If we could observe the fictitious moon, thus regularly moving in the ecliptic (say every day at midnight), and also the real moon, we might take a long series of years' observations, and sum all the excesses of it's longitude over that of the fictitious body, when there are excesses, and all the defects when there are defects. We might expect to find the one sum equal to the other ; but we are taught by the theory (which, as before seen, is exact enough to find the moon's place within a second) that the equality of these sums will not be absolutely attained in any series of years, however great, if we take the commencing point, at which X is to coincide with the fictitious body, at our own caprice. Wherever' 9 may be, there is a proper place for this fictitious moon, before or behind it, from which if we allow the former to start, the longer we go on with the series of sup posed observations, the more nearly will the excesses balance the defects ; supposing always that our series of observations stops at the end of a complete number of circuits, and not in the middle of one. This position is called the mean place of the moon, as distinguished from 9, its real place. Let us suppose it to he at v ; then if the average moon start from v, with the moon's average motion, it will at every instant of time point out what is called the mean place of the moon corresponding to the then real place. At the commencement of the present century, that is, when it was 12 o'clock at Greenwich on the night of December 31, 1800, the longitude of the average moon, or the moon's mean longitude, was (according to Burckhardt)118°17'3" ; and the mean longitude at any other time is found by adding in the proportion of for every 3d5 days, and making the necessary additive allowance for the precession of the equinoxes. [PaEcr-ssioe In the same way the node and perigee of the moon have their mean places, and, as we have seen, their mean motions. The mean longitude of the perigee. at the commencement of the century, was ; that of the exceeding node 13' 53'22 '2.
To the above must be added that these average motions, an they are called, are subject to a Might acceleration, which hardly shows itself in a century ; that of the longitude was detected by Halley from the comparison of some Chaldean eclipses with those of modern times. This acceleration would, in a century, increase the mean longitude of the moon by 11', that of the perigee by Mr , and that of the ascending node by 7".
The mean longitude being ascertained for the given time, the true longitude is found by applying a large number of corrections, as they are called, some determined from the theory of gravitation. but the larger ones, as might be supposed, detected by observation before that theory was discovered, and since confirmed by it. Into this subject it will be impossible to enter at length ; we shall therefore merely instance a few of the principal corrections for the longitude. observing that the latitude, the distance, &c., arc all determined by adding or subtract:ng a number of corrections from the results of the supposition that the moon moves uniformly in the ecliptic at her average distance from the earth.
The first correction is one which brings the motion nearer to an elliptical one, and is called the equation of the centre, It depends upon the moon's distance from her perigee, called the anomaly. The meats anomaly is the distance of the moon's mean place from that of the perigee. The mathematical exprosaion is (we give only rough constants) 6' 17' x sin (mean anomaly).
The second correction, known as the erection, and discovered by Ptolemy, M P x sin I 2 — 0) — mean anoinalyi where ( and 0 stand for the mean longitudes of the moon and sun. The variation and the annual equation (discovered by Tycho Brehd) are represented by 39'x sin 2 ( (---0) and 11' x sin (Q's mean anomaly.) Many such corrections (but those which remain, of less amount) must be added to or subtracted from the mean longitude before the true longitude can be determined.
Having thus noticed the actual motions of the moon, we proceed to the phenomena of eclipses and of the harvest-moon, as it is called. An eclipse of the moon has now lost most of its astronomical im portsnce, and can only be useful as an occasional method of finding longitude, when no better is at hand. Eclipses of the sun, observed in a particular way, may be made useful in the correction of the theory both of the sun and moon ; in this case matter is absolutely hid from view by matter, and the moment of disappearance can be distinctly perceived. But In the ease of the moon, which is eclipsed by entering the shadow of the earth, the deprivation of light is gradual ; so that it is hardly poseible to note, with astronomical exactness, the instant at which the disappearance of the planet's edge takes place.
In a lunar eclipse the first thing to be ascertained is the diameter of the earthen shadow at the distance of the moon. Suppose this shadow, that is, its section at the distance of the moon, to be repre sented by the circle whose centre is 0: it ie directly opposite to the sun, its centre is on the ecliptic, and moves in the direction of the Run's general motion, or from west to east.
which actually come into use may he represented by straight lines. Let the centre of the moon be at F: when that of the shadow is at c; and let the hourly motions of the sun (that is, of the shadow) and of the moon be o r and a o. If then we communicate to the whole system a motion equal and contrary to C F [MOTION), the shadow will be reduced to rest, and the relative motion of the moon with respect to it will remain unaltered. Take E 1:1 equal to c F, and contrary in direction ; then E L will represent the quantity and direction of the hourly motion of the moon relatively to the shadow at rest. By geometrical construction therefore N.t x and r may be ascertained, the positions of the moon 'a centre at the beginning, middle, and end of the eclipse ; and F. M, F. N, and E r, at the rate of E t. to an hour, represent the times elapsed between that of the moon being at E and the pheno mena in question. Such is the geometrical process : the one employed in practice is algebraical, and takes in several minor circumstances which it is not worth while here to notice.