declination are of the same name, but if the latitude and declination are of different names, subtract the arc taken out from 180°, the remainder is arc 2. 12. Under log. cosec. 1, and log. sec. 1, just taken out, put the following quantities:: Under log. cosec. 1 put log. cos. S. Add together log. cosec. 1 and the two logs. placed beneath it; the sum will be the log. sin. arc 3. 13. Take out the log. sec. arc 3, and put it down twice, once under log. cos. D, and again a little to the right. 14. Add together the log. sec. 1, and the three logarithms beneath it; the result is log. cos. arc 4, which find in the Tables. 15. Under arc 4 put are 2, and take the difference in all cases when the line drawn through the places of the sun at the two observations will when produced not pass through the zenith and pole (that is, the difference must be taken, if it is seen that their sum would exceed 90°), otherwise take their sum; the result is arc 5. Lastly. Under log. sec. arc 3, already taken out, put log. sec. arc 5; the sum will be the log. cosec. of the required latitude. The arrangement on the paper of the logarithms to be taken out, as directed by the rule, will be better seen in the following blank form: and it would also facilitate the working out questions in other rules of Navigation if blank forms, similar to the one now given, were constructed on thick drawing paper by the student for each rule. EXAMPLES. (156.) Oct. 11, 1845, in latitude by account 54° N., and long. 83° 15′ W., the following double altitude of the sun The run of the ship in the interval was S. by W. 15 miles, index correction + 5' 10", and height of eye above the sea was 18 feet, required the latitude at the second observation. Ans., 53° 54′ N. (157.) March 20, 1845, in latitude by account 52° 10' N., and long. 55° 15' W., the following double altitude of the sun was taken. The run of the ship in the interval was N.W. by W. 10 miles, index correction O, and height of eye 20 feet, required the latitude at the second observation. Ans., 52° 27' N. (158.) Dec. 11, 1845, the following double altitude of the sun was observed. The run of the ship in the interval was E.N.E. 25 miles, index correction 1' 50", and height of eye 16 feet, required the latitude at second observation, the latitude by account being 60° S. and long. 79° 15′ W. Ans., 56° 59′ S. (159.) Nov. 10, 1846, in latitude by account, 35° 30′ N., long. 94° 30′ E. the following double altitude of the sun was observed. The run in the interval was S.S.E. 15 miles, index correction + 4' 10", and height of eye 18 feet, required the true latitude at the second observation. Ans., 35° 31' N. (160.) Oct. 30, 1846, in latitude by account 52° 10′ N., and long. 159° 45′ E., the following double altitude of the The run of the ship in the interval was S. by W. 1 mile, index correction + 3' 50" and height of eye above the sea was 20 feet, required the true latitude at second observation. Ans., 49° 56' N. (161.) March 5, 1846, in latitude by account 60° N., and long. 46° W., the following double altitude of the sun was observed. The run of the ship in the interval was S.W. by W. 15 miles, index correction + 2′ 10′′ and height of eye 20 feet, required the true latitude at second observation. A valuable extension of this problem has recently been made by Mr. Riddle, the head master of the Greenwich schools. It consists in finding the hour angle z PM with very little additional labour, and thence apparent time at the ship. For since in the triangle z P E, sin. h = sin. arc 3, sec. lat., we have only to add to sin. arc 3, already taken out of the table, the log. sec. lat. to determine the hour angle h, which will also be ship-apparent time, if PM or what it wants of 24 hours if A.M.; by applying the equation of time we obtain mean time at the ship. If therefore we know, by means of the chronometer, mean time at Greenwich, at the same instant, we can readily find the longitude in time by the following rule. Rule XLI. Rule for finding the longitude by means of the observations of the sun for latitude by double altitude. 1. Find the equation of time for the Greenwich date. 2. To the log. sec. lat. add log. sin. arc 3 already known, the sum will be log. sin. hour angle at the middle time between the observations. 3. If P.M. at ship at the middle time, this will also be ship apparent time. If A.M., subtract the hour angle from 24 hours, the remainder is ship apparent time. 4. Apply the equation of time with its proper sign, and thus get ship mean time. 5. To the mean time shown by chronometer at the middle time between the observations (found by taking half the sum of the times by chronometer at first and second observations), apply the error of chronometer, and thus get Greenwich mean time. 6. The difference between Greenwich mean time and ship mean time is the long. in time. If the Greenwich time is the least, the longitude is east, otherwise west. |