SECTION II. MEASUREMENT OF ANGLES. 190. To Measure with the Transit a Horizontal Angle subtended by two objects.-Place the axis of the instrument directly over the point at which the angle is to be measured. This is effected by means of a plumb, suspended from the centre of the plate which forms the upper end of the tripod. Having made the limb truly level, place the 0 of the vernier at any exact degree (merely to avoid minutes and seconds in the first reading), and fasten the clamp screw Q of the vernier plate. Then, facing in the direction between the lines which subtend the angle to be measured, loosen the lower clamp and sight one of the objects very nearly, without wasting time in trying to secure perfect bisection by the cross-wires; tighten the lower clamp and make perfect bisection by the lower tangent screw. This being done, loosen the clamp-screw Q of the vernier plate, and direct the telescope to the other object; the are passed over by the 0 point of the vernier, is the measure of the angle sought; the difference of the two readings (if the 0° of the limb be not passed in turning to the second object), is the required angle; if the 0° be passed over, then add 360° to the smaller reading and subtract the greater reading from this sum. Always be careful to take both readings from the same vernier. NOTE 1.-In measuring horizontal angles, it does not matter whether the telescope has to be elevated or depressed to sight either or both of the objects, since the telescope revolves on its axis in a vertical plane, and the angles measured are always the horizontal nr-- tions of the angle. NOTE. 2.-When great accuracy is desired, a repetition, or several repetitions of the measure may be made, and a mean of the observations taken as the true measure. (See Arts. 254, 255.) In the measurement of vertical angles, it is necessary to understand, first: 191. The method of determining the index error of the vertical limb. Having leveled the horizontal limb, direct the telescope to some distinctly marked object, as the top of a chimney, and read the instrument. Revolve the telescope on its axis and turn the vernier plate 180°, and having directed the telescope to the same object again, read the instrument. If the two readings are the same, the limb is adjusted; that is, the O of the limb coincides with the 0 of its vernier, when the axis of the telescope is parallel to the horizontal limb. When the reading, found with the telescope in the first position, is greater than that obtained in the reversed position, the true elevation of the object, which is equal to a mean of the readings, may be obtained by subtracting half the difference from the first reading. If the first reading is less than the second, the half difference must be added to the first. Hence, To find the index error, take the reading of the limb when the telescope is directed to a fixed object, and then with the telescope and vernier plate both reversed. Take half the difference of these readings, and affect it with a minus sign if the first is the greater, or a plus sign if the second is the greater; this is equal to the index error. Let the operation be repeated several times, using different objects, and a mean of the errors will be more correct than the result of a single observation. Then, second: 192. Having determined the index error, let the axis of the telescope be directed to any point either above or below the plane of the limb, and read the arc indicated by the 0 of the vernier. To the arc so read apply the proper correction, if any, and the result will be the true angle of elevation or depression. The angle of elevation may be more correctly found by taking the elevation of the object, and repeating the observation with the telescope and vernier plate reversed, and then taking a mean of the readings for the angle required. 193. The true azimuth of a line or course is the angle which the vertical plane through it makes with the plane of the meridian. The azimuth of a line or course referred to some preceding course, or to any given line, is the angle made by the line or course with the prolongation of the line of reference or of a parallel to it through the angular point, the measurement being made around to the right. Thus, The azimuth of BC with AB, (Fig. 90), is the angle RBC; the azimuth of CD with AB is the angle SCD (SQ being parallel to RA). 194. To find with the transit the azimuths of several successive courses with a given first course. B R S D FIG. 90. Place the transit at B (Fig. 91) and level it; make the zero of the vernier coincide with the zero of the horizontal limb, and clamp clamp the vernier plate and direct the telescope to C; the reading will be the angle RBC, the azimuth of BC with AB. Clamp the vernier plate and remove the instrument to C; reverse the telescope on its horizontal axis, loosen the lower clamp and sight B; the horizontal limb now has its zero point in the direction of QP, or its parallel AR, as it had at B; tighten the lower clamp and revolve the telescope on its axis; unclamp the vernier plate, direct the telescope to D, and the reading will be the angle QCD; which CD makes with PQ, or its parallel AR, and is the azimuth of CD with AB. Clamp the vernier plate and remove the transit to D; reverse the telescope on its horizontal axis, loosen the lower clamp and sight C; the limb will then have its zero point in the direction TS, or its parallel AR, as it had at C and B; tighten the lower clamp and revolve the telescope on its axis; unclamp the vernier plate and direct the telescope to E; the reading will be the angle TDE, which DE makes with TS, or its parallel RA, and is the azimuth of DE with AB. Proceed in like manner with any number of successive courses. If the courses enclose a field, the reading at the last station, sighting to the first station occupied by the transit, should be 360°. The course AB, with respect to which the azimuths are taken, is called the Meridian of the Survey. 195. The magnetic bearing of a line or course, is the angle which it makes with the magnetic meridian, and its true bearing is the angle which it makes with the true meridian. In finding the area of a piece of ground, it is not necessary to have either the true or the magnetic bearing. It is sufficient to have the bearings of the several successive courses with respect to one of the courses taken as a meridian. These may be found from the azimuths as follows: W210 180° S First suppose a north and south line, and an east and west line to be drawn, and the graduation to be made from 0° to 360°, as represented in Fig. 92; let the course taken as the meridian be represented by the line NS; then when the azimuth of the second course with the first, or meridian, is less than 90°, it is the bearing, and since the course lies between N and E, the bearing is NE; when the azimuth is 90°, the bearing is due east; when the azimuth is between 90° and 180°, the course lies between S and E, the bearing with the first course, or meridian, will be SE, and may be obtained by subtracting the azimuth from 180°; when the azimuth is 180°, the bearing is due south; when the azimuth is between 180° and 270°, the course lies between S and W, the bearing is, therefore, SW, and may be obtained by subtracting 180° from the azimuth; when the azimuth is 270°, the bearing is due west; when the azimuth is between 270° and 360°, the course lies between N and W, the bearing is NW, and may be obtained by subtracting the azimuth from 360°. For example: Let AB (Fig. 93) be taken as the meridian, and let the azimuths of the several courses with AB be as in the table; then will the bearings of the several courses with thB, be as noted in the B A |