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the moon's motion being accelerated when that of the earth is retarded, and vice versû-for, when the earth is in its perihelion, the lunar orbit is enlarged by the action of the sun; therefore, the moon requires more time to perform her revolution. But, as the earth approaches its aphelion, the moon's orbit contracts, and less time is necessary to accomplish her motion,-its period, consequently, depends upon the time of the year. In the eclipses the annual equation combines with the equation of the centre of the terrestrial orbit, so that ancient astronomers imagined the earth's orbit to have a greater excentricity than modern astronomers assign to it.
The planets disturb the motion of the moon both directly and indirectly; because their action on the earth alters its relative position with regard to the sun and moon, and occasions inequalities in the moon's motion, which are more considerable than those arising from their direct action: for the same reason the moon, by disturbing the earth, indirectly disturbs her own motion. Neither the excentricity of the lunar orbit, nor its mean inclination to the plane of the ecliptic, have experienced any changes from secular inequalities; for, although the mean action of the sun on the moon depends upon the inclination of the lunar orbit to the ecliptic, and that the position of the ecliptic is sub
ject to a secular inequality, yet analysis shows that it does not occasion a secular variation in the inclination of the lunar orbit, because the action of the sun constantly brings the moon's orbit to the same inclination on the ecliptic. The mean motion, the nodes, and the perigee, however, are subject to very remarkable variations.
From an eclipse observed by the Chaldeans at Babylon, on the 19th of March, seven hundred and twenty-one years before the Christian era, the place of the moon is known from that of the sun at the instant of opposition, whence her mean longitude may be found; but the comparison of this mean longitude with another mean longitude, computed back for the instant of the eclipse from modern observations, shows that the moon performs her revolution round the earth more rapidly and in a shorter time now, than she did formerly; and that the acceleration in her mean motion has been increasing from age to age as the square of the time all ancient and intermediate eclipses confirm this result. As the mean motions of the planets have no secular inequalities, this seemed to be an unaccountable anomaly. It was at one time attributed to the resistance of an etherial medium pervading space, and at another to the successive transmission of the gravitating force; but as La Place proved that neither of these causes,
even if they exist, have any influence on the motions of the lunar perigee or nodes, they could not affect the mean motion; a variation in the mean motion from such causes being inseparably connected with variations in the motions of the perigee and nodes. That great mathematician, in studying the theory of Jupiter's satellites, perceived that the secular variation in the elements of Jupiter's orbit, from the action of the planets, occasions corresponding changes in the motions of the satellites, which led him to suspect that the acceleration in the mean motion of the moon might be connected with the secular variation in the excentricity of the terrestrial orbit; and analysis has proved that he assigned the true cause of the acceleration.
If the excentricity of the earth's orbit were invariable, the moon would be exposed to a variable disturbance from the action of the sun, in consequence of the earth's annual revolution; it would however be periodic, since it would be the same as often as the sun, the earth, and the moon returned to the same relative positions: but on account of the slow and incessant diminution in the excentricity of the terrestrial orbit, the revolution of our planet is performed at different distances from the sun every year. The position of the moon with regard to the sun undergoes a
corresponding change; so that the mean action of the sun on the moon varies from one century to another, and occasions the secular increase in the moon's velocity called the Acceleration, a name peculiarly appropriate in the present age, and which will continue to be so for a vast num> ber of ages to come; because, as long as the earth's excentricity diminishes, the moon's mean motion will be accelerated, but when the excentricity has passed its minimum, and begins to increase, the mean motion will be retarded from age to age. At present the secular acceleration is about 11" 209, but its effect on the moon's place increases as the square of the time. It is remarkable that the action of the planets thus reflected by the sun to the moon is much more sensible than their direct action, either on the earth or moon. The secular diminution in the excentricity, which has not altered the equation of the centre of the sun by eight minutes since the earliest recorded eclipses, has produced a variation of about 1°48' in the moon's longitude, and of 7° 12′ in her mean anomaly.
The action of the sun occasions a rapid but variable motion in the nodes and perigee of the lunar orbit. Though the nodes recede during the greater part of the moon's revolution, and advance during the smaller, they perform their sidereal
revolution in 6793 37953 days; and the perigee accomplishes a revolution in 3232.56731 days, or a little more than nine years, notwithstanding its motion is sometimes retrograde and sometimes direct; but such is the difference between the disturbing energy of the sun and that of all the planets put together, that it requires no less than 114755 years for the greater axis of the terrestrial orbit to do the same. It is evident that the same secular variation which changes the sun's distance from the earth, and occasions the acceleration in the moon's mean motion, must affect the nodes and perigee; and it consequently appears, from theory as well as observation, that both these elements are subject to a secular inequality arising from the variation in the excentricity of the earth's orbit, which connects them with the Acceleration, so that both are retarded when the mean motion
is anticipated. The secular variations in these three elements are in the ratio of the numbers 3, 0735, and 1; whence the three motions of the moon, with regard to the sun, to her perigee, and to her nodes, are continually accelerated, and their secular equations are as the numbers 1, 4, and 0.265, or, according to the most recent investigations, as 1, 4.6776, and 0:391. A comparison of ancient eclipses observed by the Arabs, Greeks, and Chaldeans, imperfect as they are, with modern