The longitude of the solar perigee is now 101°, that of the earth's perihelion 281°.
Longitude of the perigee at the two epochs gave the annual motion of that element, and the difference of mean longitudes gave the mean motion.
An Anomalistic month is the time in which the moon passes from perigee to perigee, &c.
The problem of determining an orbit may be regarded as coeval with Hipparchus, who, it is supposed, found the moving positions of the apogee and perigee of the moon's orbit.
In the case of the motion of the moon around the earth, assuming the gravitation of the latter to be subject to the modification in question, the annual motion of the moon's perigee should be greater by I 5" than the theoretical motion.
The sun's perigee and the earth's perihelion are so related that they differ 180° in longitude, the first being on the line from the earth toward the sun, and the second from the sun toward the earth.
If we put g for the moon's anomaly or distance from the perigee, and D for its elongation from the sun, the inequalities in question as now known are 6.29° sin g (equation of centre) +1.27° sin (2D-g) (evection).
The Babylonians knew of the inequality in the daily motion of the sun, but misplaced by to' the perigee of his orbit.
Then, by an obvious law of kinematics, the angular motion round the earth would be most rapid at the point nearest the earth, that is at perigee, and slowest at the point most distant from the earth, that is at apogee.
Thus the apogee and perigee became two definite points of the orbit, indicated by the variations in the angular motion of the moon.
The perigee, or the node.
Assuming the mean motion of the moon to be known and the perigee to be fixed, three eclipses, observed in different points of the orbit, would give as many true longitudes of the moon, which longitudes could be employed to determine three unknown quantities - the mean longitude at a given epoch, the eccentricity, and the position of the perigee.
We may conclude the ancient history of the lunar theory by saying that the only real progress from Hipparchus to Newton consisted in the more exact determination of the mean motions of the moon, its perigee and its line of nodes, and in the discovery of three inequalities, the representation of which required geometrical constructions increasing in complexity with every step.
The anomalistic month is the mean time taken by the moon in passing from one perigee to the next; the sidereal month is the mean time in which the moon makes a circuit among the stars; the tropical month is the mean time in which the moon traverses 360° of longitude; the nodical or draconic month is the mean time taken by the moon in passing from one rising node to the next; the solar month is one-twelfth of a tropical year.
By taking three eclipses separated at short intervals, both the mean motion and the motion of the perigee would be known beforehand, from other data, with sufficient accuracy to reduce all the observations to the same epoch, and thus to leave only the three elements already mentioned unknown.