Because the difference of latitude is S. and the difference of longitude is W., hence the true course from Cape St. Vincent to Funchal is S. 55° 48' W., or S.W. by W. nearly, and the distance 471.5 miles. Also, in this case the middle latitude by Workman's Table is 34° 53'; hence the course is S. 55° 47' W., and the distance 471.2 miles, as found by Mercator's Sailing. By Inspection.—In Traverse Table look for the middle latitude 35° as a course, and for 475 the difference of longitude in a distance column, immediately opposite to which, in the difference of latitude column, will be found 389.1. Then 265 (the difference of latitude) and 389.1 (the departure) being found nearly opposite each other in their respective columns, will give the course nearly 56°, and the distance 469 miles. Given the Latitude left, the True Course, and the Distance, to find the Latitude and Longitude in Example.—A ship from lat. 52° 6' N., and long 35° 6' W., sails N.W. by W.229 miles. Required her present latitude and longitude. BY CONSTRUCTION Diff.of Long Draw the line A D, and make the angle DAC equal to the course 5 points ; lay off, from A to C, the distance 229, and draw the line C B perpendicular to the line A D; then will the departure C B measure 190, and the difference of latitude A B 127 ; hence the latitude in is 54° 13', and the middle latitude 53° 9'. Now make the angle B C D equal to the middle latitude 53°, then will C D be the difference of longitude, measuring 317 miles. Mid. Lat. 53°9' с BY CALCULATION To find the Latitude in D. lat. 127 miles, or 2° 7' N. Latitude left Latitude in Sum of latitudes 2)106 19 + L cos. co. ΙΟ Middle latitude 539 Distance 229 log. 2.359835 Course 5 pts. cos. 9°744739 D. lat. 127.2 miles log. 2.104574 52 6 N. 54 13 N. log. dist. By Inspection.-By Traverse Table look for the course 5 points at the bottom of the page, over which, and opposite the distance 229 in its column, will be the difference of latitude 127.2, and departure 190-4, in their respective columns. Then Look for the middle latitude 53° as a course, and the departure 190:4 in the difference of latitude column, opposite the nearest to which, in the distance column, will be found 316, the difference of longitude. Ans. Lat. in, 54° 13' N. Long. in, 40° 231 W, Given one Latitude, the Departure, and Difference of Longitude, to find the other Latitude, the True Course, and Distance* Example.--A ship from lat. 36° 32' N. sails between the south and west until she has made 480 miles of departure, and 562' of difference of longitude. Required her present latitude, the true course, and distance run. Having drawn the line A D, make B C perpendicular to it, and equal to the departure 480 ; draw C D equal to the difference of longitude 562 meeting A D in D; then the middle latitude B C D will measure 31}°; hence the latitude in is 261°, and the difference of latitude 600 : now make A B equal to 600, and join A C, which will measure the distance 787; and the course C A B will be 371°. Diff. of Lat.600 m. * This case cannot be solved by Mercator's Sailing. 2 62 40 BY CALCULATION For Latitude in, and D. Lat. Middle latitude 31° 20' Double mid. lat. .. Log. dep. + 10 – log. d. long. Latitude left 36 32 N. = L cos. mid. lat. Latitude in 26 8 N. Departure 480 log. (+10) 12:681241 D. long. 562 log. D. lat. 2.749736 10 24 = 624m. Mid. lat. 31° 20' COS. 9.931505 To find the Distance Dist. = cosec. CO. Dep. dep. X cosec. co. .. Log. dep. + 10 -- log. d. lat. Log. dist. = log. dep.. + L cosec. co. 10. Departure 480 log. (+10) 12.681241 Departure 480 log. 2:681241 D. lat. 624 log. 2.795185 Course 37° 34' cosec. 10:214895 Course S. 37° 34' W. tan. 9.886056 Distance 787'3 miles log. 2.896136 Ans. Lat. in, 26° 8' N. Course S. 37° 34' W. Dist. 787.3 miles. tan. co. .. dist. The ordinary method of middle latitude sailing is but an approximation ; the departure actually made is not exactly equal to the arc of the middle parallel, and the principles of parallel sailing require that the departure should be reckoned in the parallel to which it truly belongs. The parallel to which it truly belongs is clearly that parallel which will give the difference of longitude actually marle; and hence should contain the secant of the true middle latitude-i.e., the middle latitude in which the departure should be reckoned-instead of the secant of the mean middle latitude. Workman's Table (see Norie's Tables) converts the mean middle latitude into true middle latitude, in the sense required. But in most cases the results must be determined by computation, as the methods by inspection and the Traverse Tables do not admit of strict accuracy, since these tables only run to whole degrees, and one place of decimals, and consequently not much is gained by interpolation. MERCATOR'S SAILING MERCATOR'S SAILING is the art of finding on a plane surface the track of a ship upon any assigned course of the compass which shall be true in latitude, longitude, and distance sailed. This method is derived from the projection of Mercator's Chart, in which the degrees of longitude are everywhere equal, the degrees of latitude increase towards the poles, and the parallels, meridians, and rhumb-lines are represented by straight lines. (See Description of Mercator's Chart.) Notwithstanding the inaccuracy of PLANE CHARTS, in which the degrees of longitude and latitude are everywhere equal, mariners were content to use these, and to do much computation on the basis of spherical trigonometry, until Gerard Mercator about the year 1556 published a chart ir which the meridians are all parallel to each other, but in order to com pensate for the expansion of the degrees of longitude he increased the distance between the parallels ; hence a chart thus constructed has obtained the name of MERCATOR'S CHART. It does not, however, appear that Mercator understood the true principles of this projection ; at least he never divulged the method on which he proceeded. In the year 1599, Edward Wright, of Caius College, Cambridge, published the true principles of Mercator's Chart, in a work entitled “The Correction of certain Errors in Navigation,” wherein he showed, by a Table of Meridional Parts, the length of the enlarged meridians in miles of the equator to every minute of latitude. From the principles of Mercator's projection there exists the following relation or, in words, the meridians being parallel, arcs of parallels of latitude are show'n as equal to corresponding arcs of the equator, each being expanded in the ratio of the secant of its latitude. The sum of the lengths of all the small portions of the meridians thus increased, reckoned from the equator, and expressed in minutes of the equator, is tabulated in the Table of Meridional Parts. By such a Table a Mercator's Chart is constructed, and the various problems of Mercator's sailing are solved on the basis of right-angled plane trigonometry. Let A B C (see the Fig., f 300) be a triangle, in which A is the course, AC the distance, A B the true (or proper) difference of latitude, and B C the departure; then, corresponding to A B, the Table of Meridional Parts, or increased latitudes, gives A D as the meridional difference of latitude (mer. diff. lat.); and completing the right-angled triangle A D E, the difference of longitude is represented by D E. The principles of plane sailing appertain to, and may be deduced from the triangle A B C; while from the triangle A D E is deduced the characteristic principle of Mercator's sailing. To find the length of the expanded meridian between any two parallels of latitude, or, as it is called, the meridional difference of latitude, the same rules are to be observed as in finding the true (or proper) difference of latitude ; that is, if the latitudes are of the same name, take the difference of their corresponding meridional parts, but if the latitudes are of contrary names, take the sum of those parts for the meridional difference of latitude. When the course is nearly east or west, that is, when there is a large difference of longitude but only a small difference of latitude, Mercator's sailing is not so suitable as middle latitude sailing. The same examples are introduced as in middle latitude sailing, for comparison of the two methods. It is recommended, when finding the compass course, to convert the true course into a New Pattern Compass course by the Table of “ Compass Equivalents," and then add all Westerly variation or deviation and subtract all Easterly variation or deviation. After applying the variation and deviation, convert it into its proper quadrant. Example.---True course S. 70° E., variation 20° W., and deviation 15° E. Find the compass course. S. 70°E. = 110° 130 Given the Latitudes and Longitudes of Two Places, to find the True Course and Distance from the one place to the Other 1. For the true Difference of Latitude.—Latitudes of the same name, take their difference ; latitudes of different names, take their sum. The result will be the true difference of latitude; then multiply the degrees by 60, and add in the miles. 2. For the Meridional Difference of Latitude.–From the Tables take out the meridional parts corresponding to the two latitudes; take their difference for latitudes of the same name ; take their sum for latitudes of different names. The result will be the meridional difference of latitude. 3. For the Difference of Longitude.-Longitudes of same name, take their difference; longitudes of different names, take their sum, and if this sum exceeds 180° subtract it from 360°. The result will be the difference of longitude ; then multiply the degrees by 60, and add in the miles. Required the true course and distance from Cape St. Vincent, in lat. 37° 3' N. and long. 9° 1' W., to Funchal, Madeira, in lat. 32° 38' N. and long. 16° 56' W. |