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 I. Suppose on the 6th of January 1822, in longitude 7° west by account, at 15.33' 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=136:38' P.M. 7° longitude W. = Oh.28' Reduced time, or time at Greenwich nearly =14h.1 Dis semi-dia. at midnight, 5th Jan. 1822=16'.20" Hor: paral. =59'.50% = 59.41" 12h; 4":: 2n. 1' : 0. 0" 12h : 9"::: 2h. I' : 0 1" D's semi-diam. at midnight = 16'.20"|Hor, parallax at midnight =594.50" D's semi-diam. at red. time = 16'.20" Hor. Parallax at red. time = 59'.49" D's augmentation (Tab. VII.)= + 7" 60 D's true semi-dia. at red. time = 16'.27" In seconds 3589 Dip. ) = } III. Di's observed altitude =240.18'.30"||Cos ) ''s apparent altitude =9.95897 )'s semi-dia. =16'.27" Horizontal parallax 3589 log = 3.55497 = + 12', 24' Parallax in altitude 3266 log = 3.51394 App. altitude ) 's centre =240.30.54" D's.correction 52.21" 32664 = 54.26 D's refr. (Tab. IV.) = 2'. 5" D's corrected altitude = 250.23'. 15' D's correction =52'.21" * The reduced time being so near to midnight, the proportions here made are superfluous, excepting to show the method of proceeding in other cases. VI. To find the true distance. alt. 249.30'.54" Log. sec. alts. 20°.38'.18" Nat. cos. 93582 63°45'33' Nat. cos. 44215 10:15168 O Common log. of 49367= 4.69344 *'s cor. alt. 45o. 8.15" Log. cos. 9.84844 D's cor. alt. 24°.23.15* Log. cos. 9.95589 4.69048. Natural number 49032 True dist. 630.13.0" Nat. cos. 45086 VII. To find the longitude. Dist. Jan. 5th 1822 at 12"=64o.26.42" || Earliest ephem. distance =649.26.42! Dist. Jan. 5th 1822 at 156=620.37'.38" | Computed distance -630.18'. O'M First difference 10.49'. 4" || Second difference 19.19.42 Then 10.49'.4" : 3h::10.13'.42" : 2h. 1'.38' True time at Greenwich = 14h. 1.38" Time at ship = 136.33'. 0" Difference = 01.28.38"– 70. 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 68o.9'.30"; the altitude of the sun's lower limb 310.48'.16"; the altitude of the moon's lower limb 23o.44'.16''; and the eye 18 feet above the level of the sea; required the true longitude. D's semi-diameter at noon, 26th April (Naut. Alm.)=16'.5", and at midnight=15'.58"; horizontal parallax at noon=59'.1", and at midnight = 58.36"; also the O's semi-diameter = 15.55". Answer. The reduced time is 2h.16'; y's true semi-diameter=16'.11"; horizontal parallax=58'.57"; apparent altitude of the D's centre 23°.56'.24"; and the correction of her altitude=51'46". The apparent altitude of the O's centre=32°.0'.8"; correction 1'23"'; apparent distance of the 'and D's centres 68.9.41.36"; true distance 68'.18'.47", and the true longitude =1059.37' East. (Naut. Alm. distance at noon being = 67°.6.15", and at III-689.44.1".) EXAMPLE III. Suppose on the 10th of July 1822, in longitude 116° W., by account, the following observations to be taken: Required the true longitude, the eye being 21 feet above the level of the sea. D's semi-diameter at noon, July 10th (Naut. Alm.) = 15'.44", and at midnight 15'.49"; horizontal parallax at noon =57'.46", at midnight 58'.4"; also the O's semi-diameter = 15.46". Answer. The reduced time is 5h.15'.33" July 10th; D's true semi-diameter 15'.52"; horizontal parallax 57'.53"; apparent altitude of her centre 19o.42.52"; and correction of her altitude 51'51". The apparent altitude of the sun's centre is 45°.32.29"; correction 50"; apparent distance of centres 1069.47'.13"; true distance 105°.51'.29", and the true longitude 116°.O'.2" W. (Naut. Alm. distance on the 10th at Ilth = 1070.2.33", and at VID=1050.28'.18”.) EXAMPLE IV. Alt. * Suppose on the 6th of May 1822, in longitude 720.10^E. by account, the following observations to be taken: Alt. )'s Dist. of * from Spica me D's far limb. 12”.37'.10" 190.10'. 20 200.39'.40 319.35.45" 31. 34. 10 31. 32. 50 31. 31. 35 81. 30. 40 Required the true longitude? the eye being 18 feet above the level of the sea. D's semi-diameter at noon, May 6th (Naut. Alm.) = 14'.42", and at midnight the same; horizontal parallax at noon 53'.58", and at midnight 53.57". Answer. The reduced time is 76.52.39", 's true semi-diameter=14'.48"; horizontal parallax=53'.58", apparent altitude of the D's centre = 18°.56.30", and her correction = 48'.19". App. altitude of the star 209.10',56", correction 2.35"; App. distance of centres 31°18'.12"; true distance 310.12.2017 The true longitude 729.9' East. (Naut. Alm. distance at V[b= 30°.16'.54", and at IXb= 310.45'.25"). (W) In all the preceding examples the watch is supposed to have been previously regulated; when that is not the case, the error of the watch must be found from observations of the altitudes of the sun or of a star, taken" either before or after that of the distance, as directed in Problems XIV. or X.V., pages 316 and 319. Or if the sun or star be sufficiently. distant from the meridian, the mean of the sun's or star's alti-tudes, taken at the same time as the distance is taken, together with the latitude of the place and the sun's declination, &c. may be used to correct the watch; with this corrected time proceed as before. EXAMPLE V. At sea, April 26th 1822, in latitude 47°39' N. and longitude 1789.20.30" West from Greenwich, by account, at 45.23' P.M. per watch not previously regulated; suppose the observed altitude of the sun's lower limb to be 250.57.10”, and that of the moon's lower limb 46°22'.32", and at the same time the distance of the sun's and moon's nearest limbs to be 750.33.37"; required the longitude? The eye being 18 feet above the level of the sea. Dis semi-diameter at midnight, April 26th (Naut. Alm.)= 15.58", and at noon April 27th=15'.51"; horizontal parallax at midnight = 58'.36"; and at noon April 27th=58'.11"; also the D's semi-diameter=15'.55'. Answer. The reduced time is 16h.16'.22"; D's true semidiameter 16.7"'; horizontal parallax 58'.28"; apparent altitude of her.centre 460.34'.36'', and correction of her altitude 39'.16". The apparent altitude of the sun's centre is 26°:9'.2"; correction 1'.48”; apparent distance of centres 760.5'.39"; true distance 75.51.20". The sun's declination (Naut. Alm.) on the 26th=130:25".19'' N. on the 27th=130.44.36' N. which reduced to the time and place of observation (B. 270.)=13° 38'.23" N.; with this declination, the O's true altitude 260.7'.14"; and the co-latitude 42°21', find the correct apparent time at ship (F. 316.)= 4h.24'.6"; and hence the true longitude is 1770.51'.45" west. (Naut. Alm. distance at XVb=750.11'16', and at XVIII = 76o.47'.7":) CHAP. XIII. OF THE FLUXIONAL ANALOGIES OF SPHERICAL TRIANGLES.* PROPOSITION I. (Plate IV. Fig. 3.) (X) A preparatory Proposition. CONSTRUCT a general figure as at Prop. XIX. Book III. Chap. I. (L. 151.) Thus, let a be the pole of the circle HGFE; F the pole of ABH; Ethe pole of cGI; and cthe poleof EDI. * See Cote's “ De Æstimatione Errorum in mixta Mathesi.” Cambridge, 1722. Simpson's Fluxions, vol. ij. page 278, et seq.- Crakelt's translation of Mauduit's Trigonometry, Chap. v. page 164,&c.— Traité de Trigonométrie, par M. Cagnoli, Chap. xix. page 310, &c -La Lande's Astronomy, Paris, 1792, vol. iii. page 588, et seq; or art. 3997 to 4051, &c. &c. |