.270 1.2371 .275 1.2436 .250 1.2114 .485 1.9169 1.9253 .950 2.3335 1.9337 1.9506 1.9599 .970 2.3714 .280 1.2501 .515 .285 1.2567 .520 .290 1.2634 .525 .295 1.2700 530 1.6097 .755 1.9845 .985 2.4002 1.9931 990 2.4098 To find the Length of the Curve of a Right Semi-Ellipse. Proceed with the foregoing table by the rule for ascertaining the lengths of circular arcs, page 76. EXAMPLE.-What is the length of the curve of the arch of a bridge, the span being 70 feet, and the height 30.10 feet? 30.10÷70.430= per table, 1.4627, and 1.4627×70=102.3890, the length re quired. When the Curve is not that of a Right Semi-Ellipse, the height being half of the Transverse Diameter. RULE.-Divide half the base by twice the height; then proceed as in the foregoing example, and multiply the tabular length by twice the height, and the product will be the length required. EXAMPLE. What is the length of the profile of arch (it being that of a semi-ellipse), the height measuring 35 feet and the base 60 feet? 60-230-35×2.428, the tabular length of which is 1.4597. Then, 1.4597x35x2=102.1790, the length required. NOTE. When the quotient is not given in the column of heights, divide the difference between the two nearest heights by .5; multiply the quotient by the excess of the height given and the height in the table first above it, and add this sum to the tabular area of the least height. Thus, if the height is 118, To find the Area of a Zone by the above Table. RULE 1-When the zone is greater than a part of a semicircle, take the height on each side of the diameter of the circle, of which it is a part; divide the heights by the diameter; find the respective quotients in the column of heights, and take out the areas opposite to them, multiplying the areas thus found by the square of the diameter or chord, and the products, added together, will be the area required. NOTE. When the quotient is not given in the column of heights, divide the difference between the two nearest heights by 5, and multiply the quotient by the excess between the height given and the height in the table first above it, and add this sum to the tabular area of the least height. Thus, if the height is .333, .30416-30790.00374÷5.000748×3 (excess of 333 over 330) = .002244+.30416 =.306404, the area for 333. EXAMPLE,-What is the area of zone, the diameter of the circle being 100, and the heights respectively 20 and 10, upon each side of it? 20÷100.200, and 200, per table, .19453×10021945.3. RULE.-When the zone is less than a semicircle, proceed as in rule 1 for one height. EXAMPLE.-What is the area of a zone, the longest chord being 10, and the height 4? 4÷10.400.35182x10235.182 Ans. To find the Solidity of a Cube-fig. 23. RULE.-Multiply the side of the cube by itself, and that product again by the side, and this last product will be the solidity. To find the Solidity of a Parallelopipedon-fig. 24. RULE.-Multiply the length by the breadth, and that product by the depth, and this product is the solidity. OF REGULAR BODIES. To find the Solidity of any Regular Body. RULE.-Multiply the tabular solidity in the following table by the cube of the linear edge, and the product is the solidity. TABLE of the Solidities of the Regular Bodies when the Linear Edge is To find the Solidity of Cylinders, Prisms, and Ungulas—figs. 25, 26, and 27. RULE.-Multiply the area of the base by the height, and the product is the solidity. To find the Solidity of an Ungula, fig. 28, when the section passes obliquely through the cylinder, abcd. RULE.-Multiply the area of the base of the cylinder by half the sum of the greater and less heights a e, cf of the ungula, and the product is the solidity. When the Section passes through the base of the Cylinder and one of its sides-fig. 29, abc. RULE. From of the cube of the right sine ad, of half the arc ag of the base, subtract the product of the area of the base, and the cosine df of said half arc.* Multiply the difference thus found by the quotient of the height, divided by the versed sine, and the product is the solidity. When the Section passes obliquely through both ends of the Cylinder, adce-fig. 30. RULE. Find the solidities of the ungulas á dce and db c, and the difference is the solidity required (conceiving the section to be continued till it meets the side of the cylinder). NOTE. For rules to ascertain the solidity of conical ungulas, see Ryan's Bonnycastle's Mensuration, page 136 (1839). To find the Solidity of a Cone or Pyramid-figs. 31 and 33. RULE.-Multiply the area of the base by the height cd, and the product will be the content. To find the Solidity of the Frustrum of a Cone-fig. 32. RULE. Divide the difference of the cubes of the diameters a b, c d of the two ends by the difference of the diameters; this quotient, multiplied by .7854, and again by of the height, will give the solidity. To find the Solidity of the Frustrum of a Pyramid-fig. 34. RULE. Add to the areas of the two ends of the frustrum the square root of their product, and this sum, multiplied by of the height a b, will give the solidity. * If the height of the base be less than radius, otherwise add them. |