duces on the lunar orbit, and the diminution of the moon's periodic time, will be kept in a continual state of fluctuation, increasing as the sun approaches its perigee, and diminishing as it recedes. And this is consonant to fact, the observed difference between a lunation in January (when the sun is nearest the earth) and in July (when it is farthest) being no less than 35 minutes.

(560.) Another very remarkable and important effect of this cause, in one of its subordinate fluctuations, (extending, however, over an immense period of time,) is what is called the secular acceleration of the moon's mean motion. It had been observed by Dr. Halley, on comparing together the records of the most ancient lunar eclipses of the Chaldean astronomers with those of modern times, that the period of the moon's revolution at present is sensibly shorter than at that remote epoch; and this result was confirmed by a further comparison of both sets of observations with those of the Arabian astronomers of the eighth and ninth centuries. It appeared from these comparisons, that the rate at which the moon's mean motion increases is about 11 seconds per century,—a quantity small in itself, but becoming considerable by its accumulation during a succession of ages. This remarkable fact, like the great equation of Jupiter and Saturn, had been long the subject of toilsome investigation to geometers. Indeed, so difficult did it appear to render any exact account of, that while some were on the point of again declaring the theory of gravity inadequate to its explanation, others were for rejecting altogether the evidence on which it rested, although quite as satisfactory as that on which most historical events are credited. It was in this dilemma that Laplace once more stepped in to rescue physical astronomy from its reproach, by pointing out the real cause of the phænomenon in question, which, when so explained, is one of the most curious and instructive in the whole range of our subject,—one which leads our speculations further into the past and future,

and points to longer vistas in the dim perspective of changes which our system has undergone and is yet to undergo, than any other which observation assisted by theory has developed.

(561.) If the solar ellipse were invariable, the alternate dilatation and contraction of the moon's orbit, explained in art. 559., would, in the course of a great many revolutions of the sun, at length effect an exact compensation in the distance and periodic time of the moon, by bringing every possible step in the sun's change of distance to correspond to every possible elongation of the moon from the sun in her orbit. But this is not, in fact, the case. The solar ellipse is kept (as we have already hinted in art. 536., and as we shall very soon explain more fully) in a continual but excessively slow state of change, by the action of the planets on the earth. Its axis, it is true, remains unaltered, but its excentricity is, and has been since the earliest ages, diminishing; and this diminution will continue (there is little reason to doubt) till the excentricity is annihilated altogether, and the earth's orbit becomes a perfect circle; after which it will again open out into an ellipse, the excentricity will again increase, attain a certain moderate amount, and then again decrease. The time required for these evolutions, though calculable, has not been calculated, further than to satisfy us that it is not to be reckoned by hundreds or by thousands of years. It is a period, in short, in which the whole history of astronomy and of the human race occupies but as it were a point, during which all its changes are to be regarded as uniform. Now, it is by this variation in the excentricity of the earth's orbit that the secular acceleration of the moon is caused. The compensation above spoken of (which, if the solar ellipse remained unaltered, would be effected in a few years or a few centuries at furthest in the mode already stated) will now, we see, be only imperfectly effected, owing to this slow shifting of one of the essential data. The steps of restoration are no longer identical with, nor equal to, those

of change. The same reasoning, in short, applies, with that by which we explained the long inequalities produced by the tangential force. The struggle up hill is not maintained on equal terms with the downward tendency. The ground is all the while slowly sliding beneath the feet of the antagonists. During the whole time that the earth's excentricity is diminishing, a preponderance is given to the action over the re-action; and it is not till that diminution shall cease, that the tables will be turned, and the process of ultimate restoration will commence. Meanwhile, a minute, outstanding, and uncompensated effect is left at each recurrence, or near recurrence, of the same configurations of the sun, the moon, and the solar and lunar perigee. These accumulate, influence the moon's periodic time and mean motion, and thus becoming repeated in every lunation, at length affect her longitude to an extent not to be overlooked.

(562.) The phænomenon of which we have now given an account is another and very striking example of the propagation of a periodic change from one part of a system to another. The planets have no direct, appreciable action on the lunar motions as referred to the earth. Their masses are too small, and their distances too great, for their difference of action on the moon and earth, ever to become sensible. Yet their effect on the earth's orbit is thus, we see, propagated through the sun to that of the moon; and what is very remarkable, the transmitted effect thus indirectly produced on the angle described by the moon round the earth is more sensible to observation than that directly produced by them on the angle described by the earth round the sun.

(563.) The dilatation and contraction of the lunar and planetary orbits, then, which arise from the action of the radial force, and which tend to affect their mean motions, are distinguishable into two kinds ; the one permanent, depending on the distribution of the attracting matter in the system, and on the order which each planet holds in it; the other periodic, and which operates in length of time its own compensation. Geo

meters have demonstrated (it is to Lagrange that we owe this most important discovery) that, besides these, there exists no third class of effects, whether arising from the radial or tangential disturbing forces, or from their combination, such as can go on for ever increasing in one direction without self-compensation; and, in particular, that the major axes of the planetary ellipses are not liable even to those slow secular changes by which the inclinations, nodes, and all the other elements of the system, are affected, and which, it is true, are periodic, but in a different sense from those long inequalities which depend on the mutual configurations of the planets inter se, Now, the periodic time of a planet in its orbit about the sun depends only on the masses of the sun and planet, and on the major axis of the orbit it describes, without regard to its degree of excentricity, or to any other element. The mean sidereal periods of the planets, therefore, such as result from an average of a sufficient number of revolutions to allow of the compensation of the last-mentioned inequalities, are unalterable by lapse of time. The length of the sidereal year, for example, if concluded at this present time from observations embracing a thousand revolutions of the earth round the sun, (such, in short, as we now possess it,) is the same with that which (if we can stretch our imagination so far) must result from a similar comparison of observations made a million of years hence.

(564.) This theorem is justly regarded as the most important, as a single result, of any which have hitherto rewarded the researches of mathematicians. We shall, therefore, endeavour to make clear to our readers, at least the principle on which its demonstration rests; and although the complete application of that principle cannot be satisfactorily made without entering into details of calculation incompatible with our objects, we shall have no difficulty in leading them up to that point where those details must be entered on, and in giving such an insight into their general nature as will render it evident what must be their result when gone through.

(565.) It is a property of elliptic motion performed under the influence of gravity, and in conformity with Kepler's laws, that if the velocity with which a planet moves at any point of its orbit be given, and also the distance of that point from the sun, the major axis of the orbit is thereby also given. It is no matter in what direction the planet may be moving at that moment. This will influence the excentricity and the position of its ellipse, but not its length. This property of elliptic motion has been demonstrated by Newton, and is one of the most obvious and elementary conclusions from his theory. Let us now consider a planet describing an indefinitely small arc of its orbit about the sun, under the joint influence of its attraction, and the disturbing power of another planet. This arc will have some certain curvature and direction, and, therefore, may be considered as an arc of a certain ellipse described about the sun as a focus, for this plain reason, that whatever be the curvature and direction of the arc in question, an ellipse may always be assigned, whose focus shall be in the sun, and which shall coincide with it throughout the whole interval (supposed indefinitely small) between its extreme points. This is a matter of pure geometry. It does not follow, however, that the ellipse thus instantaneously determined will have the same elements as that similarly determined from the arc described in either the previous or the subsequent instant. If the disturbing force did not exist, this would be the case; but, by its action, a variation of the elements from instant to instant is produced, and the ellipse so determined is in a continual state of change. Now, when the planet has reached the end of the small arc under consideration, the question whether it will in the next instant describe an arc of an ellipse having the same or a varied axis will depend, not on the new direction impressed upon it by the acting forces, for the axis, as we have seen, is independent of that direction, not on its change of distance from the sun, while describing the former arc,- for the elements of that arc