have been in all the cases right-angled at C; in the first two cases supposed, the angle at B would have been 60°; and in the latter case, the angle A would 37° 49′ 18′′, and that at B would = 52° 10′ 42′′.

=

... sin. C

a

a

Log.(a = 325)

(113.) Given the base AC = 1003, AB=918, and the vertical angle ABC = 60° 43′; to find the other angles and the side BC.

(114.) Given the base AC=158.7, the side BC= 200, and the angle at A=71° 37′; to find the other angles and the side AB.

(115.) Given the angle at A = 38° 12', the angle at C = 48° 24′, and the side AC=1250; to find the angle at B, and the sides AB and BC.

(116.) Given the angle at A = 56° 15′, that at B=61° 12′, and the side AC1105; to find the angle at C, and the sides AB and BC.

(117.) If AB=432, BC=528, and ▲ A= 60°; what are the angles at B and C, and the side AC?

(118.) If AB=593, AC=486, and ▲ C = 80° 45′ ; what are the angles at A and B, and the side BC?

(119.) If AB=738, BC=439, and the angle at C = 59° 18′; find the angles at A and B, and the third side AC.

(120.) If AC = 490, BC=365, and the angle at A = 40° 30′ ; find the angles at B and C, and the third side AB.

(121.) The sum of the three sides of a triangle is 1105, one side is 420, and the angle opposite it is 63°; find the other two angles, and the other two sides. (Apply Euc. I. 20.)

(122.) The difference of two sides of a triangle is 40, and the third side is 583, also the angle opposite this third side is 76° 30′; find the other angles, and also the other sides.

E

CHAPTER III.

OF ASTRONOMY AND GEOGRAPHY, SO FAR AS NECESSARY TO THE NAVIGATOR.

137. The land and water of this earth make up a body of a spherical or globular form, or so nearly so that it is called the TERRAQUEOUS Globe.

138. This globe moves round its axis from west to east in about 24 hours, and thereby causes the celestial bodies to revolve apparently from east to west in the same time; whence the vicissitudes of day and night.

139. The two points in which the axis, on which the earth revolves, meets the surface, are called the poles of the earth. If the same axis be produced both ways to meet the celestial sphere, it will meet it in two opposite points, called the CELESTIAL POLES. The pole of the earth lying towards the north is called the NORTH or ARCTIC POLE, and that towards the south is called the SOUTH or ANTARCTIC POLE.

140. If a plane be supposed to pass through the earth's centre at right angles to its axis, it will intersect its surface in a great circle, called the EQUATOR; and if the plane of the equator be produced to meet the celestial sphere, it will trace out on it a circle, called the EQUINOCTIAL, which will divide the earth and heavens into two halves or hemispheres; that towards the north being called the NORTHERN HEMISPHERE, and that towards the south being called the SOUTHERN HEMISPHERE.

141. A GREAT CIRCLE is the intersection of a plane passing through the centre with the circumference of the sphere.

142. Great circles passing through the poles of the earth are called MERIDIANS, and that which passes through any place is called the meridian of that place. These planes being produced to the heavens pass through the celestial poles, and trace out on the celestial sphere the CELESTIAL MERIDIANS. They are called meridians, because when the sun is on the meridian of any place, it is noon or mid-day at that place.

143. If a plane be supposed to touch the surface of the earth in any point upon which a spectator stands, and to be produced to the heavens, it will there make a circle, called the HORIZON, which separates the visible from the invisible parts of the celestial sphere. This horizon is, properly speaking, called the sensible horizon; the TRUE or RATIONAL HORIZON is traced out by a plane passing through the centre of the earth, and parallel to the former plane, or sensible horizon; it divides the earth and heavens into hemispheres, called the UPPER and LOWER HEMISPHERES. These two horizons, however, when produced to the region of the fixed stars, sensibly coincide; but at the distance of the Sun, Moon, and Planets, their distance causes what is called PARALLAX. (See Note on page 73.)

144. Since the earth revolves on its axis from west to east, it is plain that a spectator upon its surface, together with his horizon, in the axis of which he always is, must move the same way; consequently those celestial bodies towards the east that were before in the Lower Hemisphere, will gradually come into the Upper Hemisphere of the spectator, and thereby become visible, his horizon being depressed below them; while those in the west that were at first visible will gradually disappear, the horizon being elevated above them. In this way arises the apparent motion of the heavenly bodies by which they are said to rise and set, and by which they appear to describe circles round the pole, parallel to the equinoctial, which are greater or less, according as they are more or less distant from the nearest pole.

145. When a celestial body comes first into view on the eastern side of the horizon, it is said to rise; and when by

its apparent motion it comes to the meridian, it is said to CULMINATE, being at its greatest height; and lastly, when it begins to disappear on the western horizon, it is said to set.

146. If through the centre of the earth there be drawn a line perpendicular to the plane of the horizon, and produced on both sides to the heavens, it will there mark out the two poles of the horizon; the one which is directly over the spectator's head is called the ZENITH, and the opposite one, which is invisible to him, being directly under his feet, is called the NADIR.

147. Vertical or azimuth circles are great circles passing through the zenith and nadir. Among these there are two principal ones-namely, the meridian, which passes through the zenith and nadir and the poles of the celestial sphere, and so cuts both the horizon and equinoctial at right angles; the points in which it cuts the horizon are called the NORTH and SOUTH points. The other principal vertical is called the PRIME VERTICAL, and cuts at right angles the meridian in the zenith and nadir, and the horizon in two opposite points, called the EAST and WEST points. The north, south, east, and west are called the four CARDINAL POINTS.

148. The MERIDIAN of any place is a semicircle passing through that place, and terminating at the poles of the equator.

149. The LATITUDE of any place is that portion of the meridian which is contained between the equator and the given place, and is either north or south, according as the given place is in the northern or southern hemisphere.

150. The PARALLEL of LATITUDE of any place is a circle passing through that place parallel to the equator.

151. The DIFFERENCE of LATITUDE between any two places is an arc of a meridian intercepted en their corresponding