of new or full Moon, on account of the inequality of the Sun's motion, is three hours 48 minutes 28 seconds and that is, when the Sun's anomaly is either 3 signs 1 degree, or 8 signs 29 degrees; sooner in the first case, and later in the last. In all other signs and degrees of anomaly, the difference is gradually less, and vanishes when the anomaly is either nothing or six 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 distance, deduced from the difference of the Sun's apparent diameter at these times, is as 983 to 1017.

The Moon's orbit is dilated in winter, and contracted in summer; therefore the lunations are long. er in winter than in summer. The greatest difference is found to be 22 minutes 29 seconds; the lu nations increasing gradually in length while the Sun is moving from his apogee to his perigee, and decreasing in length while he is moving from his perigee to his apogee.-On this account the Moon will be later every time 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 of the mean to the true syzygy (see Table VII.) This equational difference is to be subtracted from the time of the mean syzygy when the Sun's anomaly is less than six signs, and added when the anomaly is more. At the greatest, it is 4 hours 10 minutes 57 seconds, viz. 3 hours 48 minutes 28 seconds,

The word syzygy signifies both the conjunction and opposition of the Sun and Moon.

on account of the Sun's unequal motion, and 22 minutes 29 seconds, on account of the dilatation of the Moon's orbit.

This compound equation would be sufficient 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 quite equable in it.-But the Moon's orbit is more elliptical than the Sun's, and her motion in it 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 minutes 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 2 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 eccentric, when the Sun is in the same sign and degree either with the Moon's apogee or perigee; shortest of all, or least eccentric, when the Sun's distance from the Moon's apogee is either three signs or nine 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 eccentricity, her apogeal distance from the Earth's centre is to her perigeal distance from it, as 1067 is to 933; when least eccentric, as 1043 is to 957; 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 six signs, it becomes equal in quantity to the Moon's mean anomaly at the time of full Moon.Therefore, a table may be constructed so as to answer all the various inequalities depending on the different eccentricities of the Moon's orbit in the syzygies; and called The second equation of 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 six signs, and subtracted when the anomaly is more.

There are several other inequalities in the Moon's motion, which sometimes bring on the true syzygy a little 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 minutes 58 seconds; and from the latter, 1 minute 34 seconds.

Having described the phenomena arising from the inequalities of the solar and lunar motions, we shall now shew the reasons of these inequalities.

In all calculations relating to the Sun and Moon, we consider the Sun as a moving body, and the Earth as a body at rest; since all the appearances are the same, whether it be the Sun or the Earth that moves. But the truth is, that the Sun is at rest, and the Earth moves round him once a year, in the plane of the ecliptic. Therefore, whatever sign and degree of the ecliptic the Earth is in, at any given time, the Sun will then appear to be in the opposite sign and degree.

The nearer that any body is to the Sun, the more it is attracted by him; and this attraction increases as the square of the distance diminishes; and vice

versá.

The Earth's annual orbit is elliptical, and the Sun is placed in one of its focuses. The remotest point

of the Earth's orbit from the Sun is called The earth's aphelion; and the nearest point of the Earth's orbit to the Sun, is called The Earth's perihelion.When the Earth is in its aphelion, the Sun appears to be in its apogee; and when the Earth is in its perihelion, the Sun appears to be in its perigee.

As the Earth moves from its aphelion to its perihelion, it is constantly more and more attracted by the Sun; and this attraction, by conspiring in some degree with the Earth's motion, must necessarily accelerate it. But as the Earth moves from its perihelion to its aphelion, it is continually less and less attracted by the Sun; and as this attraction acts then just as much against the Earth's motion, as it acted for it in the other half of the orbit, it retards the motion in the like degree.The faster the Earth moves, the faster will the Sun appear to move; the slower the Earth moves, the slower is the Sun's apparent motion.

The Moon's orbit is also elliptical, and the Earth keeps constantly in one of its focuses.-The Earth's attraction has the same kind of influence on the Moon's motion, as the Sun's attraction has on the motion of the Earth: and therefore, the Moon's motion must be continually accelerated while she is passing from her apogee to her perigee; and as gradually retarded in moving from her perigee to her apogee.

At the time of new Moon, the Moon is nearer the Sun than the Earth is at that time, by the whole semidiameter of the Moon's orbit; which, at a mean state, is 240,000 miles; and at the full, she is as much farther from the Sun than the Earth then is. Consequently, the Sun attracts the Moon more than it attracts the Earth in the former case, and less in the latter. The difference is greatest when the Earth is nearest the Sun, and least when it is farthest from him. The obvious result of this is, that as the Earth is nearest to the Sun in winter,

Tt

and farthest from him in summer, the Moon's or bit must be dilated in winter, and contracted in

summer.

These are the principal causes of the difference of time, that generally happens between the mean and true times of conjunction or opposition of the Sun and Moon. As to the other two differences, viz. those which depend on the difference between the anomalies of the Sun and Moon, and upon the Sun's distance from the lunar nodes, in the syzygies, they are owing to the different degrees of attraction of the Sun and Earth upon the Moon, at greater or less distances, according to their respective anomalies, and to the position of the Moon's nodes with respect to the Sun.

If ever it should happen, that the anomalies of both the Sun and Moon were either nothing or six signs, at the mean time of new or full Moon, and the Sun should then be in conjunction with either of the Moon's nodes, all the above-mentioned equations would vanish, and the mean and true time of the syzygy would coincide. But if ever this circumstance did happen, we cannot expect the like again in many ages afterward.

Every 49th lunation (or course of the Moon from change to change) returns very nearly to the same time of the day as before. For, in 49 mean lunations there are 1446 days 23 hours 58 minutes 29 seconds 25 thirds, which wants but 1 minute 30 seconds 34 thirds of 1477 days.

In 2953059085108 days, there are 100000000000 mean lunations exactly: and this is the smallest number of natural days in which any exact num ber of mean lunations will be completed.