Sun's declinat. at mean noon, Greenwich Sun's declinat. at the middle time Hence the north polar distance at the middle time = 78° 33′ 3′′.1 (=rs In the triangle zss' (P representing the perimeter), log. tan. zss' (28° 46′ 35′′′) In the triangle Psz, log. cos. Psz (27° 53′ 6′′) 9.9463973 -log. cotan. zs (48 34 44.5) 9.9456033 =log. tan. sq: = 45° 3' 9" 10.0007940 PS = 78 33 3 Pq= 33 29 54 9.9211149 log. cos. zs (48° 34′44′′.5) 9.8205877 +log. cos. Pq (33 29 54) +log. cos. sq (45 3 9) co. ar. 0.1 509133 =log. cos. PZ = 38° 39′ 17′′ 9.8926159 The required co-latitude. The same example by the indirect method in the "Requisite Tables," as above (art. 354.). Interval between the observations in time Half interval (= "half elapsed time ") log. half elapsed time (from the tables) log. secant. lat. by account (suppose 51° 30′) +log. secant. sun's declination (11° 26′ 56′′) (as whole numbers) =log. half difference ZPS, ZPS' ho. =2 55 12 as above = 1 27 36 = 0.42830 = 0.21458 = 10.20585 | 9696, log. 3.98659 4.62947-0 h. 49′ 12′′ 1 27 36 (designated L, and found in the col. headed "middle time.") =Smaller angle, zPs' in time, or the angle corresponding to O ho. 38 24 = 3.14625 Log. of the number represented by N = 2.93077 Whence N = 853 +Nat. sin. sun's altitude at s' (=49° 20′ 10′′) = 75854 -Nat. sin. sun's merid. alt. (=50 5 30) = 76707 358. In determining the longitude of a station or ship by means of the distance of the moon from the sun or a star, it is necessary to have the altitudes of the two luminaries; and since the circumstances may be such as to prevent the altitudes from being observed, the following problem, by which, when the latitude of the station and the time there are known, one or both may be computed, is introduced. PROB. IX. To find the altitude of the sun, the latitude of the station and the time of day being given. From the given time there must be obtained the sun's horary angle and also his declination, or polar distance; and then, in the triangle ZPS, there are given PZ the colatitude, ZPS the hour angle, and Ps the north polar distance; to find zs, the true distance of the sun from the zenith. Let fall zt perpendicularly on PS; then, by Spher. Trigo., in the triangle Pzt (art. 62. (ƒ')), Rad. cos. P=cotan. PZ tan. Pt, and in the triangle P z § (3 Cor. art. 60.), Cos. PZ cos. ts=cos. Pt cos. ZS. Z Thus zs, the complement of the sun's true altitude, is found. But the rule frequently given in treatises of navigation is obtained from the formula cos. Zs=cos. ZPS sin. ZP sin. PS + cos. PS cos. PZ (art. 60.), by transforming it, as before shown (art. 354. (B)), into cos. ZS cos. (PS-PZ)-vers. sin. ZPS sin. PS sin. PZ. The term cos. (PS-PZ) is the sine of the sun's meridional altitude, and log. vers. sin. ZPS is designated log. rising, which may be found in the "Requisite Tables" (tab. XVI.) for the given hour angle in time: therefore, the natural number corresponding to log. rising+log. cosin. latitude + log. cosin., sun's declination being represented by N, we have the sine of the sun's meridian altitude-N sine of the required altitude. The processes are exactly similar when the celestial body is the moon or a star. The zenith distance or altitude found by either of the methods is not affected by parallax or refraction; and since, in the problem for finding the longitude, the apparent zenith distance or altitude is required, it becomes necessary to apply to the computed zenith distance, for example, the difference between the sun's refraction and parallax. Now the sun's refraction is greater than his parallax, so that the sun appears always elevated above his true place; therefore that difference must be subtracted from the value of zs above found in order to have the apparent zenith distance. The apparent place of a fixed star or planet is also higher than the true place; but the apparent place of the moon is lower, because the moon's parallax is greater than her refraction; therefore the difference between her refraction and her parallax in altitude must be added to the computed value of zs in order to have the apparent zenith distance of the moon. Ex. 1. At Sandhurst, Sept. 4. 1843, at 9 ho. 35′ 55′′ A. M., mean time, the longitude of the station from Greenwich being about 3', in time, and its latitude equal to 51° 20′ 33′′; it is required to find the sun's altitude. Time before noon at ho. / 54.8 9 36 49.8; hence the apparent time at Greenwich is 9 ho. 39′ 49′′.8, or 2 ho. 20′ 10′′.2 before apparent noon. 12 2 23 10.2 35° 47' 33", the sun's hour angle (=ZPS). The sun's declination apparent noon, Gr.7 20 44.2 (N. Naut. Alm.) +Variation for 2 ho. 20' 10".2 2 9.3 Sun's declination at the given instant 7 22 53.5 90 Sun's north polar distance (=Ps) - 82 37 6.5 In the spherical triangle ZPS, log. cos. ZPS (35° 47′ 33′′) 9.9090956 - log. cotan, PZ (38 39 27 ) 0.0969459 log. cos. PZ + log. cos. ts + log. cos. Pt, co. ar. =log. cos. z s = 52° 55′ 39", the true zenith distance of the sun's centre. The same example, by the rule in the "Requisite Tables." Time before noon as above 2 ho. 23' 10", log. rising 4.27612 +Log. sin. Ps (82° 37′ 6′′) +Log. sin. PZ (38 39 27) 9.99638 or nat. sin. sun's meridian alt. 60282 Nat. cos. 52° 55′ 40′′, the sun's true zenith distance. Ex. 2. At Sandhurst, April 13. 1840, at 10 ho. 5′ 25′′.6 P. M. mean time, or 10 ho. 5′ 4′′.3 apparent time, the altitude of ß, Geminorum (Pollux), was required. =49° 31′ 0′′, the star's true zenith distance. Ex. 3. At Sandhurst, April 13. 1840, at 10ho. 5′ 25′′.6 mean time, or 10ho. 5' 4".3 apparent time, the altitude of the moon's centre was required. Computation of the moon's right ascension and declination. The estimated difference of longitude between the station and Greenwich being 3', as above, the mean time at Greenwich was 10 ho. 8′ 25′′.6. ho. The moon's right asc. at 10 ho. mean time Gr. 11 23 57.2 (Naut. Alm.) +Variation for 8' 25".6 since 10 ho. Moon's right asc. at the given instant Moon's declin. at 10 ho. mean time Gr. -Variation for 8' 25".6 since 10 ho. 15.7 11 24 12.9 2 26 3.8 (N. Naut. Alm.) 2 5.9 Moon's declinat. at the given instant -= 2 23 57.9 Moon's north polar distance (= PS) 90 = 87 36 2.1 = In the triangle ZPS (s in the figure, representing the true place of the moon), 359. The longitude of a station may be determined by various methods which will be presently explained, but that of a ship which is beyond the sight of land can be ascertained astronomically only by what are called lunar observations, or by obtaining the mean time of the day or night from the observed altitude of a celestial body, and comparing it with the time shown by a chronometer which is supposed to express, at every instant, the mean solar time at Greenwich. The principle on which the processes are founded is the same in all cases, and consists in a determination of the time at Greenwich and at the station or ship at the same physical instant; the difference between those times being equal to the difference, in longitude, between Greenwich and the ship. This is obvious; for since apparent time at any place is expressed by the angle (in time) between the meridian of the place and a horary circle passing through the sun at the instant, the difference between the apparent times, and consequently between the mean times, at two places at the same instant, must be equal to the angle, in time, between the meridians of the two places. Whatever phenomenon, therefore, constitutes a signal which may be observed at two places at the same instant becomes a means of determining the difference between the longitudes of those places; and many such phenomena present themselves in the heavens, as the eclipses of the sun and moon, the immersions and emersions of Jupiter's satellites, &c. The moon's proper motion from west to east, is more rapid than the like motion in the sun or a planet, and the fixed stars have no such motion; hence, the angular distance between the moon and any other celestial body varying continually, the occurrence |