number of units of volume, in a rectangular parallelopipedon, is equal to the number of superficial units in its base multiplied by the number of linear units in its altitude, and SO on. MENSURATION OF PLANE FIGURES. To find the area of a parallelogram. 95. From the principle demonstrated in Prop. V., we have the following RULE. in Book IV., Multiply the base by the altitude; the product will be the area required. EXAMPLES. 1. Find the area of a parallelogram, whose base is 12.25, and whose altitude is 8.5. Ans. 104.125. 2. What is the area of a square, whose side is 204.3 feet? 3. How many square yards are whose base is 66.3 feet, and altitude Ans. 41738.49 sq. ft. there in a rectangle, 33.3 feet? Ans. 245.31 sq. yd. 4. What is the area of a rectangular board, whose length is 12 feet, and breadth 9 inches? 93 sq. ft. 5. What is the number of square yards in a parallelogram, whose base is 37 feet, and altitude 5 feet 3 inches?' Ans. 21 To find the area of a plane triangle. 96. First Case. When the base and altitude are given. From the principle demonstrated in Book X, Prop. VI., we may write the following RULE. Multiply the base by half the altitude; the product will be the area required. EXAMPLES. 1. Find the area of a triangle, whose base is 625, and altitude 520 feet. Ans. 162500 sq. ft. 2. Find the area of a triangle, in square yards, whose base is 40, and altitude 30 feet. Ans. 663. 3. Find the area of a triangle, in square yards, whose base is 49, and altitude 25 feet. Ans. 68.7361. Second Case. When two sides and their included angle are given. Let ABC represent a plane tri angle, in which the side BC = α, and the angle AB = C, B, are given. From A draw AD perpendicular to BC; this will be the altitude of the triangle. From For BaD mula (1), Art. 37, Plane Trigonometry, we have, AD c sin B. Denoting the area of the triangle by Q, and applying the rule last given, we have, hence, we may write the following RULE. Add together the logarithms of the two sides and the logarithmic sine of their included angle; from this sum subtract 10; the remainder will be the logarithm of double the area of the triangle. Find, from the table, the number answering to this logarithm, and divide it by 2; the quotient will be the required area. EXAMPLES. 1. What is the area of a triangle, in which two sides a and b, are respectively equal to 125.81, and 57.65, and whose included angle C, is 57° 25′? .. Ans. 2Q 6111.4, and ૨ = 3055.7 Ans. 2. What is the area of a triangle, whose sides are 30 and 40, and their included angle 28° 57' ? Ans. 290.427. 3. What is the number of square yards in a triangle, of which the sides are 25 feet and 21.25 feet, and their included angle 45° ? Ans. 20.8694. LEMMA. To find half an angle, when the three sides of a plane triangle are given. 97. Let ABC be a plane triangle, the angles and sides being de noted as in the figure. We have (B. IV., P. XII., XIII.), a2 = b2 + c2 = 2c. AD. When the angle A is acute, we have (Art. 37), DC. (1.) AD b cos A; when obtuse, AD' b cos CAD'. But as CAD' is the supplement of the obtuse angle A, cos CAD' = cos A, and AD' = - b cos A. Either of these values, being substituted for AD, in (1), If we add 1 to both members, and recollect that 1 + cos▲ = 2 cos2 4 (Art. 66), Equation (4), we have, Substituting in (3), and extracting the square root, the plus sign, only, being used, since A < 90°; hence, The cosine of half of either angle of a plane triangle, is equal to the square root of half the sum of the three sides, into half that sum minus the side opposite the angle, divided by the rectangle of the adjacent sides. By applying logarithms, we have, log cos 4 = [logs + log (sa) + (a. c.) log b + (a. c.) log c]. (A.) If we subtract both members of Equation (2), from 1, and recollect that 1 2 sin2 4 == cos A = 2 sin2 4 (Art. 37), we have, Substituting in (5), and reducing, we have, a+b+c= 8, we have, and, a − b + c = The sine of half an angle of a plane triangle, is equal to the square root of half the sum of the three sides, minus one of the adjacent sides, into the half sum the half sum minus the other adjacent side, divided by the rectangle of the adjacent sides. Third Case. To find the area of a triangle, when the |