2. By resection. Place the table in position at A (Fig. 37), and draw a line in the direction of C, as in the former case; then remove the instrument to C', place it in position by the line drawn from a, make the edge of the ruler pass through b, and turn the alidade about b until B is in the line of sight. A line drawn along the edge of the ruler will intersect the line from a in c. 3. By radiation. Place the table in position at A (Fig. 38), and draw a line from a toward C, as in the former cases. Measure AC, and lay off ac to the same scale as ab. To plot a field ABCD..... 1. By radiation. Set up the table at any point P, a and mark p on the paper over P. B Draw indefinite lines from p to ward A, B, C ..... Measure PA, and lay off pa, pb, ..........., to a suitable scale, and join a and b, b and c, c and d, etc. 2. By progression. Measure Set up the table at A, and draw a line from a toward B. AB, and plot ab to a suitable scale. Set up the table in position at B, and in like manner determine and plot bc, etc. 3. By intersection. Plot one side as a base line. Plot the other corners by the method of intersection, and join. 4. By resection. Plot one side as a base line. Plot the other corners by the method of resection, and join. The Three Point Problem. Let A, B, C represent three field stations plotted as a, b, c, respectively (Fig. 39); it is required to plot d representing a fourth field station D, visible from A, B, and C. Place the table over D, level and orient approximately by the declinatoire. Determine d by resection as follows: Make the edge of the ruler pass through a and lie in the direction aA, and draw a line along the edge of the ruler. In like manner, draw lines through b toward B and through e toward C. If the table were oriented perfectly these lines would meet at the required point d, but ordinarily they will form the triangle of error, ab, ac, bc. In this case, through a, b, and ab; a, c, and ac; and b, c, and bc, respectively, draw circles; these circles will intersect in the required point d. For at the required point the sides ab, ac, be must subtend the same angle as at the points ab, ac, bc, respectively. Hence, the required point d lies at the intersection of the three circles. mentioned. The plane-table may now be oriented accurately. NOTE. The three point problem may be solved by fastening on the board a piece of tracing paper and marking the point d representing D, after which lines are drawn from d toward A, B, and C. The tracing paper is then moved until the lines thus drawn pass through a, b, c, respectively, when by pricking through d the point is determined on the plot below. CHAPTER III. TRIANGULATION.* § 27. INTRODUCTORY REMARKS. Geographical positions upon the surface of the earth are commonly determined by systems of triangles which connect a carefully determined base line with the points to be located. Let F (Fig. 40) represent a point whose position with reference to the base line AB is required. Connect AB with F by the series of triangles ABC, ACD, ADE, and DEF, so that a signal at C is visible from A and B, a signal at D visible from A and C, a signal FIG. 40. at E visible from A and D, and a signal at F visible from D and E. In the triangle ABC, the side AB is known, and the angles at A and B may be measured; hence, AC may be computed. In the triangle ACD, AC is known, and the angles at A and C may be measured; hence, AD may be computed. In like manner DE and EF or DF may be determined. DF, or some suitable line connected with DF, may be measured, and this result compared with the computed value to test the accuracy of the field measurements. * In preparing this chapter the writer has consulted, by permission, recent reports of the United States Coast and Geodetic Survey. |