other signs and degrees of anomaly, the difference is gradually less, and vanishes when the anomaly is either 0, or 6 signs. The Sun is in his apogee on the 30th. of June, and in his perigee on the 30th. of December, in the present age; so that he is nearer the earth in our winter than in our summer. The proportional difference of the Sun's apparent diameter at these times, is as 983 to 1017. mer. The Moon's orbit is dilated in winter, and contracted in sumThe greatest difference is found to be 22 minuets 39 seconds; the lunations increasing gradually in length while the Sun is moving from his apogee, and decreasing in length while he is moving from his perigee to his apogee. On this account the Moon will be later in coming to her conjunction with the Sun, or being in opposition to him,* from December till. June, and sooner from June to December, than if her orbit had continued of the same size all the year round. As both these differences depend on the Sun's anomaly, they may be fitly put together into one table and called The annual, or first equation for reducing the mean to the true syzygy.† (See Table VII.) This equational difference is to be subtracted from the time. of mean New or Full Moon when the Sun's anomaly is less than 6 signs, and added when the anomaly is more. At the greatest, it is 4 hours 10 minuets 57 seconds, Viz. 3' hours 48 minuets 28 seconds, on account of the Sun's unequal motion, and 22 minuets 29 seconus, on account of the dilation. of the Moon's orbit. This compound equation would be snfficient for reducing the mean time of New or Full Moon to the true time, if the Moon's orbit were of a circular form, and her motion exactly equable in it. But the Moon's orbit is more elliptical than the Sun's, and her motion in it is so much the more unequal. The difference is so great, that she is sometimes in conjunction with the Sun, or in opposition to him, sooner by 9 hours 47 minuets 54 seconds, than she would be if her motion were equable; and at other times as much later. The former happens when her mean anomaly is 9 signs 4 degrees, and the latter when it is signs 26 degrees. See Table IX. At different distances of the Sun from the Moon's apogee, the figure of the Moon's orbit becomes different. It is longest of all, or most excentrick, when the Sun is in *The term conjunction, when it respects the relation of the Moon to the Sun, signifies New Moon, or Change; and opposition is used to signify the place of the Moon at her full. The word syzygy siguifies both the conjunction and opposi tion of the Sun and Moon. the same sign and degree either with the Moon's apogee or perigee; shortest of all, or least excentrick, when the Sun's distance from the Moon's apogee is either 3 signs or 9 signs; and at a mean state when the distance is either 1 sign 15 degrees, 4 signs 15 degrees, 7 signs 15 degrees, or 10 signs 15 degrees. When the Moon's orbit is at its greatest excentricity, her apogeal distance from the Earth's centre is to her perigeal distance from it, as 1067 is to 933; when least excentrick, as 1043 is to 457, and when at the mean state, as 1055 is to 945. But the Sun's distance from the Moon's apogee is equal to the quantity of the Moon's mean anomaly at the time of New Moon, and by the addition of 6 signs, it becomes equal in quantity to the Moon's mean anomaly at the time of Full Moon. Therefore, a table may be constructed as to answer all the various irregualities depending on the different excentricities of the Moon's orbit in the syzygies; and called The second equation for reducing the mean to the true syzygy, (See Table IX.) and the Moon's anomaly, when equated by Table VIII. may be made the proper argument for taking out this second equation of time, which must be added to the former equated time, when the Moon's anomaly is less than 6 signs, and subtracted when the anomaly is more. There are several other inequalities in the Moon's motion, which sometimes bring the true syzygy a litle sooner, and at other times keep it back a little later than it would otherwise be; but they are so small, that they may be all omitted except two; the former of which (See Table X.) depends on the difference between the anomalies of the Sun and Moon in the syzygies and, the latter, (see Table XI.) depends on the Sun's distance from the Moon's nodes at these times. The greatest difference arising from the former, is 4 minuets 58 seconds; and from the latter, I minuet 34 seconds. Besides the tables already mentioned, there are various others annexed in the following, to facilitate the labour of astronomical calculations, and will be treated of in their proper place. |