EXAMPLES. 1. In an oblique-angled spherical triangle ABC are given, A= 32° 26′ 6′′, B=130° 5′ 22′′, and the side a 44° 13′ 42′′; to determine the other parts. b has two values, because the sine of B is greater than that of A. shall take the acute value. We II. To find the Side c. As cos. (A. B) 48°49′ 37′′ aritb. comp. 0.1815543 : cos. (A+B) 81 15 44 :: tan. (a + b) 64 14 : tan.c 9.1815890 7 10.3163591 2. Given A: = 103° 59′ 57′′, B=46° 18′ 7′′, and a = 42° 8′ 48′′; to find the angle C. C = 36° 7′52′′. 3. Given A=17° 46′ 16′′, B=151° 43′ 52′′, and a=37° 48′; to find the remaining sides, b being obtuse, b = 108°, c = 74° 30′. SCHOLIUM. Previously to closing this second part it may be worth while to remark, that if, in the foregoing investigations, we consider the radius of the sphere, upon which the triangles concerned are described, to be infinite, then, as any finite portion of the spheric surface may be considered as a plane, the spherical triangles will become plane triangles, and the sines and tangents of their sides will become identical with the sides themselves; so that all the foregoing rules and formulas, înto which cosines, cotangents, secants, or cosecants, of the sides do not enter, are applicable as well to plane as to spherical triangles. *** Before proceeding to the following part, the student may consult note (B), at the end, wherein is examined the unsound doctrine laid down by Professor Vince, at page 43 of his Trigonometry, (third edition,) respecting the tangent and secant of an arc of 90°. PART III. APPLICATION OF PLANE AND SPHERICAL TRIGONOMETRY TO THE PRINCIPLES OF NAVIGATION AND NAUTICAL ASTRONOMY. (62.) HAVING in the two preceding parts of the present treatise pretty fully explained and illustrated the principles of plane and spherical trigonometry, we shall now, for the purpose of showing the practical utility of these principles, apply them to the solution of one of the most important mathematical problems that has ever engaged the attentio of man, viz. to determine the place of a ship at sea. When a ship sails from any known place, and a correct account is kept of her various directions, and rates of sailing, her situation at any time may be readily ascertained by the rules of plane trigonometry, and the solution of the problem from these data belongs to Navigation. But it is impossible to measure a ship's course and the distance sailed exactly; so that after a long passage it would be unsafe to compute the place of the ship from the ship's reckoning. In such cases, therefore, the solution must be effected from other data, independent of the ship's account; these are furnished by astronomical observation, and the computation is performed by the rules of spherical trigonometry; the problem then becomes one of Nautical Astronomy. We shall devote a distinct chapter to each of these important branches. L 110 CHAPTER I. THE PRINCIPLES OF NAVIGATION. Definitions. (63.) 1. The earth is very nearly spherical. Navigation it may be considered as perfectly so. For the purposes of of its diameters, called its axis, in about twenty-four hours. This rotation is from the west towards the east, causing the heavenly bodies to have an apparent motion from the east towards the west. 2. The great circle, whose poles are the extremities of the axis, is called the equator. The poles of the equator are called also the poles of the earth; the one being the north pole, and the other the south pole. 3. Every great circle which passes through the poles, and which, therefore, cuts the equator at right-angles, is called a meridian circle. Through every place on the surface of the earth such a great circle is supposed to be drawn; it is the meridian of the place. It is expedient for the purposes of Geography and Navigation to fix upon one of these meridians as a first meridian, from which the meridians of other places are measured. The English have fixed upon the meridian of Greenwich Observatory for the first meridian. 4. The longitude of any place is the arc of the equator, intercepted between the meridian of that place and the first meridian; the longitude, therefore, is the measure of the angle between the two meridians. The longitude is east or west, according as the place is situated on the right or on the left of the first meridian, when we look towards the north pole. 5. The difference of longitude between two places is the arc of the equator intercepted between the meridians of those places, or the measure of the angle which they include; hence, when the longitudes of the places are of the same denomination, that is, either both east or both west, the difference is found by subtracting the one from the other; but when they are of contrary denominations the difference is found by adding the one to the other. 6. The latitude of a place is its distance from the equator, measured on the meridian of the place. Latitude, therefore, is north or south, according to the pole towards which it is measured, and cannot exceed 90°. 7. The small circles drawn parallel to the equator are called parallels of latitude. The arc of a meridian, intercepted between two such parallels, drawn through any two places, measures the difference of latitude of those places: when the latitudes are of the same denomination the difference of latitude is found by subtraction, but when the denominations are not the same the difference of latitude is found by addition, like difference of longitude. 8. The horizon of any place is an imaginary plane, conceived to touch the surface of the earth at that place, and to be extended to the heavens; such a plane is called the sensible horizon, and one parallel to it, but passing through the earth's centre is the rational horizon of the place. A line drawn across the horizon and through the place, in the plane of its meridian, is the meridian of the horizon, or the north and south line; the horizontal line through the same point, and perpendicular to this, is the east and west line. Besides the North, South, East, and West, points thus marked on the boundary of the horizon, this boundary is conceived to be subdivided into other intermediate points, corresponding to the divisions in the circle at the top of next page. 9. The course of a ship is the angle which her track makes with the meridians; so long as this angle remains the same, the ship is said to sail on the same rhumb line, or loxodromic curve. The magnitude of the angle or the course is indicated by the mariner's compass. 10. The Mariner's compass consists of a circular card, whose circumference is divided into thirty-two equal parts, called points, and each of these are subdivided into four equal parts, called quarter points; across this card is fixed a slender bar of magnetized steel, called the needle; the tapering extremities of which point to two diametrically opposite divisions of the card. These opposite divisions are marked N. and S., corresponding to the north and south poles, or ends, of the |