Ex. 2. Required the solidity of the parabolic conoid, whofe altitude is 21; and the diameter of its base 12. Anf. 1187.5248. Ex. 3. Required the solidity of a paraboloid, whose height is 30, and base diameter 40. Anf. 18849.65 N. B. The paraboloid conoid is = its circumscribing cylină der. PROBLEM XXV. To find the solidity of the fruftum of a paraboloid. RULE. Multiply the sum of the squares of the diameters of the two ends by .7854, and this product by the altitude for the folidity, EXAMPLE, I. Required the content of the frustum of a paraboloid, the greatest diameter being 30, the least 24, and the altitude 18. 30*=900 1476 .7854 5904 7380 11808 10332 1159.2504 9 10433.2536 Ex. 2. Required the solidity of the frustum of a paraboloid, the diameter of the greater end being 60, of the less 48, and length 18. Anf. 41733.0144. Ex. 3. Required the folidity of the frustum of a parabolic conoid, whose diameters are 58 and 30, and the height 36. Anf. 60281.0208. PROBLEM XXVI. To find the folidity of a parabolic spindle. RULE. Multiply the square of the middle diameter by .7854, and the product by the length, and 7, the last product, will be the folidity. Note. The parabolic spindle is equal to 1's, the circumscribing cylinder. EXAMPLE I. Required the solidity of a parabolic spindle, whose length is 18, and middle diameter 6. Ex. 2. Required the folidity of a parabolic spindle, whose length is 40, and middle diameter 10. Anf. 1675.52. Ex. 3. Required the folidity of a parabolic spindle, whose length is 100, and middle diameter 1o. Anf. 16755.2. PROBLEM XXVII. To find the solidity of the middle fruftum of a parabolic spindle. RULE. Add into one fum 8 times the square of the greatest diameter, 3 times the square of the least diameter, and 4 times the product of the two diameters; multiply the sum by the length of the frukum, and the product again by .05236 for the folidity. EXAMPLE I. Required the folidity of the middle frustum of a parabolic {pindle, the length being 20, the greatest diameter 16, and the least 12. 16'=256x8=2048 12'=144X3= 432 16X12 =192*4= 768 3248 20 64960 .05236 389760 194880 129920 324800 3401.3056c Ang . ) Ex. 2. Required the solidity of the fruftum of a parabolic spindle, whofe length is 10, the diameters being 8 and 6. Anf. 425.1632. Ex. 3. Required the solidity of the middle frustum of a parabolic spindle, whose length is 30 feet, and diameter 16 and 20. Anf. 8243.5584. SURVEYING, SURVEYING URVEYING of Land is considered to have been the primitive part of Geometry, and consists of three principal parts, viz, The taking of the dimensions, and making the necessary observations on the ground ; —the laying down the same in a map or drawing on paper or vellum ;-and the finding the content or area thereof. The instruments commonly used in surveying of land, are, the gunter's chain, a case of instruments, a set of plotting scales, the theodolite, and plain table. The gunter's chain, whether Scots or English, is divided in to 100 links. The English chain is 66 feet, and the Scots 74; consequently a link of the English chain is 7.92 inches, and that of the Scots 8.88 inches : likewise the English chain is divided into 4 poles or perches, each 164 feet, and the Scots chain into 4 falls, each 184 feet. 10 square chains are i acre, either Scots or English ; and 4 Scots acres are nearly equal to $ English miles. |