height of eye 12 feet, time by chronometer 11h 58m 29°, slow on October 13th, 1871, 1h 52m 53, and losing daily 69.

Ex. 14. 1872, February 11th, at about 9h A.M. at ship, latitude 53° 12' N., longitude by account 5° 30′ W., observed altitude sun's L.L. 12° 10', index correction — 49′′, height of eye 12 feet, time by chronometer 10h 22m 228 fast on December 1st, 1871, 38m 59%, and gaining daily 8.3.

Ex. 15. 1872, October 26th, at about 8h 30m A.M. at ship, latitude 28° 10′ N., longitude by account 62° 45′ W., observed altitude sun's U.L. 25° 32′ 20′′, height of eye 17 feet, time by chronometer oh 54m 6s, fast on September 4th, 30m 6s, and losing daily 25.

Ex. 16. 1872, February 6th, at about 4h 20m P.M. at ship, latitude 6° 58′ N., longitude by account 69° E., observed altitude sun's U.L. 21° 43′ 40′′, height of eye 18 feet, time by chronometer 11h 40m 263, slow on January 20th, 17m 42a, and losing daily 54.

Ex. 17. 1872, May 1st, at about 4h 20m P.M. at ship, latitude 21° 8' N., longitude by account 140° 45′ E., observed altitude sun's L.L. 28° 5' 30", index correction +2′ 50′′, height of eye 16 feet, when a chronometer showed 6h 50m 2984, which was 10m 12s slow for mean noon at Greenwich, December 31st, 1871, and gaining 35-3 daily.

Ex. 18. 1872, April 21st, at about 3h 50m P.M. at ship, latitude at noon o° 19' N., longitude account 179° 40′ E., observed altitude sun's U.L. 32° 21′ 10′′, index correction 1' 10", height of eye 12 feet, time by chronometer 3h 44m 1s, slow on November 14th, 1871, 9m 7, and gaining daily 16, course since noon S. W. by W. (true), distance 56 miles required the longitude at the time of observation and also at noon.

Ex. 19. 1872, August 21st, at about 8h 20m A.M. at ship, latitude at noon o° 20' S., longitude by account 179° 50' W., observed altitude sun's L.L. 33° 49′, index correction+2' 10", height of eye 15 feet, time by chronometer 8h 14m 0s, slow on March 13th, 4m 40%, and losing daily 1925, course S. W. by W., 36 miles.

Ex. 20. 1872, March 20th, at about 8h A.M. at ship, latitude o° longitude by account 88° 30' W., observed altitude sun's L.L. 28° 50' 10", index correction + 1', height of eye 23 feet, time by chronometer 2h om, fast on Greenwich mean noon February 28th, 2m 88, and gaining daily 0o 3.

Ex. 21. 1872, June 15th, at about roh om A.M. at ship, latitude 29° 10' S., longitude by account 165° E., observed altitude sun's L.L. 30° 40', height of eye 25 feet, time by chronometer 10h 59m 208, fast on Greenwich mean time 4m 35% on March 20th, and losing daily 80.

Ex. 22. 1872, March 21st, at about 9h 1om A.M. at ship, latitude 41° 32' S., longitude by account 92° 40′ W., observed altitude sun's L.L. 32° 59', height of eye 20 feet, time by chronometer 3h 28m, fast on Greenwich mean time 1m 40$ on December 20th, 1871, and losing daily oo'9.

• When the latitude in at noon is given, the latitude in at time of observation must be found by means of the course steered and distance sailed. The diff. of lat. from noon is to be named North or South, according as the ship at the time of observation is North or South of noon. When the longitude

is found as above, the diff. of long. between the ship at the time of observation and noon must be applied to find the longitude at noon. The diff. long. is to be named East or West, according as the ship at the time of observation is East or West of its position at noon.

VARIATION BY AN AZIMUTH.

THE AZIMUTH OF A CELESTIAL BODY is the arc of the horizon intercepted between the North or South point (according as it is the North or South pole, which is elevated), and the circle of azimuth passing through the place of the body. Or it may be defined to be:-The angle at the zenith contained between the vertical circle passing through the elevated pole (the meridian), and the vertical circle passing through the object. Azimuths, usually reckoned from the North or South point, eastward and westward, from 0° to 180°, and sometimes the intersection of the horizon with that part of the meridian which is on the polar side of the zenith is taken as the zero point.

Azimuth and altitude are the horizon co-ordinates for describing the points of the celestial concave relatively to the position of an observer on the earth's surface. When a body is in the horizon, the element used to define its position is the "amplitude," which is the compliment of the azimuth in this case.

TRUE AZIMUTH is the bearing of an object from the true North or South point, and is the azimuth found by calculation from the observed altitude or hour-angle of the body. It is in general simply called "the azimuth," but it is thus qualified as the "true azimuth" to distinguish it from the magnetic azimuth, which is the bearing of the object from the compass North or South point, and which is found by direct observation with an instrument carrying a magnetic needle. The difference between the true and magnetic azimuth gives the correction (variation and deviation combined).

Given the latitude, altitude, and declination of an object, to find the true azimuth.

RULE LXX.

1°. Add together the polar distance, the latitude, and the altitude, take half the sum, and take the difference between the half sum and the polar distance.

2°. Add together the log. sec. of latitude, the log. sec. of altitude, the log. cosines of the half sum and remainder; the sum (rejecting tens) is log. sine square of true azimuth (Table 69, Raper). Or half the sum of the four logs. is the log. sine of half of the true azimuth, which take out of the table (Table 24, Norie), and double it: the result is the true azimuth.

3. Mark the true azimuth N. when the latitude is S., but mark it S. when the latitude is N.; also mark it E. when it is A.M., or the altitude is increasing, but W. when it is P.M., or the altitude is decreasing.

(a) When latitude is o, if declination is N. reckon the azimuth from the South; if declination is S., reckon the azimuth from the North.

(b.) When both latitude and declination are o, the object moves on the prime vertical, or is E. while the altitude is increasing, and W. while the altitude is decreasing.

EXAMPLES.

Ex. 1.

Given the latitude 47° 46′ S.; declination 22° 27′ 22′′ (or polar distance 67° 32′ 38′′); true altitude 26° 44′ decreasing, or (being P.M.)

The logs. are taken out in these examples only to the nearest minute, when the seconds are less than 30" reject them, but if more than 30" increase the minutes (') by one, and the arc corresponding to the half sum of the four logs is taken out to the nearest half-minute.

In each of the following examples it is required to find the true azimuth:- (The sign+means A.M., and the sign

means P.M.)

Given the true bearing and compass bearing, to find the variation.

RULE LXXI.

1o. Reckon the true and magnetic azimuth from the same point of the compass, and take their difference for the variation, except when one azimuth is reckoned towards the East, and the other is reckoned towards the West, in which case take the sum.

(a) If one of the azimuths be expressed from the North, and the other from the South, take either of them from 180°, and it will then be reckoned from the same point as the other (see examples 7 and 8).

(b) If the bearing by compass be reckoned from East or West towards North or South, take it from 90°, and reverse the position of the letters, or add 90°, and it will then be expressed from the opposite point to that from which it is reckoned when taken from 90° (see example 5).

(c) When the magnetic azimuth is either East or West, it is reckoned as 90° from North or South, according as the true azimuth is North or South.