greatest altitude was taken. Log. rising = Nat. co-versed sine of the greatest alt. = 4.368450 703190. Log.ratio=0. 226446 Sun's mer.zen.dis. 71:54 0N. Nat.V.S.=689322 = Latitude of ship= 49:56:38" north. And, since this latitude differs so much from that by account, it will be necessary to repeat the operation. only 52 seconds from the last, it may, therefore, be esteemed as the true latitude. Note. The correct latitude, by spherical trigonometry, is 49:56:0 north. Example 2. At sea, April 14th, 1825, in latitude 43:47 S., by account, and longitude 60:25 E., at 23 20:40: apparent time, the observed altitude of the sun's lower limb was 35°54', and at 2'10" 10: apparent time, April 15th, the observed altitude of that limb was 28:42:15", and the bearing of the sun's centre, by azimuth compass, N.W. & N.; the height of the eye above the level of the horizon was 24 feet, and the ship's course during the elapsed time S.W., at the rate of 9 knots an hour; required the latitude of the ship at the time of observation of the greater altitude? Sun's bearing at 2d observation = N.W. N., or 3 points. Contained angle = S.W.. Time elapsed between the observations = or = 4 points. As 19: 2:49:30:: 25:26" the distance made good between the observations. Now, to course 74 points, and distance 25 miles, the difference of latitude is 3.7 miles, the reduction of altitude; which is additive to the least altitude, because the contained angle is less than 8 points, and the observation was made in the afternoon. First observed altitude of the sun's lower limb = 35:54 0N. 15.58" } Diff. = 4.42 +11.16 Second observed altitude of sun's lower limb = 28:42:15" Latitude by account = 43:47' 0"Log. sec. 10. 141486 Sun's mer. z. dist. 53:15' 4"S.Nat.ver. S.=401691 Lat. of the ship = 43:33:30 S. But, since this latitude differs so much from that by account, it becomes necessary to repeat the operation. Time from noon when the greatest alt. was taken = 0:3748! 4.132610 Nat. co-vers. sine of the greatest altitude = 411255 Log.ratio=0.146102 Natural number = Log. rising = Sun's mer.z.dis.=53:14:30 S. Nat.V. S.=401561 Sun's red. dec. 9.41.34 N. Latitude 43:32:56 south. And, since this latitude only differs 34 seconds from the last, it may be considered as being the latitude of the ship at the time of observation of the greater altitude. The correct latitude, however, by spherical trigonometry, is 43:29:30% south: hence the method by double altitudes, even after repeating the operation, differs from the truth by 3 minutes and 26 seconds. Note. The method of finding the latitude by double altitudes, being a very tedious and indirect operation, and generally a very inaccurate one, unless the limitations pointed out in the remarks (page 342) are strictly attended to, no notice, therefore, would have been taken of it in this work, were it not for the purpose of giving the most ample illustration of the general use of the Tables. And, notwithstanding what has been said in favour of double altitudes by theoretical writers, this method of finding the latitude at sea is evidently far from being one of the most advantageous in practical navigation for the operation, besides being rather circuitous, requires a considerable portion of time to go through with it correctly; and, after all, it frequently happens, that although every seeming precaution has been taken, the mariner's hopes are disappointed in the result. We will now proceed to a more direct and universal method of finding the latitude, either at sea or on shore. PROBLEM VIII. Given the Altitudes of two known fixed Stars observed at the same instant, at any Time of the Night, to find the Latitude of the Place of Observation, independent of the Latitude by Account, the Longitude, or the Apparent Time. In the preceding problems for finding the latitude (the two last excepted), the meridional altitudes of the celestial objects were the principal elements under consideration : however, since it frequently happens that, in conse quence of the interposition of clouds, or other causes, the altitudes of the heavenly bodies cannot always be taken at their respective times of transit, the present problem is, therefore, proposed, which possesses the peculiar advantage of enabling the mariner to determine the position of his ship, with respect to latitude, by the altitudes of two known fixed stars, observed at the same instant and at any hour of the night, either before or after their passing the meridian, and independent of the latitude by account, the longitude, or the apparent time of observation. Nor will the mariner, in this method, be subjected to the necessity of repeating the operation, or of puzzling himself with a variety of cases and corrections, in finding an approximate latitude. RULE. Let the altitudes of two stars be observed, at the same moment, whose computed spherical distance asunder is given in Table XLIV.; and let those observed altitudes be reduced to the true by Problem XVII., page 327. Take the right ascensions and declinations of the two stars, and also their computed spherical distance, from Table XLIV., and let these be reduced, respectively, to the night of observation. Let the star which is adjacent, or nearest to the elevated pole, be distinguished by the letter A, and that which is remote, or farthest, by the letter R.-Now, To the log. sine of the tabular distance between the two stars, add the log. secant of the declination of the star A, and the log. half-elapsed time of the difference of right ascension; the sum, rejecting 20 from the index, will be the log. half-elapsed time of arch the first. From the natural co-versed sine of the altitude of the star A, subtract the natural co-versed sine of the sum of the tabular distance between the stars and the altitude of the star R, and find the logarithm of the remainder; to which add the log. co-secant of the tabular distance, and the log. secant of the altitude of the star R;-the sum of these three logarithms, abating 20 in the index, will be the log. rising of arch the second; the difference between which and arch the first, will be arch the third. To the log. rising of arch the third, add the log. co-sines of the declination and altitude of the star R, and the sum, abating 20 in the index, will be the logarithm of a natural number; which, being added to the natural versed sine of the difference between the altitude and declination of the star R, when its polar distance is less than 90%, or to that of their sum when it is more than 90%, the sum will be the natural co-versed sine of the latitude. |