regarding the height of the observer as an equivalent for that of the point observed on the boat? Ans. 79 and 71 yards, very nearly. NOTE. When an object, such as a boat, is spoken of as a point observed, some particular point in the boat is understood; and, when there are two observations, they must both be directed to the same part of the object. 12. Wanting to determine the Sun's altitude (that is, its angular elevation), at a particular instant, and having no better instrument at hand, I placed a pole upright, 18 feet high, and found that the shadow of its top fell on a point 23 feet 4 inches from the pole (horizontal measurement), and 37 inches higher than its base. What is the altitude computed from these data? Ans. 33° 27'. 13. At the Vernal Equinox, at noon, in Latitude 55°, the shadow of an upright tree, growing on level ground, was 96 feet in length; and since it was a fir tree, tapering acutely towards the top, the outermost point of the shadow corresponded to the highest point of the tree. What was the tree's height? Ans. 66 feet, very nearly. NOTE. At either the Vernal or the Autumnal Equinox, the Sun shines directly over the Equator, or at right angles to the Earth's axis. 16. A spire stands on a base 35 feet square. I take a station higher than the base, but opposite the middle of one side, and 20 yards distant from that side measured horizontally. From that station the elevation of the vane is observed by a common quadrant (not Hadley's) to be 56° 34'. Then, directing the sights of the quadrant horizontally, I mark the point which they strike on the upright wall, and find it 23 feet above the ground. What is the height of the vane? 17. A tower stands on the brow of a uniformly sloping acclivity. I place a common quadrant at the foot of the slope and find the angular elevation of a point on the top of the perpendicular wall of the tower, to which I am directly opposite, 41° 15'. Then, guided by a plumb line, I mark two other points on the wall directly beneath that observed at the top,-the one at the base, the other 5 feet higher, corresponding to the height of the quadrant,-and find the elevation of the latter 18° 52'. I then suspend the plummet from the quadrant to the ground, and ascertain the distance from the plummet to the mark at the base of the wall, measured on the slope, to be 384 feet. What is the height of the wall from its base? Ans. 199.5 feet. 18. Placing a quadrant on a plane but not level ground, 90 feet from the base of an upright corner of a building, the distance being measured on the sloping ground, as described in Exercise 17, I take the angle of the summit of the corner, and find it 39° 27'. I then bring the telescope of the quadrant to a horizontal position, turned towards the corner, and, marking the point to which it is now directed, I find the point thus marked to be 26 feet above the base, the height of the quadrant being 5 feet. What is the height of the wall? Ans. 98.01 + feet. 19. From the brow of a steep headland a ship was observed in the roads beneath. The angle of depression of the ship, at the water line, was found to be 9° 28′; and that of the shore immediately below, in the direction of the ship, 72° 40'. What was the distance of the ship from the point of observation, and what from the shore, the perpendicular height of the precipice being known to be 254 feet, and the surface of the sea being regarded as a plane? Ans. 1,544 and 1,444 feet, nearly. 20. Two streets meet at an acute angle. The one lies N 51° W, and the other S 48° W. The distance from the corner to a druggist's shop door in the first street is 315 yards; and the distance from the same corner to a surgeon's door in the other street is 406 yards. What is the distance in a straight line from the surgeon's door to the druggist's? Ans. 473 yards. 21. From a vessel at anchor two rocks are observed to the westward, the one (A) bearing WNW; and the other (B), W by S, from the vessel. The rocks, being well known, are laid down on the chart, from which it is found that the former bears NNE from the latter, distant 645 yards. What are their respective distances from the vessel? Ans. A is distant 965, and B, 1,161 yards, nearly. 24. A castle-wall rises perpendicularly from the shore of a lake, its base being washed by the water. A rock stands opposite, which is known to rise 50 feet above the level of the water. On the top of this rock I place my theodolite, 5 feet high, and take the angular elevation of the top of the wall, 10° 41', and the depression of its base, 13° 25'. What is the height of the wall? 26. In a trigonometrical survey a base line AB is measured of 10 chains, and the following angles are taken with the theodolite : : ZABC 48° 23', <CDE=64 10, B C E ZDCE 61 57, The ground is every where level. DE is required. NOTE. On this system the great National Survey of the British Isles is conducted. A base is measured, with the utmost accuracy, as the foundation of the whole survey, and afterwards angles only are observed, triangle being added to triangle, till, after many such operations, one of the computed sides is also measured as a proof of the accuracy of all the intermediate operations. 27. In order to ascertain E F R the height and position of a hill a flag (P) is placed on the top; and, from a station (E) on the plain below, the angle of elevation of the flag is found to be 20° 49'. A second station (F) on the plain, directly between the first station and the flag, is ascertained to be on the same level with the first and 213 feet distant from it. The elevation of the flag at the second station is observed to be 26° 12'. The distance of the flag from the nearer station, and the height of the hill above the plain, are required, the flag and the eye of * In practice, it is usual to measure the third angle also, of each triangle, to serve as a check on the others; but, in this place, it is better to allow the scholar to find it. the observer being at the same height above the ground on which they stand. Ans. Distance, 807; height, 356+, feet. NOTE. When it is said that the second station is directly between the first station and the flag, it is not meant, of course, that they are all three in the same straight line, but that they are in the same vertical plane. This is determined practically by observing if a plumb line, suspended at the first station, appear to the eye to cut both the flag and the second station at once, or by using the vertical hair in the telescope instead of the plumb-line. 28. In an operation similar to the last, the angles of elevation observed at the two stations were 6° 25′ and 9° 14'; and the base (that is, the distance between the stations) was exactly 10 chains. The height of the hill is required in feet. Ans. 241 feet. 29. In a third operation similar to the two preceding, the two observed angles of elevation were 5° 59′ and 9° 26'. The base was again 10 chains, but was found, in this instance, not to be level but inclined at an angle of 3° 4′ to the horizon, the more distant station being the higher. The height of the hill's summit above each of the stations is required. Ans. 247 and 283 feet, nearly. 30. The height of a fourth hill is to be determined in the same manner as the last and under similar circumstances. A base line is measured of 12 chains. The angles of elevation of the top of the hill at the two stations are 4° 23′ and 7° 40'. It is found by levelling that the ground at the station furthest from the hill is 17 feet lower than at the nearer station, and 93 feet higher than the level of a lake near it. What is the height of the top of the hill above the lake? Ans. 211+ feet. P 31. Desiring to know the height of the upright corner of a wall (Pr), standing on the brow of a uniformly sloping bank (er); but, having only a sextant with me, and no artificial horizon or any other means of taking an angle of elevation, I proceeded thus:-The wall being accessible, I placed a mark, R, upon it, 5 feet in height, equal to the height of my eye above the ground. I then measured 20 feet E F R r f down the bank, from r to f, and then, standing at f, I took the angle PFR, and found it 70°. I next measured 80 feet further, in the same direction, from f to e, and found the angle PER, 40°. What is the height of the corner? Ans. 103-feet. NOTE. In the preceding exercises, the height of an object is obtained by means of a base pointing directly towards the object, and by observing vertical angles only. But unless the object is elevated at a considerable angle (even as seen from the more remote station), or unless the ground, on which the base is measured, rises considerably as it recedes from the object, that method is not susceptible of great accuracy; for, except in the cases mentioned, the angle which the base subtends at the object (that is, the angle EPF in the two preceding diagrams) is so small, that a minute error in that angle will produce a comparatively large error in the computed distance and height of the object. It is better, therefore, to select a base fronting the object—that is, lying across the direction in which the object is seen, as in the following exercises-and to observe the horizontal and vertical angles with a theodolite, or the oblique angles with a sextant. If the observation is made at sea, or if an artificial horizon is used, the angles of elevation may also be taken with the sextant. When the theodolite is used, the base must be level, or its inclination must be ascertained. It is otherwise when the angles are taken with the sextant: the obliquity of the base is then immaterial, and entirely disregarded. E 32. It is required to determine the height of a very high tree, PR. It stands in the midst of broken ground and brushwood, so that a base line cannot be measured up to the foot of the tree, or in any line directly receding from it; but there is a clear tract in the direction EF, and the top of the tree can be seen from the two points E and F, but the R foot of the tree from E only: these two points are therefore taken as our stations. The base EF of 116 feet is not level; but that is of no consequence, since the angles are to be taken with the sextant. The ground at E is just so much lower than that on which the tree grows that when |