Answer, ambiguous, A C being 89° 47′ 6′′, 4 C 53° 35′ 16′′, and ▲ B 119° 30′ 31"; or AC 36° 19′ 48′′, ▲ C 126° 24′ 44′′, and 4iB 31° 7' 59".

9. Given A B 81° 26' 37", 4 A 40° 18′ 22′′, and 4 B 67° 3′ 29′′, to find the other parts?

Answer, B C 40° 50′ 55′′, ▲ C 102° 2′ 36′′, and A C 68° 36′ 48′′. 10. Given A C 47° 28′ 32", 4 A 39° 16', and 4 C 98° 21', to find the other parts?

Answer, B C 33° 31′ 58′′, A B 59° 42′ 56′′, and 4 B 57° 36′ 31′′. 11. Given A B 128° 13′ 47′′, 4 C 131° 11′ 12′′, and B C 77° 25′ 11′′, to find the other parts ?

Answer, A C 84° 29′ 24′′, 4 A 69° 14′, and ▲ B 72° 28′ 46′′. 12. Given A C 97° 18′ 39′′, A B 86° 53′ 46′′, and B C 89° 21′ 37′′, to find the other parts ?

Answer, A 88° 57′ 20′′, 4 B 97° 21′ 26", 13. Given B C 74° 16′, 4 B 46° 34', and

other parts?

and 4 C 86° 47′ 17′′.

C 81° 20′, to find the

Answer, A C 44° 34′ 42′′, ▲ A 84° 46′ 35′′, and A B 72° 50′ 43′′. 14. Given C 77° 22′ 21′′, ≤ A 71° 33′ 30′′, and ▲ B 57° 53′ 55′′, to find the sides?

Answer, A B 61° 14′, B C 58° 27′, and A C 49° 33'. 15. Given A B 80° 12′ 21′′, B C 50° 36′ 39′′, and ▲ B 68° 40′ 26′′, to find the other parts ?

Answer, A C 67° 21' 40", 4 A 51° 15' 50", and C 91° 58′ 1′′. 16. Given A C 100° 21′ 30′′, B C 97° 18′ 22′′, and 4 B 100° 28′ 48", to find the other parts ?

Answer, A B 19° 22′ 2′′, ▲ C 19° 21′ 33′′, and ▲ A 82° 31′ 14". 17. Given A B 43° 22′ 13′′, B C 31° 26′ 41′′, and 4 A 12° 16′, to find the other parts ?

Answer, ambiguous, A C 73° 7' 35", C 16° 14' 27", and 4 B 115° 0′ 31′′; or A C 12° 17′ 39′′, ▲ C 163° 45′ 33′′, and 4 B 47° 1′ 36′′.

18. Given

A 59° 29′ 6′′, ▲ B 54° 39′ 32′′, and A B 68° 32′ 46′′, to find the other parts?

Answer, C 92° 6′ 11′′, A C 49° 26′ 18′′, and B C 53° 21′ 17′′. 19. The perpendicular C D falling within the triangle, given A D 6° 25′ 33′′, B D 42° 54′ 46′′, and ▲ A 82° 33′ 53′′, to find the other angles and the sides?

Answer, B 51° 33′ 4′′, B C 56° 13' 24", and A C 41° 2' 0". 20. The perpendicular C D falling within the triangle, given A D, 32° 54′ 16′′, 4 A 56° 18′ 40′′, and 4 B 39° 10′38′′, to find the sides, and the angle A C B, D B being obtuse?

Answer, A C B 135° 47′ 56′′, A B 123° 4′ 56′′, B C 90° 8′ 17′′, and A C 49° 23′ 41′′.

21. The perpendicular C D falling without the triangle, given A D

27° 36′ 3′′, BD 79° 49′ 3′′, and C D 15° 40′ 21′′, to find the sides and angles of the triangle A B C ?

Answer, A 148° 48′ 6′′, ▲ B 15° 54′ 38′′, 4 C 24° 32′ 52′′, A B 52° 13′, B C 80° 12′, and A C 31° 26'.

22. Given A C 53° 12′ 19′′, B C 80° 4′ 52", 4 A 93° 40′ 58′′, and ZB acute, to find the other parts ?

Answer, B 54° 13′ 10′′, 4 C 70° 25′ 32′′, and AB 68° 26' 29". 23. Given A B 110°, B C 116°, and A C 116°, to find the angles? Answer, A 134° 9′ 5′′, 4 B 134° 9′ 5′′, and ▲ C 131° 23′ 48′′. 24. Given A C 96° 31′ 47′′, A B 19° 28′ 3′′, and

the other parts?

C 10°, to find

Answer, ambiguous, 4 A 30° 50′ 53′′, B 148° 49′ 29′′, and B C 79° 48′ 5′′; or ▲ A 151° 26′ 59′′, ▲ B 31° 10′ 31′′, and B C

113° 27′ 23′′.

25. Given A B 112o 26′ 21′′, A C 57° 29′16′′, and ▲ A 87° 46′ 50′′, to find the other parts?

Answer, B C 100° 4′ 37′′, ▲ B 58° 51′ 12′′, and ▲ C 110° 16′ 16′′. 26. Given B C 40° 16′, A C 28° 1′, and ▲ C 53° 39′ 20′′, to find the other parts?

Answer, A B 31° 24', 4 A 87° 46′ 13", and B 46° 34' 5". 27. Given ▲ A 45° 26′ 12′′, ▲ B 41° 11′ 6′′, and ▲ C 134° 54′ 27′′, to find the sides?

Answer, A B 112° 16′ 52′′, B C 68° 34′ 13′′, and A C 59° 21′ 18′′. 28. Given B 129° 48′ 0′′, 4 A 38° 1′ 29′′, and A B 49° 22′ 4′′, to find the other parts ?

Answer, C 35° 39′ 56′′, B C 53° 18′ 6′′, and A C 90° 1′ 10′′. 29. Given A B 53° 16′ 32′′, B C 48° 26′ 39′′, and A C 51° 36′ 30′′, to find the angles?

Answer, A 62° 18′ 11′′, ▲ B 68° 1′ 55′′, and ▲ C71° 30′ 57′′. 30. Given A B 90°, B C 90°, and A C 90°, to find the angles? Answer, each angle is 90o. 31. Given A B 90°, B C 90°, and A C 41° 3′ 10′′, to find the angles? Answer, A 90o, ▲ B 90°, and C 41° 3′ 10′′. 32. Given A B 86° 12′ 52′′, B C 79° 38′ 21′′, and A C 58° 39′ 16′′, to find the angles?

Answer, A 80° 10' 10", ▲ B 58° 48′ 36′′, and C 91° 52′ 42′′.

ELEMENTARY PRINCIPLES

OF

NAVIGATION.

NAVIGATION is the art of conducting a ship from one place to another. That this art may be rightly understood, or practised with advantage, it is necessary that the navigator should be acquainted with the form of the earth, the relative situation of the lines conceived to be drawn upon its surface, and be furnished with correct charts of such parts of it as he may have occasion to visit, as well as with tables, in which the situations of the most remarkable sea coasts, islands, rocks, shoals, &c. are accurately described: and he must also understand the use and application of such instruments as are necessary to determine the direction in which the ship is steered, and the distance which she sails; and be further possessed of sufficient mathematical skill to deduce, from the data which these instruments furnish, the situation of the ship at any time; and to find the direction and distance of any place to which it may be required that the ship should be taken.

That the earth, in its general figure, is a round body is evident from various considerations. If it were flat, then in clear weather, though distant objects upon its surface might appear small, they would still be within the limit of view; but it is uniformly observed, in every part of the earth, that to whatever quarter a ship sails, after she has proceeded a few miles to sea, she is gradually lost sight of, appearing as it were to sink in the waters, or to be hidden behind their convexity; the lower parts disappearing first, and the higher in succession. Now the figure of the object on which this appearance uniformly takes place must necessarily be round.

In lunar eclipses, which are caused by the moon's passing through the shadow of the earth, it is always observed that the bounding line of the shadow on the face of the moon is a curve line; the earth therefore, which casts that shadow, must be a round body. To these and many other considerations it may be added, that several cele brated navigators, by proceeding forward always, or as nearly as circumstances admitted, in the same direction, have actually arrived at the place which they sailed from, and have thus sailed round, or circumnavigated the earth.

But though the figure of the earth is nearly spherical, it is not exactly so. It revolves round one of its diameters once in a day; and this revolution produces an effect upon its figure, which, in nice observations, becomes very apparent. It is flattened towards the extremities of the axis of rotation, but so slightly, that in computing the place of a ship, from the distance which she has gone, and the

direction in which she has sailed, the earth may be safely considered as a sphere.

The diameter round which it revolves is called the axis, and the extremities of that diameter the poles of the earth. That to which we in Europe are nearest is called the north pole, and the other the south pole.

Great circles passing through the poles are called meridians; the great circle, equidistant from both poles, and which therefore cuts the meridians at right angles, is called the equator, the equinoctial, or the line; and less circles, whose planes are parallel to the plane of the equator, are called parallels of latitude. The meridian passing over any place is called the meridian of that place; and the portion of a meridian intercepted between a place and the equator, is called the latitude of that place; and it receives the denomination of north or south, according as the place is on the north or south side of the equator.

It is customary to call the meridian of some remarkable place the first meridian, and the angle included between the first, and any other meridian, is called the longitude of that other meridian, or of any place over which the meridian passes. And as the angle included between two great circles is measured by the arc which they intercept on another great circle, whose pole is at the point of their intersection, the longitude of a place may also be defined to be the arc of the equator intercepted between the first meridian and the meridian of that place, and it is considered as east or west according as the place is situated towards the east or west of the first meridian. English geographers and seamen refer to the meridian of the Royal Observatory at Greenwich as the first meridian, Frenchmen to that of the Observatory at Paris, &c.

The difference of latitude between any two places, is an arc of a meridian intercepted between the parallels of latitude on which the places are situated; and the difference of their longitudes is the angle at the pole included between their meridians, or the arc of the equator which those meridians intercept.

Hence when the latitudes or the longitudes of two places are of the same denomination, the difference of their latitudes or of their longitudes will be found by subtracting the less from the greater; but when they are of different denominations, by taking their sum.

A curve that cuts every meridian which it meets at the same angle, is called a rhumb line; the angle which the rhumb line makes with the meridian, is called the course between any two places through which the rhumb passes; and the arc of a rhumb line intercepted between two places is called their nautical distance.

The meridian distance which a ship has made, is an arc of the parallel on which the ship is, intercepted between the meridian left and the meridian arrived at; and the departure which a ship makes

in sailing upon a rhumb line, is the sum of all the intermediate meridian distances, computed on the supposition that the distance is divided into indefinitely small equal parts.

The parallel of latitude, which is 23° 28′ north of the equator, is called the tropic of cancer; and that which is 23° 28′ south of the equator is called the tropic of capricorn. The parallel of latitude, which is 23° 28′ from the north pole, is called the arctic circle; and that which is at the same distance from the south pole, is called the antarctic circle. These four circles divide the surface of the earth into five parts, called zones. The part included between the tropics is called the torrid or burning zone, from the intense heat produced by the vertical action of the sun's rays. Those included between the poles and the arctic and antarctic circles, are called frigid or frozen zones, from the great cold arising from the periodical absence of the sun, and the obliquity with which his rays at all times meet the surface of those parts of the earth. The two remaining parts are called temperate zones, from their enjoying the advantages of an intermediate state between the extremes of heat and cold which prevail in the torrid and frigid zones.

The equator bisects the torrid zone, and also divides the whole surface of the earth into two equal parts; that in which the north pole is situated being called the northern hemisphere, and the other the southern hemisphere.

With respect to the magnitude of the earth, it has been found, by various admeasurements, that it is nearly equal to a sphere of 7916 English miles in diameter, or 24869 miles in circumference. Hence a geographical or a nautical mile, which is the 21600th of 360°, is nearly 6079 English feet.

a

For the sake of illustration, let us suppose that in the annexed figure, P is the north pole, TK Q the equator, or a great circle, every part of which is a quadrant distant from P; PT, P H, P K, &c. great circles passing through P, and of course cutting the equator at right angles; A I, b B, R S, &c. arcs of smaller circles parallel to the equator, and therefore cutting the meridians at right angles; A E a curve cutting every meridian which it meets, as PK, PL, PM, &c. at the same angle. Then P H, P K, &c. produced till they meet at the opposite pole, are called meridians; A I, b B, R S, &c. continued round the globe, are called parallels of latitude; AE is called the rhumb line, passing through A and E; the length of AE is called the nautical distance from A to E; and the angle ba B, or any of its equals; c BC,

R

B

A

K L M

N