The Equator is a great circle circumscribing the earth, every point of which is equally distant from the poles; thus dividing the globe into two equal parts, called hemispheres: that towards the North Pole is called the northern hemisphere, and the other, the southern hemisphere.-The equator, like all other great circles, is divided into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes; each minute into 60 equal parts, called seconds, and so on.

The Meridian of any place on the earth is a great circle passing through that place and the poles, and cutting the equator at right angles.-Every point on the surface of the sphere may be conceived to have a meridian line passing through it; hence there may be as many meridians as there are points in the equator.-Since the First Meridian is merely an imaginary circle passing through any remarkable place and the poles of the world; therefore it is entirely arbitrary.-Hence it is that the British reckon their first meridian to be that which passes through the Royal Observatory at Greenwich: the French esteem their first meridian to be that which passes through the Royal Observatory at Paris; the Spaniards that which passes through Cadiz, &c. &c. &c.

Every meridian line may be said, with respect to the place through which it passes, to divide the surface of the sphere into two equal parts, called the eastern and western hemispheres.

The Latitude of any place on the earth is that portion of its meridian which is intercepted between the equator and the given place; and is named north or south, according as the given place is in the northern or southern hemisphere. As the latitude begins at the equator, where it is nothing, and is reckoned thence to the poles, where it terminates; therefore the greatest latitude any place can have, is 90 degrees.

The Difference of Latitude between two places on the earth is an arc of the meridian intercepted between their corresponding parallels of latitude; showing how far one of them is to the northward or southward of the other:-The difference of latitude between two places can never exceed 180 degrees.

The Longitude of any place on the earth is that are or portion of the equator which is contained between the first meridian and the meridian of the given place; and is denominated east, or west, according as it may be situated with respect to the first meridian.-As the longitude is reckoned both ways from the first meridian (east and west) till it meets at the same meridian on the opposite part of the equator; therefore the longitude of any place can never exceed 180 degrees.

The difference of Longitude between two places on the earth is an arc of the equator intercepted between the meridians of those places; showing how far one of them is to the eastward or westward of the other :-The difference of longitude between two places can never exceed 180 degrees.

When the latitudes of two places on the earth are both north or both south; or their longitudes both east or both west, they are said to be of the same name.-But, when one latitude is north and the other south; or one longitude east and the other west; then they are said to be of different

names.

The Horizon is that great circle which is equally distant from the zenith and nadir, and divides the visible from the invisible hemisphere; this is called the rational horizon.-The sensible horizon is that which terminates the view of a spectator in any part of the world.

The Mariner's Compass is an artificial representation of the horizon :-it is divided into 32 equal parts, called points; each point consisting of 11:15.-Hence the whole compass card contains 360 degrees; for 11:15 multiplied by 32 points = 360 degrees.

A Rhumb Line is a right line, or rather curve, drawn from the centre of the compass to the horizon, and obtains its name from the point of the horizon it falls in with.-Hence there may be as many rhumb-lines as there are points in the horizon.

The Course steered by a ship is the angle contained between the meridian of the place sailed from, and the rhumb-line on which she sails; and is either estimated in points or degrees.

The Distance is the number of miles intercepted between any two places, reckoned on the rhumb line of the course; or it is the absolute length that a ship has sailed in a given time.

The Departure is the distance of the ship from the meridian of the place sailed from, reckoned on the parallel of latitude at which she arrives; and is named east or west, according as the course is in the eastern or western hemisphere.

If a ship's course be due north or south, she sails on a meridian, and therefore makes no departure :-hence the distance sailed will be equal to the difference of latitude.

If a ship's course be due east or west, she sails either on the equator, or on some parallel of latitude; in this case since she makes no difference of latitude, the distance sailed will, therefore, be equal to the departure.

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.