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1st. The extent from radius 8 points, to the complement of the course 5 points, on the line marked SR, will reach from the distance 300, to the difference of latitude 249.4, on the line of numbers.

2dly. The extent from the radius 4 points, to the course 3 points, on the line marked TR, will reach from the meridional difference of latitude 327, to the difference of longitude 218.5, on the line of numbers.

BY INSPECTION.

As in Case I. Plane Sailing, find the course at the top or bottom of the tables, either among the points or degrees, and in that page, opposite the distance, will be found the difference of latitude and departure in their respective columus. 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 109 nearly, in the departure column; which, multiplied by two, gives 218, the difference of longitude, nearly.

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.

Latitude left.. 49° 57' N.

Latitude in ... 47 44 N.

Meridional parts 3470
Meridional parts 3268

Diff. of latitude 2 13 133 miles. Mer. diff. of lat. 202 miles.

BY PROJECTION.

Draw the meridian AEB, on which take AE equal to the proper difference of latitude 133 miles, and AB equal to the meridional difference of latitude 202 miles; make the angle BAC equal to the course 39°, and draw ED, BC, perpendicular to AB, cutting ADC in D and C; then will AD be the distance 171.1 miles, and BC the difference of longitude 163.6 miles.

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To find the difference of longitude.
As radius 45°.
10.00000
Is to merid. diff. of latitude 202. 2.30535
So is tangent course 39°....... 9.90837

To the difference of long. 163.6 2.21379

Difference of longitude....

30° 0′ W.

2 44 W.

BY GUNTER.

1st. The extent from the complement of the course 51°, to the radius 90°, on the sines, will reach from the proper difference of latitude 133, to the distance 171.1, on the line of numbers.

2dly. The extent from radius 45°, to the course 39°, on the line of tangents, wil reach from the meridional difference of latitude 202, to the difference of longitude 163.6, on the line of numbers.

BY INSPECTION.

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 202, found in the latitude column, stands 163.6, 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 north-westerly, until she is in the latitude of 41° N.; required the course steered, and longitude in.

Latitude left
Latitude in

....

37° N.
41° N.

Meridional parts. 2393

Meridional parts. 2702

Difference of lat. 4°: 240 miles. Merid. diff. of lat. 309 miles.

BY PROJECTION.

Draw the meridian ABC; make AB equal to the proper difference of latitude 240, and AC equal to the meridional difference of latitude 309 miles; draw BD and CE perpendicular to ABC; with an extent equal to the distance 300 in your compasses, and one foot in A, as a centre, describe an arc cutting BD in D; draw AD, and continue it to cut CE in E, and it is done; for the angle BAD is equal to the course of 36° 52′, BD is the departure, and CE is the difference of longitude 231.7 miles.

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st. The extent from the distance 300, to the proper difference of latitude 240, on the line of numbers, will reach from the radius, or 90°, to 53° 8', the complement of the course on the line of sines.

2dly. The extent from radius 45°, to the course 36° 52', on the line of tangents will reach from the meridional difference of latitude 309 to the difference of longituds 231.7, on the line of numbers.

BY INSPECTION.

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 154.5 (the nearest to which is 154.1), stands 116.2 in the departure column, which doubled gives the difference of longitude 232.4.

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.

BY PROJECTION.

A

F

Distance

course

D

Departure

Draw the meridian ABC, and at a distance from it equal to the departure 160 miles, draw the line FD parallel to ABC; make the angle BAD equal to the course 6 points; draw AD to cut FD in D; from D let fall upon AB the perpendicular DB; then will AD be the distance 173.2 miles, AB the difference of latitude 66.3 miles; hence we have both latitudes, and the meridional difference of latitude 104 miles; make the line AC equal thereto, and draw CE perpendicular to AC, meeting AD continued in E; then will CE be the difference of longitude 251.1 miles.

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B

с

Difference Longitude

Latitude left.. 50° 10' S. Mer. parts 3490
Diff. of latitude 1 06 S.

Latitude in... 51 16 S. Mer. parts 3594

Merid. difference of latitude 104

To find the difference of longitude. As radius 4 points....

10.00000 Is to the merid. diff. of lat. 104. 2.01703 So is tangent course 6 points.. 10.38278 To diff. of long. 251 = 4° 11′ E. 2.39981 Longitude left 30 00 E.

......

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1st. The extent from the course 6 points, to radius 8 points, on the line marked S. R. will reach from the departure 160, to the distance 173.2, on the line of numbers.

2dly. The extent from radius 4 points, to the complement of the course 2 points, on the line marked T. R., will reach from the departure 160, to the difference of latitude 66.3, on the line of numbers.

3dly. The same extent (from the radius 4 points to the course 6 points on the line marked T. R. will reach from the meridional difference of latitude 104, to the difference of longitude 251, on the line of numbers.

BY INSPECTION.

As in Case III. Plane Sailing, find the course either in Table I. or Table II., and the departure in its column, corresponding to which will stand the distance and 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 25° W., sails south-easterly 215 miles, making 167 miles departure; required the course steered, and the latitude and longitude in.

BY PROJECTION.

A

Distance

Draw the meridian ABC, and on any point of it draw BD perpendicular thereto, and make it equal to the departure 167 miles; with an extent equal to the distance 215 miles in your compasses, and one foot on D, as a centre, describe an arc to cut AB in A; join AD; then will AB be the proper difference of latitude 135.4 miles, and the angle BAD will be the course 50° 58′; hence we have the other latitude, and the meridional difference of latitude, to which make AC equal, and draw CE parallel to BD, meeting AD produced in E; then will CE be the difference of longitude 250.4 miles.

Diff Lat.

Departure

D

C

Difference Longitude

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1st. The extent from the distance 215, to the departure 167, on the line of numbers, wi. reach from the radius 90°, to the course 50° 58', on the line of sines.

2lly. The extent from radius 90°, to the complement of the course 39° 02′, on the line of sines, will reach from the distance 215, to the difference of latitude 135.4, on the line of numbers.

3dly. The extent from the radius 45°, to the course 50° 58', on the line of tangents, will reach from the meridional difference of latitude 203, to the difference of longitude 250.4, on the line of numbers. Or, the extent from the proper difference of latitude 135.4, to the departure 167, will reach from the meridional difference of latitude 203, to the difference of longitude 250.4, on the line of numbers.

BY INSPECTION.

Find the course and difference of latitude, as in Case V. Plane Sailing, by seeking in Table II., 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 meridiona 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 165.3. Find in this table one half the meridional difference of latitude 101.5, oppose to which, in the departure column, stands 125.1; this doubled gives 250.2, for the difference of longitude, nearly.

Having explained the method of calculating single courses by Middle Latitude and Mercator's Sailing, it now remains to explain the method of calculating compound courses. To do this, you must construct a traverse table, and find the difference of latitude and departure for each course and distance, as in Traverse Sailing, and from thence the whole difference of latitude, departure, and latitude in; with the departure and latitudes, find the difference of longitude and longitude in, as in Case II. of Middle Latitude or Mercator's Sailing.

This method is exact enough for working any single day's work at sea, except in high latitudes, where it will be a little erroneous; in this case the difference of longitude and longitude in, may be calculated for every single course and short distance; but in general this nicety in calculation may be neglected.

To illustrate the method of working compound courses, we shall here work an example by Middle Latitude and Mercator's Sailing.

A ship from Cape Henlopen, in the latitude of 38° 47' N., longitude 75° 5 W., sails the following true courses, viz. E. by S. 20 miles, E. N. E. 15 miles, S. E. 26 miles, south 16 miles, W. S. W. 6 miles, N. W. 10 miles, and east 30 miles; required her latitude and longitude.

By constructing the traverse table with these courses and distances, it appears that the ship has made 27.8 miles of southing, and 69.3 miles of easting; and by subtracting the southing from the latitude of Cape Henlopen, there remains the latitude in 38° 19' N.

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EXAMPLE.

TRAVERSE TABLE.

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Dist.

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E. by S.
E. N. E. 15
S. E.
South.

20

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26

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16

16.0

W. S. W.

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N. W.
East.

10

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30

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Diff. of lat... 27.8

By inspection of Table II. it appears that the difference of latitude 27.8, and departure 69.3, correspond to a course of 68° nearly, and a distance of 75 miles; and in the same page of the table, opposite to the meridional difference of latitude, found in the column of latitude, stands the difference of longitude 89 miles in the departure column; this being subtracted from the longitude of Cape Henlopen, 75° 5′ W., leaves the longitude in 73° 36′ W., by Mercator's Sailing. Or, with the middle latitude 38° 33′ to 39°, as a course, find the departure 69.3, in the latitude column, opposite to which is 89 in the distance column, which is the difference of longitude by Middle Latitude Sailing; consequently the longitude in is 73° 36′ W., as above.

Thus we see that such examples are performed as in Traverse Sailing, and Case II. of Mercator's or Middle Latitude Sailing, either by inspection, as above, or by the

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