the flag, the difference of the angle, in minutes, is denoted by the distance of the broad arrow in the vernier from the 360o, or the zero point of the other plate, which distance, as in the case of the vernier of the theodolite, is read, by observing which line of division in the vernier, reading to the left or the right (as the broad arrow is to the left or the right of the 360°), first coincides with some line in the graduated circle. In the circumferentor, the broad arrow of the vernier is placed in the centre, and if the distance from the 360° exceed half a degree, it is requisite to carry on the observation, as to which line first coincides, all round the plate, so as to end at the 360o. In some circumferentors the vernier is erroneously marked, as, in taking observations, the 15 may become 45, and the 45, 15, they should have been marked 13, and 13, in the same way as in the theodolite, the vernier is marked 18, 48. CHAP. II. To find the true bearing of a line, the magnetic bearing and the variation being given. By the variation of the compass, is always meant the variation of the needle, which is the only variable, east or west, from the true north, which is ever constant. RULE.-Mark upon paper, the relative positions of the given line and the magnetic north, which represents the north end of the needle, then observe, whether the variation be easterly or westerly; if easterly, the true north will be to the left of the magnetic; and if westerly, to the right; place this also on paper in its proper relative position. The angle made between this last line (of the true north) and the given line, is the angle of bearing; the expression of this angle depending, of course, whether by the variation, it may or may not have been moved into a different quadrant, being often changed from the magnetic N.W. to the true S. W., or N.E. EXAMPLES. 1. Let the line AB bear N. 24° E., and the variation be 234o W.; required the true bearing of the line. AB, bearing N. 24° E., the needle is on the left of the line, but the needle bears west, the true north is on the right of the needle, therefore the line and the true north, being both on the right of the needle, the line being at the greater angle, the difference of the angles is the bearing of the line eastwards, N. 0° 30′ E. 2. Let the reverse bearing of AB be S. 24° W., with the same variation, what is the true bearing? Ans. S. 0° 30' W., which is the reverse of the former. 3. Let AB bear S. 12° 45' E., and the variation be 24° W.; required the true bearing? Now the north end of the needle is to the left of the true north, therefore : EXAMPLE 1. Given magnetic bearing AB, S. 34° 20′ E., and variation 3° 27' W. Required the true bearing. Ans. S. 37° 47' E. EXAMPLE 2. Given AB, N. 74° 15' E., and variation 12° 45′ W. Required the true bearing. Ans. N. 61° 30' E. EXAMPLE 3. Given AB, N. 11° 30′ W., and variation 13° 20' E. Required the true bearing. Ans. N. 1° 50′ E. EXAMPLE 4. Given AB, S. 78° 30′ 11° 30' E. Required the true bearing. W., and variation Ans. Due west. 、 EXAMPLE 5. Given AB, S. 77° 35' E., and variation 16° 20′ W. Required the true bearing. Ans. N. 3° 55' E. EXAMPLE 6. Given AB, N. 72° 36' W., and variation 8° 24' E. Required the true bearing. Ans. N. 81° 0' W. The Bearings of two Lines being given, to determine the angle between them. RULE. First let both these lines run northwards or southwards. If they run on the same side of north or south, whether eastward or southwards, this angle will be the difference of their angles of bearing; if on different sides, it will be their sum. Next, let one run north and the other south. If they both run on the same side of the meridian, this angle will be the supplemental angle of the sum of the angles of bearing. If one be on the east, and the other on the west of the meridian, this angle will be the supplemental angle of the difference of their angles of bearing. In the interior angles of a polygon, as an angle may exceed 180°, the required angle might be the difference between the angle calculated as above, and 360°. EXAMPLE 1. Given AB, N. 16° W., and AC, N. 12° E., to find the angle between them. 16° 12° 28° angle between them. EXAMPLE 2. Given AB, N. 16° W., and AC, S. 12° W., to find the angle between them. Angle = supplement of sum = 180° (16° 12.) 152 = angle between them. EXAMPLE 3. Given AB, N. 84° 20′ W., and AC, S. 49° 51′ E., to find the angle between them. Here the angle the supplement of their difference. 180° · (84° 20′ — 49° 51′) = 180° = = angle required. 34° 29' 145° 31' EXAMPLE 4. Given AB, S. 25° E., and AC, S. 16° E.; required the angle between them. Angle their difference 25° -16° 9° angle required. |