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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
* 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
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.
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.
To reduce two-pole chains and links to four-pole chains
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.
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.
Multiply the chains and links by 4. Thus 13.64 four