EXAMPLE VI The apparent distance of the moon's centre from that of the sun was 70°.8'.23" at a time when the apparent altitude of the sun's centre was 22°.14′.55", and the apparent altitude of the moon's centre 80°.52′.22′′; the sun's correction was 2′.11", and. the moon's correction 8.47"; required the true distance of their centres? Answer. 70°.7.56".. EXAMPLE VII. The apparent distance of the sun and moon's centre was 630.5.46", the apparent altitude of the sun 45°.9.12", that of the moon 24°.29.40"; the sun's correction 57", and the moon's. 47.47"; required the true distance of their centres ? Answer. 62°.44. EXAMPLE VIII The apparent distance of the sun and moon's centres was 720.21.40", the apparent altitude of the moon 19°.19′, that of the sun 25°.16'; the moon's correction being 50'.40", and the sun's 1'.53"; required the true distance of their centres? Answer. 72°.3.51" PROBLEM XVIII. (U) The latitude of a place and its longitude by account, the distance between the sun and the moon, or the moon and a star* in the NAUTICAL ALMANAC,. being given, to find the correct longitude. *The principal stars used for finding the longitude at sea are the following: a Arietis in the head of Aries. This is a small star, without the zodiac, and cannot be readily found and applied by the generality of persons; 22° to the right hand of the Pleiades. appears about 2. Aldebaran in the Bull's eye is easily distinguished by its largeness, colour, and position to the other stars, being half way between the Pleiades and the star which forms the western shoulder of Orion. 3. a Pegasi or Markab, about 44° to the right hand of Arietis, a line drawn in imagination from the Pleiades through Arietis will pass through a Pegasi.—The constallation Pegasus is very remarkable, the three principal stars in it, with the head of Andromeda, form a large square, of which the four corner stars are all of the second magnitude. 4. Pollux, a little northward of Aldebaran, and about 45° towards the left hand, there are two stars nearly together: the right hand one is Castor, the left hand one Pollux. 5. Regulus, about 38° S. E. of Pollux, is easily distinguished by being the southern-most of four bright stars resembling the letter Z inverted, it lies to the north-east of Aldebaran. Z I. Turn the longitude, by account, into time (X. 265 and Z. 266.) and add it to the astronomical time* where you are if west longitude, but subtract it from that time if east, and you will have the astronomical time at Greenwich nearly, which call reduced time. II. In page VII. of the Nautical Almanac, for the given month and day, take out the moon's semi-diameter, and horizontal parallax, for the noon and midnight between which the reduced time falls, subtract the less semi-diameter from the greater, and the less parallax from the greater. Then, as 12 hours are to the first difference, so is the reduced time to a fourth number, which must be added to the moon's semi-diameter if the tables be increasing, but subtracted if they be decreasing; to the sum or difference add the augmentation (Table VII.), and you will have the moon's true semi-diameter at reduced time. Again, As 12 hours are to the second difference, so is the reduced time to a fourth number, which must be added to or subtracted from the horizontal parallax at the nearest noon, or midnight, preceding the reduced time, according as the tables are increasing or decreasing, and it will give the horizontal parallax at reduced time. (See Example II. page 272.) III. Clear the observed altitude of the moon of dip (TableV.) and semi-diameter †, and you have the apparent altitude of her centre: to the cosine of the moon's apparent altitude, add the Mogarithm of the horizontal parallax at reduced time in seconds; the sum, rejecting 10 from the index, will be the logarithm of 6. Spica, or a Virginis, a white sparkling star, about 54° south-east of Regulus. 7. Antares. The arc of a great circle passing through Regulus and Spica Virginis -east-south-east, or to the left hand of Regulus southward, will pass through Antares, which is about 45° from Spica Virginis. 8. Fomalhaut, about 45° south of a Pegasi. 9. a Aquila, 47° westward, or to the right hand of a Pegasi. The distance between the moon and the sun, and between the moon and the nine stars above described (being near her path), are given in the VIIIth, IXth, Xth, and XIth pages of the Nautical Almanac, to every three hours apparent time, by the meridian of Greenwich. The observer who uses the Nautical Almanac, and certainly no observer ought to be without it, is under the necessity of taking the distance of one or other of the above stars from the moon. The distances calculated in the Nautical Almanac afford, perhaps, the readiest method of knowing the star from which the moon's distance ought to be observed. For the sextant being fixed to that distance, and the moon found upon the horizon glass, there is nothing more to do than to look to the east or west of the moon, according as the distance corresponds to the VIIIth and IXth, or Xth and XIth pages of the Nautical Almanac, guiding the sextant in a line with the moon's shortest axis. * By a late order from the admiralty, the Nautical Day begins at midnight, in consequence of which the Navy Logs correspond with the civil reckoning; but the tables in the Nautical Almanac are adapted to Astronomical time, see the note, Jage 267. + Viz. take their difference and add it to the observed altitude of the moon's lower limb; or take their sum and subtract it from the observed altitude of the moon's upper limb, according as the lower or upper limb has been observed. the moon's parallax in altitude in seconds (T.96.), from which take the refraction of the moon in altitude (Table IV.), the remainder will be the moon's correction. The moon's correction must always be added to the apparent altitude to obtain the true altitude. IV. Additional Preparation for the Sun and Moon. Clear the observed altitude of the sun of dip and semidiameter *, and you have the apparent altitude of his centre. From the refraction of the sun's altitude (Table IV.) take his parallax in altitude (Table VI.), and you have the correction of the sun's altitude; which must always be subtracted from the apparent altitude to obtain the true altitude. V. To the observed distance of the sun and moon's nearest limbs, add their semi-diameters at reduced time, and the sum will be the apparent distance of their centres. IV. Additional Preparation for the Moon and a Star. From the star's observed altitude take the dip of the horizon (Table V.), the remainder will be its apparent altitude. The refraction of a star (Table IV.) is the correction of its altitude, and must always be subtracted from the apparent altitude to obtain the true altitude. V. To the observed distance of the moon from a star, add the moon's semi-diameter at reduced time, the sum will be the apparent distance; if the farthest limb was observed, subtract the semi-diameter. VI. To find the true distance. With the apparent altitudes, their corrections, and the apparent distance, find the true distance by the general rule. (T. 333 and 334.) Note. If the watch has not been previously regulated, the true time must now be found with the mean altitude of the sun or star, and the latitude of the ship, as in Prob. XIV. and XV. taking care to calculate the sun's declination to the reduced time. (B. 270.) VII. To find the longitude. In page VIII, IX, X, or XI, of the Nautical Almanac for the given month and day, look for the computed distance between the moon and the other object; if you find it there exactly, the time at Greenwich stands at the top of the column; but if you do not find it exactly, take the nearest distance to it both less and greater; take their difference, and likewise the difference between the computed distance and the earliest Ephemeris distance. * Viz. take their difference, and add it to the observed altitude of the O's lower limb; or take their sum, and subtract it from the observed altitude of the O's upper limb, according as the lower or upper limb has been observed. Then, as the first difference is to 3 hours, so is the second difference to a fourth number, which being added to the time standing over the earliest Ephemeris distance, will give the true time at Greenwich. The difference between the ship's time and the time at Greenwich (turned into degrees) will be the longitude of the place, where the altitude of the object was taken for determining the true time. If the ship's time be greater, the longitude is east; if less, west. EXAMPLE 1. Suppose on the 6th of January 1822, in longitude 7° west by account, at 1.33' A. M. per watch well regulated, the distance of the moon's farthest limb from the star Regulus to be 64°.2'; the altitude of the moon's lower limb 24°.18',30"; the star's altitude 45°.13'.15"; and the eye 18 feet above the surface of the sea, required the true longitude? I. Time per watch at ship 5th of January 13.33' P.M. 7° longitude W. Reduced time, or time at Greenwich nearly =Ob.28′ 14h.1 II. De's semi-dia. at midnight, 5th Jan. 1822-16.20" Hor. paral. 12h 9: 2h. I': 0′ 1′′ =59′.50" D''s semi-diam. at midnight = 16'.20"|| Hor, parallax at midnight D's semi-diam. at red. time =16′.20" Hor. parallax at red. time D's true semi-dia. at red, time = 16′.27" III. =9.95897 =240.18'.30"|| Cos D's apparent altitude D's observed altitude App. altitude's centre D's corrected altitude The reduced time being so near to midnight, the proportions here made are superfluous, excepting to show the method of proceeding in other cases. Diff. of true alts. 19°.45'.0" Nat. cos. 94118 True dist. 63°.13'.0" Nat. cos. 45086 VII. To find the longitude. Dist. Jan. 5th 1822 at 12h=64°.26'.42" 4.69344 9.84844 9.95589 4.69048 Then 1°.49'.4" : 3h::10.13.42′′ : 2h. 1′.38" Difference 0.28′.38′′ 7°. 9′.30′′ the longitude west of Greenwich, because Greenwich time exceeds the ship's time. EXAMPLE II. Suppose on the 26th of April 1822, in longitude 105° E. of Greenwich, by account, at 9h.16" P.M. the distance between the sun and moon's nearest limbs to be 68°.9'.30"; the altitude of the sun's lower limb 31°.48'.16"; the altitude of the moon's lower limb 23°.44'.16"; and the eye 18 feet above the level of the sea; required the true longitude. |