PROBLEM IV. Given the Latitude of a Place, and the Moon's Declination, to find the Times of her Rising and Setting. RULE. Take, from page VI. of the month in the Nautical Almanac, the moon's transit, or passage over the meridian of Greenwich, on the given day, and also her declination. Let the time of transit be reduced to the meridian of the given place by Table XXXVIII.; to which apply the longitude, in time, by addition or subtraction, according as it is west or east ; and the sum, or difference, will be the corresponding time at Greenwich: to this time, let the declination be reduced by Table XVI., or by Problem II., page 80;then, With this reduced declination, and the latitude of the given place, find the moon's semidiurnal arch, or the time of half her continuance above the horizon, by Problem II., page 128, and, thence, the approximate times of rising and setting, in the same manner precisely as if it were a fixed star that was under consideration: call these the estimated times of rising and setting. To the estimated times of rising and setting, thus found, let the longitude, in time, be applied by addition or subtraction, according as it is west or east; and the sum, or difference, will be the corresponding times at Greenwich. To these times respectively, let the moon's declination be reduced by Table XVI., or by Problem II., page 80; with which, and the latitude, find the moon's semidiurnal arch at each of the estimated times. To the respective semidiurnal arches, thus found, apply the corrections corresponding thereto, and the retardation of the moon's transit (Table XXXVIII.) by addition, and the correct semidiurnal arches will be obtained. Now, the semidiurnal arch answering to the estimated time of rising, being subtracted from the moon's reduced transit, will leave the approximate time of her rising at the given place; and that corresponding to the estimated time of setting, being added to the moon's reduced transit, will give the approximate time of her setting at the said place. Example 1. Required the times of the moon's rising and setting, Jan. 17th, 1824, in latitude 51:29. north, and longitude 78:45: west of the meridian of Greenwich? Moon's transit over merid. of Greenwich on the given day is 1334′′ 0: Corr. fr. Tab. XXXVIII., ans. to retard. 53, and long. 75: west + 10.39 App. time of moon's transit reduced to the given meridian 13:44:39: Longitude 78:45 west, in time = Moon's dec. red. to Gr. time, by Table XVI., is 10:25:30 N. Semidiurnal arch, in Table L., answering to lat. 51:29?N., and declination 10:25 N., is Estimated time of the moon's rising Estimated time of the moon's setting 5.15. 0 18:59:39: 6454 0: 13.44.39 6:50 39' 20:38:39: Greenwich time past noon of the given day Moon's dec. reduced to Greenwich time, is 12:10:53′′N. Time, in Table L., ans. to lat. 51:29. N. and dec. 12:11 N., is 7 3" 0: Correction, Table XXXVIII., ans. to 53 and 7:3" = + 15. 0 7:18 0: Moon's dec. reduced to Greenwich time, is 8:41:11′′N. Time, in Table L., answ. to lat. 51:29 N. and dec. 8:41 N., is 6:44" 0: Correction, Table XXXVIII., ans. to 53 and 6:44′′ Example 2. Required the approximate times of the moon's rising and setting, January 20th, 1824, in latitude 40:30: north, and longitude 80 degrees east of the meridian of Greenwich? Moon's transit over the merid. of Greenwich on the given day is 16 6 0: Cor. fr. Tab. XXXVIII., ans. to retard. 49' and long. 80: east - 10.32 Moon's dec. red. to Green. time, by Table XVI., is 5:55:40′′S. Seminocturnal arch, in Table L., answering to lat. 40:30:N. 10:35 28: and dec. 5:56:S. 6:20", subtracted from 12, leaves 540" 0: Moon's dec. reduced to this time, is 4:30:49 S. Time, in Table L., answering to lat. 40:30: N., and dec. Moon's correct semidiurnal arch at rising Approximate time of moon's rising 10:15 28: 5.20. 0 4:55 28: 5:45 0: + 11. 0 Time, in Table L., answering to lat. 40:30 N. and dec. 7:18.S., is 6:25, which, subtracted from 12, leaves Corr. Table XXXVIII., ans. to 49 and 5:35" Moon's correct semidiurnal arch at setting Moon's reduced transit Approximate time of moon's setting Remark. The approximate times of the moon's rising and setting may be reduced to the respective apparent times by the following rule; viz., * Find the sum and the difference of the natural sine of the latitude and the natural co-sine of the declination at the estimated times of rising and setting (rejecting the two right-hand figures from each term), and find 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 constant log. 1. 1761, and the proportional log. of the difference between the horizontal parallax and the sum of the horizontal refraction and dip of the horizon: the sum of these three logs., abating 4 in the index, will be the proportional log. of a correction, which, being added to the approximate time of rising and subtracted from that of setting, the respective apparent times of rising and setting will be obtained: thus, Let it be required to reduce the approximate times of rising and setting, as found in the last example, to the respective apparent times, the dip of the horizon being 4:50" Note.-The moon's horizontal parallax computed to the reduced estimated time of rising, is 59'6", and that at the reduced time of setting 58:40% 59:6% - 37:50% (33% + 4:50′′) = 21:16" Prop. log. Apparent time of moon's rising =10' 1"20: This is the proportional log, of 12 hours esteemed as minutes. 58:40 37:50: (33: +4:50%) 20:50 Prop. log. Apparent time of moon's setting = 21:39:37: Note. The direct method of solving this and the three preceding Problems, by spherical trigonometry, is given in some of the subsequent pages of this work. TABLES LI. AND LII. For computing the Meridional Altitude of a Celestial Object. Since it frequently happens, at sea, that the meridional altitude of the sun, or other celestial object, cannot be taken, in consequence of the interposition of clouds at the time of its coming to the meridian; and since it is of the utmost importance to the mariner to be provided at all times with the means of determining the meridional altitude of the heavenly bodies, for the purpose of ascertaining the exact position of his ship with respect to latitude, these Tables have therefore been carefully computed; by means of which the meridional altitude of the sun, or any other celestial object whose declination does not exceed 28 degrees, may be very readily obtained to a sufficient degree of accuracy for all nautical purposes, provided the altitude be observed within certain intervals of noon, or time of transit, to be governed by the meridional zenith distance of the object: thus, for the sun, the number of minutes and parts of a minute contained in the interval between the time of observation and noon, must not exceed the number of degrees and parts of a degree contained in the object's meridional zenith distance at the place of observation. And since the meridional zenith distance of a celestial object is expressed by the difference between the latitude and the declination when they are of the same name, or by their sum |