Examples for Practice

Example 1.-June 28th, 1890; lat. by D.R. 51°; long. 105° 46′ W.; the observed meridian altitude of the Moon's lower limb being 53° 10′ bearing north; height of eye 20 feet; required the latitude.

Ans. Lat. 51° 13′ 35′′ S.

Example 2.-September 28th, 1890; lat. by D.R. 32°; long. 178° 30′ E.; the observed meridian altitude of the Moon's lower limb being 54° 50'; observer north of the moon; eye 25 feet; required the latitude.

Ans. Lat. 31° 53′ 18′′ N.

Example 3.--April 6th, 1890; a.m. at ship; lat. by D.R. 13° 20′; long. 81° W.; observed meridian altitude of MOON's lower limb 69° 40′ 30′′; zenith north of moon; index error 1′ 40′′ to subtract; eye 24 feet; required the latitude.

Ans. Lat. 13° 28′ 35′′ N.

Example 4.-March 9th, 1890; a.m. at ship; lat. by D.R. 40°; long. 105° 30′ E.; the observed meridian altitude of Moon's lower limb being 49° 24′ bearing north; height of eye 23 feet; required the latitude.

Ans. Lat. 39° 51′ 11′′ S.

Latitude by a Meridian Altitude BELOW the Pole.

When a star's declination and the latitude of a place are of the same name, both N., or both S., if the declination is greater than co-latitude; or, put otherwise, if the star's polar distance is less than the latitude of place; such stars are never below the horizon of the observer, and are called circumpolar stars; hence such stars pass the meridian both above and below the pole.

Similarly, when the latitude is higher than 661°, the sun is above the horizon throughout the whole 24 hours during part of the summer months of the hemisphere.

Also, for the reason given above, when the moon's declination and latitude of the place are of the same name, during a part of every month the moon's altitude can be taken both above and below the pole when the polar distance is less than the latitude of observer.

N.B. In an observation below the pole, the lowest altitude is the meridian altitude, and the latitude can be found as follows :—

RULE. To the true altitude of the heavenly body add its polar distance (90° Decl.); the sum will be the latitude of the same name as the declination.

Obs. The altitude is to be corrected in the usual way; but, as it will be low, note the state of the barometer and thermometer, and apply the necessary correction to the refraction (Table "Correction of Mean Refraction").

Obs. For the Sun.-The time being apparent midnight, the Greenwich date, apparent time, will be 12h. plus longitude W., but 12h minus longitude E.; for which Greenwich apparent time correct the sun's declination.

Obs. For the Moon.-Use the lower meridian passage, and correct it in the same way as the time of the upper passage.

Example.-June 28th, 1890; at 12h. p.m., in longitude 40° E., the meridian altitude of the sun's lower limb below the pole was 6° 30′; height of the observer's eye being 20 feet; required the latitude.

Example.-July 14th, 1890; at about 3h. 20m. a.m. (twilight), the altitude of the star DUBHE (a Ursa Maj.) when on the meridian below the pole, was 21° 14′, height of the observer's eye being 19 feet, and the index error +1' 30"; required the latitude.

By Table "Apparent Time of Principal Stars passing the Meridian of Greenwich" the time of Dubhe's passing the meridian above the pole is 3h. 20m. p.m.; therefore 12h. after that time, or about 3h. 20m. a.m., it will be on the meridian below the pole.

Example 1.-July 18th, 1890; at 12h. p.m., in long. 75° W., the altitude of the sun's lower limb, when on the meridian below the pole, was 8° 26′ 20′′, the height of the observer's eye being 19 feet, and the index error + 3′ 15′′ ; required the latitude.

Ans. Lat. 77° 42′ 2′′ N.

Example 2.-August 19th, 1890; at about 2h. 20m. a.m., the star a Crucis being on the meridian below the pole, its altitude was observed to be 17° 32′ 10", the height of the observer's eye being 26 feet, and the index error 2′ 25′′; required the latitude. Declination of the star 62° 29′ 37′′ S. Ans. Lat. 44° 52′ 3′′ S.

Degree of Dependence.-In fine weather an observation of the sun when on the meridian should not be in error 2'. A star or planet taken at twilight might have the same error, no more; but on a dark night it might be 3′ or 4'; there is always more or less uncertainty about the moon. The accuracy of the approximate result will then depend on the accuracy with which the various corrections are made.

LATITUDE BY THE REDUCTION TO THE

MERIDIAN

When there is a probability that the meridian altitude may be lost, owing to clouds, or other causes, altitudes of the sun may be taken near noon and the times noted by watch, regulated to local time from the a.m. observations for time, and reduced by a correction due to the difference of longitude in the interval.

This observation for latitude should be limited to altitudes taken within a given time from the meridian (see Table below); for unavoidable errors occur in the time as determined at sea, and the error in the latitude produced by an error in the time is considerable when the observations are made outside the prescribed limits.

Thus the term near the meridian" has a specific signification; and speaking in general terms, the number of minutes in the time from noon should not exceed the number of degrees in the sun's meridian zenith distance; or the number of minutes of time in the Meridian distance should never exceed the number of minutes of arc in the reduction. The meridian distance should not exceed the limits in the following table, which is computed to give the number of minutes of meridian distance, when an error of half a minute in the time will produce an error of 1' in the reduction.

To use the Table.-With the approximate latitude in the side column and the declination at the top, having regard to the precept as to name, find the time; this is the time from noon, or meridian distance in time, within. the limits of which you must keep when finding the latitude by the reduction to the meridian. See also Ex-Mer. Table in Norie's Table

The necessary elements for the solution of the problem are:

The hour angle, which in the case of the sun is the apparent time from

noon

The declination of the object.

The altitude, and by some methods the latitude by account.

The following methods are those best adapted to the computation of the reduction:

METHOD I.—For all practical purposes at sea, you do not require the logarithms to more than four places when using this method of obtaining the reduction.

Formula-Sin. r = hav. hx cos. l x cos. d x co-sec. (l±d), where 7 i the reduction, h the mer. dist., / the latitude, and d the declination.

zenith distance obtained from latitude by D.R. and declination.

Add together the following:

Log. hav. meridian distance.

Log. co-sine of the latitude by dead reckoning.

Log. co-sine of the declination, and

Log. co-secant of the meridian zenith distance by dead reckoning.

NB. For the zenith distance deduced from dead reckoning proceed as follows-With latitude and declination of the same name, take their difference; with latitude and declination of different names, take their sum; the result will be the zenith distance by dead reckoning.

The sum of the logs. (rejecting tens from the index) will be the Log. sine of half the reduction, which take out, and multiply by 2, for the whole reduction, to be added to the true altitude off the meridian.

The reduced altitude taken from 90° gives the zenith distance, to which apply the declination as in the usual problem of finding the latitude by meridian altitude.

Obs. 1 The investigation of method I. admits of two other practical solutions for example, if, to the form already given the log. of 2, which is 0.301030, be added as a constant, or, if instead of log. haversine of meridian distance the log. rising be taken and the index increased by 5, then the result will be the log. sine of the reduction without the necessity of "multiplying by 2," as stated above.

Obs. 2.—Or again, if, to the form already given, the constant 5-61546 be added, the result will be a log., the nat. number of which will be the reduction in seconds of arc, to be added (as before) to the true alt.

If the observation is made below the pole, the reduction is to be subtracted from the true altitude, for the reduced altitude, which, added to the object's polar distance, gives the latitude.

METHOD II.-Navigators have long been accustomed to use the log.