When the course is 4 points, or 45 degrees, the difference of latitude and departure are equal.

When the course is less than 4 points, or 45 degrees, the difference of latitude exceeds the departure; but when it is more than 4 points, or 45 degrees, the departure exceeds the difference of latitude.

Note. Since the distance sailed, the difference of latitude, and the departure form the sides of a right angled plane triangle; in which the hypothenuse is represented by the distance; the perpendicular, by the difference of latitude; the base, by the departure; the angle opposite to the base, by the course; and the angle opposite to the perpendicular, by the complement of the course; therefore any two of these five parts being given, the remaining three may be readily found by the analogies for right angled plane trigonometry.

These being premised, we will now proceed to the following Introductory Problems.

PROBLEM I.

Given the Latitudes of Two Places on the Earth, to find the difference of

Latitude.

RULE.

When the latitudes are of the same name; that is, both north, or both south, their difference will be the difference of latitude; but when one is north and the other south, their sum will express the difference of latitude.

Note. The same Rule is to be observed in finding the meridional difference of latitude between two places.

Note. In finding the difference of latitude, or the difference of longitude between two places (when any of the sailings are under consideration), it will be sufficiently exact to take out the latitudes and longitudes from Table LVIII. to the nearest minute of a degree, as above.

PROBLEM II.

Given the Latitude left and the difference of Latitude, to find the
Latitude in.
RULE.

When the latitude left and the difference of latitude are of the same name their sum will be the latitude; but when they are of contrary denominations, their difference will be the latitude required :-This latitude will always be of the same name with the greater quantity.

Given the Longitudes of Two Places on the Earth, to find the difference

of Longitude.

RULE.

When the longitudes are of the same name: that is, both east, or both west, their difference will express the difference of longitude; but when one is east and the other west, their sum will be the difference of longitude. If the sum of the longitudes exceed 180:, subtract it from 360°, and the remainder will be the difference of longitude.

Given the Longitude left and the difference of Longitude, to find the

Longitude in.
RULE.

When the longitude left and the difference of longitude are of the same name, their sum will be the longitude in; should that sum exceed 180°, subtract it from 360; and the remainder will be the longitude in, of a contrary name to the longitude left.-But, when the longitude left and the difference of longitude are of contrary names, their difference will be the longitude in, of the same name with the greater quantity.

Remarks. If a ship be in north latitude sailing northerly, or in south latitude sailing southerly, she increases her latitude, and therefore the difference of latitude must be added to the latitude left, in order to find the latitude in :-but, in north latitude sailing southerly, or in south latitude, northerly, she decreases her latitude; therefore the difference of latitude subtracted from the latitude left will give the latitude in :-should the difference of latitude be the greatest, the latitude left is to be taken from it; in this case the ship will be on the opposite side of the equator with respect to the latitude sailed from.-Again,

If a ship be in east longitude sailing easterly, or in west longitude sailing westerly, she increases her longitude; therefore the difference of longitude added to the longitude left will give the longitude in; should the sum exceed 180, the ship will be on the opposite side of the first meridian with respect to the longitude sailed from.-But, in east longitude sailing westerly, or in west longitude sailing easterly, she decreases her longitude, and therefore the difference of longitude is to be subtracted from the longitude left, in order to find the longitude in ;-should the difference of longitude be the greatest, the longitude left is to be taken from it; in this case the ship will, also, be on the opposite side of the first meridian with respect to the longitude sailed from. These remarks will appear evident on a comparison with the above Examples.

SOLUTION OF PROBLEMS IN PARALLEL SAILING.

Parallel Sailing is the method of finding the distance between two places situate under the same parallel of latitude; or of finding the difference of longitude corresponding to the meridional distance, when a ship sails due

east or west.

PROBLEM I.

Given the Difference of Longitude between two Places, both in the samé Parallel of Latitude, to find their Distance.

RULE.

As radius, is to the co-sine of the latitude; so is the difference of longitude, to the distance.

Example.

Required the distance between Portsmouth, in longitude 1:6 W., and Green Island, Newfoundland, in longitude 55:35. W.; their common latitude being 50:47: N.?

In the right-angled triangle A B C, where the hypothenuse A C represents the difference of longitude between the two given places, the angle A the latitude of the parallel of those places, and the base A B their meridional distance given the side AC 3269 miles, and the angle A= 50:47, to find the side A B. Hence, by right-angled plane trigonometry, problem I., page 171,

As radius=

90: 0 0 Log. co-secant = Is to the diff. of long. AC 3269 miles Log. = So is the lat.=the angle A=50:47 0 Log. co-sine =

To the merid. dist. AB = 2066. 8 miles Log. =

To find the Meridional Distance by Inspection in the general Traverse

Table:

Note. This case may be solved by Problem I., page 107, as thus: To latitude 50 as a course, and one-eleventh of the difference of longitude (viz. 297. 2) as a distance, the corresponding difference of latitude is 190.9; and to latitude 51°, and distance 297. 2, the difference of latitude