120:10:50%

81.59.55 Log. co-secant, less radius=0.004248. 70. 11.45 Log. co-secant, less radius=0.026477

272.22.30

136:11:15"

54. 11. 20 Log. sine =

Remainder, by A B = 65.59.30 Log. sine =

Half the angle A =

63: 5 S Log. sine =

9.950210

Which being doubled, shows the angle A to be 126:10:16"; the same as by the former rule.

PROBLEM VI.

Given the Three Angles of a Spherical Triangle, to find the Sides.

RULE.

Add the three angles together and take half their sum; find the difference between the half sum and the angle opposite to the required side, which call the remainder.-Then,

To the log. co-secants, less radius, of the other two angles, add the log. co-sines of the half sum, and the remainder; half the sum of these four logs. will be the log. sine of half the required side.

One side being thus found, the remaining sides may be computed by Rule 3. Problem I., page 198.

Example.

In the spherical triangle A B C, let the angle A be 125:16:25; the angle B 84:20:50%, and the angle C 72:40:15"; required the sides B C, A B, and A C?

Р

Half the side BC=, 62:37:13 Log. sine =

The double of which gives 125:14:26%, for the whole side B C.

To find the Side A B :

9.948402

Remark. The required side of a spherical triangle (when the three angles are given,) may be also found by the following general rule; viz.,

Add the three angles together and take half their sum; find the difference between the half sum and each of the angles comprehending the required side, and note the remainders.-Then to the log. co-secants less radius, of those angles, add the log. co-sines of the two remainders: half the sum of these four logs, will be the log. co-sine of half the required side.

Thus, to find the side BC in the last example.

THE RESOLUTION OF PROBLEMS IN NAVIGATION BY LOGARITHMS; AND, ALSO, BY THE GENERAL TRAVERSE TABLE.

Lest the mariner should feel some degree of disappointment in not finding a regular course of navigation in this work: the author thinks it right to remind him, that his present intention carries him no farther than merely to show the proper application of the Tables to some of the most useful parts of the sciences on which he may touch :-it being completely at variance with the plan of this work, to enter into such parts of the sciences as could reasonably be dispensed with, without entirely losing sight of their principles. Hence it is, that the cases of plane sailing, usually met with in books on navigation, will not be noticed in this.-However, since it is not improbable that this volume may fall into the hands of persons not very deeply versed in nautical matters; it therefore may not be deemed unnecessary to give a few introductory definitions, &c. for their immediate guidance, previously to entering upon the essentially useful parts of the sailings.

NAVIGATION is the art of conducting a ship, through the wide and pathless ocean, from one part of the world to another.-Or, it is the method of finding the latitude and longitude of a ship's place at sea; and of thence determining her course and distance from that place, to any other given place.

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 arc 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.