ed from the globe, but preserving its motion of rotation. Under these circumstances, the nodes of this ring would have a retrograde motion along the ecliptic, which would be to the similar motion of the nodes of the lunar orbit, in the ratio of a sidereal day to the sidereal revolution of the moon. But as this ring is in truth attached to the sphere beneath, it must communicate a considerable part of its motion to the rest of the earth, and the effect being thus divided between the two masses, of which the latter is much the greatest, will be much diminished. Newton inferred that it was diminished by a factor amounting to 3ds of the square of a semicircle whose radius is unity. The true salution is, that the precession of the earth is equal to that of the ring multiplied by the ratio of the mass of the ring to that of the earth, and by the factor.

The precession obtained by considering the whole excess of the terrestrial spheroid over a sphere whose diameter is equal to the equatorial axis, as accumulated in a ring at the equator, is obviously too great. Newton finally discovered that this was to be corrected by reducing the precession thus obtained to ths, and this is the true amount.

The sun, however, does not act constantly in the plane of the equator, but in that of the ecliptic; a correction is therefore required on account of this circumstance, and this Newton applied, by multiplying the last result by the square of the co-sine of the obliquity, in which he was also right.

Newton, therefore, except in the error of his first reduction, had reached the true theory of so much of the precession as is due to the action of the sun.

The moon also exerts a similar action, and the precession is due to their united effects. Newton finds the lunar precession by multiplying the solar precession by the ratio of the attractions of the two bodies. This he concluded, from observations of the tides, to be 1 to 63ds, and reduced it, from other considerations, to 14.4815. But this is far too great, as has been more recently ascertained, the true multiplier being no more than 24d.

The calculation is affected by the nature of the terrestrial spheroid, which increases in density from the surface to the centre. Hence the results for a homogeneous sphere are untrue. Newton was aware of the probability of this being the nature of earth, but mistook the manner in which it ought to influence the precession.

The precession of the equinoxes is effected by an irregularity. This grows out of the inclination of the orbit of the moon to the ecliptic, and is called the nutation of the earth's axis. This irregularity may be represented, by supposing, that while the mean position of the earth's axis revolves around the pole of the ecliptic, the real position of the pole describes a small ellipse

around the mean place, in a time that is equal to an entire revolution of the lunar nodes.

The phenomena of precession and nutation, throw light upon the constitution of the terrestrial spheroid, both as respects its comparative density and its oblateness. The latter can be shown not to exceed 17th part, and the calculations of Bouvard, Burg, and Burkhardt, have giventh for this amount.

The law of nutation was first determined by observation alone; it had been remarked by Newton, but was only fully exhibited by the observations of Bradley. D'Alembert investigated the physical cause, and included it among the general phenomena of gravitation. Euler also applied himself to this subject, and simplified the analysis.

But these investigations still left various circumstances included in the problem unexamined, and the full solution of these was reserved for Laplace.

The precession is farther changed by the fluidity of the sea, by its currents, and by those of the atmosphere. It is also influenced by the effect the flattening of the earth produces upon the obliquity of the ecliptic, and the length of the year. If the sun and moon were the only bodies that acted upon the earth, the obliquity of the ecliptic would be constant; but each of the planets produces an effect similar to the precession, which causes a retrograde motion in the intersection of the plane of the ecliptic with its own orbit. The joint effect of these, is to produce a variation in the inclination of the earth's equator to the ecliptic. This is an inequality which has a very long period; the obliquity of the ecliptic has been diminishing since the date of the earliest observations; it is now doing so at the rate of 52" in a century. After a course of ages it will again increase, and the plane oscillate on each side of a mean position. The secular variation of the ecliptic is now about its maximum, and after the 22d century of our era, it will again decrease. The total change of the obliquity cannot exceed five and a half degrees. The retrograde motion of the intersection of the ecliptic with the orbits of the planets, gives a small direct motion to the equinoctial points, which must be deducted from that produced in a retrograde direction, by the sun and moon.

The variation in the obliquity of the ecliptic, causes, in the axis of the earth, motions similar to nutation; these affect the action both of the sun and moon, but are extremely slow.

All these causes, affecting the rate of the precession of the equinoxes, cause a variation in the length of the tropical year, which has grown shorter since the observations of Hipparchus; this, like all the other variations, is periodic. It is a most important question, whether the secular inequalities in the motion of the earth and moon are capable of producing a permanent change

in the position of the earth's axis of rotation. Laplace has ascertained that such a change is wholly insensible. The lunar orbit is subject to a nutation corresponding to that of the earth's axis, whose coefficient, like that of the earth, depends upon the oblateness of the terrestrial spheroid.

The precession of the equinoxes causing a falling back of the equinoctial points, at the rate of about 50" per annum, the latter will perform a complete revolution in the ecliptic in a period of upwards of 25,000 years. A knowledge of this fact has thrown considerable light on questions of chronology, and might, had the ancient observations been more precise, have settled many disputed eras in history.

The ancients did not fail to remark, that the moon always presents the same face to the earth, and so far from being surprised at it, they considered it as a necessary consequence of the revolution of one body around another. The view of the planetary bodies by the telescope, shows that this is far from a general rule; nor is it a necessary consequence. This false impression threw difficulties in the way of Copernicus, when he attempted to explain the motion of the earth around the sun, but Kepler remarked, that it was sufficient to represent all these phenomena, that the axis of the earth should continue parallel to itself during its whole annual revolution. But when this remark of Kepler is applied to the case of the moon, the motion of this body becomes very difficult to explain, inasmuch as the time of the rotation must correspond exactly with that of the revolution around the earth, of which it was impossible, at that period, even to suspect the cause. Still, however, although it be true, in general terms, that the moon always presents the same face to us, there is a slight inequality which is called libration, in consequence of which, the hemisphere that is presented to us, varies continually. Galileo ascertained that there was a variation in this respect, which depended upon the parallax in altitude of the moon, and on her latitude. Riccioli discovered a libration in longitude, but, with Hevelius, failed in giving a full explanation of its cause. Newton finally seized upon the true reason, which is this; that the moon moves uniformly around her axis, while her motion in her orbit is irregular. He, however, supposed that the axis of the moon's rotation was perpendicular to the ecliptic. D. Cassini determined the true position of this axis, and thus completed the explanation of the appearances. This result of observation has since been confirmed by the calculations of Mayer, Lalande, Bouvard, and Arago, and shows that the inclination of the lunar equator to the ecliptic is constant, and that. its nodes coincide with those of the moon's orbit. This subject does not possess sufficient importance to induce

us to follow Laplace through the history of the investigation of the physical causes of the libration of the moon. They also are intimately connected with the inequalities of the lunar motion, which we shall pursue at some length in their proper place.

The third chapter of the XIV th book treats of Saturn's rings. The whole theory of their motion and equilibrium is due to Laplace himself.

A ring of a fluid substance, and such, we must suppose the rings of Saturn to have been originally, will maintain itself in equilibrio around a planet, by virtue of the mutual attraction of its particles combined with a rotary motion; provided the generating figure of the ring be an ellipse, whose greater axis is directed towards the planet. The duration of the rotation must be the same as that of a satellite, whose distance from the centre of the planet is the same as the distance of the centre of the figure that generates the ring. Laplace concluded, from his theory, that this revolution must occupy about ten hours, and this was afterwards ascertained to be the fact by the observations of Herschell.

Laplace next remarked, that if the ring were perfectly similar in all its parts, the centres of the planet and ring would repulse each other mutually the moment they ceased to coincide, which must necessarily take place, in consequence of the influence of the other bodies of the system. The centre of the ring would, in this case, describe a curve convex to the centre of the planet, until the ring touched the surface of the planet, when the two bodies would unite. It is therefore necessary, for the stability of the equilibrium of a ring, that its sections should be dissimilar, and that its centre of gravity should not coincide with its centre of magnitude.

Saturn has two rings, which being at different distances from the planet, ought to assume different motions of precession under the action of the sun's attraction. Did nothing oppose these motions, the relative position of the planes of the two rings must be constantly changing. Observation, however, shows that there is no such inequality; there must therefore be some cause which maintains these rings nearly in the same plane, although the sun act continually to cause them to deviate from it. Laplace finds this cause in the flattening of the spheroid of Saturn, produced by its rapid motion of rotation; and here again the result of the theory of gravitation was in advance of observation. His analysis shows, that if the rings be but little inclined to the equator of the planet, the flattening at his poles will tend to retain them in the position whence the attraction of the sun tends to withdraw them. While the rings turn around their own centres of gravity, the latter turn around the centre of the planet, and hence will arise continual changes in the relative po

sitions of the planes of the two rings; but these are periodic, and confined within small limits. Observation proves that such changes do actually take place.

The XVth book, treats of the motion of the planets, and comets. Kepler, after having fruitlessly attempted to reconcile the motion of the planet Mars, to the hypothesis of a circular orbit, as assumed by Copernicus, discovered at last that it moved in an ellipse, of which the sun occupies one of the foci, around which the radius vector of the planet describes areas proportioned to the times. Kepler, however, failed in pointing out the probable cause, and Bovelli was the first to imagine that the planetary motions arose from the action of an original force of projection, combined with a tendency towards the centre of the sun. The laws of the central forces which govern bodies moving in circles, were demonstrated by Huygens; and Newton, Halley, Wren, and Hooke, found that these laws, if applied to the planetary bodies, showed that another law of Kepler, namely, that the squares of the times of the planets' revolutions are proportioned to the cubes of their mean distances, was consistent with a tendency of the planets to the sun, varying in the inverse ratio of their distances from that body.

These investigations were confined to circular orbits, and the discovery of Kepler showed that such did not represent the true path of the planets. It might therefore be well doubted, whether, when a planet is transported to the orbit of another planet, the former would be affected by the attraction of the sun in the same degree as the latter. To solve this doubt, it became necessary that it should be demonstrated, that the same planet, at its various distances from the sun, is always affected by an attraction varying inversely as the squares of the distances. It is in the solution of this question, that we find the origin of physical astronomy, or the mechanics of the heavenly bodies, and it was attempted in vain, by Halley, Wren, and Hooke.

Newton persevered, and performed what his former associates had failed to do. He showed, first, that the law of the areas described by the radius vector of a planet, necessarily produces a tendency in that planet towards the centre of the sun. He next concluded that the ellipticity of the planets' orbits requires this tendency to be in the inverse ratio of the squares of the distances; he finally demonstrated, from the law that the squares of the periodic times are proportioned to the cubes of the greater axes, that the tendency towards the sun varies from one planet to another, only in consequence of their different distances. The three laws determined by Kepler, from a collection of observations, were therefore shown by Newton to be immediate consequences of an attractive force residing in the centre of the sun, and varying in the inverse ratio of the squares of the distances.