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, occa sions corresponding changes in the motions of the satellites, which led him to suspect that the acces leration 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 acce= leration.

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 re turned 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 number 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, 0.735, 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, 46776, and 0.391. A comparison of ancient eclipses observed by the Arabs, Greeks, and Chaldeans, imperfect as they are, with modern

observations, perfectly confirms these results of analysis. Future ages will develop these great inequalities, which at some most distant period will amount to many circumferences. They are indeed periodic; but who shall tell their period? Millions of years must elapse before that great cycle is accomplished; but such changes, though rare in time, are frequent in eternity.'

The moon is so near, that the excess of matter at the earth's equator occasions periodic variations in her longitude, and also that remarkable inequality in her latitude already mentioned as a nutation in the lunar orbit, which diminishes its inclination to the ecliptic when the moon's ascending node coincides with the equinox of spring, and augments it when that node coincides with the equinox of autumn. As the cause must be proportional to the effect, a comparison of these inequalities, computed from theory, with the same given by observation, shows that the compression of the terrestrial spheroid, or the ratio of the difference between the polar and equatorial diameters, to the diameter of the equator, is 35. It is proved analytically that, if a fluid mass of homogeneous matter, whose particles attract each other inversely as the square of the distance, were to revolve about an axis as the earth does, it would assume the form of a spheroid whose compression

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, 46776, and 0.391. A comparison of ancient eclipses observed by the Arabs, Greeks, and Chaldeans, imperfect as they are, with modern