This was a new proof of the reality of the principle of gravitation, which, how ever, was rendered less conclusive by the consideration that the exact quantity of the motion of the apsides observed, did not come out from the diminution of the moon's as above assigned. There was a sort of cloud, therefore, which hung over this point of the lunar theory, to dissipate which, required higher improvements in the calculus than it was given to the inventor himself to accomplish. It was not so with respect to another motion to which the plane of the lunar orbit is subject, a pheno menon which had been long known in consequence of its influence on the eclipses of the sun and moon. This was the retrogradation of the line of nodes, amounting to 3' 10" every day. Newton showed that the second of the forces into which the solar action is moved being exerted, not in the plane of the moon's orbit, but in that of the ecliptic, inclined to the former at an angle somewhat greater than five degrees, its effect must be to draw down the moon to the plane of the ecliptic sooner than it would otherwise arrive at it; in consequence of which, the intersection of the two planes would approach, as it were, toward the moon, or move in a direction opposite to that of the moon's motion, or become retrograde. From the quantity of the solar force, and the inclination of the moon's orbit, Newton determined the mean quantity of this retro gradation, as well as the irregularities to which it is subject, and found both to agree very accurately with observation.
Another of the lunar inequalities,—that discovered by Tycho, and called by him the Variation, which consists in the alternate acceleration and retardation of the moon in each quarter of her revolution, was accurately determined from theory, such as it is found by observation ; and the same is true as to the annual equation, which had been long confounded with the equation of time. With regird to the other inequalities, it does not appear that Newton attempted an exact determination of 'them, but satisfied himself with this general truth, that the principle of the sun's disturbing force led to the supposition of inequalities of the same kind with those actually observed, though whether of the same exact quantity it must be difficult to determine. It was reserved, indeed, for a more perfect state of the calculus to explain the whole of those irregu larities, and to deduce their precise value from the theory of gravity. Theory has led to the knowledge of many inequalities, which observation alone would have been un able to discover.
While Newton was thus so successfully occupied in tracing the action of gravity among those distant bodies, he did not, it may be supposed, neglect the consideration of its effects on the objects which are nearer us, and particularly on the Figure of the Earth. We have seen that, even with the limited views and imperfect information which Copernicus possessed on this subject, he ascribed the round figure of the earth and of the planets to the force of gravity residing in the particles of these bodies. Newton,
on the other hand, perceiv6d that, in the earth, another force was combined with gra• vity, and that the figure resulting from that combination.could not be exactly spheri cal. The diurnal revolution of the earth, he knew, must produce a centrifugal force, which would act most powerfully on the parts most distant from the axis. The amount of this centrifugal force is greatest at the equator, and being measured by the momentary recess of any point from the tangent, which was known from the earth's rotation, it could be compared with the force of gravity at the same place, measured in like manner by the descent of a heavy body in the first moment of its fall. When Newton made this comparison, he found that the centrifugal force at the equator is the 289th part of gra vity, diminishing continually as the cosine of the latitude, on going from thence to ward the poles, where it ceases altogether. From the combination of this force, though small, with the force of gravity, it follows, that the line in which bodies actually gravitate, or the plumb-line, cannot tend exactly to the earth's centre, and that a true. horizontal line, such as is drawn by levelling, if continued from either pole, in the plane of a meridian all round the earth, would not be a circle but an ellipse, having its greatest axis in the plane of the equator, and its least in the direction of the axis of the earth's rotation. Now, the surface of the ocean itself actually traces this level as it extends from the equator to either pole. The terraqueous mass which we call the globe, must therefore be what geometers call an oblate spheroid, or a solid generated by the revolution of the elliptic meridian about its shorter axis.
In order to determine the proportion of the axes of this spheroid, a problem, it will readily be believed, of no ordinary difficulty, Newton conceived, that if the waters at the pole and at the equator were to communicate by a canal through the interior of the earth, one branch reaching from the pole to the centre and the other at right angles to it, from the centre to the circumference of the equator, the water in this canal must be in equilibrio, or the weight of fluid in the one branch just equal to that in the other. Including, then, the consideration of the centrifugal force which acted on one of the branches but not on the other, and considering, too, that the figure of the mass being no longer a sphere, the attraction must not be supposed to be directed to the centre, but must be considered as the result of the action of all the particles of the spheroid on the fluid in the canals ; by a very subtle process of reasoning, Newton found that the longer the two canals must be to the shorter as 230 to 229. This, therefore, is the ratio of the radius of the equator to the polar semiaxis, their difference amounting, according to the dimensions then assigned to the earth, to about 17T, English miles.