Example 2.

Let the latitude be 20:40, south, and the declination of a celestial object 30:29, north; required the corresponding semi-diurnal arch?

20:40 south, log. tangent. .9.576576

.30. 29 north, log. tangent. . 9.769860

The present Table has been computed agreeably to the first example; but as in most nautical computations, it is not absolutely necessary that the semi-diurnal arch should be determined to a greater degree of accuracy than the nearest minute; the seconds have, therefore, been rejected, and the nearest minute retained accordingly.

Since the Table for finding the time of the rising or setting of a celestial object (commonly called a Table of semi-diurnal and semi-nocturnal arcs,) is scarcely applied to any other purpose, by the generality of nautical persons, than that of merely finding the approximate time of the rising or setting of the sun; the following problems are, therefore, given for the purpose of illustrating and simplifying the use of this Table; and of showing how it may be employed in determining the apparent times of the rising and setting of all the celestial objects whose declinations come within its limits.

PROBLEM I.

Given the Latitude and the Sun's Declination, to find the Time of its Rising or Setting.

RULE.

Let the sun's declination, as given in the Nautical Almanac, be reduced to the meridian of the given place by Table XV., or by Problem I., page 76; then,

Enter the Table with this reduced declination at top, or bottom, and the latitude in either of the side columns; under or over the former, and opposite to the latter, will be found the approximate time of the sun's setting when the latitude and declination are of the same name; or that of its rising when they are of contrary names.-The time of setting being taken from 12 hours will leave the time of rising, and vice versa, the time of rising being taken from 12 hours will leave that of setting.

Note.-Proportion must be made, as usual, for the excess of the minutes of latitude and declination above the next less tabular arguments.

Example 1.

Required the approximate times of the sun's rising and setting July 13, 1824, in latitude 50:48, north, and longitude 120 degrees west?

Sun's declination July 13th. per Nautical

var. of dec. 8.58%, and long. 120: W. - 2.59%

Note.-Twice the time of the sun's setting will give the length of the day; and twice the time of its rising will give the length of the night.

*

Example 2.

Required the approximate times of the sun's rising and setting October 1st, 1824, in latitude 40:30 Sun's declination October 1st. Almanac, is Correction from Table XV., answering to var. of dec. 23:20, and long. 105: E.

north, and longitude 105 degrees east?
per Nautical

3:16 6 south.

6.48"

The nearest minute of declination is sufficiently exact for the purpose of finding the

approximate times of the rising and setting of a celestial object.

Remark.

Since the times of the sun's rising and setting, found as above, will differ a few minutes from the observed, or apparent times in consequence of no notice having been taken of the combined effects of the horizontal refraction and the height of the observer's eye above the level of the sea, by which the time of rising of a celestial object is accelerated, and that of its setting retarded; nor of the horizontal parallax which affects these times in a contrary manner; a correction, therefore, must be applied to the approximate times of rising and setting, in order to reduce them to the apparent times. This correction may be computed by the following rule; by which the apparent times of the sun's rising and setting will be always found to within a few seconds of the truth,

.

Rule. To the approximate times of rising and setting, let the longitude, in time, be applied by addition or subtraction, according as it is west or east, and the corresponding times at Greenwich will be obtained to these times, respectively, let the sun's declination be reduced by Table XV., or by Problem I., page 76; then,

Find the sum and the difference of the natural sine of the latitude, and the natural co-sine of the declination (rejecting the two right hand figures from each term), and take out the common log. answering thereto, rejecting also the two right hand figures from each :-now, to half the sum of these two logs. add the proportional log. of the sum of the horizontal refraction and the dip of the horizon diminished by the sun's horizontal parallax, and the constant log. 1. 1761*; the sum of these three logs., abating 4 in the index, will be the proportional log. of a correction; which being subtracted from the approximate time of rising, and added to that of setting, the apparent times of the sun's rising and setting will be obtained.

Thus,-Let it be required to reduce the approximate times of the sun's rising and setting, as found in the last Example, to the respective apparent times; the horizontal refraction being 33; the dip of the horizon 5:15", and the sun's horizontal parallax 9 seconds.

The sun's declination reduced to the approximate time of rising, is 3:3.37%, and to that of setting 3:14.58% south.

*This is the proportional log. of 12 hours esteemed as minutes.

Note. In this method of reducing the approximate to the apparent time of rising or setting, it is immaterial whether the latitude and declination be of the same, or of contrary names :-nor is it of any consequence whether the declination be reduced to the approximate times of rising and setting or not, since the declination at noon will be always sufficiently exact to determine the correction within two or three seconds of the truth, on account of its natural co-sine being only required to four places of figures-this will appear evident by referring to the above example, where,

although there is a difference of 11:21" between the reduced declinations. at the approximate times of rising and setting; yet this difference has no sensible effect on the correction corresponding to those times.

PROBLEM II.

Given the Latitude of a Place and the Declination of a fixed Star, to find the Times of its Rising and Setting.

RULE.

Let the right ascension and declination of the star, as given in Table XLIV, be reduced to the given day; then, from the right ascension of the star, increased by 24 hours if necessary, subtract that of the sun, at noon of the given day; and the remainder will be the approximate time of the star's transit, or passage over the meridian; from which, let the correction answering thereto and the daily variation of the sun's declination (Table XV.,) be subtracted, and the apparent time of the star's transit will be obtained.

If much accuracy be required, and the place of observation be under a meridian different from that of Greenwich, a correction depending on the longitude and variation of the sun's right ascension (Table XV.,) must be applied to the time of transit :-this correction is subtractive in west, and additive in east longitude; the time being always reckoned from the preceding noon: now,

Enter Table L., with the declination at top or bottom, and the latitude in the side column; and in the angle of meeting will be found the semidiurnal arch, or the time of half the star's continuance above the horizon, when the latitude and declination are of the same name; but if these elements are of different names, the time, so found, is to be subtracted from 12 hours, in order to obtain the half continuance above the horizon: then this half continuance * being applied by subtraction and addition to the apparent time of transit, will give the approximate times of the star's rising and setting.

Example 1.

At what times will the star a Arietis rise and set January 1st. 1824, in latitude 50°48 north?

* In strictness the semi-diurnal arch, or half continuance above the horizon ought to be corrected by subtracting therefrom the proportional part (Table XV.,) corresponding to it and the variation of the sun's right ascension for the given day.