PART IV.

ASTRONOMICAL PROBLEMS.

EXPLANATIONS OF THE TABLES.

THE Tables which form a part of this work, and which are employed in the resolution of the following Problems, consist of Tables of the Sun, Tables of the Moon, Tables of the Mean Places of some of the Fixed Stars, Tables of Corrections for Refraction, Aberration and Nutation, and Auxiliary Tables.

The Tables of the Sun, which are from XVII to XXXIV inclusive, are, for the most part, abridged from Delambre's Solar Tables. The mean longitudes of the sun and of his perigee for the beginning of each year, found in Table XVIII, have been computed from the formulæ of Prof. Bessel, given in the Nautical Almanac of 1837. The Table of the Equation of Time was reduced from the table in the Connaissance des Tems of 1810, which is more accurate than Delambre's Table, this being in some instances liable to an error of 2 seconds. The Table of Nutation (Table XXVII) was extracted from Francœur's Practical Astronomy. The maximum of nutation of obliquity is taken at 9.25. The Tables of the Sun will give the sun's longitude from the mean equinox within a fraction of a second of the result obtained immediately from Delambre's Tables, as corrected by Bessel. The Tables of the Moon, which are from XXXV to LXXXV inclusive, are abridged and computed from Burckhardt's Tables of the Moon. To facilitate the determination of the hourly motions in longitude and latitude, the equations of the hourly motions have all been rendered positive, like those of the longitude. Some few new tables have been computed for the same purpose. The longitude and hourly motion

in longitude will very rarely differ from the results of Burckhardt's Tables more than 0".5, and never as much as 1". The error of the latitude and hourly motion in latitude will be still less. The other tables have been taken from some of the most approved modern Astronomical Works. (For the principles of the construction of the Tables, see Chap. X.)

Before entering upon the explanation of each of the tables, it will be proper to define a few terms that will be made use of in the sequel.

The given quantity with which a quantity is taken from a table, is called the Argument.

The angular arguments are expressed in some of the tables according to the sexagesimal division of the circle. In others, they are given in parts of the circle supposed to be divided into a 100, a 1000, or 10000 &c. parts.

Tables are of Single or Double Entry, according as they contain one or two arguments. The Epoch of a table, is the instant of time for which the quantities given by the table are computed. By the Epoch of a quantity, is meant the value of the quantity found for some chosen epoch, from which its value at other epochs is to be computed by means of its known rate of variation.

Table I, contains the latitudes, and longitudes from the me ridian of Greenwich, of various conspicuous places in different parts of the earth. The longitudes serve to make known the time at any one of the places in the table, when that at any of the others is given. The latitude of a place is an important element in various astronomical calculations.

Table II, is a table of the Elements of the Orbits of the Planets with their secular variations, and serves to make known the elements at any given epoch different from that of the table. From these the elliptic place of the planets at the given epoch may be computed.

Table III, is a similar table for the Moon.

Tables IV, V, VI, VII, require no explanation.

Table VIII, gives the mean Astronomical Refractions; that is, the refractions which have place when the barometer stands at 30 inches, and the thermometer of Fahrenheit at 50°.

Table IX, contains the corrections of the Mean Refractions for

+1 inch in the barometer, and -1° in the thermometer, from which the corrections to be applied, at any observed height of the barometer and thermometer, are easily derived.

Table X, gives the Parallax of the Sun for any given altitude on a given day of the year; for reducing a solar observation made at the surface of the earth to what it would have been, if made at the centre.

Table XI, is designed to make known the Sun's Semi-diurnal Arc, answering to any given latitude, and to any given declination of the sun; and thus the time of the sun's rising and setting, and the length of the day.

Table XII, serves to make known the value of the Equation of Time, with its essential sign, which is to be applied to the apparent time to convert it into the mean. If the sign of the equation taken from the table be changed, it will serve for the conversion of mean time into apparent. This table is constructed for the year 1840.

Table XIII, is to be used in connection with Table XII, when the given date is in any other year than 1840. It furnishes the Secular Variation of the Equation of Time, from which the proportional part of its variation in the interval between the given date and the epoch of Table XII is easily derived.

Table XIV, contains certain other Corrections to be applied to the equation of time taken from Table XII, when its exact value, to within a small fraction of a second, is desired.

Table XV, gives the Fraction of the Year, corresponding to each date. This table is useful, when quantities vary by known and uniform degrees, in deducing their values at any assumed time from their values at any other time.

Table XVI, is for converting Hours, Minutes, and Seconds into decimal parts of a Day.

Table XVII, is for converting Minutes and degree into the decimal division of the same.

for the conversion of minutes and seconds of parts of an hour.

Seconds of a

It will also serve

time into decimal

The last two tables will be found frequently useful in arithmetical operations.

Table XVIII, is a table of Epochs of the Sun's Mean Longitude, of the Longitude of the Perigee, and of the Arguments for

finding the small equations of the Sun's place. They are all calculated for the first of January of each year, at mean noon on the meridian of Greenwich. Argument 1, is the mean longitude of the Moon minus that of the Sun; Argument II, is the heliocentric longitude of the Earth; Argument III, is the heliocentric longitude of Venus; Argument IV, is the heliocentric longitude of Mars; Argument V, is the heliocentric longitude of Jupiter; Argument VI, is the mean anomaly of the Moon; Argument VII, is the heliocentric longitude of Saturn; and Argument N, is the supplement of the longitude of the Moon's Ascending Node. Argument I, is for the first part of the equation depending on the action of the Moon. Arguments I and VI, are the arguments for the remaining part of the lunar equation. Arguments II and III, are for the equation depending on the action of Venus; Arguments II and IV, for the equation depending on the action of Mars; Arguments II and V, for the equation depending on the action of Jupiter; and Arguments II and VII, for the equation depending on the action of Saturn. Argument N, is the argument for the Nutation in longitude: it is also the argument for the Nutation in right ascension, and of the obliquity of the ecliptic.

Table XIX, shows the Motions of the Sun and Perigee, and the variations of the arguments, in the interval between the beginning of the year and the first of each month.

Table XX, shows the Motions of the Sun and Perigee, and the variations of the arguments, for Days and Hours.

Table XXI, gives the Sun's Motions for Minutes and Seconds. Tables XVIII to XXI, make known the mean longitude of the Sun from the mean equinox at any moment of time.

Table XXII. Mean Obliquity of the Ecliptic for the beginning of each year contained in the table. It is found for any intermediate time by a simple proportion.

Tables XXIII and XXIV, furnish the Sun's Hourly Motion and Semi-diameter.

Table XXV, is designed to make known the Equation of the Sun's Centre. When the equation has the negative sign, its supplement to 12s. is taken. This is to be added along with the other equations of longitude, and 12s. are to be subtracted from the sum. The signs of the argument are given both at the head

and foot of the columns. The numbers in the table are the values of the equation of the centre, or of its supplement, diminished by 46".1. This constant is subtracted from each value, to balance the different quantities added to the other equations of the longitude, in order to render them affirmative. The epoch of this table is the year 1840.

Table XXVI, gives the Secular Variation of the Equation of the Sun's Centre, from which the proportional part of the variation in the interval between the given date and the year 1840, may be derived.

Table XXVII, is for the Nutation in Longitude and Right Ascension, and of the Obliquity of the Ecliptic. The nutation in longitude and in right ascension, serve to transfer the origin of the longitude and right ascension from the mean to the true equinox. And the nutation of obliquity serves to change the mean into the true obliquity.

Tables XXVIII to XXXIII, give the Equations of the Sun's Longitude, due respectively to the attractions of the Moon, Venus, Jupiter, Mars, and Saturn.

Table XXXIV is for the variable part of the Sun's Aberration. The numbers have all been rendered positive by the addition of the constant 0".3.

Table XXXV, contains the Epochs of the Moon's Mean Longitude, and of the Arguments for finding the equations which are necessary in determining the True Longitude and Latitude of the Moon. They are all calculated for the first of January of each year, at mean noon on the meridian of Greenwich. The Argument for the Evection is diminished by 30'; the Anomaly by 2°; the Argument for the Variation by 9°; and the Supplement of the Node is increased by 7'. This is done to balance the quantities which are added to the different equations in order to render them affirmative.

Tables XXXVI to XL, inclusive, give the Motions of the Moon, and the variations of the arguments for Months, Days, Hours, Minutes, and Seconds; and, together with Table XXXV, are for finding the Moon's Mean Longitude and the Arguments at any assumed moment of time.

Tables XLI to LIII, inclusive, give the various Equations of the Moon's Longitude. It is to be observed, with respect to Table