By the Sliding Rule. Set the circular gage-point, for the proposed integer, on D, to half the altitude on C, and opposite to the diameter of the segment's base on D, note the number on C; also without shifting the slide, note the number on C opposite to the altitude on D. Add these two numbers, and one-third part the latter, together, and the sum will be the content. EXAMPLE. Let ABCD be a segment of a globe, whereof the altitude is 20 inches, and BC the diameter of the base 80 inches; the content is required in wine gallons, ale gallons, and malt bushels. A B D SOLUTION BY THE PEN. 20 altitude Multiply by 20 altitude 400 square of the altitude 40x40 × 3 = 4800 thrice the sq. of half BC. 5200 sum Multiply by 20 altitude 104000 product Consequently it will be, W.G. sph. divisor 441.18) 104000 (235-73 W.G. A. G. sph. divisor 538-58) 104000 (193∙10 A. G. M. B. sph. divisor 4106-2) 104000 (25-32 M.B. PROBLEM XX. To determine the Content of the Frustum of a Globe. RULE. To two-thirds the square of the altitude, add half the square of the diameter of the frustum's base, and half the square of the top diameter; then multiply the sum by the altitude, and divide the product by the circular divisor for the proposed integer. EXAMPLE. Let ABCD be the frustum of a sphere, the altitude being thirty inches; the diameter of the base, CD, 70 inches; and the top-diameter, AB, 45 inches: the content is required in wine gallons, ale gallons, and malt bushels. Square of the altitude 900 brought over Divide by 3) 1800 product Quotient 600 two-thirds the sq. of the alt. 2450 half the square of CD. 70 x 702 = 45 x 452 = 1012.5 half the square of AB. 4062.5 sum. Again, 4062.5 sum of the 3 numbers Multiply by 30 altitude W.G. cir. divisor 294)121875 (414-54 wine gallons. A. G. cir. divisor 359)121875 (339.48 ale gallons. M. B. cir. divisor 2738)121875 ( 44.51 malt busbels. PROBLEM XXI. To determine the Content of the Frustum of a Paraboloid. RULE. By the Pen. To the square of the diameter of the base, add the square of the diameter of the top, and multiply the sum by half the altitude: then divide the product by the circular divisor for the proposed integer, and the quotient will be the content. By the Sliding Rule. 4 Set the circular gage point, for the proposed integer, on D, to half the altitude on C, and opposite to the two diameters respectively on D, take the content on C. The sum of the two contents will be the content required EXAMPLE. Let ABCDEFG be the frustum of a paraboloid, whereof the greater diameter, AC, is 56 inches; the less diameter, EG, 32 inches; and the altitude iF 36 inches: the content is required in wine gallons, ale gallons, and malt bushels. |