By Calculation. (114° 24′ + 32° 15′) = = 33° 21'. The angle C 180 Since the angle B is greater than 90°, its sine cannot be found in the Tables, but by taking it from 180°, its supplement 65° 36′ is obtained, the sine of which is also the sine of the angle B. See DEFINITIONS. 2. In a plane triangle are given AB = 60 chains, BC = 95.12 chains, and the angle C = 33° 21'; required the side A C, and the angles A and B. B side A B = 60, as radius, and centre B, describe the arc a a, cutting A C in two points A, A'; then either A B C or A'B C is the triangle required, and the angle A may be either acute or obtuse. This is called the ambiguous case, as both triangles answer the required conditions. By measuring the sides and angles, there results A C= 50.03 or 108.87 chains, angle A = 119° 22′ or 60° 38', angle B = 27° 17′ or 86° 1'. The sum of the angles C and A As sin. C 33° 21' = 9.74017 = subtracted from 180°, leaves the AB = 60 1.77815 = 9.66124 : AC 50.03 = 1.69922 3. In a plane triangle A B = 98, and B C = 95.12 chains, and angle C33° 21'; required the other parts. 4. In a plane triangle are given two angles 79° 23′ and 54° 22′, and a side 1250 links opposite the first angle, to find the other parts. Ans. Sides 1033.6 and 918.7 links. CASE II. Given two Sides and their included Angle, to find the rest. RULE. -As the sum of the given sides: their difference :: co-tangent of half the included angle tangent of half the difference of the required angles. This angle, added to half the complement of the included angle, gives the greater required angle, and subtracted, gives the lesser. The other side is then found by Case I. = 1. Given the side A B = 9800 links, the side B C 9512, and their included angle B = 114° 24', to find the other side and angles. By Construction. Make A B = 9800 links, and the angle B = B 2. Given the two sides 103 and 126 chains, and their contained angle 56° 30′ to determine the triangle. Ans. The angles 72° 20′ and 51° 10′, the side 110.3 chains. 3. Two sides of a triangle are 34500 links and 17407, and their included angle 37° 20′; required the other angles and side. Ans. The angles 27° 4′ and 115° 36', the side 23200 links. CASE III. Given the three Sides to find the Angles. RULE I. From half the sum of the three sides subtract the side opposite the angle sought; add the logarithms of the half sum and remainder, and increase the index of the sum by 20; from the sum thus increased, subtract the sum of the logarithms of the sides containing the angle sought; the remainder, divided by 2, is the log. co-sine of half the angle sought. RULE II.-From half the sum of the three sides subtract each of the sides containing the angle sought; add the logarithms of the two remainders, and increase the index of the sum by 20; also from half the sum of the three sides subtract the side opposite the angle sought, and add the logarithms of the half sum and remainder; then half the difference between these logarithmic sums is the log. tangent of half the angle sought. The remaining angles may be found by Case I. NOTE. — Rule II. is to be preferred when the angle sought is very small or near 180°; since the co-sine, in these cases, as given by Rule I., varies so imperceptibly as to make the exact magnitude of the angle doubtful. 1. Given the three sides A C = 98, A B = 162.34 and B C = 95.12 chains, to find the angles of the triangle. The remaining angles may be found by Case 1. 2. In a plane triangle are given the three sides 7000, 10400, and 14202 links, to find the angles. Ans. 27° 59', 44° 12', and 107° 49'. 3. When the sides of the triangle are 2253, 2240, and 2400 links, what are the angles? Ans. 57° 27′, 57° 59′, and 64° 34′. THE MEASUREMENT OF HEIGHTS AND DISTANCES. THE measurement of heights and distances depends on the rules of Plane Trigonometry already given, and on the use of instruments for taking angles. Horizontal and vertical angles are usually taken with a Theodolite, and oblique angles with a Sextant. THE THEODOLITE. Two of the principal parts of a theodolite are the circular brass plates A and B, which turn one upon the other horizontally on the vertical axis C. The circumference of the lower plate B is divided into 360°, and these again into half degrees. At the extremities of a diameter of the upper plate A are fixed two veniers for reading off the minutes in the divisions of the plate B. This plate has its clamp-screw H, and its adjusting-screw I, which also clamp and adjust the whole instrument. To the plate A are fixed two spiritlevels d, d at right angles to each other; and a compass concentric with it. It has also its clamp and adjusting screw, (not shown in the figure). These two plates are used to take horizontal angles. The vertical semicircle M turns on an axis supported by the frame K L, resting on the plate A; its axis passing perpendicularly through |