MERCATOR'S SAILING. 280. Mercator's Sailing, like middle latitude sailing, relates to finding the difference of longitude a ship makes when sailing on any oblique rhumb, and is a perfectly general and rigorously true method, which the other is not. Mercator's sailing is characterised by the use of the Table of Meridional Parts, and the chart constructed by means of it called Mercator's Chart. With the assistance of this Table, the rules of plane trigonometry suffice for the solution of all the problems. In the triangle ACB let A be the course, AB the distance, AC the true difference of latitude, CB the departure; then corresponding to AC, the Table of Meridional Parts gives AC, the meridional difference of latitude, and completing the rightangled triangle ACB, CB will be the difference of longitude. In addition, then, to the three canons of plane sailing which can be deduced from the triangle ACB, the triangle AC'B' gives the characteristic canon of Mercator's Sailing (since CB'= AC tang. A) diff. long. mer. diff. lat. X tang. course. 281. Given the latitudes and longitudes of two places, to find the course and distance between them. RULE LXXI. 1o. Find the true difference of latitude, according to Rule XLII, page 106. 2°. Find the meridional difference of latitude, Rule XLIII, page 107. 3°. Next find the difference of longitude, Rule XLVI, page 109. 4°. To find the course.-From the log. of diff. of longitude (increasing its index by 10), subtract the log. of meridional diff. of lat.: the remainder is the tangent of course, which take out of the tables, and place before it the letter of diff. of lat., and after it the letter of diff. of long. NOTE.-Be careful to remember that when the sum of the longitudes exceeds 180°, it must be taken from 360°, and the course must be named the same as the longitude left. 5°. To find the distance.-To the secant of course (rejecting 10 from the index), add the log. of diff. of lat.: the sum is the log. of distance, the natural number corresponding to which find in the tables. EXAMPLES. Mer. parts 3970 Ex. 1. Required the course and distance from Tynemouth Light to the Naze of Norway. Lat. Tynemouth 55° 1' N. Long. Tynemouth 1°25′ W. Lat. Naze 57 58 N. Mer. parts 4291 Long. Naze 7 2 E. 8 27 60 Ex. 3. Required the course and distance from Cape Bajoli to Cape Sicie. Ex. 4. Required the course and distance from Cape Formosa to St. Helena. Ex. 5. Required the course and distance from Bahia to Fernando Po. This question worked to the nearest minute of arc gives course N. 54° 43′ W., and distance 10455 miles. Ex. 7. Required the course and distance from Cape East, New Zealand, to Cape Horn. Lat. Cape East 37° 42′ 8. 18 17 60 Diff. of lat. 1097 S. Diff. long. 6844 Log. (+ 10) 13.835310 Mer. diff. lat. 1627 Course 76° 37′ 0′′ Log. 3211388 Parts for 39" 345 Diff. of lat. 1097 Log. 3'040207 3.676067 53 935)36400(39 2805 8350 8415 Course S. 76° 37′ 39′′ E. EXAMPLES FOR PRACTICE. Required the course and distance from A to B in each of the following 282. To find the latitude and longitude in, having given the latitude from, the longitude from, and the course and distance between the two places by Traverse Table and meridional parts.* RULE LXXII. 1o. With given course and distance enter the Traverse Table and take out the corresponding true difference of latitude, Rule LXIII, page 184, from which and latitude from find latitude in, as in Rule XLIV, page 108, and then meridional difference of latitude, as in Rule XLIII, page 107. 2°. At the given course look in the column of the true difference of latitude for the meridional difference of latitude; the corresponding departure will be the difference of longitude, from which and the longitude from find the longitude in, as in Rule XLVII, page 110. EXAMPLES. Ex. 1. A ship from lat. 55° 1′ N., long. 1° 25′ W., sails S.S.E. † E., 246 miles: required the lat. in and long. in. Entering Traverse Table II, with course S. 2} points E., and distance 246, we obtain diff. lat. 217.0, and dep. 116.0. * The general method of solution by "meridional parts," is from the formula:— True diff. lat. dist. X cos. course. .. log. true diff. lat. = log. dist. + log. cos. course -- 10. The course 2 points, and half mer. diff. lat. 181.5 (in diff. lat. column), the nearest found in the Table is 1817, the corresponding departure is 971, which multiplied by 2 (having divided mer. diff. lat. by 2) gives diff. long. 194'2 miles. Ex. 2. A ship from lat. 42° 36′ S., long. 178° 43′ E., makes diff. lat. 178''1 S., and dep. 240' 2 E.: find lat. in and long. in. Course 4 points, and half mer. diff. lat. 124 (in diff. lat. column), give in dep. column 1671, which doubled is 334 2, the diff. long. Ex. 3. From lat. 50° 48′ N., and long. 1° 10' W., sailed S. 41° E., 275 miles: required the lat. in and long. in. In the Traverse Table at the distance 275, and course 41°, the corresponding true diff. lat. is 207*5, or 3° 27'5, which being subtracted from 50° 48′ N., the lat. in is 47° 20′5 N.; taking out the mer. parts for 50° 48′, and 47° 20′5, the mer. diff. lat. is found to be 317, to half which as a true diff. lat., and the course 14°, the dep. is 1378, twice which is 275'6,—that is, the diff. long. is 4° 36′ E.: hence the long. in is 3° 26′ E. Ex. 4. From lat. 50° 30′ N., and long. 37° 55′ W., sailed S.W. & S., until arrived at lat. 52° 15' N. Lat. from 50° 30' N. Course 3 points, and mer. diff. lat. in diff. lat. column, give in dep, column 125'4, which is the diff. long. Lat. in 52 15 N. Mer. parts 3521 EXAMPLES FOR PRACTICE. For examples for practice in this problem take those given in middle latitude sailing at page 199. REMARKS ON MIDDLE LATITUDE AND MERCATOR'S SAILINGS. 283. "The difference of longitude found by middle latitude is true at the equator, and very nearly true for short distances in all latitudes, especially when the course is E. or W. In high latitudes, when the distance is great and the course oblique, the error becomes considerable: but the result may be made as accurate as we please by sub-dividing the distance run into small portions, and finding the difference of longitude for each portion separately. The difference of longitude deduced by middle latitude sailing is too small: |