7.

CHAP. XVI.

(ON THE MOON'S MOTION FROM OBSERVATION AND IT'S PHÆNOMENA.

(202.) THE moon being the nearest, and after the sun, the most remarkable body in our system, and also useful for the division of time, it is no wonder that the ancient astronomers were attentive to discover it's motions; and it is a very fortunate circumstance that their observations have come down to us, as from thence it's mean motion can be more accurately settled, than it could have been by modern observations only; and it moreover gave occasion to Dr. Halley, from the observations of some ancient eclipses, to discover an acceleration in it's mean motion. The proper motion of the moon, in it's orbit about the earth, is from west to east; and from comparing it's place with the fixed stars in one revolution, it is found to describe an orbit inclined to the ecliptic; it's motion also appears not to be uniform; and the position of the orbit, and the line of it's apsides are observed to be subject to a continual change. These circumstances, as they are established by observation, we come now to explain.

To determine the Place of the Moon's Nodes.

(203.) The place of the moon's nodes may be determined as in Art. 185, or by the following method.

In a central eclipse of the moon, the moon's place at the middle of the eclipse is directly opposite to the sun, and the moon must also then be in the node; calculate therefore the true place of the sun, or, which is more exact, find it's place by observation, and the

opposite point will be the true place of the moon, and consequently the place of it's node.

Ex. M. Cassini, in his Astronomy, p. 281, informs us, that on April 16, 1707, a central eclipse was observed at Paris, the middle of which was determined to be at 13h. 48' apparent time. Now the true place of the sun, calculated for that time, was 0. 26°. 19′ 17"; hence, the place of the moon's node was 6. 26°. 19. 17". The moon passed from north to south latitude, and therefore this was the descending node.

(204.) To determine the mean motion of the nodes, find (203) the place of the nodes at different times, and it will give their motion in the interval; and the greater the interval, the more accurately you will get the mean motion. Mayer makes the mean annual motion of the nodes to be 12°. 19'. 43",1.

On the Inclination of the Orbit of the Moon to the Ecliptic.

(205.) To determine the inclination of the orbit, observe the moon's right ascension and declination when it is 90° from it's nodes, and thence compute it's latitude (114), which will be the inclination at that time. Repeat the observation for every distance of the sun from the earth, and for every position of the sun in respect to the moon's nodes, and you will get the inclination at those times. From these observations it appears, that the inclination of the orbit to the ecliptic is variable, and that the least inclination is about 5o, which is found to happen when the nodes are in quadratures; and the greatest is about 5°. 18', which is observed to happen when the nodes are in syzygies. The inclination is also found to depend upon the sun's distance from the earth.

On the mean Motion of the Moon.

(206.) The mean motion of the moon is found from observing it's place at two different times, and

you get the mean motion in that interval, supposing the moon to have had the same situation in respect to it's apsides at each observation; and if not, if there be a very great interval of the times, it will be sufficiently exact. To determine this, we must compare together the moon's places, first at a small interval of time from each other, in order to get nearly the mean time of a revolution; and then at a greater interval, in order to get it more accurately. The moon's place may be determined directly from observation, or deduced from an eclipse.

(207.) M. Cassini, in his Astronomy, p. 294, ob, serves, that on September 9, 1718, the moon was eclipsed, the middle of which eclipse happened at Sh. 4', when the sun's true place was 5s. 16°. 40'. This he compared with another eclipse, the middle of which was observed at 8h. 32'. on August 29, 1719, when the sun's place was 5o. 5°. 47'. In this interval of 354d. 28' the moon made 12 revolutions and 349°. 7' over; divide therefore 354d. 28' by 12 revolutions +349°. 7'. part of a revolution, and it gives 27 d. 7h. 6' for the time of one revolution. From two eclipses in 1699, 1717, the time was found to be 27d. 7h. 43'. 6".

(208.) The moon was observed at Paris to be eclipsed on Sept. 20, 1717, the middle of which eclipse was at 6h. 2'. Now Ptolemy mentions, that a total eclipse of the moon was observed at Babylon on March 19, 720 years before J. C. the middle of which happened at 9h. 30', at that place, which gives 6h. 48′ at Paris. The interval of these times was 2437 years (of which 609 were bissextiles) 147 days wanting 46′; divide this by 27d. 7h. 43′. 6′′, and it gives 32585 revolutions and a little above. Now the difference of the two places of the sun, and consequently of the moon, at the times of observations, was 6s. 6°. 12'. Therefore, in the interval of 2437y. 174d. wanting 46', the moon had made 32585 revolutions 6$. 6°. 12', which gives 27d. 7h. 43'. 5" for the mean time of a revolution. This determination is very exact, as the

moon was at each time very nearly at the same distance from it's apside. Hence, the mean diurnal motion is 13°. 10'. 35", and the mean hourly motion 32'. 56". 27". M. de la Lande makes the mean diurnal motion 13°. 10'. 35",02784394. This is the mean time of a revolution in respect to the equinoxes. The place of the moon at the middle of the eclipse has here been taken the same as that of the sun, which is not accurate, except for a central eclipse; it is sufficiently accurate, however, for this long interval.

(209.) As the precession of the equinoxes is 50",25 in a year, or about 4" in a month, the mean revolution of the moon in respect to the fixed stars must be greater than that in respect to the equinox, by the time the moon is describing 4" with it's mean motion, which is about 7". Hence, the time of a sidereal revolution of the moon is 27d. 7h. 43′. 12′′.

(210.) Observe accurately the place of the moon for a whole revolution as often as it can be done, and by comparing the true and mean motions, the greatest difference will be double the equation. If two observations be found, where the difference of the true and mean motions is nothing, the moon must then have been in it's apogee and perigee (168). Mayer makes the mean excentricity 0,05503568, and the corresponding greatest equation 6°. 18'. 31′′6. It is 6°. 187. 32" in his last Tables, published by Mr. Mason, under the direction of Dr. Maskelyne.

(211.) To determine the place of the apogee, from M. Cassini's observations, we have the greatest equation=5°. 1'. 44"5; therefore (171), 57°. 17'. 48"8: 2o. 30'. 52′′25 :: AC 100000: CS 4388 for the moon's excentricity at that time*. Now (Fig. p. 101.) let v be the focus in which the earth is situated; then

=

*The excentricity of the moon's orbit is subject to a variation, it being greatest when the apsides lie in syzygies, and least when in quadratures.

(169) supposing QSP to be the mean anomaly, as QvP is the true anomaly, their difference SP is the equation of the orbit, which equation is here 37'. 50′′,5; and as PS= Pr, the angle vr S=18′. 55′′,25; hence, (Trigonometry, Art. 128) vS=8776 vr= 200000: sin. vrS= 18'. 55",25: sin. vSr, or QSr,= 7°. 12. 20", from which take vrS-18'. 55",25, and we have QuP=6°. 53'. 25" the distance of the moon from it's apogee; add this to 25. 19°. 40', the true place of the moon, and it gives 25. 26°. 33'. 25" for the place of the apogee on December 10, 1685, at 10h. 38'. 10" mean time at Paris. This therefore may be considered as an epoch of the place of the apogee.

To determine the mean Motion of the Apogee.

(212.) Find it's place at different times, and compare the difference of the places with the interval of the time between. To do this, we must first compare observations at a small distance from each other, lest we should be deceived in a whole revolution; and then we can compare those at a greater distance. The mean annual motion of the apogee in a year of 365 days is thus found to be 40°. 39'. 50", according to Mayer. Horrox, from observing the diameter of the moon, found the apogee subject to an annual equation of 12°,5.

(213.) The motion of the moon having been examined for one month, it was immediately discovered that it was subject to an irregularity, which sometimes amounted to 5 or 6°, but that this irregularity disappeared about every 14 days. And by continuing the observations for different months, it also appeared, that the points where the inequalities were the greatest, were not fixed, but that they moved forwards in the heavens about 3° in a month, so that the motion of the

1

moon, in respect to it's apogee, was about less than

120

it's absolute motion; thus it appeared that the apogee