Then, as the less sum is to the greater, so is the ordinate to the length of the curve nearly. When the transverse, v e, the conjugate, d f, and the abscissæ, v b and b e, are given, to find the ordinate, a b, fig. 21. RULE.-As the transverse diameter is to the conjugate, so is the square root of the product of the two abscisse to the ordinate required. Note.-In the hyperbola, the less abscissæ added to the axis gives the greater; and the greater abscissa subtracted from the axis gives the less. When the transverse and conjugate diameters, and the ordinate, are given, to find the abscissæ. RULE. To the square of half the conjugate add the square of the ordinate, and extract the square root of that sum. Then, as the conjugate diameter is to the transverse, so is the square root to half the sum of the abscissæ. To this half sum add half the transverse diameter for the greater abscissa, and subtract it for the less. When the transverse diameter, ordinate, and abscissa, are given, to find the conjugate. RULE.-As the square root of the product of the two abscissæ is to the ordinate, so is the transverse diameter to the conjugate. When the conjugate diameter, the ordinate, and the two abscissæ, are given, to find the transverse diameter. To this root add the half conjugate when the less abscissa is used; and subtract it when the greater abscissa is used; reserving the sum or difference. Then, as the square of the ordinate is to the product of the abscissa and conjugate, so is the reserved sum or difference to the transverse. MENSURATION OF SOLIDS. OF CUBES AND PARALLELOPIPEDONS. Fig. 23. Fig. 24. b 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; the last product will be the solidity of the given cube. To find the Solidity of a Parallelopipedon, fig. 24. RULE.-Multiply the length, breadth, and depth or altitude, continually together, or, in other words, multiply the length, a b, by the breadth, a c, and that product by the depth or altitude, c d; this will give the required solidity. To find the Solidity of Cylinders and Prisms. RULE.-Multiply the area of the base by the height of the cylinder or prism, and the product will give the solid content. To find the Convex Surface of a Cylinder. RULE.-Multiply the circumference by the length of the cylinder; the product will be the convex surface required. To find the Convex Surface of a Right Cone, or Pyramid, fig. 27. RULE.-Multiply the perimeter, or circumference of the base, by the slant height, or length of the side of the cone, and half the product will be the surface. To find the Convex Surface of a Frustum of a Right Cone, or Pyramid, fig. 28. RULE.-Multiply the sum of the perimenters of the two ends by the slant height or side of the frustum, and half the product will be the surface required. To find the solidity of a Cone, or Pyramid, figs. 27 and 29. RULE.-Multiply the area of the base by the height, c d, and one-third of the product will be the content. To find the Solidity of the Frustum of a Cone, fig. 28. RULE.-Divide the difference of the cubes of the diameters of the two ends by the difference of the diameters; this quotient, multiplied by 7854 and again by one-third for the height, will give the solidity. To find the Solidity of the Frustum of a Pyramid, fig. 30. RULE.-Add to the areas of the two ends of the frustum the square root of their product, and this sum, multiplied by one-third of the height, will give the solidity. OF WEDGES AND PRISMOIDS. To find the Solidity of a Wedge, fig. 31. RULE. To the length of the edge of the wedge add twice the length of the back; multiply this sum by the height of the wedge, and then by the breadth of the back; one-sixth of the product will be the solid content. |