane of the ecliptic are found to vary with time, the movement of the nodes taking place in retrograde order. The amount of the retrogradation can be determined approximately by computing the longitude of the moon when she is in one of the nodes, and again when, in her revolutions about the earth, she is next in the same node, and taking the difference between the longitudes (the interval in time being also known); but it is evident that the movement will be determined more accurately if the computation be made for times very distant from each other. In this manner it may be ascertained that the retrogradation of either of the nodes to the amount of 360 degrees, or the time in which either of the nodes performs a revolution about the earth, from a point in the ecliptic at any distance from the place of the vernal equinox to a point at an equal distance from that place, is accomplished in about 18 years (=6798 days, 12 hours, 57 minutes, 52 seconds). 212. The place of the node being determined for any given instant, and the latitude ME (fig. to art. 209.) being found for the same time; the arc MN, or the angular distance of the moon from the node, on her orbit, may be found by trigono metry in the right angled spherical triangle MEN. If a series of such distances be computed from the observed right ascensions and declinations of the moon when on the meridian, and the intervals, in time, between the transits be also observed; it is plain that the daily angular motion of the moon in her orbit about the earth may be determined. It should be remarked that the longitude of the moon may be converted into the distance on her orbit, from the point Y, and the converse, by the rules of spherical trigonometry, as in art. 189., or by the formula for 7 a in art. 190.: in either case, for these purposes, 7 must be made to represent the distance Y M on the orbit, a the longitude y E, and ✪ the angle MYE. -- 213. The relative distances of the moon from the earth may be ascertained approximately from the observed angular magnitudes of the moon's diameter; or the absolute distances may be ascertained by means of her horizontal parallaxes, the latter being determined by the method explained in art. 160. Thus, let it be supposed that the earth is a sphere, and that M, the centre of the moon, is in the horizon of an observer at s (fig. to art. 154.); the angle CSM will be a right angle, and the angle SMC the horizontal parallax; therefore (Pl. Trigo., art. 57.) sin. SMC radius:: SC: CM; hence sc the semidiameter of the earth being known, CM the required distance will be found, (about 237,000 miles). Then, by means of the moon's daily angular motions in her orbit, and her daily distances from the earth, the figure of the moon's orbit may be determined by a graphical construction, as that of the earth was supposed to be determined (art. 194.). It would thus be found that the orbit is nearly elliptical, having the earth in one of its foci; and from the properties of the ellipse, the excentricity of the orbit may then be approximatively ascertained. The places of the perigeum and apogeum, and the equation of the centre, sometimes called the first inequality, may evidently be determined by methods similar to those which have been described in the account of the earth's orbit. 214. A series of the daily longitudes of the sun and moon being obtained from observations, there may be found by inspection, and by proportions founded on the observed daily motions of the two luminaries in longitude, two successive times at which the sun and moon had the same longitude, or were in conjunction in longitude (times of new moon), or two successive times at which the longitudes of the sun and moon differed in longitude by 180 degrees, that is, two successive oppositions (times of full moon); the two times, in either case, synodical revolution of the moon. and the interval between constitutes the period of a Let this synodical period, 360° in days, be represented by s; then will be the excess of the moon's mean daily motion in longitude above that of the sun or earth. But the moon's motion being subject to great irregularities, the mean movement deduced from a single synodical period can only be considered as a first approxima tion to that element. In order to obtain it with greater correctness, the recorded time of an ancient eclipse of the moon, at which time the sun and moon were nearly in opposition, should be compared with the time of a modern eclipse, the moon being nearly in the same part of her orbit with respect to the points of apogee or perigee: the difference between these times divided by the number of synodical revolutions which have taken place in the interval, the number being found from the approximate time of one revolution determined as above, will give the accurate mean time of a synodical revolution. 215. Ptolemy has stated in the Almagest the occurrence of three eclipses of the moon, which were observed by the Chaldeans in the years 721, 720, and 719 B. C.; and, from the time elapsed between those eclipses and one which happened in the year 1771 of our era, Lalande determined the time in which a synodical revolution was accomplished. The time of such revolution has also been found by a comparison of the Chaldean eclipses with three of those which were observed at Cairo by Ibn Junis between the years 997 and 1004, and also by comparisons of eclipses which have been observed within the two last centuries; and from a combination of the results, not only has the time of a revolution been determined, but certain differences have been found in the durations of the revolutions from whence it is ascertained that the mean motion of the moon has for many ages experienced an acceleration which varies with time. La Place has determined that, in the present age, the synodical revolution is performed in 29 days, 12 hours, 44 minutes, 2.8 seconds. 216. The sidereal period of the moon, or the time of her revolution from one fixed star to the same, may be found thus. The number of days between the times of two observed eclipses of the sun or moon, at an interval of many years, is known, and from the approximate length of a synodical revolution, the number of such revolutions in the same interval is also known. Then, since the angular space described by the moon about the earth between two successive con junctions or oppositions of the sun and moon exceeds 360 degrees by the angular motion of the sun during the same revolution; it follows that if n be the number of synodical revolutions, and m be the number of degrees described by the sun with his mean sidereal movement in the whole time between the observed eclipses, we shall have n 360°+m for the number of degrees described by the moon in that time: let N be the number of days in the time, then n 360°+m: 360° :: N : N'; where N' is the number of days during which the moon would describe the circuit of the heavens from any fixed star to the same: thus the time of a sidereal revolution of the moon about the earth is found to be 27.321661 da., or 27 da. 7 ho. 43 m. 11.5 sec. The period of a tropical revolution of the moon may from thence be immediately deduced: for let p (=375 sec.) be the amount of precession during one sidereal revolution of the moon (= 50′′.2 annually); then 360° : 360° — p:: 27.321661 da. : 27.321582 da., or 27 da. 7 ho. 43 m. 4.68 sec. This is the periodical lunar month, or the time of her revolution from one equinox to the same. 217. The places of the moon's nodes, found as above mentioned by the determinations of her place when in the ecliptic, or when her latitude is zero, are observed to vary with time; and this movement, which takes place in retrograde order, can be found by computing the longitudes of the nodes at times very distant from each other. By comparing together many times when the moon was in, or at equal distances from the nodes, it has been found that a tropical revolution of the nodes (a revolution from one equinox to the same) takes place in 6788.54019 da., and a sidereal revolution, in 6793 da. .42118. 218. The duration of an anomalistic revolution of the moon may be found by taking the interval between two times when she is equally distant from, and very near the point of apogeum or perigeum; and from thence the place of her perigeum may be found, as the perihelion point of the earth's orbit was obtained. On comparing the computed places of the moon's perigeum at different times, it will be found that the major axis of the orbit has, during one sidereal revolution of the moon about the earth, a movement in direct order equal to 3° 2′31′′-6; and hence the period of an anomalistic revolution is equal to 27.5546 da. It may be remarked that these periods are seldom now obtained from observation; astronomers confining themselves chiefly to the determination of a correct value of the mean tropical motion of the moon by means of her longitudes observed at great intervals of time. The duration of a mean revolution being already known very nearly, the number (N) of complete circumferences described by the moon in any interval between the times of two observed longitudes will be known: consequently N. 360° added to the difference between the observed longitudes will give the whole number of degrees described by the moon during the interval; and this sum, divided by the number of days in that interval, the years being considered as Julian years (=365.25 days each), such being the nature of the years in astronomical tables, will give the mean daily tropical motion: in the present age this is found to be equal to 13° 10' 34".896. A formula has been investigated by astronomers for determining the correction which must be applied to the mean motions obtained from the lunar tables in order to reduce them to their value at a given time: the correction thus found and applied, produces results which accord very nearly with the results of observation within 1000 or 1200 years before or since the epoch for which the tables are computed. The variation of the moon's mean motion affects the periodical times of the tropical, sidereal, and other revolutions of the moon, and renders it necessary to apply continually to those times certain small corrections in order to obtain their true values. 219. In seeking the longitude of the moon for any given time, the mean longitude is first obtained from the tables as if the moon revolved uniformly about the earth in a circular orbit coincident with the plane of the ecliptic; then the application of the equation of the centre, and the corrections for the motions of the perigeum and nodes, give the place of the moon as if the projection of her orbit on that plane were a perfect ellipse. But the moon is subject to many inequalities of motion from the perturbations produced by the mutual attractions of the bodies composing the solar system; and physical astronomy both assigns the several causes of these inequalities and determines the effects due to each. Some of these inequalities are, however, of sufficient magnitude to be capable of being detected by comparisons of observations made in particular positions of the moon; and accordingly they were discovered and their values computed before the theory of gravitation was employed to account for the phenomena of the solar system. 220. The greatest of these inequalities is that which is |