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tion of the moon is ascertained by comparing ancient with modern observations; for if the ancient observations be assumed as observed longitudes of the moon, a calculation of her place for the same epoch from the lunar tables will render the acceleration manifest, since these tables may be regarded as data derived from modern observations.
An eclipse of the moon observed by the Chaldeans at Babylon, on the 19th of March, 721 years before the Christian era, which began about an hour after the rising of the moon, as recorded by Ptolemy, has been employed. As an eclipse can only happen when the moon is in opposition, the instant of opposition may be computed from the solar tables, which will give the true longitude of the moon at the time, and the mean longitude may be ascertained from the tables. Now, if we compare this result with another mean longitude of the moon computed from modern observations, the difference of the longitudes augmented by the requisite number of circumferences will give the arc described by the moon parallel to the ecliptic during the interval between the observations, and the mean motion of the moon during 100 Julian years may be ascertained by dividing this arc by the number of centuries elapsed. But the mean motion thus computed by Delambre, Bouvard, and Burg, is more than 200" less than that which is derived from a comparison of modern observations with one another. The same results are obtained from two eclipses observed by the Chaldeans in the years 719 and 720 before the Christian era. This acceleration was confirmed by comparing less ancient eclipses with those that happened recently; for the epoch of intermediate observations being nearer modern times, the differences of the mean longitudes ought to be less than in the first case, which is perfectly confirmed, by the eclipses observed by Ibn-Junis, an Arabian astronomer of the eleventh century. It is therefore proved beyond a doubt, that the mean motion of the moon is accelerated, and her periodic time consequently diminished from the time of the Chaldeans.
Were the eccentricity of the terrestrial orbit constant, the term 3 m2 f(e12 — ē2) dv
would be united with the mean angular velocity of the moon;
but the variation of the eccentricity, though small, has in the course of time a very great influence on the lunar motions. The mean motion of the moon is accelerated, when the eccentricity of the earth's orbit diminishes, which it has continued to do from the most ancient observations down to our times; and it will continue to be accelerated until the eccentricity begins to increase, when it will be retarded. In the interval between 1750 and 1850, the square of the eccentricity of the terrestrial orbit has diminished by 0.00000140595. The corresponding increment in the angular velocity of the moon is the 0.0000000117821th part of this velocity. As this increment takes place gradually and proportionally to the time, its effect on the motion of the moon is less by one half than if it had been uniformly the same in the whole course of the century as at the end of it. In order, therefore, to determine the secular equation of the moon at the end of a century estimated from 1801, we must multiply the secular motion of the moon by half the very small increment of the angular velocity; but in a century the motion of the moon is 1732559351.544, which gives 10".2065508 for her secular equation. Assuming that for 2000 years before and after the epoch 1750, the square of the eccentricity of the earth's orbit diminishes as the time, the secular equation of the mean motion will increase as the square of the time: it is sufficient then during that period to multiply 10".2065508 by the square of the number of centuries elapsed between the time for which we compute and the beginning of the nineteenth century; but in computing back to the time of the Chaldeans, it is necessary to carry the approximation to the cube of the time. The numerical formula for the acceleration is easily found, for since
3mf (e- é2)dv
is the acceleration in the mean longitude of the moon, the true longitude of the moon in functions of her mean longitude will contain the term
e being the eccentricity of the terrestrial orbit at the epoch 1750. If then, t be any number of Julian years from 1750, by article 480,
is the eccentricity of the earth's orbit at any time t, whence the acceleration is
10."1816213. T+0".018538444 . T3,
T being any number of centuries before or after 1801.
In consequence of the acceleration, the mean motion of the moon is 7' 30" greater in a century now than it was 2548 years ago.
Motion of the Moon's Perigee.
749. In the first determination of the motion of the lunar perigee, the approximation had not been carried far enough, by which the motion deduced from theory was only one half of that obtained by observation; this led Clairaut to suppose that the law of gravitation was more complicated than the inverse ratio of the squares of the distance; but Buffon opposed him on the principle that, the primordial laws of nature being the most simple, could only depend on one principle, and therefore their expression could only consist of one term. Although such reasoning is not always conclusive, Buffon was right in this instance, for, upon carrying the approximation to the squares of the disturbing force, the law of gravitation gives the motion of the lunar perigee exactly conformable to observation, for e' being the eccentricity of the terrestrial orbit at the epoch, the equation c = √1-p-p'e when reduced to numbers is c = 0.991567, consequently (1 − c) v the motion of the lunar perigee is 0.008433 . v; and with the value of c in article 735 given by observation, it is 0.008452. v, which only differs from the preceding by 0.000019. In Damoiseau's theory it is 0.008453. v, which does not differ much from that of La Place. The terms depending on the squares of the disturbing force have a very great influence on the secular variation in the motion of the lunar perigee; they make its value three times as great as that of the acceleration for the secular inequality in the lunar perigee is
·S(e1a — e2)ndt,
or, when the coefficient is computed, it is
and has a contrary sign to the secular equation in the mean motion.
The motion of the perigee becomes slower from century to century, and is now 8'.2 slower than in the time of Hipparchus.
Motion of the Nodes of the Lunar Orbit.
750. The sidereal motion of the node on the true ecliptic as determined by theory, does not differ from that given by observation by a 350th part; for the expression in article 727 gives the retrograde motion of the node equal to 0.0040105v, and by observation
the difference being 0.00001125. Mr. Damoiseau makes it
The secular inequality in the motion of the node depends on the variation in the eccentricity of the terrestrial orbit, and has a contrary sign to the acceleration. Its analytical expression gives
· ƒ (e12 – ē2) dv = 0.735452 §m3 ƒ(e12 —e2)dv. 2√√1+9
As the motion of the nodes is retrograde, this inequality tends to augment their longitudes posterior to the epoch.
751. It appears from the signs of these three secular inequalities, as well as from observation, that the motion of the perigee and nodes become slower, whilst that of the moon is accelerated; and that their inequalities are always in the ratio of the numbers 0.735452, 3.00052, and 1.
752. The mean longitude of the moon estimated from the first point of Aries is only affected by its own secular inequality; but the mean anomaly estimated from the perigee is affected both by the secular variation of the mean longitude, and by that of the perigee; it is therefore subject to the secular inequality-4.00052 m2(e12 – e2) dv more than four times that of the mean longitude. From the preceding values it is evident that the secular motion of the moon with regard to the sun, her nodes, and her perigee, are as the numbers 1; 0.265; and 4; nearly.
753. At some future time, these inequalities will produce variations equal to a fortieth part of the circumference in the secular motion of the moon; and in the motion of the perigee, they will amount to
no less than a thirteenth part of the circumference. They will not always increase depending on the variation of the eccentricity of the terrestrial orbit they are periodic, but they will not run through their periods for millions of years. In process of time, they will alter all those periods which depend on the position of the moon with regard to the sun, to her perigee, and nodes; hence the tropical, synodic, and sidereal revolutions of the moon will differ in different centuries, which renders it vain to attempt to attain correct values of them for any length of time.
Imperfect as the early observations of the moon may be, they serve to confirm the results that have been detailed, which is surprising, when it is considered that the variation of the eccentricity of the earth's orbit is still in some degree uncertain, because the values of the masses of Venus and Mars are not ascertained with precision; and it is worthy of remark, that in process of time the developement of the secular inequalities of the moon will furnish the most accurate data for the determination of the masses of these two planets.
754. The diminution of the eccentricity of the earth's orbit has a greater effect on the moon's motions than on those of the earth. This diminution, which has not altered the equation of the centre of the sun by more than 8'.1 from the time of the most ancient eclipse on record, has produced a variation of 1°8′ in the longitude of the moon, and of 7°.2 in her mean anomaly.
Thus the action of the sun, by transmitting to the moon the inequalities produced by the planets on the earth's orbit, renders this indirect action of the planets on the moon more considerable than their direct action.
755. The mean action of the sun on the moon contains the inclination of the lunar orbit on the plane of the ecliptic; and as the position of the ecliptic is subject to a secular variation, from the action of the planets, it might be expected to produce a secular variation in the inclination of the moon's orbit. This, however, is not the case, for the action of the sun retains the lunar orbit at the same inclination on the orbit of the earth; and thus in the secular motion of the ecliptic, the orbit of the earth carries the orbit of the moon along with it, as it will be demonstrated, the change in the ecliptic affecting only the declination of the moon. No perceptible