ON FINDING THE LATITUDE BY REDUCTION TO THE MERIDIAN. 346. 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 altitude 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 RULE CI. 47). 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 24h, 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.t 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. 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 56s (see Ex. 1 following): then the time from noon is 16m 568 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 168 (see Ex. 2 following): then we have 24h Om os 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 LXXVIII, page 227). 5°. Take out of Nautical Almanac, page I, the declination, and reduce it to the Greenwich date (Rule LXXIX, page 230). 6°. Correct the observed altitude of sun's upper or lower limb, and so get the true altitude of sun's centre (Rule LXXXIV, page 242). Method I. 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 t 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 oh im o is 9'97860, 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 is the index of the log. rising of 10m o3, 10m 5a, 10m 10a and of 10m 15o, and changes to 2 somewhere between 10m 15 and rom 203. 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 LXXXV, 4°, page 244). 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 LXXXV, 5°). 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 sine2 § t. 347. We may also compute directly the reduction of the observed altitude to the meridian altitude by the following: Method II. 1°. Add together the following logarithms : 2 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 tens 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. 2°. Add the reduction to the true altitude: the result is the meridian altitude.† 3°. Having the meridian altitude; find the latitude as by the method of meridian altitudes (Rule LXXXV, page 244). NOTE.-This Method (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. 2o. 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. • If we use the constant log. 0301030 (this is log. of 2) instead of that given above, viz., 5615455, 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. At the Liverpool Local Marine Board Examinations the candidate is expected to solve this problem by means of Towson's Ex-Meridian Tables. EXAMPLES. Ex. 1. 1882, 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 + ' 58′′, height of eye 20 feet, time by watch January 2d oh 48m 22, which was found to be 29m 16a fast on apparent time at ship, difference of longitude 32'4 miles to West: required the latitude by reduction to meridian. Ex. 2. 1882, 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 watch 6d oh 4m 4, found to be 28m 47" fast on app. time at ship, diff. of long. made to East 29.8 miles since error of watch on app. time at ship was determined: required the latitude by reduction to meridian. |