CHAP. VIII. Of finding the Longitude at Sea or Land, BY An Observation of the Distance between the Moon and a Star, not used in the Nautical Almanac. INTRODUCTION. THE distances between the Moon and the Sun, and ten of the brightest fixed stars nearest the Moon's påth, are given in the Nautical Almanac, when the Moon is in a proper position with respect to those objects. It may, however, happen, that some other star is in a more favourable position, for observation, than any of those given in the Ephemeris; the ship's longitude might therefore be determined by such an observation, when perhaps it would otherwise be impossible. The difference of longitude between the Moon and the star with which it is to be compared, must not, however, be less than a certain quantity, otherwise the Moon's relative motion will be too slow, for the purpose of determining the longitude with that degree of precision with which it may be ascertained when the Moon is compared with one of the stars employed in the Ephemeris. Several other elements enter also into this method, which makes it necessary to treat of it in a particular manner. PROBLEM To find that Quantity which the Difference between the Longitudes of the Moon and the given Star must exceed, that the Moon's relative Motion may not be too much diminished. RULE. Enter Table XVII. with the difference between the latitudes of the Moon and the proposed star, if both latitudes are of the same name; but with their sum, if of contrary names, and find the corresponding quantity. 2D2 quantity. Now, if this quantity is less than the difference between? the longitudes of the Moon and given star, their distance, may be observed, for the purpose of finding the longitude of the place of observation; but if it is greater than that difference, the Moon's relative motion will be too slow to derive any benefit for the above purpose, from such an observation. EXAMPLE. Is it proper to compare the Moon with a Orionis, in order fo determine the longitude, November 21, 1792, about 10h. P. M. reduced time? Now in Table XVII. opposite to the diff. of latitude 17° 7′ is 33° 48'; which, being less than the diff. of longitude 105° 48', therefore shows that this star may, in the present case, be employed for the purpose of finding the longitude. PROBLEM Given the true Distance between the Moon and a fixed Star, together with the Latitude of each, to find the Moon's Longitude. RULE. To the true distance, add the latitudes of the Moon and star, and find the difference between the half sum and distance. Now to the log. secants of the latitudes of the Moon and star, add the log. co sines of the half suin and difference, if the latitudes are of the same name, or the log. sines if of a contrary name; half the sum of these four logs. will be the log co-sine, or sine of half the difference of longitude, according as the latitudes of the Moon and star are of the same, or of a different name. To the apparent longitude of the star, add the difference of longitude, if the Moon be east of the star, otherwise subtract it, and the sum or remainder will be the true longitude of the Moon. REMARK. It will be sufficient in most cases to take out the latitudes of the Moon and star to the nearest minute. EXAMPLE. Let the true distance between a Orionis and the Moon's center be 105° 105° 26' 14", November 21, 1792, at 10 hours reduced time. Required the Moon's longitude? 6 Difference of long. 105 46 Mean longitude of a Orionis, Nov. 21, 1792, Aberration Apparent longitude of a Orionis Difference of longitude, the D being west of Moon's longitude Sine PROBLEM III. Given the True Longitude of the Moon, to find the Apparent Time at Greenwich. RULE. From the Nautical Almanac, take four longitudes of the Moon, two of which immediately preceding, and two following the given longitude. Find the difference between each pair successively; find also the second difference, and let their mean be taken. Now, to the constant log. 2.857332, add the ar-co. of the log. of the variation of the Moon's longitude in 12 hours, reduced to seconds, and the log. of the difference between the given and preceding longitudes in seconds; the sum, rejecting radius, will be the log. of the approximate time in minutes, to be reckoned from the preceding noon or midnight. Take the equation of second difference from Table xxxvII.answering to the approximate time, and the mean second difference, with which enter Table xxxvIII. at the top, and find the equation corresponding thereto, and the Moon's motion in longitude in 12 hours, in the side column; which being applied to the approximate time, by addition or subtraction, according as the first difference of the Moon's motion is increasing, or decreasing, will give the apparent time at Greenwich. EXAMPLE. November 21, 1792, the longitude of the Moon deduced from observation was 11s 10° 5' 50". Required the apparent time at Greenwich? 20d. at midn. 10° 27° 22′ 0′′ D's 21 at noon 11 4 18 52 Diff.6° 56' 52′′ Given the Latitude and estimate Longitude of the Place of Observation, the Distance between the Moon and a fixed Star, together with the Altitude of each, to find the true Longitude of that Pluce. RULE. With the given latitude, the corrected altitude, and declination of the star, compute the apparent time of observation by Prob. vii. page 133. Reduce this time to the meridian of Greenwich, and find the Moon's semidiameter and horizontal parallax agreeable thereto. With the true and apparent altitudes, and the apparent central distance, compute the true distance by Prob. 1. page 150; with which, and the latitudes of the Moon and star, find the difference of longitude by Prob 11. page 204, and from thence find the apparent time at Greenwich, by the last problem. Now the difference between the apparent time at the place of observation, and that at Greenwich, will be the longitude of the place in time, as formerly. EXAMPLE. |