To Determine the Distance to an Inaccessible Point.-Let it be required to determine the distance from the point B to an inaccessible point P, Fig. 4. Measure BC in any convenient direction and run a line A'D' parallel to BC. Measure AD, the distance between the points where the lines PB and PC intersect A'D'. Measure also AB. Then, \B EXAMPLE.-If, in Fig. 4, BC= 100 ft., AB= 52.4 ft., and AD =124.2 ft., what is the distance BP? SOLUTION. Substituting these values in the preceding equation, To Determine the Distance Between Two Points Invisible From Each Other.-Let it be required to find the distance between two points A and B, Fig. 5, that are invisible from each other. First run a random line AD' in such a manner that it will pass as near B as can be estimated. From B drop a perpendicular BD on AD' and compute the required distance AB by the formula AB= √AD2+BD2 EXAMPLE.-If, in Fig. 5, the distance AD is 206.1 ft. and the distance BD is 35.1 ft., what is the distance from A to B? SOLUTION.-Here AD=206.1 and BD=35.1; therefore, substituting in the preceding formula, AB= √206.12+35.12 =209.1 ft. Survey of a Closed Field.—If a closed field is to be surveyed without the aid of an angle-measuring instrument, the area is divided into triangles by means of diagonals, which are measured on the ground. The area of each triangle may then be determined by the formula A = √s(s-a) (s—b) (s—c), in which a, b, and c represent the three sides and s represents a+b+c 2 half of their sum, or When obstacles make it impossible to measure directly the diagonals of a field, as, for instance, the diagonal BE, Fig. 6, EXAMPLE.-In Fig. 6, let the lengths of the sides be as follows: AB=320 ft., BC=217 ft., CD=196 ft., DE=285 ft., and EA=304 ft. It is required to calculate the length of the diagonal BE by means of a tie-line. SOLUTION.-Let the line BA be prolonged 100 ft. beyond A; that is, make AF100 ft. Then, AG must be equal to Let the length of GF, as found by measurement, be 125 ft. Precision. In chain surveying, an error of 1 in 500 is generally permissible, and should not be exceeded; that is, two measurements of the same line should not give results differing by more than 1 ft. for every 500 ft. measured. If, however, the chaining is done carefully, and the ground is not rough, the error need not exceed 1 in 800 or 1,000. ANGULAR SURVEYING COMPASS SURVEYING The compass used in surveying consists essentially of a magnetic needle supported freely on a pivot at the center of a horizontal graduated circle. To this circle is attached a pair of sights. The needle and graduated circle are enclosed in a brass case having a glass cover, and the whole is attached to a tripod, or Jacob's staff, by a ball-and-socket joint and is leveled by FIG. 1 means of the plate levels. Fig. 1 illustrates the type of compass in general use. Adjustments of the Compass.-Besides several conditions that are attended to by the instrument maker, the following are indispensable for accurate work: 1. The plane tangent to the level bubbles when at the centers of their respective tubes must be perpendicular to the vertical axis of the socket. 2. The two ends of the needle and the pivot must be in the same vertical plane. 3. The needle pivot must be in the center of the graduated circle. A new compass made by a good manufacturer always satisfies these conditions, as the instrument is sold by the maker in perfect adjustment. Rough usage, however, a fall, or a hard blow may throw the compass out of order, and it is necessary that the surveyor should know how to test and readjust it. To Adjust the Plate Levels.-Bring the bubbles to the centers of the level tubes by moving or rotating the plate carefully by means of the ball-and-socket joint; then revolve the compass horizontally through 180°; that is, turn it end for end. If the bubbles remain in the centers of the tubes, the levels are in adjustment. But if in turning the compass end for end, either bubble runs toward one end of the tube, lower that end and raise the opposite end sufficiently to bring the bubble half way back, by means of small screws that attach the ends of the tube to the plate; then bring the bubble to the center by moving the plate as before. Repeat the operation until both bubbles remain in the centers of the tubes in every position of the compass. To Straighten the Needle.-Level the compass and turn it so that the north end of the needle points exactly toward or cuts some prominent graduation mark of the needle circle, and note the exact reading of the south end of the needle. In order to read either end of the needle accurately, the eye should be directly above a line in the prolongation of the opposite end of the needle. Then reverse the compass end for end and turn it so that the south end of the needle cuts the same graduation mark, and observe whether the north end reads the same as the south end did before reversing. If it does not read the same, correct one-half the error by bending the needle carefully, and repeat the operation, using different graduation marks, until exact reversals are obtained. To Center the Needle Pivot.-Having, if necessary, straightened the needle, turn the compass so that the north end of the needle will exactly cut some prominent graduation mark, and observe whether the south end exactly cuts the opposite graduation mark. If it does not, find the position of the needle that shows the greatest difference in the readings of its opposite ends; then remove the needle from the pivot and bend the pivot carefully at right angles to this position an amount equal to one-half the error. Repeat the operation until the needle cuts accurately all opposite graduation marks. Use of the Compass.-By means of the compass the angle between any line and the direction of the needle, or the magnetic meridian, can be measured directly. This angle is called the magnetic bearing of the line. The angle between two lines can be determined by either subtracting or adding their bearings, as the case may require. A rough sketch, showing the relative positions of the two lines with reference to the meridian, will enable one to determine by inspection the required arithmetical operation. P3 P2 W- - N 0--- -E P Bearings are reckoned from 0° to 90° and indicate the amount by which a line is east or west of north or south. In Fig. 2, in which NS represents the magnetic needle, N and Sbeing, respectively, the north and south ends, the line OP1 makes, with the north half of the needle, an angle of 60°. As the line lies between the north point N and the east point E, its bearing is 60° northeast or 60° to the east of north. This is indicated by the notation N 60° E. Similarly, the bearings of OP2, OP3, and OP1 are, respectively, N 42° W, S 70° W, and S 50° E. S FIG. 2 To determine the magnetic bearing of a line, turn the compass, after it has been set and leveled, until the line SN, Fig. 3, which is the line of the sights, coincides with the line OP whose bearing is to be determined, the observer's eye being at the slit near S. The north end of the needle FG is then pointing to |