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. Course 30° 20' 48" Log. 3.082785 Log. 3*146782 Distance 1402 Required the course and distance from Cape East, New Zealand, to Ex. 7. Cape Horn. Required the course and distance from A to B in each of the fol 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 XXXVII. 1o. With given course and distance enter the Traverse Table and take out the corresponding true difference of latitude, from which and latitude from, find latitude in and then meridional difference of latitude, as in Rule XXI, page 52. 2o. At the given course look in the column of the true difference of latitude for the meridional difference latitude; the corresponding departure will be the difference of longitude, from which and the longitude from find the longitude in, as in Rule XXV, page 56. EXAMPLES. Ex. I. 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, 6,0)21,7 Lat, in Lat. left 55° 1′ N. Mer. parts 3970 Mer. parts 3607 Rule XXI, 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 97.1, which multiplied by 2 (having divided mer. diff. lat. by 2), gives diff. long 1942 miles.. 6,0) 19,4'2 3° 14' Ex. 2. Long. left 1°25′ W. A ship from lat. 42° 36′ S., long. find lat, in and long. in, The ship being 1° 25′ W., or 85' West of Greenwich, must evidently be in East longitude, after having sailed 194 miles to the Eastward. 178° 43′ E., sails S.E. & E., 299 miles Course 4 points, and dist. 299, give diff. lat. 178.1, dep. 240'2. 6,0)17,8'1 Lat. left 42° 36′ S. Mer. parts 2830 2° 58' Lat. in 2 58 S. 45 34 S. Mer. parts 3078 Mer diff. lat. 2)248 124 * 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 - ΙΟ .. log. diff. long.log. mer. diff. lat. + log. tang. course - IO. Course 4 points, and half mer. diff. lat. 124 (in diff. lat, column), gives in dep. column 167 1, 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 41°, the dep. is 137.8, twice which is 275.6, that is, the diff. long. is 4° 36′ E.: hence the long. in is 3° 26' E, From lat. 50° 30′ N., and long 37° 55′ W., sailed S.W. & S., until arrived Ex. 4. at lat. 52° 15′ N. Lat, from 50° 30' N. REMARKS ON MIDDLE LATITUDE AND MERCATOR'S SAILING. "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 : an estimate of the error for places on the same side of the equator may be formed by the help of a few cases. Suppose the course 4 points or 45°, and the difference of latitude 10 or 600; then if this difference of latitude is made good in any latitude below 30°, the error of the difference of longitude will not exceed 2'; if made good below the parallels of 40° and 50°, the error will be about 3'; and between 60° and 70° about 19', or of a degree. For smaller distances the errors will be much less, and for greater distances much greater, as they vary in much more rapid proportion than the distances. It has been observed before that when the course is large, the difference of longitude should be found by middle latitude in preference to Mercator's Sailing; because, although the latter is mathematically correct in principle, yet a small error in the course may, when the course is large, produce a considerable error in the difference of longitude. The reason of this is easily shown. In middle latitude sailing we convert the departure into difference of longitude. The process increases the departure in a proportion which is less than 2 to 1 in all latitudes below 60°; and exceeds 3 to 1 in all latitudes beyond 70°. The error of the departure, increased in the same proportion, becomes thus the error of difference of longitude. Now when the course is nearly E. or W., the departure is nearly the same as the distance, and an error of some degrees in the course does not affect the departure sensibly; hence in this case the error of the difference of longitude depends on that of the distance alone. But in Mercator's Sailing, on the other hand, we convert the meridional difference of latitude into difference of longitude, and the process, when the course is large, converts a given meridional difference of latitude into a difference of longitude much greater than itself; and thus increases the error of the meridional difference of latitude in the same proportion. Thus, for example, at the course 80°, the difference of longitude exceeds the meridional difference of latitude in the proportion of 6 to 1; at the course 86° this proportion is 11 to 1. Now when the course is large, a slight change in it sensibly affects the difference of latitude and also the meridional difference of latitude, which is deduced directly from it. In high latitudes the meridional parts vary rapidly, and the error of difference of longitude is aggravated accordingly; hence the precept more especially demands attention in high latitudes."-Raper's Practice of Navigation, pp. 103-104. THIS is the process of finding the ship's place at noon-that is, its latitude and longitude, having given the latitude and longitude at the noon preceding, or a departure taken since, the compass courses and distances run in the interval, the leeway (if any), variation and deviation (if any), direction and rate of current (if any), &c., &c. RULE XXXVIII.* 1o. Correct each course for leeway and variation (see Rules XXVI to XXIX, pages 58 to 66), which arrange in the tabular form as in the example following. Add together the hourly distances sailed on each course, and insert the same in the table, opposite the true course. (a) When a departure has been taken, consider the opposite to the bearing as a course, which correct for variation, and insert in the table as an actual course, with the distance of the object as a distance. The departure course is generally put down in the table as the first course. The current (b) The set of a current is to be corrected for variation, and inserted in the table as a course; the drift being taken as a distance. course is generally inserted in the table as the last course. 2o. Take out of the Traverse Tables (Table I or II, Raper or Norie) the difference of latitude and departure to each course and distance (see page 75), and proceed to find the difference of latitude and departure made good as directed in Rule XXXII, page 78, Traverse Sailing. 3°. Find the course and distance made good, see Rule XXXVI, page 89. 4. Find the latitude in by applying the difference of latitude to the latitude from. If a departure has been taken, the difference of latitude is to be applied to latitude of the point of land; if otherwise, to yesterday's latitude. 5°. To find the difference of longitude by Middle Latitude Sailing. (c) Find the middle latitude as directed, Rule XXIII, page 54. * Nearly the entire process of computing the Day's Work has already been given, and if the learner has thoroughly mastered the rules laid down in the preceding pages, he will find no difficulty in working the Day's Work without reference to them. |