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, (a. c.) log sin † (A − B) + log sin § (A+B) + log tan ♣ (a−b) −10 = log tanc; C = 39° 23′ 23′′, to find C, A, and Ans. A = 120° 59' 47", B = 33° 45′ 03′′, B. c = 43° 37' 38" b = 44° 13′ 45′′, and 3. Given a = 84° 14' 29", 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 Case 1П. Applying logarithms to Proportions (12) and (13), Art. 83, and to Proportion (11), Art. 83, we have, (a. c.) log sin (a − b) + log sin (a + b) + log tan } (A — B) −10 = log cot C. The application of these formulas is sufficient for the solution of all cases. Ans. C = 64° 46′ 24′′, a = 70° 04′ 17′′, b = 63° 21′ 27′′. 2. Given A = 34° 15′ 03′′, c = 76° 35' 36", B 42° 15′ 13", to find C, ɑ, and b. and Ans. C 121° 36' 12", a = 40° 0' 10", b = 50° 10′ 30′′. 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. 1. Given a = 74° 23', b = 35° 46′ 14′′, and c = 100° 39′, to find A, B, and C. Applying logarithms to Formula (3), Art. 81, we have, we cos 4 10+ [log sin slog sin (8 — a) +(a. c.) log sin b + (a. c.) log sin c20]; Using the same formula as before, and substituting B for A, b for a, and a for b, 186 69° 37' 53", log sin 18 and recollecting that we have, .. B = 12° 15′ 43′′, and B 24° 31' 20'. Using the same formula, substituting C for A, с for a and a for C, recollecting that 18 c = 4° 45′ 07", We have, 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 cosa [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" A = 109° 55′ 42′′, B 116° 38' 33", 2. Given C = 120° 43' 37", to find b, and C. and Ans. a = 98° 21′ 40′′, b = 109° 50′ 22′′, c = 115° 13′ 28′′. MENSURATION. 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 |