Construction. Draw AB = 34, and perpendicular to it, make BC = 100; join AC and bisect it in D, and draw DE perpendicular to AC, meeting BC in E; then AE = CE = the part broken off. * Calculation. 1. In the right-angled triangle ABC, we have AB = 34 and BC = 100, to find the angle C = 18° 47'. 2. In the right angled triangle ABE, we have AEB = ACE + CAE=2 ACE = 37° 34' and AB = 34, to find AE = 55.77 feet, one of the parts; and 100 - 55.77 = 44.23 feet the other part. PRACTICAL QUESTIONS. 1. At 85 feet distance from the bottom of a tower, the angle of its elevation was found to be 52° 30': required the altitude of the tower. Ans. 110.8 feet. 2. To find the distance of an inaccessable object, I measured a line of 73 yards, and at each end of it took a * DEMONSTRATION. In the triangles AED, DEC, the angle ADE CDE, the side AD=CD, and DE is common to the two triangles, therefore (4.1) AE =CE. Note. This question may be neatly solved in the following manner, without finding either of the angles. Thus, draw DF perpendicular to BC, then (31.3 and cor. 8.6) FC: DC :: DC : CE; .consequently DC2 AC AB2 + BC2 CE and FC = · į BC; there4 ABP + BC2 342 + 1002 1156 + 10000 11156 fore CE = 2BC 200 200 200 FC; but DC2 the angle of position of the object and the other end, and found the one to be 90°, and the other 61° 45'; required the distance of the object from each station. Ans. 135.9 yards from one, and 154.2 from the other. 3. Wishing to know the distance between two trees C and D, standing in a bog, I measured a base line AB = 339 feet; at A the angle BAD was 100° and BAC 36° 30'; at B the angle ABC was 121° and ABD 49°; required the distance between the trees. Ans, 6971 feet. 4. Observing three steeples A, B and C, in a town at a distance, whose distances asunder are known to be as follows, viz. AB= 213, AC = 404, and BC = 262 yards, I took their angles of position from the place D where I stood, which was nearest the steeple B, and found the angle ADB = 130.30', and the angle BDC = 29° 50'. Required my distance from each of the three steeples. Ans. AD = 571 yards, BD = 389 yards, and CD = 514 yards. 5. A May-pole, whose top was broken off by a blast of wind, struck the ground at 15 feet distance from the foot of the pole: what was the height of the whole Maypole, supposing the length of the broken piece to be 39 feet? Ans. 75 feet. 6. At a certain" place the angle of elevation of an inaccessible tower was 26° 30'; but measuring 75 feet in a direct line towards it, the angle was then found to be 51° 30': required the height of the tower and its distance from the last station. Ans. Height 62 feet, distance 49.. 7. From the top of a tower by the sea side, of 143 feet high, I observed that the angle of depression of a ship’s bottom, then at anchor, was 35o; what was its distance from the bottom of the wall? Ans. 204.2 feet. • 8. There are two columns left standing upright in the ruins of Persepolis; the one is 64 feet above the plane, and the other 50; In a right line between these stands an ancient statue, the head of which is 97 feet from the summit of the higher, and 86 from that of the lower column; and the distance between the lower column and the centre of the statue's base is 76 feet: required the distance between the top of the columns. Ans. 157 feet. SURVEYING. SURVEYING is the art of measuring, laying out, and dividing land. MEASURING LAND. Preliminary Definitions, Observations, &c. The instrument used for measuring the sides of fields, or plantations, is a GunTER'S Chain, which is 4 poles or 66 feet in length, and is divided into 100 equal parts or links; consequently the length of each link is 7.92 inches: also 1 square chain is equal to 16 square perches, and 10 square chains make an acre. When the land is uneven or hilly, a four-pole chain is too long to be convenient, and the measures cannot be taken with it as accurately as with one that is shorter. Surveyors therefore generally make use of a chain that is two poles in length and divided into 50 links. The measures thus taken are, for the sake of ease in the cal. culation, reduced either to four-pole chains or to perches. The following rules shew the method of making these, and some other reductions. L To reduce two-pole chains and links to four-pole chains and, links. RULE. 1. If the number of chains be even, divide them by 2, and to the quotient annex the given number of links. Thus, in 16 two-pole chains and 37 links, there are 8 four-poie chains and 37 links. Or because each link is the hundredth part of a four-pole chain, the four-pole chains and links may be written thus 8.37 four-pole chains. 2. If the number of chains be odd, divide by 2 as before, and for the 1 that is to carry, add 50 to the given number of links. Thus in 17 two-pole chains and 42 links, there are 8 four-pole chains and 92 links, or 8.92 four-pole chains. To reduce two-pole chains and links, to perches and deci mals of a perch. RULE. a Multiply the links by 4 and the chains by 2. If the links when multiplied by 4, exceed a hundred set down the excess and carry 1 to the chains. Thus 17 two-pole chains and 21 links = 34.84 perches; also 15 two-pole chains and 38 links = 31.52 perches. To reduce four-pole chains and links, to perches and deci mals of a perch. RULE. Multiply the chains and links by 4. Thus 13.64 four |