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61. To find the area of a piece of ground in the form of a circle.
RULE.-Measure the radius AC: then
multiply the square of the radius by
3.1416 (Mens., Art. 105).
To find the area of a circular piece of land, of which the diameter is 25 ch.
Ans. 49 A. 0 R. 14 P.
62. To find the contents of a piece of ground in the form of
RULE.-Measure the semi-axes AE, CE. Then multiply them together, and their product by 3.1416.
A F EG B
To find the area of an elliptical piece of ground, of which the transverse axis is 16.08 ch., and the conjugate axis 9.72 ch. Ans. 12 A. 1 R. 4 P.
NOTE 1.-The following is the manner of tracing an ellipse on the ground, when the two axes are known.
From C, one of the extremities of the conjugate axis as a centre, and AE, half the transverse axis, as a radius, describe the arc of a circle cutting AE in the two points F and G: these points are called the foci of the ellipse.
Then, take a tape, the length of which is equal to AB, and fasten the two ends, one at the focus F, the other at the focus G. Place a pin against the tape and move it around, keeping the tape tightly stretched: the extremity of the pin will trace the curve of the ellipse.
NOTE 2.-In determining the contents of ground, in the examples which have been given, the linear dimensions have been taken in chains and decimals of a chain.
If the linear dimensions were taken in terms of any other unit, they may be readily reduced to chains. For, a chain is equal to 4 rods, equal to 22 yards, equal to 66 feet. Hence,
1st. Rods may be reduced to chains and the decimal of a chain, by dividing by 4.
2d. Yards may be reduced to chains and the decimal of a chain, by dividing by 22.
3d. Feet may be reduced to chains and the decimal of a chain, by dividing by 66.
NOTE 3.-If it is thought best to calculate the area, without reducing the linear dimensions to chains, the result can be reduced to acres:
1st. By dividing it by 160, when it is in square rods (Art. 50).
2d. By dividing it by 4840, when it is in square yards (Art. 52).
3d. By dividing it by 43560, when it is in square feet
63. THE AXIS of the earth is the immovable diameter about which it revolves; and the poles are the points in which the axis meets the surface.
64. Any plane passing through the axis of the earth is called a meridian plane; and its intersection with the surface is called a meridian line, or simply a meridian.
65. All the meridians converge towards the poles, but they vary so little from parallelism, within the narrow limits of surveys made with the compass, that they may, without sensible error, be regarded as parallel straight lines.
66. If a magnetic needle be suspended freely, and allowed to settle to a state of rest, a vertical plane passed through its axis is called the plane of the magnetic meridian; and its intersection with the surface of the earth is called the magnetic meridian, or sometimes, a North and South line. A line perpendicular to a North and South line, is called an East and West line.
67. A line traced, or measured on the ground, is called a Course; and the angle which this line makes with the magnetic meridian, passing through the point of beginning, is called the Bearing.
Thus, if we start from the point A, and measure in the direction AB, the line AB is the course, and the angle NAB, is the bearing.
When the course, like AB, falls between the north and east
points, and makes an angle of 46° with the meridian, the bearing is read, north 46° east, and is written, N. 46° E.
When the course, like AC, falls between the north and west points, and makes with the meridian an angle of 30°, the bearing is read, north 30° west, and is written, N. 30° W.
'When the course, like AD, falls between the south and west points, and makes an angle with the meridian of 70°, the bearing is read, south 70° west, and is written, S. 70° W.
When the course, like AF, falls between the south and east points, and makes with the meridian an angle of 70°, the bearing is read, south 70° east, and is written, S. 70° E.
A course which runs due north, or due south, is designated by the letter N, or S; and one which runs due east, or due west, by the letter E, or W.
68. If, after having passed over a course, the bearing is taken to the back station, this bearing is called the back sight, or reverse bearing.
69. The perpendicular distance between the east and west lines, drawn through the extremities of a course, is called the northing or southing, according as the course is run towards the north or south. This distance is also called the difference of latitude, or simply the latitude, because it shows .the distance which one end of the course is north or south of the other.
Thus, in running the course from A to B, AC is the difference of latitude, north.
70. The perpendicular distance between the meridians passing through the extremities of a course, is called the departure of that course, and is east or west, according
as the course lies on the east or west side of the meridian passing through the point of beginning.
71. It is found convenient, in explaining the rules for surveying with the compass, to attribute to the latitudes and departures, the algebraic signs, and -.
We shall, therefore, consider every northing as affected with the sign +, and every southing as affected with the sign We shall also consider every easting as affected with the sign +, and every westing as affected with the sign.
72. The meridian distance of a point is its perpendicular distance from any assumed meridian. Thus, if the distance be estimated from the meridian NS, BC will be the meridian distance of the point B.
73. The meridian distance of a line, is the meridian distance of its middle point, and is east or west, according as this point lies on the east or west side of the assumed meridian. Thus, FG drawn through the middle point of AB, is the meridian distance of the line AB.
The sign will always be given to the meridian distance of a point or line, when it lies on the east of the assumed meridian, and the sign, when it lies on the west.
74. This instrument (Pl. 2, Fig. 2), consists of a compass-box DCE, a magnetic needle, a brass plate AB, from twelve to fourteen inches long, two plain sights, AF and BG, one of which is more fully shown in Fig. 3; and a stand, which is sometimes a tripod, and sometimes a single staff, pointed with iron at the lower end, so that it may be placed firmly in the ground.
The open sights, AF and BG, are placed at right angles to the plate AB, and fastened to it firmly by the screws a and b. In each sight there is a large and small aperture or slit; the larger aperture being above the smaller in one of the sights, and below it in the other. A hair or thread of silk is drawn