To find the solidity of a middle frustum of an elliptic, a circular, or an hyberbolic spindle : Add together the squares of the middle and the end diameter, and the square of double the quarter diameter; multiply the sum by the length of the frustum, and the product by .1309, and the result will be the solidity, nearly. To find the solidity of a middle frustum of a parabolic spindle : To twice the square of the middle diameter add the square of the end diameter, and from the sum subtract four-tenths of the square of the difference between these two diameters; multiply the remainder by the length, and the product by .2618. EXAMPLES. 1. What is the solidity of the middle frustum of an elliptic spindle, the middle, quarter, and end diameters being 32, 30.157, and 24 inches, and the length 40 inches? Ans. 27424.8 cubic inches. 2. What is the solidity of the middle frustum of a circular spindle, its middle, end, and quarter diameters being 16, 15.0788, and 12 inches, and its length 20 inches? Ans. 3430.707 inches. 3. Find the solidity of a middle frustum of an hyperbolic spindle, whose length is 14, its middle diameter 12, the quarter diameter 11.705, and end diameter 10.8. Ans. 1481.96516. 4. What is the solidity of a middle frustum of a parabolic spindle, the middle and end diameters being 16 and 12, and its length 30? Ans. 5101.958. 5. What is the solidity of a middle frustum of a parabolic spindle, whose length is 18, and diameters 18 and 10 inches? Ans. 3404.235 inches. Casks usually approach the form of the frustum of a parabolic spindle more nearly than any other solid. See pp. 165 and 166. 57. IRREGULAR SOLIDS. To find the solidity of an irregular solid of an oblong form:Find the areas of several equidistant sections perpendicular to the line which measures the length of the solid, and divide the sum of the areas by the number of equidistant sections, and multiply the quotient by the length. Or:-Multiply the half sum of the areas of each of the two proximate sections by the distance between them, and the sum of the several products will be the solidity. When the solid is not great, and is very irregular, immerse it in water in some vessel of a regular form; then take out the body, and measure the capacity of that portion of the vessel which is contained between two positions of the surface of the water before and after the body was removed, and the result will be the solidity of the solid. EXAMPLES. 1. Find the area of an oblong solid, whose length is 100 feet, and the areas of five equidistant sections, 50, 55, 70, 80, and 82 square feet. Ans. 6740 cubic feet. 2. What is the solidity of an oak tree of irregular form, the lengths of four portions of it being respectively 8, 5, 6, and 7 feet, and the areas of the middle sections of each part 10, 8, 7, and 5 square feet? Ans. 197 cubic feet. 3. An irregular mass of copper ore being immersed in water in a cylinder, whose diameter is 10 inches, the water is found to rise 6 inches higher when the solid is introduced; its solidity is required. Ans. 471.24 cubic inches. 4. An irregular mass of California gold being immersed in a square vessel 12 inches on a side, the water in the vessel rises 8 inches; its volume is required. Ans. 96 cubic inches. T58. GAUGING. Gauging is the art of measuring the dimensions, and computing the capacity of any vessel, or any portion of it. When the capacity of a vessel is known in cubic inches, the number of wine, ale, and English imperial gallons may be found by dividing the capacity by 231, by 282, and by 277.274. See table Iv. 12. The divisors for circular areas, or for cylindrical vessels, are found by dividing the number of cubic inches in the measure of capacity by .785398, or .7854; and the square roots of the divisors are the GUAGE POINTS on the line D. Thus 231, 282, and 277.274 being divided by .7854, the quotients are 294.118, and 359.05, and 353.04; and the gauge points are 17.15, and 18.95, and 18.79. In like manner the gauge points may be found for bushels, barrels, &c. Thus, if we multiply the square of the diameter of a cylinder by its length or altitude, the dimensions being taken in inches, and divide the product by the circular divisors, as 294.118, 359.05, &c., the quotients will be the number of gallons, bushels, barrels, &c., which the vessel contains. If the circular divisors are increased in the ratio of 2 to 3, the results are spherical divisors; and the square roots of the spherical divisors are the spherical gauge points.—For conical vessels the divisors are three times those for cylinders; and the gauge points are the square roots of the divisors. The following table exhibits the divisors and gauge points for vessels of various forms, when their dimensions are taken in inches. To gauge vessels having rectangular bases by the sliding rule-Find the side of an equal square, (as directed under example 9, ¶ 15,) and then use the gauge points for vessels with square bases. To gauge vessels having elliptical bases :-Find the diameter f an equal circle, (as directed under example 1, ¶ 28,) and then use the gauge points for cylindric vessels. |