The greatest value of the parallax is 1° 1′ 29′′.32, which happens when the moon is in perigee and opposition; the least, 58′ 29′′.93, happens when the moon is in apogee and conjunction.

E

742. With m= Mr. Damoiseau finds the constant part of the

equatorial parallax equal to 3431".73.

743. The lunar parallax being known, that of the sun may be determined by comparing the coefficients of the inequality

122.014 (i + 1) sin (v — mv)

in the moon's mean longitude with the same derived from observation. In the tables of Burg, reduced from the true to the mean longitude, this coefficient is 122".378; hence

which is the mean parallax of the sun in the parallel of latitude, the square of whose sine is .

Burckhardt's tables give 122".97 for the value of the coefficent, whence the solar parallax is 8".637, differing very little from the value deduced from the transit of Venus. This remarkable coincidence proves that the action of the sun upon the moon is very nearly equal to his action on the earth, not differing more than the three millionth part.

744. The constant part of the lunar parallax is 3432".04, by the observations of Dr. Maskelyne, consequently the equation

of whose sine is; if R', the mean radius of the earth, be assumed as unity, the mean distance of the moon from the earth is 60.4193 terrestrial radii, or about 247583 English miles.

745. As theory combined with observations with the pendulum, and the mensuration of the degrees of the meridian, give a value of the lunar parallax nearly corresponding with that derived from astronomical observations, we may reciprocally determine the magnitude of the earth from these observations; for if the radius of the

earth be assumed as the unknown quantity in the expression in article 646, it will give its value equal to 20897500 English feet.

'Thus,' says La Place, an astronomer, without going out of his observatory, can now determine with precision the magnitude and distance of the earth from the sun and moon, by a comparison of observations with analysis alone; which in former times it required long voyages in both hemispheres to accomplish.'

746. The apparent diameter of the moon varies with its parallax, for if P be the horizontal parallax, R' the terrestrial radius, r the radius vector of the moon, D her real, and A her apparent diameters; then

a ratio that is constant if the earth be a sphere. It is also constant at the same point of the earth's surface, whatever the figure of the earth may be.

thus if be multiplied by the moon's apparent semidiameter, the corresponding horizontal parallax will be obtained.

Secular Inequalities in the Moon's Motions.

747. It has been shown, that the action of the planets is the cause of a secular variation in the eccentricity of the earth's orbit, which variation produces analogous inequalities in the mean motion of the moon, in the motion of her perigee and in that of her nodes.

The Acceleration.

748. The secular variation in the mean motion of the moon denominated the Acceleration, was discovered by Halley; but La Place first showed that it was occasioned by the variation in the eccentricity in the earth's orbit. The acceleration in the mean mo

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 observa'tions.

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 m2f (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

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,