PROBLEM I. 16. 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. 15A=60R=2400P First, of which is 49.11. . 12P; hence, 15A OR 12P=2412P; the square root 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. 24P. 1764 1760 square chains, First, = 176A 1R 24P=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. 17. To lay out a given quantity of land in a rectangu lar form, having one of the sides of the rectangle 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 the sides being given, acres in a rectangular form, one of and equal to 80 rods. First, 24042400 square chains 38400 square rods. Then, 80)38400(480 rods; which is the required side of the rectangle. 18. A great number of similar problems might be proposed. The solution of them does not, however, properly belong to surveying. The laying out of the ground, and 7 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. SECTION III. SURVEYING WITH THE COMPASS. DIVIDING LAND. 1. Before considering the principles involved in the method of surveying now to be explained, it will be ne cessary 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 which the magnetic needle is poised. This needle, if allowed to turn freely around the point of support, will settle to a state of rest: the direction which it then indicates, is 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°. The 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-box. 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 0 point of this vernier coincides with the 0 point of the graduated arc HI, the line of the compass-box marked NS, lies in the plane of the sights. screw L, Now, supposing the 0 of the vernier to coincide with the 0 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 until the needle points exactly to the line which marked the whole degrees: the space passed over by the 0 of the vernier, shows the parts of a degree that are to be added to give the true reading. SURVEYING WITH THE COMPASS. 2. The line about which the earth revolves is called its axis; and the two points in which the axis meets the surface of the earth, are called the poles. plane, and its intersection with the surface is called a meridian line or a meridian. 4. 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 error, be regarded as parallel straight lines. 5. 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. 6. 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 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. W D N B E F S When the course, like AB, falls between the north and east points, 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, the bearing is read, north 30° west, and is written N. 30° W. When the course, like AF, falls between the south and east points, the bearing is read, south 70° east, and is writ ten S. 70° E. When the course, like AD, falls between the south and west points, the bearing is read, south 70° west, and is written S. 70° W. A course which runs due north, or due south, is desig pated by the letter N or S; and one which runs due east, 7. 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. 8. 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 of the points is north or south of the other. Thus, in running the course from A to B, AC is the difference of latitude, north. 9. The perpendicular distance be- W tween 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 N H C B T E G A S the east or west side of the meridian passing through the point of beginning. east. Thus, in running the course AB, CB is the departure, 10. It will be 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 11. The meridian distance of a point is its perpendicular distance from an assumed meridian. Thus, if the distance be estimated from the meridian NS, BC will be the meri. dian distance of the point B. 12. The meridian distance of a line is the meridian distance of its middle point, and is east or west, according as |