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

Inspection :

To the course 3 points, and half the difference of latitude = 128, the distance is 154, and the departure 85.5. Now, 154 x 2 = 308 miles, is the required distance. Again, to middle latitude 12%, and departure 85.5, in a latitude column, the corresponding distance is 87; and to latitude 13, and departure 85, 5, the distance is 88: hence, to middle latitude 12:38, and departure 85.5, the distance is 87; the double of which = 175 miles, is the difference of longitude, as required.

PROBLEM IV.

Given the Latitude and Longitude of the Place sailed from, the Course, and the Departure; to find the Distance sailed, and the Latitude and Longitude of the Place come to.

With the course = A. and the departure B C, the distance may be found by trigonometry, Problem II., page 172; as thus:

=

With the course A, and the departure BC, the distance is found by trigonometry, Problem II., page 172; as thus:

With the middle latitude BCD, and the departure B C, the difference of longitude is found by trigonometry, Problem II., page 172.

To find the Distance, Difference of Latitude, and Difference of Longitude, by Inspection :

=

To course 5 points, and one-fourth of the departure 117, the distance is 141, and the difference of latitude 78. 3. Now, 141 x 4 = 564 miles, the distance, and 78.3 x 4 = 313.2, or 513, the difference of latitude; whence the latitude in, is 4897 N., and the middle latitude 45:30. Again, to middle latitude 45%, and one-fourth the departure = 117, in a latitude column, the distance is 166; and to middle latitude 46, and departure 117, the distance is 168: hence, to middle latitude 45:30, and departure 117, the difference of longitude is 167 x 4 = 668 miles; nearly the same as by calculation.

PROBLEM V.

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

Example.

A ship from St. Agnes, Scilly, in latitude 49:54. N., and longitude 6:19. W., sailed 320 miles between the south and west, and then, by observation, was found to be in latitude 45:8. N.; required the course, and the longitude come to?

C

M. lat. 47: 31:

Diff. long.

Difference of latitude =

4:46 286 miles. Sum = 95: 2:

Middle latitude = 47:31:

To find the Course A:

=

With the distance A C, and the difference of latitude = AB, the course may be found by trigonometry, Problem III., page 174; as thus:

With the course, middle latitude, and distance, the difference of longi

tude is found by the 1st analogy, page 222; as thus:

The course is S. 26:39: W., or S.S.W. W., nearly.

To find the Course and Difference of Longitude by Inspection

-

To half the distance = 160, and half the difference of latitude = 143, the course nearest agreeing is 27, and the departure 72.6. Now, to middle latitude 47: as a course, and departure 72. 6, in a latitude column, the distance is 105; and to middle latitude 48, and departure 72.6, the distance is 108: hence, the difference of distance to 1 of latitude, is 3 miles; therefore, 3 x 3160 1'. 5, which, added to 105, makes 106.5 this, being multiplied by 2, gives 213 miles the difference of longitude.

=

PROBLEM VI.

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

The distance = AC, and the departure B C, being given, the course may be found by trigonometry, Problem III., page 174; as thus:

90: 0' 0"

To find the Difference of Latitude = AB:

=

The course A, and the distance, being thus known, the difference of latitude may be computed by trigonometry, Problem III., page 174.

As radius =

To find the Difference of Longitude = CD:

The middle latitude = angle B CD, and the departure B C, being given, the difference of longitude may be found by trigonometry, Problem II., page 172; as thus:

To find the Course, Difference of Latitude, and Difference of Longitude, by Inspection:

The distance 280, and departure 186, are found to agree between 41: and 42%, and the corresponding difference of latitude 208. 1: whence the middle latitude is 41:46. Now, to middle latitude 41°, and departure 186, in a latitude column, the corresponding distance is 247; and to latitude 429, and departure 186, the distance is 250: hence, the difference of distance to 1 of latitude, is 3 miles; and 3' x 4660: 2'.3, which, added to 247, gives 249.3=the difference of longitude, as required; which nearly agrees with the result by calculation.