The annuity which $ 1, or £1 will purchase for any number of years to come, from 1 to 40.

Yrs. 4 per cent. | 5 per cent. | 6 per ceni. | Yrs.

ར་་་་་་་འ་བ་་་་་ ར་་་་་་་་་་་་་་་་་་་་ས

བ་པར་འར་འའ་

བ་་་་་་བ་་ར་ར་་ན་་་

PRACTICAL ASTRONOMY.

Containing a number of Astronomical Tables, and an easy method of calculating the times of NEW AND FULL MOONs, and ECLIPSES by them.

OF ASTRONOMICAL TABLES AND THEIR CONSTRUCTION.

IN constructing tables for computing, at any given instant, the places of the Sun, Moon, and Planets, the first step is to determine, from a series of accurate observations, the time in which those bodies describe a space of 360 degrees, or perform a complete revolution round the Sun, or primary Planet.

When this important element is exactly ascertained, we can easily find, by simple proportion, the space which any Planet describes in any number of years, months, days, hours, minutes, and seconds, upon the supposition that it moves uniformly, or describes equal spaces in equal time, in the circumference of a circle.

But as it has been found from a long series of observations, that all the bodies of the solar system move in eliptical orbits round the Sun, or their primary Planet, placed in one of the fo ci, we must next determine the form of their orbits, or the nature of the ellipse.* which they describe.

The diameters of the Sun and Moon therefore, subtend different angles at different times, as they are nearer, or more remote from the observer's eye. This proves that the Sun and Moon are constantly changing their distances from the Earth; and they are once at their greatest, and once at their least distance from it, in little more than a complete revolution.

The gradual differences of these angles are not what they would be, if the luminaries moved in circular orbits, the Earth

An ellipse is a curvilinear figure of an oblate oval form, having two centres called Foci, or Foucuses: The Sun is in the focus of the Earths orbit, and the Earth is in or near that of 'the Moon's orbit.

being supposed to be placed at some distance from the centre of the orbit, and the centre of the Earth to be in the lower focus. of each orbit.

The fartherest point of each orbit from the Earth's centre is called the APOGEE, and the nearest point is called the PERIGEE. These points are diametrically opposite to each other.

Astronomers divide each orbit into 12 equal parts called SIGNS; each sign into 30 equal parts cailed DEGREES; each degree into 60 equal parts called MINUETS; each minuet into 60 equal parts called SECONDS.

The distance of the Sun or Moon from any given point of its orbit, is reckoned in signs, degrees, minuets, and seconds. Herein is meant the distance that the luminary has moved through from any given point: and not the space it is short of it in coming round again, though it be ever so little.

The distance of the Sun or Moon from its apogee at any given time, is called its MEAN ANOMALY: therefore when the body is in its apogee, its anomaly is 0, and in its perigee, it is signs.

The motion of the Sun and Moon are observed to be continually accellerated from their apogee to their perigee, and as gradually retarded from their perigee to their apogee; moving, with the greatest velocity when the anomaly is 0, and with the least, when the anomaly is 6 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 motions 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 6 signs; and in the latter case, more.

The Moon's orbit crosses the ecliptick in two opposite points, which are called her Nodes; and the time in which she revolves, from the Sun to the Sun again, (or from change to change) is called a LUNATION, and would always consist of 29 days, 12 hours, 44 minuets, 3 seconds, 2 thirds, and 58 fourths, if the motions of the Sun and Moon were always equable. Hence, 12 mean lunations contain 354 days, 8 hours, 48 minuets, 36 seconds, 35 thirds, and 40 fourths, which is 10 days, 21 hours, 11 minuets, 23 seconds, 24 thirds, and 20 fourths, less than the length of a common JULIAN YEAR, consisting of 365 days 6 hours; and 13.

*The point of the ecliptick in which the Sun or Moon is at any moment, of time is called the PLACE of the Sun or Moon at that time..

mean lunations contain 383 days, 21 hours, 32 minuets, 39 seconds, 38 thirds, and 38 fourths, which exceeds the length of a common JULIAN YEAR, by 18 days, 15 hours, 32 minuets, 39 seconds, 38 thirds, and 38 fourths.

The mean time of New Moon being found for any given year and month, as suppose for March 1850, New Style, if this mean New Moon happens 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 1851, after abating 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; observing always to subtract 365 days in common years, and 366 days in leap-years, from the sum of this addition.

Thus, A. D. 1850, New Style, the time of mean New Moon in March, was the 12th day, at 22 hours and 11 seconds past the noon of that day (viz. at 11 seconds past X in the morning of the 13th day, according to common reckoning.) To this we must add 12 mean lunations, or, 354 days, 8 hours, 48 minuets, 36 seconds, 35 thirds, and 40 fourths, and the sum will be 367 days, 6 hours, 48 minuets, 47 seconds, 35 thirds and 40 fourths; from which subtract 365 days, because the year 1851 is a common year, and there will remain 2 days, 6 hours, 48 minuets, 47 seconds, 35 thirds and 40 fourths, for the time of mean New Moon in March, A. D. 1851.

Now to find the mean time of New Moon in March A. D. 1852, we must add 13 mean lunations to the mean time of New Moon in the next preceeding year, (because it happened before the 11th day) and the sum will be 386 days, 4 hours 21 minuets 27 seconds 13 thirds and 18 fourths; from which subtract 366 days, because the year 1852 is a leap-year, and there will remain 20 days 4 hours 21 minuets 27 seconds 13 thirds and 18 fourths, to be set down for the time of mean New Moon, in March, A. D. 1852.

In this manner was the first two of the following tables constructed to seconds, thirds, fourths; and then written out to the nearest second. The reason why Astronomers choose to begin the year with March, is 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 distance from the ascending node of the Moon's orbit, are set down in Table III, from one to 13 mean lunations.

The numbers, for 12 lunations, being added to the radical anomalies of the Sun and Moon, and to the Sun's mean distance

from the Moon's ascending node, at the mean time of New Moon in March 1850, (Table II.) will give their mean anomalies, and the Sun's mean distance from the node, at the time of New Moon in March 1851; and being added for 13 lunations to those for 1851, will give them for the time of mean New Moon in March 1852. And so on as far as you please to continue the table, (which is here carried on from 1752, to the year 1900,) always rejecting 12 signs when their sum exceeds 12, and setting down the remainder as the proper quantity.

If the number of years 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 minuets 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 co 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 centural years; and then (for want of room) written out only to the nearest minuet, 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 given year in the 18th. century; and the mean time of any New or Full Moon in any month of the year after March, by means of the third and fourth tables, within the same limits, as shown 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 observed that their places are never the same as they would be by equable motions, except when their mean anomalies are either 0, or 6 signs; and that their mean places are always forwarder than than their true places, while their anomalies are less than 6 signs; and their true places are forwarder than the mean, while the anomaly is more.

Hence it is evident, that while the Sun's anonaly is less than 6 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 6 signs. The greatest difference that can possibly happen between the mean and true time of New or Full Moon, on account of the inequality of the Sun's motion, is 3 hours 48 minuets 28 seconds; and this 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