PROBLEM VII.

Given both Latitudes and Departure; to find the Course, Distance, and Difference of Longitude.

Here, the difference of latitude = A B, and the departure B C, being given, the course is found by trigonometry, Problem IV., page 175; as thus:

With the course, thus found, and the difference of latitude AB, the distance may be computed by trigonometry, Problem IV., page 175: hence,

Hence, the course is S. 51:5! E., or S.E. E., nearly, and the distance 560.4 miles.

To find the Difference of Longitude = CD:

With the middle latitude = B CD, and the departure B C, the difference of longitude is found by trigonometry, Problem IV., page 175; as thus:

To find the Course, Distance, and Difference of Longitude, by

Inspection:

218,

Half the difference of latitude = 176, and half the departure are found to agree nearest at 51: under or over distance 280: hence, 280 x 2 = 560 miles, is the distance. Again, to middle latitude 37 as a course, and departure 218, in a latitude column, the corresponding distance is 273; and to latitude 38: and departure 218, the distance is 277: hence, the change of distance to 1 of latitude, is 4 miles. Now, 4 x 51? + 603'.4, which, added to 273, gives 276. 4; and this, being multiplied by 2, gives 552. 8 miles; which very nearly corresponds with the result by calculation.

PROBLEM VIII.

Given one Latitude, Departure, and Difference of Longitude; to find the other Latitude, Course, and Distance.

To find the Middle Latitude = the Angle BCD:-.

With the departure = BC, and the difference of longitude = CD, the angle of the middle latitude may be found by trigonometry, Problem III., page 174; as thus:

With the difference of latitude A B, and the departure B C, the course may be found by trigonometry, Problem IV., page 175; as thus :

With the angle of the course, thus found, and the difference of latitude A B, the distance may be computed by trigonometry, Problem IV., page

Hence, the course is S. 46:54. W., or S.W. W. nearly, and the distance 747.8 miles.

To find the Latitude come to, Course, and Distance, by Inspection in the general Traverse Table:

One-fourth of the difference of longitude 2234, taken as distance, and one-fourth of the departure = 136. 5, in a latitude column, will be found to agree between 52 and 53. Now, to latitude 52°, and distance 223, the difference of latitude is 137.3, which is 0'. 8 more than 136.5; and to latitude 53, and distance 223, the difference of latitude is 134. 2, being 2'.3 less than 136.5: hence, the difference of meridional distance to 1 of latitude is 0'. 8 + 2'. 3 = 3′. 1: therefore, as 3'.1: 0.8 :: 60: : 16., which, added to 52: (proportion being made for the quarter of a mile in the distance), gives the middle latitude = 52:18: hence, the latitude come to is 56:34: S., and the difference of latitude 511 miles. Again, to one-fourth of the difference of latitude 127.75, and one-fourth of the departure 136. 5, the course is 47°, and the distance 187; which, multiplied by 4, gives 748 miles the whole distance.

=

PROBLEM IX.

Given the Distance, Difference of Longitude, and Middle Latitude;

to find the Course and both Latitudes.

To find the Angle of the Course = A:

The course may be found by the 3d analogy, page 222, as thus:

To find the Difference of Latitude A B:

The difference of latitude may be found by the 8th analogy, page 222, as

SOLUTION OF PROBLEMS IN MERCATOR'S SAILING.

Mercator's Sailing is the method of finding, on a plane surface, the motion of a ship upon any assigned point of the compass, which shall be true in latitude, longitude, and distance sailed.

Mariners, generally speaking, solve all the practical cases in Mercator's Sailing by stated rules, called canons, which they early commit to memory, and, ever after, employ in the determination of a ship's place at sea. Those canons, certainly, hold good in most cases; but since they are destructive of the best principles of science, inasmuch as that they have a direct tendency to remove from the mind every trace of the elements of trigonometry, the very doctrine from which they were originally deduced, and on which the whole art of navigation is founded, the following observations and consequent analogies are, therefore, submitted to the attention of naval people, under the hope that they will serve as an inducement to the substitution of the rules of reason for the rules of rote; and thus do away with the necessity of getting canons by heart.

In the annexed diagram, let the triangle ABC be a figure in plane sailing, in which the angle A represents the course, A C the distance, A B the difference of latitude, and B C the departure. If AB be produced to D, until it is made equal to the meridional difference of latitude, and DE be drawn at right angles thereto, and parallel to BC; then the triangle ADE will be a figure in Mercator's sailing, in which the angle A represents the course, the side AD the meridional difference of E latitude, and the side D E the difference of longi

C

Depart

Diff long

A.

Curse

Diff lat

Mer diff lat

B

D

tude. Now, since the two triangles ABC and AD E are right angled,