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ON FINDING THE LATITUDE BY REDUCTION TO THE MERIDIAN.

198. The latitude of a place is most simply determined by observation of the meridian altitude of a known heavenly body. When such an observation cannot be obtained by reason of the state of the weather, the altitude of the body may often be obtained a little before or a little after its meridian passage. And if at the time of observing such an altitude near the meridian, the hour-angle of the body is known, we may find by computation very nearly the difference of altitude by which to reduce the observed to the Meridian altitude. The correction is called the "Reduction to the Meridian." This method, in point of simplicity, is little inferior to the meridian altitude, to which it is next in importance. The latitude may also be determined by a direct process, deduced from spherical trigonometry. The former is the method used in the following pages. The term "near the meridian" implies a meridian distance limited according to the latitude and declination, and also the degree of precision with which the time is known (see RAPER, Table 47).*

If the time by watch with its error on apparent time at ship is given.

RULE LXXX.

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1o. To find the Apparent Time at Ship.-To the time shown by the watch, expressed astronomically, apply the error of the watch for apparent time,† adding when the watch is slow (rejecting 24 when the sum exceeds 24", and putting the day one forward), subtracting when the watch is fast (increasing the time shown by watch by 24", if necessary, and putting the day one back).

2°. Next turn into time the difference of longitude made since the error of the watch was determined, adding when the difference of longitude is East, subtracting when the difference of longitude is West; the result is apparent time at ship when the observation was made.‡

3°. To find the Time from Noon.-If P.M. at ship the apparent time at ship is the time from noon; when it is A.M. (reckoning from the preceding noon) subtract apparent time at ship from 24"; the remainder is the time from noon.

This Table shows the limits of the method of "Reduction to the Meridian" for common practice at sea, or how long before or after the time of meridian passage the sun's altitude may be observed, so that the reduction shall not be more in error than 2' when the time is I'm in error.

+ The error of chronometer for apparent time at place should be noted when the morning sights are taken for determining the longitude. This, with the diff. long. made in the interval between this last time and the time of observing the ex-meridian altitude, will give the apparent time at ship. If the ship has not changed her meridian since the time of morning sights, the result obtained by applying the error of chronometer is, of course, the apparent time at ship.

The reason for this rule will appear on considering that if a watch is set to the time at any given meridian, it will be slow for any meridian to the eastward, but fast for any meridian to the westward, at the rate of 1m for 15' diff. long., since the sun comes to the easterly meridian earlier, and to the westerly meridian later,

EXAMPLES.

Ex. 1. Suppose it is P.M. at ship, and the watch when corrected shows January 2d oh 16m 56 (see Ex. 1 following): then the time from noon is 16m 56% past noon of the 2nd.

Ex. 2. Again, suppose it is A.M. at ship, and the watch when corrected indicates Feb. 5d 23h 37m 16 (see Ex. 2 following): then we have

24h om o
23 37 16
22 44

In this instance it is 22m 44 before noon of the 6th.

4°. To find Greenwich Apparent Time.—With apparent time at ship and longitude find Greenwich date in apparent time (Rule LVIII, page 179). 5°. Take out of Nautical Almanac, page I, the declination, and reduce it to the Greenwich date (Rule LIX, page 183).

If the time by chronometer is given, with the error on mean time at Greenwich.

(a) To the time by chronometer apply its error on Greenwich mean time, adding if slow and subtracting if fast; the result is mean time at Greenwich.

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(b) Write down the Equation of time, taken from page II of the month in the Nautical Almanac, and correct it for Greenwich mean time, and mark it "added to " or "subtracted from mean time, as directed at the heading of the column in page II of Nautical Almanac (Rule LXII, page 188); also take out the sun's declination from page II, Nautical Almanac, and correct it also for Greenwich time (Rule LIX, page 183).

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Under mean time at Greenwich, found as above in (a), write down longitude in time, adding if East, and subtracting if West: the result will be mean time at ship.

(d) To mean time at ship, just found in (c), apply the Equation of time, found as in (b): the result is apparent time at ship, and also the sun's hour-angle (or time from noon) if P.M.; but if A.M. (or more than 12h) subtract apparent time from 24", the remainder is sun's hour-angle, or time from noon required (see Exs. 1 and 2 to 3°).

6°.

To find the True Altitude.—Correct the observed altitude of sun's lower or upper limb for index error (±), dip (—), refraction (—), parallax (+), semidiameter (±), and so get the true altitude of sun's centre (Rule LXIV, page 193).

We now proceed to give three methods which are in general use for working this problem, but whichever method is adopted the foregoing rules always form the first part of the method.

Method I.-(By Norie.)

7°. Take out log. rising of time from noon (Table 29, NORIE), log. cos. declination (Table 25, NORIE), and log. cos. of latitude (Table 25, NORIE).

NOTE.-In using the natural sines and cosines to six places, it will be necessary to add 1 to the index of the log. rising, because, as given in the table, it is only adapted to five places of figures.

Caution. In the use of the table of log. rising (XXIX, NORIE), care must be taken that the correct indices are used when the minutes of the time from noon are 1, 3, 10, or 32. It

is necessary to notice that the indices in the table sometimes change in the column where they could not be inserted for want of room; this may, however, be easily known by observing that the first figure of the decimal part of the log. changes from 9 to 0.

Thus the log. rising of ch 1m os is 9'97860,

but the log. rising of oh 1m 5 is 0.04813.

The index, as given in the table, is in the form, which means that it changes from 9 to o somewhere in that line. Similarly, opposite 10m, the index is in the form, and the numerator I is the index of the log. rising of 10m os, 10m 5o, 10m 10a, and of 10m 15", aud changes to 2 somewhere between 10m 15 and 10m 20o.

8°. Take the sum of these and find the natural number corresponding thereto. (Table 24, NORIE.)

9°. To the natural number just found add the natural sine of the true altitude (Table 26, NORIE); the sum is natural cosine of meridian zenith distance, which take out of the table, and name it North when the observer is North of the sun, or when the sun bears South; but call zenith distance South when the observer is South of sun, or when it bears North (see Rule LXIV, 4°, page 193).

10°. Apply the reduced declination to the zenith distance, taking their sum if they are of the same name, but their difference if of contrary names; the result, in either case, is the latitude of the same name as the greater (Rule LXIV, 5°, page 193).

NOTE.-The foregoing Method (Method I) is only convenient when the computer is provided with a table of natural sines and cosines, as well as a table of log. versed sines, or the logarithmic value of 2 sinet.

199. We may also compute directly the reduction of the observed altitude to the meridian altitude by the following:

Method II.

Add together the following logarithms:

Constant log. 5.615455; (this is the log. of sine 1.)*

Log. cosine of latitude by account (Table 25, NORIE).

Log. cosine of declination (Table 25, NORIE).

Log. cosecant of meridian zenith distance as deduced from latitude by D.R. and declination (Table 25, NORIE).

The log. of time from noon; (this is twice the log. sine of half the hourangle). (Table 31, NORIE, and 69, RAPER.)

The sum of these logs. (rejecting tons from the index), will be the log. of the reduction in seconds ("). (Table 24, NORIE.)

The zenith distance from latitude by D.R. is found as follows:-When the latitude and declination are both of the same name, take their difference; when latitude and declination have different names, take their sum: the result in either case will be zenith distance by D.R. 2o. Add the reduction to the true altitude: the result is the meridian altitude.†

Some prefer to use the constant log. 0301030 (this is log. of 2) instead of that given above, viz., 5.615455, the sum of logs. (rejecting tens from index), will be log. sine of reduction in minutes () and seconds ("). (Table 66, RAPER, or Table 25, NORIE.) If we omit the constant altogether the sum of the other four logs. is the log. sine of half the reduction, in minutes (') and seconds ("), which must be multiplied by 2 to get the reduction.

This is only an approximate meridian altitude, in strictness a second reduction should be computed.

3o. To find the Latitude.-Having the meridian altitude subtract it from 90°, the remainder is the meridian zenith distance, which must be named of an opposite name to the bearing, that is

If bearing is North the zenith distance is S.

If bearing is South the zenith distance is N.

4°. Below the zenith distance place the reduced declination; then if zen. dist. and decl. are of same name, add; but if of different names, take the difference: the result is the latitude, of the same name as the greater.

NOTE.-This Methed (Method II) does not approximate so rapidly as the preceding (Method I), but the objection is of little weight when the observations are very near the meridian. On the other hand, it has the great advantage of not requiring the use of the Table of Natural Sines.

Method III.-(By Towson's Ex-Meridian Tables.)*

1o. Enter Table I (Towson) under nearest declination and find nearest hourangle, against which stands Augmentation I, which add to declination, at the same time take out corresponding index number in the margin.

2°. Enter Table II under true altitude and opposite index number, find Augmentation II, which add to true altitude, and thence find latitude as in meridian altitude.

EXAMPLES.

Ex. 1. 1887, January 2nd, P.M. at ship, latitude by account 52° 6' S., longitude 71° 23′ W., observed altitude sun's L.L. North of observer 60° 20′ 30′′, index correction + 2' 58', height of eye 20 feet, time by watch January 24 oh 48m 223, which was found to be 29m 16 fast on apparent time at ship, difference of longitude 32.4 miles to West: required the latitude by reduction to meridian.

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At the Liverpool Local Marine Board Examinations the candidate is expected to solve this problem by means of TowSON's Ex-Meridian Tables.

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Long. in time

Mean time at Green., Feb. 5 23 51 36

Latitude

Ex. 2. 1887, February 6th, A.M. at ship, lat. acct. 51° 58′ N., long. 105° 41′ W., obs. alt. sun's L.L. South of observer 22° 10′ 30′′, index corr. + 56′′, height of eye 22 feet, time by chronometer Feb. 6d 7h 16m 32o, which was fast on Greenwich mean time 22m 12": required the latitude by reduction to the meridian.

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Green. date (M.T.),

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Long. in time 105° 41′ W.
4

6,0)42,2 44

7 2 44

Equation of time

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App. time at ship,
Subtract from

Feb. 5 23 37 16

24 O

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