state in the octants, a maximum in the syzigies, and a minimum in the quadratures. Upon considering the influence of the solar force in diminishing or increasing the moon’s gravity to the earth, Newton saw that her distance and her periodic time must from this cause be subject to change, and in this way he accounted for the annual equation observed by Tycho. By the application of similar principles, he explained the cause of the motion of the apsides, or of the greater axis of the moon’s orbit, which has an angular progressive motion of 3° 4′ nearly in the course of one lunation; and he showed that the retrogradation of the nodes, amounting to 3′ 10″ daily, arose from one of the elements of the solar force being exerted in the plane of the ecliptic, and not in the plane of the moon’s orbit, the effect of which was to draw the moon down to the plane of the ecliptic, and thus cause the line of the nodes, or the intersection of these two planes, to move in a direction opposite to that of the moon. The lunar theory thus blocked out by Newton, required for its completion the labours of another century. The imperfections of the fluxionary calculus prevented him from explaining the other inequalities of the moon’s motions, and it was reserved to Euler, D’Alembert, Clairaut, Mayer, and Laplace to bring the lunar tables to a high degree of perfection, and to enable the navigator to determine his longitude at sea with a degree of precision which the most sanguine astronomer could scarcely have anticipated. By the consideration of the retrograde motion of the moon’s nodes, Newton was led to discover the cause of the remarkable phenomenon of the precession of the equinoctial points, which moved 50″ annually, and completed the circuit of the heavens in 25,920 years. Kepler had declared himself incapable of assigning any cause for this motion, and we do not believe that any other astronomer ever made the attempt. From the spheroidal form of the earth, it may be regarded as a sphere with a spheroidal ring surrounding its equator, one-half of the ring being above the plane of the ecliptic and the other half below it. Considering this excess of matter as a system of satellites adhering to the earth’s surface, Newton saw that the combined actions of the sun and moon upon these satellites tended to produce a retrogradation in the nodes of the circles which they described in their diurnal rotation, and that the sum of all the tendencies being communicated to the whole mass of the planet, ought to produce a slow retrogradation of the equinoctial points. The effect produced by the motion of the sun he found to be 40″, and that produced by the action of the moon 10″.