Thus, enter Tabie 1, above E. S. E., or 6 points, and seek for the departure 160, the nearest to which is 159.8; the corresponding numbers give the distance 173, and the difference of latitude 66.2 miles.

Enter Table 2 with the middle latitude 50° 43', or (51° nearly) as a course, and seek for the departure 160, in the latitude column, opposite to which, in the distance column, will be found the difference of longitude 254 miles, nearly.

CASE VII.

One latitude, distance sailed, and departure from the meridian given, to find the course, difference of latitude, and difference of longitude.

A ship in the latitude of 49° 30' N., and longitude of 25° 0′ W., sails southeasterly 215 miles, until her departure from the meridian is 167 miles; required the course steered, and the latitude and longitude the ship is in.

As in Case V, Plane Sailing, find the course by seeking in Table 2 till against the distance, in its column, is found the given departure in one of the following columns, adjoining to which, in the other column, will be the difference of latitude, which if greater than the departure the course will be at the top, but if less the course will be found at the bottom. Then take the middle latitude as a course, and find the departure in the column of difference of latitude, against which, in the distance column, will be found the difference of longitude.

Thus the dista..ce, 215, and the departure, 167, are found nearly to correspond to a course of 510, and a difference of latitude of 135.3; then with the middle latitude, 48, as a course, enter the table and seek for the departure, 167, in the latitude column; the distance corresponding, 250, is the difference of longitude nearly.

MERCATOR'S SAILING.

Art. 124. The principles of Middle Latitude Sailing already enunciated answer the requirements of the Navigator for small distances, or in cases where the difference of latitude is small in comparison with the difference of longitude; but when large differences of latitude and longitude are involved, especially in high latitudes, that method is liable to great

or, in words, the meridians being parallel, arcs of parallels of latitude are shown as equal to corresponding arcs of the equator; each being expanded in the proportion of the secant of its latitude to 1.

The lengths of small portions of the meridian thus increased, expressed in minutes of the equator, are called meridional parts; see Art. 66. The meridional parts computed for every minute of latitude

E

DLo

from 0 to 90 form the Table of Meridional Parts (Table 3) by means DL
of which a Mercator's chart may be constructed and all cases of Mer-
cator's Sailing be calculated. With the assistance of Table 3 the rules
of Plane Trigonometry answer for the solution of all the problems.

Art. 125. In the triangle ABC the angle ACB represents the course, C.; side AC represents the distance, Dist.; side BC represents the true difference of latitude, D. L.; AB represents the departure, Dep.; then, corresponding to BC, the Table of Merid. Parts, or Increased Latitudes, gives EC, the meridional difference of latitude m; and, completing the right triangle CEF, EF will represent the difference of longitude, D. Lo.

The principles of Plane Sailing can be deduced from the triangle ABC, and from the triangle FEC is derived the characteristic principle of Mercator's Sailing.

EF

Dist

Since

Tan ECF = EC

Art. 126. To find the arc of the expanded meridian intercepted between any two parallels, or the meridional difference of latitude, when both places are on the same side of the equator, subtract the meridional parts of the

lesser latitude from the meridional parts of the greater, the remainder will be the meridional difference of latitude; but if the places are on different sides of the equator, the sum of the meridional parts will be the meridional difference of latitude.

Art. 127. When the bearing is large, or near 90°; i. e., when the difference of longitude is greater than the difference of latitude, the method of Middle Latitude Sailing should be preferred to Mercator Sailing. For accuracy, the proper middle latitude should be greater than the mean of the latitudes; hence, the Table in Ärt. 122 should be used, but ordinarily the mean will suffice.

Art. 128. The various cases of Mercator's Sailing are given in the following Table, with the formulæ for their solutions; and each case is thereafter exemplified by a practical question:

The latitudes and longitudes of two places given, to find the direct course and distance between them. Required the bearing and distance from Cape Cod light-house, in the latitude of 42° 03′ N. and longitude 70° 04' W., to the island of St. Mary, one of the Western Islands, in the latitude of 36° 59' N. and longitude of 25° 10' W.

BY INSPECTION.

With the meridional difference of latitude and the difference of longitude used as difference of latitude and departure, find the course by inspecting Table 2 corresponding to those numbers; with this course and the proper difference of latitude, find the corresponding distance.

Thus, one-tenth of the meridional difference of latitude and difference of longitude are found, by interpolation, to agree nearly to a course of 81° 30'; this course and the proper difference of latitude are found to correspond to a distance of 2,165 miles, differing somewhat from the distance found by computation, owing to the neglect of a few minutes in the course.

CASE II.

Both latitudes and the departure given, to find the course, distance, and difference of longitude.

A ship in the latitude of 49° 57′ N. and longitude of 15° 16′ W. sails southwesterly until her departure is 197 miles, and then, by observation, is in the latitude of 47° 18 N.; required, her course, distance, and longitude in.

Find the course by Plane Sailing, Case VI, by seeking in the tables with the proper difference of latitude and departure till they are found to agree in their respective columns, corresponding to which will be the distance in its column, and the course will be found at the top of that column if the departure is less than the proper difference of latitude, otherwise at the bottom; with the same course find the meridional difference of latitude in the latitude column, corresponding to which, in the departure column, will be the true difference of longitude.

Thus, with the difference of latitude and departure 159 and 197, find the course 51°, and the distance 253; in the same table, opposite to half of the meridional difference of latitude 120, find the departure 148.0, which, being multiplied by 2, gives the difference of longitude 296 miles, nearly.

CASE III.

One latitude, course, and distance given, to find the difference of latitude and difference of longitude.

A ship in the latitude of 42° 30' N., and longitude of 58° 51′ W., sails S. W. by S. 300 miles; required the latitude and longitude in.

As in Case I, Plane Sailing, find the course at the top or bottom of the tables, either among the points of degrees, and in that page, opposite the distance, will be fourd the difference of latitude and departure in their respective columns. Then, in the same table, find the meridional difference of latitude, in the latitude column; corresponding to which, in the departure column, will be the difference of longitude.

Thus, under the course S. W. by S., or 3 points, and opposite the distance 300, stands the difference of latitude 249.4. Then, under the same course, find half of the meridional difference of latitude in the latitude column, against which stands 108.9, in the departure column; which, multiplied by 2, gives 217.8, the difference of longi. tude.

CASE IV.

Both latitudes and course given, to find the distance and difference of longitude.

A ship from the latitude of 49° 57′ N., and longitude of 30° W., sails S. 39° W., till she arrives in the latitude of 47° 44' N.; required the distance run, and longitude in.

As in Case II, Plane Sailing, find the course among the points or degrees, and the proper difference of latitude in its column, adjoining to which will be the distance and departure in their respective columns; then, in the same table, find the meridional difference of latitude in the latitude column, adjoining to which, in the departure column, will be the difference of longitude.

Thus, under the course 39°, and opposite the difference of latitude 133 (the nearest to which is 132.9), stand the distance 171, and the departure 107.6; in the same table, opposite the meridional difference of latitude 201.5, found in the latitude column, stands 163.2, in the departure column, which is the difference of longitude, as before.

CASE V.

Both latitudes and distance given, to find the course and difference of longitude.

A ship from the latitude of 37 N., and longitude of 32° 16′ W., sails 300 miles northwesterly, until she is in the latitude of 41° N.; required the course steered, and longitude in.

As in Case IV, Plane Sailing, seek in the table till against the distance, taken in its column, is found the given difference of latitude in one of the following columns; adjoining to it will stand the departure, which, if less than the difference of latitude, the course will be found at the top, otherwise at the bottom; in the same table find the meridional difference of latitude in the latitude column, adjoining to which, in the departure column, will stand the difference of longitude.

Thus, the distance 300, and the difference of latitude 240, are found to correspond to a course of 37°, and a departure of 180.5; and in the latitude column, opposite half the meridional difference of latitude, 153.8, stands 116 in the departure column, which doubled, gives the difference of longitude 232.

CASE VI.

One latitude, course, and departure given, to find the distance, difference of latitude, and difference of longitude.

A ship from the latitude of 50° 10' S., and longitude of 30° E., sails E. S. E. until her departure is 160 miles; required the distance sailed, and the latitude and longitude in.

As in Case III, Plane Sailing, find the course either in Table 1 or Table 2, and the departure in its column, corresponding to which will stand the distance and difference of latitude in their respective columns; in the same table find the meridional difference of latitude, in the latitude column, corresponding to which, in the departure column, will be found the difference of longitude.

Thus, over the course E. S. E., or 6 points, and against the departure 160, stands the distance 173 miles, and the difference of latitude 66.2 miles. Again, in the same table, find the meridional difference of latitude 104, in the latitude column, opposite to which, in the departure column, stands the difference of longitude 251.3 miles.

CASE VII.

One latitude, distance sailed, and departure given, to find the course, difference of latitude, and difference of longitude. A ship in the latitude of 49° 30′ N., and the longitude of 250 W., sails southeasterly 215 miles, making 167 miles departure; required, the course steered and the latitude and longitude in.

Find the course and difference of latitude, as in Case V, Plane Sailing, by seeking in Table 2, till the distance and departure are found to correspond in their respective columns, adjoining to which, in the column of latitude, will be found the true difference of latitude, which, if greater than the departure, the course will be found at the top, but if less, the course will be found at the bottom; with this course seek the meridional difference of latitude in the latitude column, adjoining to which, in the departure column, will be found the difference of longitude.

Thus, the distance 215, and the departure 167, are found to correspond to a course of about 51°, and a difference of latitude 135.3. Find in this table one-half the meridional difference of latitude 101.3, opposite to which, in the departure column, stands 125.1; this doubled gives 250.2 for the difference of longitude, nearly.

GREAT CIRCLE SAILING.

Art. 129. As has been already shown, the great advantages of Mercator's Sailing are

Ist. Being able to indicate the ship's track by merely placing a straight-edged ruler on the chart and drawing a line between the ports of departure and destination, and reading directly, by means of a parallel line to the nearest compass-rose, the course; or, if from accidental circumstances the ship is thrown off the course at any period of the voyage, to immediately determine the new one.

2d. The ability to foresee the whole of the track ahead, and thus to know whether there are such obstacles in the way as dangers, land, or shoals.

Art. 130. That method of sailing answers every purpose for short voyages or within the tropical regions; but, in the present day of long voyages and great competition, much time and distance can be saved by resorting to the Great Circle track, or to a compromise between it and the Rhumb track, called a Composite track.

The great obstacle which once existed against the practice of Great Circle Sailing, viz, the determination of