The greater angle is equal to the half sum plus the half difference, and the less is equal to the half sum minus the half difference. Hence, we have, A 27° 31' 44", and B 5° 17′ 58′′. Applying logarithms to the Proportion (13), Art. 83, we have, log tan c = log sin (A+B) + log tan (a — b) = 68° 46' 02", 2. Given α Ans. A 120° 59' 47", B = 33° 45′ 03′′, B = 33° 45′ 03′′, c = 43° 37′ 38′′. b = and B. 37° 10', and 3. Given a = 84° 14′ 29", b = 44° 13′ 45′′, and C = 36° 45′ 28′′, to find A and B. Ans. A 130° 05′ 22′′, B = 32° 26' 06".. CASE IV. Given two angles and their included side. 88. The solution of this case is entirely analogous to that of Case III. Applying logarithms to Proportions (12) and (13), Art. 83, and to Proportion (11), Art. 83, we have, log tan (a+b) = log tan (ab) = log cos (AB) + log tan c log sin (AB) + log tan c log cot C = log sin (a + b) + log tan (A – B) 10; Ans. C 64° 46′ 24′′, a = 70° 04′ 17′′, b = 63° 21′ 27′′. Ans. 121° 36′ 12′′, a = 40° 0' 10", b = 50° 10′ 30′′. CASE V. Given the three sides, to find the remaining parts. 89. The angles may be found by means of Formula (3), Art. 81; or, one angle being fonnd by that formula, the other two may be found by means of Napier's Analogies. EXAMPLES. = 35° 46′ 14′′, and c = 100° 39', 1. Given a = 74° 23', b to find A, B, and C. or, log cos A =3 [log sin s+ log sin (3s — a) Applying logarithms to Formula (3), Art. 81, we have, log cos 4 = 10 + [log sin s + log sin (48 − a) +(a. c.) log sin b + (a. c.) log sin c 20]; +(a. c.) log sin b + (a. c.) log sin c], Using the same formula as before, and substituting B for and α for b, and recollecting that b = 69° 37' 53", log sin s log sin (8-6) (a. c.) log sin a (a. c.) log sin c log cos B 9.989978 Using the same formula, substituting C for A, ... C = 67° 52′ 25′′, and C = 135° 44′ 50′′. 2. Given a = 56° 40′, b = 83° 13′, and c = 114° 30'. Ans. A = 48° 31′ 18′′, B = 62° 55′ 44′′, C= 125° 18′ 56′′. CASE VI. The three angles being given, to find the sides. 90. The solution in this case is entirely analogous to the preceding one. Applying logarithms to Formula (2), Art. 82, we have, log cos a = [log cos (SB) + log cos (S-C) + (a. c.) log sin B + (a. c.) log sin C]. In the same manner as before, we change the letters, to suit each case. EXAMPLES. 1. Given A= 48° 30', B = 125° 20', and C 62° 54'. Ans. a = 56° 39′ 30′′, b = 114° 29′ 58′′, C = 83° 12' 06". 2. Given A 109° 55′ 42′′, C = 120° 43' 37", to find a, ბ. Ans. a = 98° 21′ 40′′, b = B = 116° 38′ 33′′, and C. and 109° 50′ 22′′, c = 115° 13′ 28′′. MENSURAΤΙΟΝ. 91. MENSURATION is that branch of Mathematics which treats of the measurement of Geometrical Magnitudes. 92. The measurement of a quantity is the operation of finding how many times it contains another quantity of the same kind, taken as a standard. This standard is called the unit of measure. 93. The unit of measure for surfaces is a square, one of whose sides is the linear unit. The unit of measure for volumes is a cube, one of whose edges is the linear unit. If the linear unit is one foot, the superficial unit is one square foot, and the unit of volume is one cubic foot. If the linear unit is one yard, the superficial unit is one square yard, and the unit of volume is one cubic yard. 94. In Mensuration, the term product of two lines, is used to denote the product obtained by multiplying the number of linear units in one line by the number of linear units in the other. The term product of three lines, is used to denote the continued product of the number of linear units in each of the three lines. Thus, when we say that the area of a parallelogram is equal to the product of its base and altitude, we mean that the number of superficial units in the parallelogram is equal to the number of linear units in the base, multiplied by the number of linear units in the altitude. In like manner, the |