any given point of its orbit, is reckoned in signs, degrees, minutes, and seconds. Here we mean the distance that the luminary has moved through from any given point; not the space it is short of it in coming round again, though ever so little.

The distance of the Sun or Moon from its apogee at any given time is called its mean anomaly: so that, in the apogee, the anomaly is nothing; in the perigee, it is six signs.

The motions of the Sun and Moon are observed to be continually accelerated from the apogee to the perigee, and as gradually retarded from the perigee to the apogee; being slowest of all when the mean anomaly is nothing, and swiftest of all when it is six signs.

When the luminary is in its apogee or its perigee, its place is the same as it would be, if its motion were equable in all parts of its orbit.—The supposed equable motions are called mean; the unequable are justly called the true.

The mean place of the Sun or Moon is always forwarder than the true place*, while the luminary is moving from its apogee to its perigee; and the true place is always forwarder than the mean, while the luminary is moving from its perigee to its apogee.In the former case, the anomaly is always less than six signs; and in the latter case, more.

It has been found, by a long series of observations, that the Sun goes through the ecliptic, from the vernal equinox to the same equinox again, in 365 days 5 hours 48 minutes 55 seconds: from the first star of Aries to the same star again, in 365 days 6 hours 9 minutes 24 seconds: and from his apogee to the same again, in 365 days 6 hours 14 minutes 0 seconds. The first of these is called the solar

*The point of the ecliptic in which the Sun or Moon is at any given moment of time is called the place of the Sun or Moon at that time.

S s

year

year, the second the sidereal year, and the third the anomalistic year. So that the solar year is 20 minutes 29 seconds shorter than the sidereal; and the sidereal is 4 minutes 36 seconds shorter than the anomalistic.-Hence it appears that the equinoctial point, or intersection if the ecliptic and equator at the beginning of Aries, goes backward with respect to the fixed stars, and that the Sun's apogee goes forward.

It is also observed, that the Moon goes through her orbit from any given fixed star to the same star again, in 27 days 7 hours 43 minutes 4 seconds at a mean rate: from her apogee to her apogee again, in 27 days 13 hours 18 minutes 43 seconds: and from the Sun to the Sun again, in 29 days 12 hours 44 minutes 3 seconds. This shews, that the Moon's apogee moves forward in the ecliptic, and that at a much quicker rate than the Sun's apogee does; since the Moon is 5 hours 55 minutes 39 seconds longer in revolving from her apogee to her apogee again, than from any star to the same star again.

The Moon's orbit crosses the ecliptic in two opposite points, which are called her nodes: and it is observed that she revolves sooner from any node to the same node again, than from any star to the same star again, by 2 hours 38 minutes 27 seconds; which shews that her nodes move backward, or contrary to the order of signs, in the ecliptic.

The time in which the Moon revolves from the Sun to the Sun again (or from change to change) is called a lunation; which, according to Dr. PoUND'S mean measures, would always consist of 29 days 12 hours 44 minutes 3 seconds 2 thirds 58 fourths, if the motions of the Sun and Moon were always equable*.-Hence, 12 mean lunations contain 354 days

We have thought proper to keep by Dr. Pound's length of a mean lunation, because his numbers come nearer to the times of the ancient eclipses, than Mayer's do, without allowing for the Moon's acceleration.

8 hours 48 minutes 36 seconds 35 thirds 40 fourths, which is 10 days 21 hours 11 minutes 23 seconds 24 thirds 20 fourths less than the length of a common Julian year, consisting of 365 days 6 hours; and 13 mean lunations contain 383 days 21 hours 32 minutes 39 seconds 38 thirds 38 fourths, which exceeds the length of a common Julian year, by 18 days 15 hours 32 minutes 39 seconds 38 thirds 38 fourths.

The mean time of new Moon being found for any given year and month, as suppose for March 1700, old style, if this mean new Moon falls later than the 11th day of March, then 12 mean lunations, added to the time of this mean new Moon, will give the time of the mean new Moon in March 1701, after having thrown off 365 days.-But when the mean new Moon happens to be before the 11th of March, we must add 13 mean lunations, in order to have the time of mean new Moon in March the year following; always taking care to subtract 365 days in common years, and 366 days in leap-years, from the sum of this addition.

Thus, A. D. 1700, old style, the time of mean new Moon in March, was the 8th day, at 16 hours 11 minutes 25 seconds after the noon of that day (viz. at 11 minutes 25 seconds past IV in the morning of the 9th day, according to common reckoning). To this we must add 13 mean lunations, or 383 days 21 hours 32 minutes 39 seconds 38 thirds 38 fourths, and the sum will be 392 days 13 hours 44 minutes 4 seconds 38 thirds 38 fourths; from which subtract 365 days, because the year 1701 is a common year, and there will remain 27 days 13 hours 44 minutes 4 seconds 38 thirds 38 fourths for the time of mean new Moon in March, A. D. 1701.

Carrying on this addition and subtraction till A. D. 1703, we find the time of mean new Moon in March that year, to be on the 6th day at 7 hours

21 minutes 17 seconds 49 thirds 46 fourths past noon; to which add 13 mean lunations, and the sum will be 390 days 4 hours 53 minutes 57 seconds 28 thirds 20 fourths; from which subtract 366 days, because the year 1704 is a leap-year, and there will remain 24 days 4 hours 53 minutes 57 seconds 28 thirds 20 fourths for the time of mean new Moon in March. A. D. 1704.

In this manner was the first of the following tables constructed to seconds, thirds, and fourths; and then written out to the nearest second.-The reason why we chose to begin the year with March, was to avoid the inconvenience of adding a day to the tabular time in leap-years after February, or subtracting a day therefrom in January and February in those years; to which all tables of this kind are subject, which begin the year with January, in calculating the times of new or full Moons.

The mean anomalies of the Sun and Moon, and the Sun's mean motion from the ascending node of the Moon's orbit, are set down in Table III. from one to 13 mean lunations.-These numbers, for 13 lunations, being added to the radical anomalies of the Sun and Moon, and to the Sun's mean distance from the ascending node, at the time of mean new Moon in March 1700, (Table I.) will give their mean anomalies, and the Sun's mean distance from the node, at the time of mean new Moon in March 1701; and being added for 12 lunations to those for 1701, give them for the time of mean new Moon in March 1702. And so on, as far as you please to continue the table (which is here carried on to the year 1800), always throwing off 12 signs when their sum exceeds 12, and setting down the remainder as the proper quantity.

If the numbers belonging to A. D. 1700 (in Table I.) be subtracted from those belonging to 1800, we shall have their whole differences in 100 complete Julian years; which accordingly we find to be

4 days 8 hours 10 minutes 52 seconds 15 thirds 40 fourths, with respect to the time of mean new Moon.-These being added together 60 times, (always taking care to throw off a whole lunation when the days exceed 294) making up 60 centuries, or 6000 years, as in Table VI. which was carried on to seconds, thirds, and fourths; and then written out to the nearest second. In the same manner were the respective anomalies and the Sun's distance from the node found, for these centurial years; and then (for want of room) written out only to the nearest minute, which is sufficient in whole centuries.-By means of these two tables, we may find the time of any mean new Moon in March, together with the anomalies of the Sun and Moon, and the Sun's distance from the node, at these times, within the limits of 6000 years, either before or after any giv en year in the 18th century; and the mean time of any new or full Moon in any given month after March, by means of the third and fourth tables, within the same limits, as shewn in the precepts for calculation.

Thus it would be a very easy matter to calculate the time of any new or full Moon, if the Sun and Moon moved equably in all parts of their orbits.But we have already shewn that their places are never the same as they would be by equable motions, except when they are in apogee or perigee; which is when their mean anomalies are either nothing, or six signs and that their mean places are always forwarder than their true places, while the anomaly is less than six signs; and their true places are forwarder than the mean, while the anomaly is more.

Hence it is evident, that while the Sun's anomaly is less than six signs, the Moon will overtake him, or be opposite to him, sooner than she could if his motion were equable; and later while his anomaly is more than six signs. The greatest difference that can possibly happen between the mean and true time