We may notice, then, five distinct species of months :-1. The average sidereal month, or complete circuit of the heavens. 2. The average lunation, common month, or interval between two conjunctions with the sun. 3. The average anomalistic month, or revolution from perigee to perigee. 4. The average tropical month, or from the vernal equinox to the vernal equinox again the equinox being in retrograde motion (ParCEssiox]). p. The average nodical month, or from a node to a node of the same kind. The quantities of these months are as follows (Bally, 'Astro% Tables and Formulto ') in mean solar days :— real moon having come above the ecliptic before the last 2 but one. In the present figure the number of folds is limited, and the last joins If we compare the lunation with the common year, we shall find that 235 Inflations make 6939.69 days, while 19 years make 6939 or 6940 days, according as there are four or five leap-years in the number. Neither is wrong by a day ; consequently in 19 years the new and full moons are restored to the same days of the year This does not abso lutely follow, either from the preceding or from the method which gave it, since neither is the coincidence exact, uor are the months exactly equal. But it will generally so happen; and this is the founda tion of the Melanie Cycle. (CAureus and 31ETON, in Moo. Div.] Again, 223 lunation make 6585'322 days, and 242 nodical revolutions make 6585.357 days, so that there is only '035 of a day, or 50 minutes, difference between the two. This period of 223 lunation is the SAROS, a celebrated Chaldean period, and contains in round numbers of days 18 years and 10 days, or 18 years and 11 days, according as there are five or four leap-years. It may be worth while to express these num bers of lunatious in terms of the other months.
Metcalfe inflations make 253.999 sidereal months, 251'852 anomalistic months, and 255.021 nodical months. • lunation make 241.029 sidereal months, 238'992 anoma listic months, and 241.999 nodical months.
The rate at which the moon moves is different in different parts of the orbit. We may speak either of the rate at which she changes longitude, latitude, or distance from the earth ; and owing to the smallness of the inclination of her path to the ecliptic, her motion in longitude is nearly the same thing as her motion in her own orbit. The quickest motion is at or near the perigee, and the slowest at or near the apogee. The moon's rate of motion follows no easily obtain able law in its changes, which are different in different months. The rate of change of latitude is greatest near the nodes, and the rate of change of distance from the earth is least at the apogee and perigee, and greatest at and about the intermediate points. We have hitherto considered the apparent path of the moon among the stars : we now pass to the real orbit in space. Her average distance from the earth is 29.982175 times the equatorial diameter of the earth, which makes about 60 radii of the earth, or 237,000 miles. But the radius of the sun's body is 1114 times the radius of the earth : so that a large sphere, which, having its centre in the earth, should contain every part of the moon's orbit, would not be a quarter of the size of the sun.
Again, the sun's distance is 23,984 radii of the earth, or nearly 400 times the moon's average distance. A good idea of the relative magni tudes of the distances may be obtained as follows :—Take a ball one inch in diameter to be the suu, and another of half an inch in diameter to be the sphere which envelopes the moon's real orbit ; place these nine feet apart, and a proper idea of the distance of the sun, compared with its size and that of the moon's orbit, will be obtained.
To form a sufficient notion of the real orbit, imagine another body, directly under the moon on the plane of the ecliptic, to accompany her in her motion. Let s s e ID represent the plane of the ecliptic, in hich the sun must be, and ALB a part of the real orbit, from an ascending to a descending node; a being a position of the moon, P is the projected body on the plane of the ecliptic ; and the motion of r will be very nearly that of L, owing to the smallness of the rise of A L B above the plane of the ecliptic. The motion of the projected body will then be of the kind of which the following diagram is an exaggeration.
Suppose the moon to set out from 1 on the left, being then in apogee, and also at a node : the projected body will then describe 1 1 1, &e., until it comes to its perigee at the first 2, which is in advance of the point opposite to the apogee. But the real moon will have come to the plane of the ecliptic before it is opposite to the first 1, so that at the first 2 the moon will be below the projected orbit. The projected body then describes 2 2 2 ... up to the next apogee 3, and so on; the the first ; in the moon's orbit the number of folds is unlimited. The real relation between the greatest and least distances is slightly variible in the different folds ; one with another it may be thus stated : 5i per cent. being added to the mean distance will give the greatest or apogean distance, and subtracted, the least or perigean distance. Taking the fiction of the moving ellipse for'the moon's orbit, its eccen tricity is '0548442.
In the article GRAVITATION will be found a sketch of the producing causes of the inequality of the lunar motions, showing that they arise from the effect of the sun's unequal attraction of the earth and moon ; were it not for which, the latter would describe an ellipse round the former. In the present article we intend only to describe the motions themselves. We have pointed out both the apparent orbit in the heavens, and the real orbit : It remains to ask, In which manner is tho real orbit described ? At a given time, how is the moon's place in the heavens to be ascertained ? Returning again to the apparent orbit, we first consider motion in longitude only ; that is we ask how to find the moon's longitude at the end of a given time. Let us suppose then that, q being the apparent place of the moon in the heavens, we draw q is on the sphere perpen dicular to the ecliptic, so that at has the same longitude as ts. To con nect this figure with the last, suppose that the moon was at L when it was projected in the heavens to q, and let r be the projection of L on the ecliptic : then r will be thrown upon it in the heavens. The average it will be that of the moon, or a circuit in 27.32166 days. If then we were to suppose a fictitious moon setting out from u, and moving with this average motion, it would never be far from the point it ; which last, from the irregularity of the real moon's motion, would be sometimes before and sometimes behind the fictitious moon.