In Fig. 2, from X drop a perpendicular on the meridian, which mark X'. then in the right-angled spherical triangle P X' X, right-angled at X', given P and side P X to find side P X'.

In the same triangle, given ▲ P and side P X, to find side X' X.

In the right-angled spherical triangle Z X' X, given Z X and X'X_to find Z X'

Example. December 16th, 1890; a.m. at ship; lat. by D.R. 50° 47′ N. ; long. 20° 4′ W.; time by chronometer 8h. 33m. 42s., which was 11m. 385. fast on mean time at Greenwich; the observed altitude of Spica (a Virginis) near the meridian was 28° 6' bearing south; height of eye 24 feet; required the latitude by the reduction to the meridian.

Example.-July 3rd, 1890; a.m. at ship; lat. by D.R. 54° 2′ S.; long 60° 24′ W.; the observed latitude of a Centauri near the meridian beler the pole was 25° 1′ 50′′; height of eye 26 feet; mean time at Greenwich by chronometer (corrected for error and rate) was July 2d. 23h. 4m. 16s.: required the latitude by the reduction to the meridian.

Reduction

=

S

Fig. 4.

E

(Z X' Z X) to be subtracted from the true altitude because the altitude is least when on the meridian below the pole.

N.B. The sum of lat. and decl. subtracted from 180° is the mer. zen. dist. by D.R. for an observation below the pole; the problem re-worked may be a few seconds more exact.

Or thus: Formula

Log. nat. no. = log. rising P + log. cos. + log. cos. d - 20.

Take the difference between natural number and natural sine, because the observation is below the pole.

Examples for Practice

Example 1.-February 24th, 1890; p.m. at ship; lat. by D.R. 58° 58′ S. ; long. 56° 8′ W.; the observed altitude of the sun's lower limb when near the meridian was 39° 36′ bearing north; height of eye 26 feet; time by watch 1h. om. 20s., which had been found 20m. 4s fast on apparent time at ship; the difference of longitude made to eastward was 32' after the error on apparent time was determined; required the latitude by the reduction to the meridian.

Ans. Lat. 58° 52′ 8′′ S.

Example 2.-May 10th, 1890; at 9h. 34m. 40s. a.m. by watch at ship; lat. by D.R. 38° 8′ S.; long. 22° 40′ E.; the observed altitude of the sun's lower limb when near the meridian was 33° 32′, observer south of sun; height of eye 25 feet; the watch had been found 1h. 51m. 4s. slow on apparent time at ship, but ship had made 16' difference of longitude to eastward since the error for time had been determined; required the latitude by the reduction to the meridian.

Ans. Lat. 38° 6′ 10′′ S.

Example 3.-November 10th, 1890; p.m. at ship; lat. 59° 50′ N. ; long. 32° 46′ W.; the observed altitude of the sun's lower limb near the meridian was 11° 53′ 20′′, bearing south; height of eye 23 feet; time by chronometer 3h. 31m. 26s., which had been found 2h. 30m 56s. fast on apparent time at ship; but the ship had made 24′ difference of longitude to westward since the error for time had been ascertained; required the latitude.

Ans Lat 59° 48′ 26′′ N

Example 4.-March 16th, 1890; a.m. at ship; lat. 45° 37′ S.; long. 168° 20′ E.; observed altitude of sun's lower limb near the meridian was 45° 40′ 25′′; observer south of sun; index error of sextant 2'′ to subtract; eye 22 feet; time by chronometer 11h. 59m. 54s., which was 25m. 58s. slow on mean time at Greenwich; required the latitude.

Ans. Lat. 45° 31′ 46′′ S.

Example 5.-August 15th, 1890; a.m. at ship; lat. by D.R. 42° 50′ S. ; long. 176° 4′ W.; observed altitude of sun's lower limb near the meridian 32° 43′ 15′′ bearing north; height of eye 17 feet; time by chronometer August 15d. 11h. 20m 49s (astronomical time), which was 8m. 51s. fast on mean time at Greenwich; required the latitude by the reduction to the meridian.

Ans. Lat. 42° 37′ 50′′ S.

Example 6.-January 28th, 1890; a.m. at ship; lat 48° 3′ N.; long. 155° 16′ W.; observed altitude of sun's lower limb near the meridian was 22° 50′ 10′′ bearing south; index error 1′ 20′′ to subtract; height of eye 24 feet; time by chronometer 9h. 40m. 55s. which was 7m. 15s. slow on mean time at Greenwich; required the latitude.

Ans. Lat. 48° 12′ 14′′ N.

Example 7-October 18th, 1890; a.m. at ship; lat. 44° 52′ N.; long. 45° 44′ W.; observed altitude of Procyon (a Canis Minoris) near the meridian 50° 10′; zenith N. of star; height of eye 26 feet; time by chronometer 8h. 27m. 20s., which was 3m. 42s. fast on mean time at Greenwich: required the latitude by the reduction to the meridian.

Ans. Lat. 45° 2′ 6′′ N.

Example 8.-December 8th. 1890; a.m. at ship; lat. by account 49° 10′ N.; long 47° W; observed altitude of Sirius (a Canis Majoris) near the meridian 24° 15′ south of the observer; height of the eye 20 feet: index error of the sextant 2′ 3′′ to be subtracted. Time by chronometer 5h. Im 13s, which was correct for mean time at Greenwich; required the latitude.

Ans. Lat. 49° 10′ 11′′ N.

Example 9-June 5th, 1890; about 4h a.m. at ship; lat. by account 39° 50' N.; long. 167° 30′ E.; observed altitude of Jupiter near the meridian 32° 22′ S.; height of the eye 23 feet; time by chronometer 4h. 31m. 28s., which was 6m. 30s. slow on mean time at Greenwich; required the latitude.

Ans. Lat. 39° 50′ 30′′ N.

Example 10-December 8th, 1890; p.m. at ship; lat. by D.R. 15° 11′ N. ; long. 12° 4′ W.; the observed altitude of Dubhe (a Ursa Majoris) near the meridian below the pole was 23° 50′; height of eye 25 feet; mean time at Greenwich by chronometer (corrected for error and rate) was December 8d. 5h. 48m. 20s.; required the latitude by the reduction to the meridian. Ans. Lat. 50° 59′ 43′′ N.