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. REMARK II. In determining the content 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 chain, by dividing by 22. 3d. Feet may be reduced to chains and the decimal of a chain, by dividing by 66. REMARK III. 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. 107). 2d. By dividing it by 4840, when it is in square yards (Art. 107). 3d. By dividing it by 43560 when it is in square feet (Art. 107). OF LAYING OUT AND DIVIDING LAND. 116. The surveyor is often required to lay off a given quantity of land, in such a way that its bounding lines shall form a particular figure, viz., a square, a rectangle, a triangle, &c. He is also often called upon to divide given pieces of land into parts containing given areas, or bearing certain relations with each other. The manner of making such divisions must always depend on a judicious application of the principles of geometry to the particular case. If, for example, it were required to lay out an acre of ground in a square form, it would first be necessary to find, by calculation, the side of such a square, and then to trace, on the ground, four equal lines at right angles to each other. PROBLEM I. 117. To lay out a given quantity of land in a square form. Reduce the given area to square chains,' or square rods : then extract the square root, and the result will be the side of the required square. This square being described on the ground, will be the figure required. 1. To trace a square which shall contain 15A OR 12P First, 15 A=60 R=2400 P 12 P; hence, 15 A 0R 12 P=2412 P; the square root of which is 49.11. Therefore, if a square be traced on the ground, of which the side is 49.11 rods, it will be the required figure. 2. To trace a square which shall contain 176A 1R 24 P. First, 176 A=1760 square chains, iR= 2.5 24 P= 1.5 ; hence, 176 A 1R 24 P=1764 square chains: the square root of which is 42. Hence, if a square be traced on the ground, of which the side is 42 ch, it will be the required figure. PROBLEM II. 118. To lay out a given quantity of land in a rectangular form, having one of its sides given. Divide the given area, reduced to square chains or square rods, by the given side of the required rectangle, and the quotient will be the other side. Then trace the rectangle on the ground. 1. To lay off 240 acres in a rectangular form, one of the sides being given, and equal to 80 rods. First, 240 A=2400 square chains=38400 square rods. Then, 80)38400(480 rods; which is the required side 119. A great number of similar problems might be proposed. The solution of them does not, however, properly belong to şurveying. The laying out of the ground, and the tracing of lines, after the figure and area have been determined, are the only parts which appertain to a practical treatise. The manner of tracing lines having been already explained, it seems unnecessary to add the numerous examples often given under this head of the subject. CHAPTER IV. Of the Surveying Compass.-Of Surveying with the Compass. Of the Plane-Table. 120. Before considering the principles involved in the method of surveying now to be explained, it will be necessary to describe the instrument principally used in the field, and which is called THE CIRCUMFERENTER, OR SURVEYOR'S COMPASS. Pl. 2, Fig. 2. This instrument 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 vertically through the middle of the large aperture, as shown in Fig. 3. The compass-box DCE is circular, and generally about six inches in diameter. At the centre is a small pin, on to turn freely around the point of support, will settle to a state of rest : the direction which it then indicates, is called the magnetic meridian. In the interior of the compass-box, there is a graduated circle divided to degrees, and sometimes to half degrees: the degrees are numbered from the extremities of the diameter NS, both ways to 90°. Tlie length of the magnetic needle is a little less than the diameter of the graduated circle, so that the needle can move freely around its centre, within the circle, and its positions be noted on the graduated arc. The compass-box is turned about its centre, without moving the plate AB, by means of the milled screw L: it is fastened to the plate AB, by the screw P. In using the compass, it is important to ascertain the exact angle which may be included between the magnetic meridian and the direction that may be given to the line drawn through the eye and the sights AF and BG. To effect this, a small arc HI is described on the bar AB, having its centre at the centre of the compass-bos. This arc is divided to degrees, and sometimes to the parts of a degree. A vernier is also used, which is permanently attached to the compass-box. When the o point of this vernier coincides with the o point of the graduated arc HI, the line of the compass-box marked NS, has the same horizontal direction as the line along which the sights are directed. Now, supposing the o of the vernier to coincide with the o of the arc HI, if the end of the needle does not stand at one of the lines of division of the graduated circle, let the whole degrees be read. Then, turn the compass-box by means of the screw L, until the needle points exactly to the line which marked the whole degrees: the space passed over by the o of the vernier, shows the minutes that are to be added. OF SURVEYING WITH TIIE COMPASS. 121. The line about which the earth revolves is called its axis; and the two points in which the axis meets the surface 122. A meridian is a line traced on the surface of the earth, which would, if sufficiently produced in both directions, pass through the poles. Hence, all the meridian lines intersect each other at the two poles. The poles, however, are so distant from each other, that no sensible error will arise in supposing the meridians to be parallel; and since, in all the surveys made with the compass, the surface of the ground is regarded as a horizontal plane, the meridians are represented by horizontal and parallel lines. 123. When the compass is placed on its stand, and the needle is allowed to settle to a state of rest, the direction it assumes has been named the magnetic meridian. Although this line is different from the true meridian, yet in the surveys made with the compass, we shall take for the meridian that line which is determined by the direction of the magnetic needle. 124. If the right hand be turned towards the point where the sun rises, the direction pointed by the farthest end of the needle is called north; the direction shown by the nearest end is called south, and the line thus indicated is called a north and south line, as well as a meridian. 125. A line perpendicular to the meridian is called an east and west line : the east point being on the right hand, and the west on the left. 126. A line traced or measured on the ground, is called a course ; and the angle which this line makes with the meridian passing through the point of N beginning, is called the bearing. B W -E AB, the line AB is the course, A and the angle NAB is the bearing. S When the course, like AB, falls between the north and east points, the bearing is read, north 46° east, and is writ |