When the nodes have moved so far that one is between or nearly between the earth and sun, it is evident that the moon in moving round the earth from south to east, and then to north and west, &c. is in or near one node at the new moon, and in or near the other at the full moon; and in the highest part of her orbit when she is east of the earth at the first quarter, and in the lowest when she is west of the earth at the last quarter; and in this case, there will be an eclipse both at the New and Full Moon, for the moon will then be nearly on a level, or in the same plane with the earth and sun. At new moon, the sun and moon, as seen from the earth, always have the same longitude and are always at the same distance from the moon's node. If at new moon, the sun and moon are not more that 16o from one of the nodes, there will be an eclipse of the sun.

At the full moon, the moon is always at the same distance from one node that the sun is from the other, because the sun and moon are in opposition. If at the full moon, therefore, the sun is not more than 100 from one of the nodes, there will be an eclipse of the moon.

In calculating eclipses, the sun's distance from the node is used instead of the moon's; because the sun's place is much easier to find than the moon's; and at eclipses of the sun, the sun's distance from the node is the same as the moon's; and at eclipses of the moon, the sun's distance from one node is the same as the moon's distance from the other.

A right or straight line joining the nodes, or from one node to the other, is called the line of the nodes. When the sun at new or full moon, in this line, the sun, moon and earth are all in a right line, and if it be new moon, the moon's shadow falls upon the earth, and makes an eclipse of the sun if it be full moon, the earth's shadow falls upon the moon, and makes an eclipse of the moon.

At new moon, if the sun and moon are more than 160 from the nearest node, the moon is too high or too low in her orbit to cast any part of her shadow upon the earth, and there, will be no eclipse of the sun.

At full moon, if the sun is more than 100 from the nearest node, the moon of course is the same distance from the other node, and consequently the moon is too high or too low in her orbit to go through any part of the earth's shadow, and there will be no eclipse of the moon.

By the motion of the earth round its axis in 24 hours, the sun and moon appear to move round the earth in the same time: and by the motion of the earth round the sun in a year, the sun appears to move round the earth in the same time and every appearance is the same, whether we suppose the sun moves, and the earth remains at rest, or whether we suppose, (as is the case,) the earth moves, and the sun is

at rest.

The orbit or path, in which the moon moves round the earth once a month, is not an exact circle, but an ellipse or oval, and the earth is in one of the foci. Also the ecliptic, or orbit or path in which the earth moves round the sun once a year, is an ellipse, and the sun is in one of the foci.

Each orbit is divided into 12 equal parts called signs; each sign into 30 equal parts called degrees; each degree into 60 equal parts called minutes; each minute into 60 seconds; each second into 60 thirds, &c.

Beginning at that part of the ecliptic where the sun is, at the vernal equinox, that is, where the plane of the equator crosses the ecliptic, the names and characters rep resenting the signs, going round in the order the sun moves, are as follows:1st, Aries, 2d, Taurus, 8; 3d, Gemini, II; 4th, Cancer, ; 5th, Leo, Virgo, ; 7th, Libra,; 8th, Scorpio, m; 9th, Sagittarius, 1; 10th, Capricorn, V 11th, Aquarius, ; 12th, Pisces, .

; 6th,

As every appearance is the same, whether the earth moves, or the sun moves; and as the earth in moving round the sun, makes the sun appear to move through the signs of the ecliptic, as there is frequent occasion to speak of the sun's place in the ecliptic, which place is always the opposite point to that in which the earth is; in constructing tables of motion, the sun is considered to move in the ecliptic round the earth once a year, though the fact is otherwise, yet every appearance is the same as if this were really the case; the sun, therefore, and not the earth, is considered the moving body.

The orbits being ellipses, and not circles, the sun and moon are continually changing their distances from the earth. The mean, or average distance of the sun from the earth, is 95 millions of miles, but at the nearest point of the orbit, it is only about 93 millions; and at the farthest point about 97 millions.

The mean distance of the moon from the earth is 240,000 miles; but in that part of her orbit which is farthest from the earth, her distance is 253,000 miles; and in the part nearest the earth, the moon is only 227,000 miles from the earth. The farthest point of each orbit from the earth, is called the APOGEE; and the nearest point the PERIGEE. These points are diametrically opposite to each other, six signs apart; a right line from one to the other is called the Line of the Apsides.

When the Creator launched the planets from his hand, he gave them their centrifu gal and centripetal forces, and adjusted the one force to the other, so that the planets

should move between them, neither flying off in right lines, nor descending to their centres of gravity, but moving round them in circles or ellipses.

The nearer any object is to the eye of the spectator, the larger it appears, or the greater is the angle under which it is seen,-the farther from the eye, the less.

The sun and moon appear larger when they are in perigee than when they are in apogee; that is, their diameters subtend larger angles in perigee than in apogee; and the differences of these angles agree exactly with elliptic orbits, supposing the lower focus of each orbit to be the earth's centre. The motions of the sun and moon are observed to become continually swifter while they are moving from their apogee to the perigee; and to become continually slower from the perigee to the apogee; being slowest of all in the apogee, and swiftest of all in the perigee. The reason of this is, the nearer any body is to the sun or earth, the more it is attracted by the sun or earth; and this attraction increases as the distance becomes less.

While the sun and moon are moving from their apogee to their perigee, they are continually approaching nearer and nearer to the earth, which causes the moon to be more and more attracted by the earth; and the earth more and more attracted by the sun and this greater attraction causes the motion of the sun and moon to become swifter. But while the sun and moon are moving from their perigee to their apogee, their distance from the earth is continually increasing, and of course the attraction is constantly growing less, which causes the sun's and moon's motion to become slower. The distance of the sun or moon from its apogee, perigee, or any other given point of its orbit, is reckoned in signs, degrees, minutes, seconds, &c. Here the distance the luminary has moved through from any given point, is meant, and not the space it is short of it, in coming round again.

The point of the orbit in which the sun or moon is, at any time, is called the place of the sun or moon at that time.

The distance of the sun or moon from its apogee, at any time, is called its anomaly; so that in the apogee the anomaly is nothing; and in the perigee the anomaly is six signs. When the sun or moon is in apogee or perigee, its place is the same as it would be if its motion were equal and uniform in all parts of its orbit.

The supposed equal motions are called mean; the unequal are justly called the true. The mean place of the sun or moon is always before the true place, while the luminary is moving from its apogee to its perigee; and the true place is always before 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. These things may be made plain by a figure; thus,-Let (Fig. 1) be the earth; AWPE the orbit of the moon; A its apogee, and P its perigee. While the moon moves from A by W, towards P, it is continually getting nearer and nearer to the earth, and of course it is more and more attracted by the earth; and its motion becomes continually swifter, till it arrives at P, its least distance from the earth, where the attraction is greatest, and the moon's motion swiftest. While it moves from P by E, towards A, it is continually getting farther and farther from the earth, the attraction becomes constantly less, and the moon's motion slower, till it arrives at A, its farthest distance from the earth, when the earth's attraction is least and the moon's motion slowest.

Tables of the motion of the heavenly bodies are made as if their orbits were perfect circles, and their motion uniform; and the place of any heavenly body found from these tables is called the mean place, which is reduced to the true, or real place, by equations.

CHAP. II.-EXPLANATION OF THE TABLES,

To make tables of the motions of the sun, moon, and planets, in order to compute their places at any given instant, the first thing to be done, is to determine, from careful observations, the exact time required by the sun and moon to perform a complete revolution in their orbits, or to describe the space of 360°.

Some of the stars may be seen at any time of the day with a good telescope, and particularly at the time of an eclipse of the sun. if there are no clouds in the way. Note the exact time of the conjunction of a particular star with one of the limbs of the sun or moon; and then note how much time elapses before its next conjunction: the time thus elapsed is the time of one revolution of the sun or moon. This time may be more accurately determined by dividing the time required for several revolutions by the number of revolutions. When this important element is exactly ascertained, it can easily be found, by multiplication or division, through how much of their orbits, or how many times through their orbits, the sun and moon move in any given time; or in any number of years, days, hours, minutes, seconds, &c. It has been found, by a long series of observations, that the sun goes through the ecliptic from any fixed star to the same star again in 365d. 6h. 9m. 11.58., which is called the Sideral year; from its apogee to its apogee again, in 365d. 6h. 13m. 58.88., which is called the Anomalistic year; and from either equinox or solstice to the same again, in 365d. 5h. 48m. 493, which is called the Solstitial or Tropical year. This latter period is the true solar year, as on it the seasons depend. The reason the two first periods are longer than the solar year. is the combined attrac.ion of the sun, and moon, and planets, upon the earth.-explained under the article, Precession of the Equinoxes. As the sun, each year, performs its revolution from the equinox to the equinox again, 20 minutes 22.5 seconds sooner than from a fixed star to the same star again, it shows, that with respect to the fi ed stars, the equinoxes have a slow amua! backward motion, or from east to west, contrary to the motion of the sun and moon, and coutrary to the order of the signs of the ecliptic, which is called the "Precession of the Equinoxes." And as the sun, each year, moves round from any fixed star to the same star again 4m. 47.3 seconds sooner than from its apogee or perigee to the same again, it shows that the apogee and perigee have a small annual forward motion, or from west to east.

It is observed that the moon goes through her orbit, from either node to the same again, in 27d. 5h. 5m. 36s. from any point of the ecliptic to the same again, in 27d. 7h. 42m. 59.803s.; from any fixed star to the same ptar again, in 27d. 7h. 43m. 4.7652s.; from its apogee or perigee to the same again, in 27d. 13h. 18m. 34.8526.j and from the sun to the sun again, in 29d. 12h. 44m. 2.82339. This last period is called a lunation, and it

would always be of this length, if the motions of the sun and moon were always uniform and equal in one part of their orbit to what they were in any other part. But this is not the case; and the strong attraction of the earth and sun upon the moon varies her motion, and makes the above mean periods sometimes longer and sometimes shorter: but all the variations of the moon's motion may be accurately ascertained, and tables of them constructed; as may also the variation of the motion of the sun, or rather of the earth, caused by the attraction of the sun.

The difference between the three first of the above periods of the moon's motion, viz., her revolution with regard to the node, to a fixed star, and to her apogee, is caused by the attraction of the sun and earth; but the reason the lunation is so much longer than either of the other periods, is, that while the moon goes round the earth, from the sun to the sun again, the earth goes almost a sign forward, causing the sun to appear to go through almost a sign in the ecliptic; consequently, the moon must go almost a sign more than round the earth before she can overtake the sun. The time of the lunation is found as follows: viz.-find the time from the middle of one eclipse, to the middle of another of the same kind, many years after; divide this time by the number of lunations between the two eclipses, and the time of a mean lunation will be had. When the moon or a planet is said to be in such a sign, degree, &c., or in such a part of the ecliptic, the meaning is, that the moon or planet is in the same direction from the centre of the earth that this part of the ecliptic is.

It is frequently said, that the moon or a planet is in, or crosses, the ecliptic or the equator,-when the meaning is, that it is in, or crosses, the plane of the ecliptic or of the equator.

The moon moves round from either node to the same again, 2h. 37m. 23 9933. sooner than from any point of the ecliptic to the same again; therefore the nodes, each revolution, move backwards, or as much from east to west in the ecliptic, as the moon moves in 2h. 37m. 23.8033. This motion of the nodes in the ecliptic is about 19° 20' 26" in a year, which carries the nodes quite through, or round the ecliptic, in 18 years, 224 days, 12 hours, 57 minutes, and 52 52-125 seconds,-4 of the 18 years being leap years. The moon, in going round from her apogee or perigee to the same again, is 5h. 35m. 35.0798. longer than in going round from any point of the ecliptic to the same again; consequently the apogee and perigee, each revolution, move as much forward as the moon moves in 5h. 35m. 35.079s. This motion of the apogee and perigee is about 40° 39 49' in a year, by which the apogee moves quite round from any point in the ecliptic to the same again, in 8y. 3c9d. 11h. 24m. 8 16-25s..-2 of the 8 years being leap-years.

If the motions of the sun and moon were always uniform, and equal in one part of their orbits to what they are in any other part, it would be quite an easy thing to make tables of their motions, and compute their places at any time; but it has already been shown that their places are never what they would be if their motions were uniform, except when they are in apogee or perigee; which is when their anomalies are either nothing, or six signs: and that their mean places are always before their true places, while the anomaly is less than 6 signs; and the true places are before the mean, while the anomaly is more.

The anomalies of the sun and moon, and the sun's distance from the moon's ascending node, are the argu. ments for finding the equations by which the mean places are reduced to the true,

To construct tables of their anomalies, &c., begin by finding through how much anomaly, or how far from their apogee, the sun and moon move, and the distance the sun and moon's ascending node move from each other, while the moon completes a lunation.

Knowing the time the sun requires to move from his apogee to his apogee again; or knowing that in this time his mean motion in anomaly is 360°, it is easy to find his mean motion, in anomaly, for any other time. The same may be said of the moon. If the sun, in 365d. 6h. 13m. 58.8s., moves in anomaly 360°; how far, in anomaly, will he move in 29d. 12h. 44m, 2.8283s.? And if the sun, in 365d. 5h. 48m. 49s., moves 360° in longitude; how far, in longitude, does he move in 29d. 12h. 44m. 2.8283s.? Thus the sun's mean motion, in anomaly, in the time of one mean lunation, is found by multiplying this time by 360, and dividing the product by the time required by the sun to move from his apogee to his apogee again. And the moon's mean anomaly, for one lunation, is found by dividing the same product by the time required by the moon to move from her apogee to her apogee again. The sun's mean motion in longitude, for one lunation, is found by dividing the same product by the time required by the sun to move from one equinox to the same again. The distance the node moves in one lunation, is found by dividing the same product by the time required by the node to move entirely round through the moon's orbit from some point to the same again; which is 6798 days, 4h. 52m. 523. The distance the sun and node move from each other in one lunation, (as the node moves backwards,) is found by adding the distance the node moves, to the sun's mean motion in longitude for one lunation. Thus, Table 1. is made as follows:-first set down the time of one mean lunation; then set down the sun and moon's mean motion in anomaly, and the distance the sun and moon's ascending node move from each other in this time: multiply all these numbers by 2, then the same numbers by 3, by 4, and so on to 13, and then by 24, and all the other numbers in this column,-and the time, and the sun and moon's mean motion in anomaly, and the mean distance the sun and moon's ascending node move from each other in so many lunations, will be had. The whole number of days in the time up to 9 years, is inserted; after that, only the excess over complete centuries of Julian years. In making these calculations, when the signs amount to more than 12, divide them by 12, and set down only the remainder as the proper quantity. The numbers for the half lunation are found by dividing those for one lunation by two.

The sun and moon's mean anomalies, and the sun's mean distance from the moon's ascending node, at the time of mean new moon, have many times been determined; from which, the same at the time of any other mean new moon may be easily found.

To make Table 2, fix upon some particular year and month to begin at; and observe, calculate, or take from tables already calculated, the astronomical time of mean new moon for this year and month, and the sun and moon's mean anomalies, and the sun's mean distance from the moon's ascending node at this time. Table 2, is begun with the month of March of the year 1800; and the time of mean new moon for that Imonth and year, and the mean anomalies of the sun and moon, and the sun's mean distance from the moon's ascending node at the said time, were found to be as put down at the head of the table.

Now, to find the time of mean new moon, and the sun and moon's mean anomalies, and the sun's mean distance from the moon's ascending node, for March, 1801, add the numbers from Table 1 for 12 lunations, to the numbers for March, 1800; and rejecting 12 signs when they occur, and setting down the remainder, and subtracting 365 from the days,-the time of mean new moon, and the sun and moon's mean anomalies, and the sun's mean distance from the moon's ascending node for March, 1801, will be had. Again, add in the same manner the numbers for 12 lunations, to the numbers for March, 1801, and subtract 365 days,-and the numbers for March, 1802, will be obtained. To get the numbers for 1803, add 13 lunations to 1802; for if but 12 lunations be added, the numbers for only February will be obtained. Add 12 lunations to 1803, and subtract 366 days instead of 365, as 1804 is a leap-year, and the numbers for March, 1804, will be obtained. In this manner the table may be continued as far as you please, adding 12 lunations when the new moon happens after the 11th day of the month, and 13 when it happens before, always subtracting 365 days for common years and 366 for leap-years. Thus the table was constructed, and then written out to every tenth year. Table 3, similar to 2; only calculated for the Old Style.

Table 4, page 19, is merely the number of days from the beginning of March to the beginning of each of the other months.

The mean new or full moon never coincides with the true, except when the anomalies of both the sun and moon are either nothing or six signs, and the sun is then in conjunction with one of the moon's nodes but this does not take place twice in a thousand years.

has been before observed, the unequal motion of the sun and moon, is the reason their mean and trus aces differ: for the same reason, the time of mean syzygy differs from the time of true syzygy. The first difference, or equation, between the mean and true syzygy, is caused by the sun's unequal motion, and the variation of his distance from the earth and moon; this depends on his anomaly.

When the sun's anomaly is less than six signs, that is, while he is moving from his apogee to his perigee, his mean place being before his true place, it is evident that the moon will overtake him, or be opposite to him, sooner, or in less time, than she could if his motion were uniform and equal; and when his anomaly is more than six signs, that is, while he is moving from his perigee to his apogee, his true place being before his mean place, the moon will require more time to overtake him, or be in opposition to him, than she would if his motion were uniform.

The greatest difference that can happen between the time of mean and true new or full moon, on account of the sun's unequal motion, and unequal distance from the earth and moon, is 4h. 10m. 57s. This takes place when the sun's anomaly is either 3 signs 1 degree, or 8 signs 29 degrees. The true syzygy is sooner than the mean in the first case, and later in the last. In all other signs and degrees of anomaly, the differ ence or equation is gradually less, and vanishes when the anomaly is either nothing, or six signs. This equation is computed in Table 5, page 16, for every sign and degree of the sun's anomaly, by putting 4h. 10m. 575. against 3 signs 1 degree, and 8 signs 29 degrees. Against every other degree the equation is proportionably less; and is nothing against 0 signs 0 degrees. It is called the annual or first equation of the mean to the true syzygy. 0, 1, 2, 3, 4, and 5 signs, are at the top of the table, and their degrees are at the left hand, and are reckoned downward. 6, 7, 8, 9, 10, 11 signs are at the bottom of this table, and their degrees are at the right hand, and are reckoned upwards. This is the case with several other tables.

Another, and the greatest difference, between the mean and true syzygy, is caused by the unequal motion of the moon in her eccentric orbit, and depends on her anomaly; and as her anomaly is affected by the sun's, before it is used to find this difference it must be equated by a quantity or equation depending on the sun's mean anomaly, and calculated in Table 6, page 17.

The moon's motion is more unequal than the sun's, as her orbit is more eccentric. Like the sun, while she is moving from her apogee to her perigee, her motion being continually accelerated, her mean place is before her true place; and this causes the mean syzygy to happen before the true. While she is moving from her perigee to her apogee, het motion becomes continually slower, and her true place is before her mean place; and the mean syzygy is after the true.

This difference between the mean and true syzygy caused by the eccentricity of the moon's orbit, and the moon's unequal motion, at greatest, is 9h. 47m. 54s.; this is when the moon's equated anomaly is either 2 signs 26 degrees, or 9 signs 4 degrees; at all other signs and degrees of anomaly, it is gradually less, and is nothing when the moon's anomaly is 0, or 6 signs.

This difference is computed for every sign and degree of the moon's equated anomaly, in Table 7, page 17, and is called the second equation of the mean to the true syzygy. It is to be added to the mean syzygy when the moon's equated anomaly is less than 6 signs, and subtracted when more.

Another small difference or equation between the mean and true syzygy, depends on the difference between the sun's and moon's anomaly; at greatest it is 4m. 59s.: this is when the remainder found by subtracting the moon's equated anomaly from the sun's mean anomaly, is either 3, or 9 signs. If the moon's anomaly to be subtracted, exceeds the sun's, add 12 signs to the sun's anomaly, and then a remainder will be had. This remainder is the argument for the equation. For every 5 degrees of this argument, the equation is calculated in Table 8, page 19 it is called the third equation of the mean to the true syzygy, and is to be subtracted from the mean syzygy, when the argument is less than 6 signs, and added when more.

"The fourth and last equation of the mean to the true syzygy, depends on the sun's distance from the moon's ascending node at the time. At greatest, it is only im. 34s. ; this is when the sun's distance from the moon's ascending node is either 1 sign 15 deg., or 7 signs 15 deg.; or 4 signs 15 deg,; or 10 signs 15 deg. i to be added to the mean syzygy in the two first cases, and subtracted in the two latter. This equation is calculated for every 5 degrees of the sun's distance from the moon's ascending node in Table 9, page 21. Table 10, page 18, for finding the sun's Lon. and anomaly, at any time, is formed as follows:-Knowing the time the sun requires to move from one equinox to the same again; or knowing that in this time the sun moves 360d. in longitude, multiply 365 by 360, and divide the product by 365d. 5h. 48m. 49s., (the solar or solsticial year,) and the quotient will be the sun's mean motion in longitude for a common year of 365 days; which, d vided by 365, will give his mean motion for one day. The mean motion for one year, multiplied by 2, gives the same for 2 years; and the mean motion for 1 year added to that of 2 years, gives the same for 3 years; but to find the mean motion for 4 years, as every fourth year has one day more than a common year, the mean motion for 1 year and 1 day must be added to that for 3 years: thus, the table may be carried on at pleasure. Having the mean motion for one day, the same for any number of days may be found by multiplication; and for any part of a day, as an hour, a minute; or a second, by division. The motions for months are found by taking 0 for the motions for January for common years, because January is the beginning of the year; by taking the motions for 31 days for the motions for February for common years; by taking the motions for 59 days for the motions for March; 90 days for April; 120 days for May; 151 days for June; 181 for July; 212 for August; 243 for September; 273 for October; 301 for November, and 334 for December, either common or leap years. From the motions for January, (or from 12 signs.) subtract the motions for one day, and the motions for January of leap year will be had; from the motions for February, subtract the motions for one day, and the motions for February of leap year will be had. The motions for January and February are taken thus for leap years, because the additional day does not come in till the last of February; but the motions for this day are added to the motions for leap years, as if the day came in at the beginning of the year.

To find the sun's mean anomaly, proceed in the same manner as to find his mean longitude, only instead of 365d. 5h. 48m. 49s. take 365d. 6h. 13m. 58.8s. the anomalistic year.

The distance of the equinox and apogee from any particular star, and the sun's longitude and anomaly at any time, may easily be found by any person having accurate instruments, and knowing how to use them. Having the mean longitude and anomaly at any particular time, the same may be found for any other time, by adding or subtracting the mean motion in longitude and anomaly for the interval.

The longitude and anomaly for the noon of the last day of any year, is taken for the longitude and anomaly of the next year; because to find the longitude and anomaly for any particular month and day of any year, it is convenient to add the longitude and anomaly for that month and day to the longitude and anomaly for the year. Thus to find the iongitude and anomaly for the noon of January 1st, of any year, it is plain that the longitude and anomaly for one day, must be added to the longitude and anomaly for the noon of the 31st December preceding: that is, for the longitude and anomaly for the year, add the longitude and anomaly for one day, and the longitude and anomaly for Jan. 1st will be had. So to the longitude and anomaly for the year add the longitude and anomaly for 15 days, and the longitude and anomaly for Jan. 15th will be had. And to the longitude and anomaly for the year, add the longitude and anomaly for February, and for 20 days, and the longitude and anomaly for the noon of February 20th of that year will be had; and so on for any other month, day, &c. of any year.

As a leap year has one more day than a common year, the longitude and anomaly for the leap years are given for the 1st day of January instead of for the day before, (the 31st of December preceding :) therefore in finding the longitude and anomaly for any time in either the month of January or February of a leap year, the rule for Problem 6 may be followed, or the longitude and anomaly for the given month of a common

year, and for a number of days one less than the given day of the month, may be added to the longitude and anomaly for the given year.

The distance the sun and moon's ascending node move from each other in hours, minutes, and seconds, is annexed to this Table; being used in calculating eclipses.

Table 11, page 22, is an equation depending on the sun's mean anomaly, used to reduce the sun's mean place to the true place. The principle has already been explained.

Table 12, page 15, is on the same principle as Table 11. It is formed by adding the annual equation of the Moon's Node, to the equation of the Sun's centre; that is, the numbers in Table 32, to those in Table 11. Table 13, page 22, shows the sun's declination to every degree of his true longitude.

Table 14, page 17. Angle of the moon's orbit with the ecliptic in eclipses; which depends on her distance from her Node, and her hourly motion from the sun.

Table 15, page 23; The moon's latitude in eclipses; depending on her distance from her Node, and the angle of her orbit with the ecliptic.

Table 16, page 16. The horizontal parallax of the moon; and the semidiameters and hourly motions of the sun and moon, depending on their respective anomalies.

Table 17, p. 18; Reduction of the ecliptic to the equator; or difference between the sun's longitude and right ascension; or between the longitude and right ascension of any point of the ecliptic, found by the sun's longitude, or the longitude of the points of the ecliptic. The reason of this difference may readily be seen, and the difference found by an artificial globe, in which it will appear that while the longitude is between 0 and 3, and between 6 and 9 signs, the longitude is greater than the right ascension, and at all other times, less.

Table 18, p. 20; Equation of time. This is computed by finding for every degree of the sun's longitude, the corresponding anomaly, and taking the equation for this anomaly from Table 11, and changing it into time, which gives one part of the equation; then change Table 17 into time for the other part; the sum or difference of these parts is the equation. Thus 'Table 18 is formed; every equation in it being the sum or difference of two other equations.

Table 19, page 19, is formed by reckoning four minutes of time to one degree of longitude; and in this proportion for any greater or less quantity.

Table 20, p. 16, is formed by observing through a lunar month how much the tide is accelerated or retarded by the action of the sun as the sun, when it passes the meridian from 0 to 8 hours before the moon, (that is, when the moon passes the meridian before 8 o'clock, either morning or evening.) accelerates the time of ligh water, and retards it at other times. The equation in this table depends also in part upon the moon's distance from the earth; the time of high water happening earliest when the moon is nearest the earth; and the moon is nearest the earth when her horizontal parallax is greatest.

Table 21, page 19, shows the time (found by observation) to be added to the time the moon is on the meri. dian, to find the time of high water as unaltered by the sun. The perpendicular rise of the water in feet at some of the places is also given; and also the latitude and longitude of the places from Greenwich observatory.

Table 22, page 19, the right ascension and declination of some of the principal fixed stars.

Table 23, page 24, dip, or depression of the horizon, at different heights above it: that is, this table contains the angle between a level line, and a line from the eye of the observer to the visible unobstructed horizon; when the eye is at different heights in feet above the horizon.

Table 21, page 18, dip or angle between a level line, and a line to the surface of the earth, or water, at different distances from the observer.

Table 25, p. 20, augmentation of the altitude of a celestial body, caused by the refraction of the earth's atmosphere at different altitudes of the body above the horizon. The state of the atmosphere affects this a little, the refraction being greatest when the thermometer is lowest. This Table is calculated for 55 deg. of Fahrenheit; but it is within a few seconds of the truth at any state of the thermometer.

Table 26, p. 20, the semi-diurnal or semi-nocturnal arc of the sun at different declinations and at different latitudes on the earth: if the declination and latitude are of the same name, this Table shows the semi-diur nal arc; but if the declination and latitude are of different names, it shows the semi-nocturnal arc. By cor recting, it also answers for the semi-diurnal or semi-nocturnal arc of any other celestial body, or rather for half the time of continuance of a celestial body above or below the horizon. But the semi-diurnal arc may be more accurately found by trigonometry, as follows, viz. to the tangent of the latitude of the place, add the tangent of the declination of the sun: subtract the radius from the sum, the remainder is the cosine of the semi-diurnal arc, when the latitude of the place and the declination are of different names, that is, one north, and the other south and the remainder is the cosine of the semi-nocturnal arc, when the lati tude and declination are of the same name; that is, both north, or both south. Subtract the semi-nocturnal arc from 12 hours, and the remainder is the semi-diurnal arc. When the declination of a planet, star, or the moon, is more than 23 deg. 25m. its semi-diurnal arc, or half time of its continuance above the horizon, must be calculated in the latter way.

Table 27, p. 21, change of the moon's right ascension from the sun's; a correction to be applied to the time the moon is on the meridian at any place, to find the time she is on the meridian of any other place. This Table also contains the correction to be added to the sun's semi-diurnal arc, to find the moon's semi-diurnal

arc.

Table 28. p. 23, time answering to a change of 33m. of altitude at the horizon; or the time the sun, or any heavenly body, (on account of the refraction of the earth's atmosphere,) appears in the horizon, before it is actually there; corresponding to the declination of the sun, or body, at the head of the Table, and the lat tude of the place in the left hand column.

TABLES of the moon's mean motion are made in the same manner as those of the sun. The moon's pe riod, or time in which she makes one complete revolution being known. To find her mean motion in longitude for one year of 365 days, multiply 360 degrees by 365, and divide the product by the time of her tropical revolution, the quotient is her mean motion in longitude for 365 days; divide this by 365, and her mean motion in longitude for one day will be had. Her mean motion in anomaly for 365 days, is found by multiplying 360 deg. by 365, and dividing the product by the time of her anomalistic period, or revolution; divide this quotient by 365, and her mean motion in anomaly for one day will be had. The retrograde motion of the moon's node in 365 days, is found by multiplying 360 deg. by 365, and dividing the product by the time of the revolution of the node; divide this quotient by 365, and the motion of the node in one day will be had.

The mean motions for one year multiplied by 2, give the same for 2 years; and the mean motions for one year, multiplied by 3, give the same for 3 years; to the mean motions for 3 years, (because every 4th year is leap year,) add the mean motions for one year and one day, and the mean motions for 4 years will be had. In this manner a table may be made and carried as far as you please.

The motion in longitude and anomaly, and the motion of the node for one hour, is found by dividing the motions for one day by 24. Divide the motions for one hour by 60, and the motions for one minute will be had; divide the motion for one minute by 60, and the motion for one second will be had, and so on. The mean motions for any number of years, days, hours, minutes, seconds, &c. are found by multiplying the mean motions for one, by that number. The mean motions for months are found in the same manner as those of the sun. Thus Table 29, page 21, is formed.

Table 30, page 18, is formed by observing or calculating the moon's mean Lon. and anomaly, and the mean longitude of the node at noon on the 31st day of December 1793, these are taken for her mean longitude and

2