either to the sum or to the difference of the effects of the two luminaries. At the time of new and full moon the two spheroids will have their axes coincident, and the height of the tide, which will then be a _spring_ one, will be equal to the sum of the elevations produced in each spheroid considered separately, while at the first and third quarters the axes of the spheroids will be at right angles to each other, and the height of the tide, which will then be a _neap_ one, will be equal to the difference of the elevations produced in each separate spheroid. By comparing the spring and neap tides, Newton found that the force with which the sun acted upon the waters of the earth was to that with which the sun acted upon them as 4.48 to 1;—that the force of the moon produced a tide of 8.63 feet;—that of the sun one of 1.93 feet;—and both of them combined, one of 10½ French feet,—a result which in the open sea does not deviate much from observation. Having thus ascertained the force of the moon on the waters of our globe, he found that the quantity of matter in the moon was to that in the earth as 1 to 40, and the density of the moon to that of the earth as 11 to 9. The motions of the moon, so much within the reach of our own observation, presented a fine field for the application of the theory of universal gravitation. The irregularities exhibited in the lunar motions had been known in the time of Hipparchus and Ptolemy. Tycho had discovered the great inequality called the _variation_, amounting to 37′, and depending on the alternate acceleration and retardation of the moon in every quarter of a revolution, and he had also ascertained the existence of the annual equation. Of these two inequalities Newton gave a most satisfactory explanation. The action of the sun upon the moon may be always resolved into two, one acting in the direction of the line joining the moon and earth, and consequently tending to increase or diminish the moon’s gravity to the earth, and the other in a direction at right angles to this, and consequently tending to accelerate or retard the motion in her orbit. Now, it was found by Newton that this last force was reduced to nothing, or vanished at the syzigies or quadratures, so that at these four points the moon described areas proportional to the times. The instant, however, that the moon quits these positions, the force under consideration, which we may call the tangential force, begins, and it reaches its maximum in the four octants. The force, therefore, compounded of these two elements of the solar force, or the diagonal of the parallelogram which they form, is no longer directed to the earth’s centre, but deviates from it at a maximum about 30 minutes, and therefore affects the angular motion of the moon, the motion being accelerated in passing from the quadratures to the syzigies, and retarded in passing from the syzigies to the quadratures. Hence the velocity is in its mean