nearest to which is 132,9) stand 171 and 107.6, these doubled give the distance 842 and departure 215.2; and in the same table opposite the half mer. diff. of lat. 198 found in the latitude column, stands 160.5 in the dep. column, which doubled gives the difference of longitude 321 miles, nearly

as before.

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 arch cutting BD in D; draw AD. which continue 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.

1st. 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 longitude 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 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

CASE VI.

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

Draw the meridian ABC, and at a distance from it equal to the departure 957 miles draw the line FD parallel to ABC; make the angle BAD equal to the course 6 points, draw AĎ to cut FD in D; from D let fall upon AB the perpendicular DB; then will AD be the distance 1036 miles, AB the difference of latitude 396 miles; hence we have both latitudes, and the meridional difference of latitude 667 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 1610 miles.

1st. The extent from the course 6 points to radius 8 points on the line marked S. R. will reach from the departure 957 to the distance 1036 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, 957 to the difference of latitude 396 on the line of numbers.

3dly. The same extent (from the radius points to the course 6 points on the line marked T. R.) will reach from the meridional difference of latitude 667, to the difference of longitude 1610, 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 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 one-fifth of the departure 191.4 stand 79.2 and 207, which multiplied by 5 give the difference of latitude 396 miles, and the distance 1035 miles; then in the latitude column find a tenth of the meridional difference of latitude 66.7, the nearest to that is 66.6, against which, in the departure column, stands 160.8, which multiplied by 10 gives 1608, the difference of longitude.

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 southeasterly 645 miles, making 500 miles departure; required the course steered, and the latitude and longitude in?

BY PROJECTION.

Dist 645

Dep 500

D

Draw the meridian ABC, and on any point of it draw BD perpendicu- Ao lar thereto and make it equal to the departure 500 miles, with an extent equal to the distance 645 miles in your compasses, and one foot on D as a centre describe an arch to cut AB in A, join AD; then will AB be the pro- B per difference of latitude 407.5 miles, and the angle BAD will be the course 50° 49'; 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, 722,6 miles.

To find the course.

As the distance 645

Is to the radius 900
So is the departure 500

To sine of course 500 49'

To find the diff. of long.

As radius 450
Is to the mer diff lat 589
So is tang. course 500 49'

C

XLong

2 69897 So is co-sine course 500 49'

9 88941 To diff. 1. 407,5-60 48' S.
Lat. left

E

BY LOGARITHMS.

To find the diff. of lat.

9.80058

2.61014

49 30 N.

Mer. diff. lat. 2.85891 Long. left

589

25° 00′ W.

To diff. long. 72.6

To diff. long 722,7

BY GUNTER.

1st. The extent from the distance 645 to the departure 500 on the line of numbers, will reach from the radius 90° to the course 50° 49' on the line of sines.

2dly. The extent from radius 90° to the complement of the course 39 11' on the line of sines, will reach from the distance 645 to the difference of latitude 407.5 on the line of numbers.

3dly. The extent from the radius 45° to the course 50° 49' on the line of tangents, will reach from the mer. diff. of lat. 589 to the difference of longitude 722.6 on the line of numbers. Or, the extent from the proper difference of latitude 407.5 to the departure 500, will reach from the meridional difference of latitude 589 to the difference of longitude 722.7 on the line numbers.

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 one-third of the distance 215, and one-third of the departure 166,7 are found nearly to correspond to a course of 51 degrees, and a difference of latitude of 135,3, which multiplied by 3, gives the true difference of latitude 406 nearly. Then one-fourth of the meridional difference of latitude 147, in the latitude column, is found nearly to correspond to the departure 181.9; this multiplied by 4, gives 727.6 the difference of longitude nearly. Having explained the method of calculating single courses by Middie 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.

EXAMPLE.

A ship from Cape Henlopen, in the latitude of 38° 47' N. longitude 75° 10' 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.

Cape Henlopen's latitude 380 47′ N.
Latitude in
38 19 N.

Sum of latitudes
Middle latitude

77 6 38 33

By inspection of Table II. it appears that the difference of latitude 27.8 and departure 69.3 correspond to a course of 63° 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 subtracted from the longitude of Cape Henlopen 75° 10′ W. leaves the longitude in 73° 41' 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° 41' 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 scale of logarithms.

Having gone through the necessary problems in Mercator's Sailing, we shall now show how Mercator's Chart may be constructed by means of the Table of Meridional Parts.

To construct a Mercator's Chart to commence at the Equator.

Suppose it was required to construct the Chart in the plate prefixed to this work which begins at the equator, and reaches to the parallel of 50 degrees; and contains 95 degrees of longitude west from the meridian of Greenwich? Draw the line AD representing the equator, then take from any scale of equal parts the number of minutes contained in 95 degrees, viz. 5700, which set off from A to D; subdivide this line into 95 equal parts representing degrees of longitude. Through A and Ddraw the lines AB, DC perpendicular to AD, and make each of them equal to 3474 which are the meridional parts, corresponding to 50 degrees. Join BC which must be subdivided in the same manner as the line AD; and through the corresponding points of the lines AD BC must be drawn (at the distance of 10° or 20°) the lines parellel to AB, representing meridians of the earth; these lines must be numbered 0, 10, 20, &c. beginning at the line AB which represents the meridian of Greenwich. Set off in like manner upon the meridians AB, DC, (beginning from the equator AD) the meridional parts corresponding to each degree of latitude from 0° to 50°; and through the corresponding points (at the distance of 10° or 20°) draw lines parallel to the equator AD, to represent the parallels of latitude. Then the upper part of the chart will represent the north, the lower the south, the right hand the east, and the left hand the west (which is generally supposed in charts, unless the contrary is expressly mentioned.)

If the Chart does not commence at the equator, but is to serve for a certain portion of the globe contained between two parallels of latitude on the same side of the equator; you must draw the meridians as directed in the last example; then subtract the meridional parts of the least latitude of the chart from the meridional parts of the other latitudes, and set off these differences on the extreme meridians, draw lines through the corresponding points, and they will be the parallels of latitude on the chart.

If the chart is to be bounded by parallels of latitude on different sides of the equator, you must draw a line representing the equator, and perpendicular to it draw the lines to represent the meridians, continuing them on both sides of the equator; then set off the parallels of latitude on both sides of the equator, in the same manner as in the first example.

Take from the Table of latitudes and longitudes of places the latitude and longitude of each particular place contained within the bounds of the chart, and lay a rule over its latitude and another crossing that over its longitude; the point where these meet will represent the proposed place upon the chart. The most remarkable point of a sea coast being thus laid down, lines may be drawn from point to point which will form the outlines of the sea coast, islands, &c. to which may be annexed the depths of water expressed in common Arabian numbers, the time of high water on the full and change days expressed in Ronan numbers: the setting of the tide expressed by an arrow and whatever else may be thought convenient for the chart to contain.

This chart is not to be considered as a just representation of the earth's surface for the figures of islands and countries are distorted towards the poles, as is evident from the construction; but the degrees of latitude and longitude are increased in the same proportion, so that the bearings between places will be the same on the chart as on the globe; and as the meridians are right lines, it follows, that the rhumbs, which form equal angles with the meridians, will be straight lines, which render this projection of the earth's surface much more easy and proper for the mariner's use than any other. Having the latitude and longitude of a ship or place, to find the corresponding point on the chart.