See Gauge Points for the Equilateral Triangular Prism, ¶ 35; and for the Regular Solids, ¶ 39. EXAMPLES. 1. Find the content, in pounds, of hard soap, of an oblong form, its length being 201, its breadth 60 inches, and thickness 1 inch. Ans. 444.36 pounds. 2. Find the content, in imperial gallons and bushels, of a vessel with a square bottom, each side being 30 inches, and its depth 40 inches. Ans. 129.83 and 16.229. 3. What is the content, in bushels and imperial bushels, of a cylindric vessel, whose diameter is 48 inches and depth 64 inches ? Ans. 53.85, and 52.2. 4. Find the content, in wine, ale, and imperial gallons, of a conical vessel, the diameter of the base being 27 inches, and its height 60 inches. Ans. 49.2; 40.2; and 41.3 gallons. 5. What is the content of a vessel, its form being that of a regular hexagonal pyramid, one side of the base being 40 inches, and its height 72 inches, in bushels and imperial bushels? Ans. 46.3 and 44.98. In the above example, place one-third of the altitude over the gauge points for bushels and imperial bushels in the hexagonal prism, viz. 28.77 and 29.22, and over the side found on D, will be found the contents on the line C. 6. What is the capacity of a hollow sphere in bushels, its inner diameter being 72 inches? Ans. 91, nearly. To gauge or find the capacity of stills, brewing-vessels, &c. :— Divide the vessel into small portions by planes parallel to the base; find the areas of the middle sections of these portions, and multiply these areas by the corresponding depths of the portions to which they belong; the products are the cubic contents of the portions, and the sum of these products is the whole content: divide the whole content by the number corresponding to the required measure or weight, and the result is the required content. EXAMPLE. Find the content, in ale and wine gallons, of a flat-bottomed copper, the mean diameters at the middle of four portions of it being 54.4, 51.9, 49.6, and 47.3, and the depth of the respective portions, 12, 10, 10, and 10 inches. Ans. 271.6, and 334.12. When the bottom of a vessel is concave or convex, the content of the bottom portion may be most easily and accurately found by measuring the quantity of water required to fill it up, till the bottom is covered. CASK GAUGING. Casks are usually divided into four varieties:-The first variety is the middle frustum of a spheroid; the second, the middle frustum of a parabolic spindle; the third, two equal frustums of a paraboloid united at their bases; and the fourth, two equal conic frustums united at their bases. When casks are much curved, they are considered to belong to the first variety; when less curved, to the second; when still less, to the third; and when the staves are straight from the bung to the head, to the fourth variety. 1. To find the content of a cask of the first variety :— To twice the square of the bung diameter, add the square of the head diameter; multiply the sum by the length of the cask, and divide the product by 882.354 for wine, and by 1077.15 for ale, and by 1059.11 for imperial gallons:-That is, divide by the divisors for conical vessels, and the quotient is the content. 2. To find the content of a cask of the second variety : To twice the square of the bung diameter, add the square of the head diameter, and from the sum subtract four-tenths of the square of the difference of these diameters; multiply the remainder by the length, and divide the product by the divisors for conical vessels, and the quotient will be the content. 3. To find the content of a cask of the third variety :— Add the square of the bung diameter to that of the head diameter; multiply the sum by the length, and divide the product by 588.236 for wine, and by 718.106 for ale, and by 706.0724 for imperial gallons. 4. To find the content of a cask of the fourth variety: Add together the product of the bung and head diameters, and their squares; multiply the sum by the length, and divide the product by the divisors for conical vessels, and the quotient will be the content. EXAMPLES. 1. What is the content of a cask of the first variety, whose bung diameter is 32 and its head diameter 24 inches, and its length 40 inches, in wine, ale, and imperial gallons? Ans. 118.95; 97.44; and 99.1. To gauge a cask of the first variety by the sliding rule: Add seven-tenths (or .7) of the difference between the head and bung diameters to the head diameter, and the sum will be the MEAN DIAMETER of the cask, nearly; then place the length of the cask over the gauge points for cylindric vessels, and over the mean diameter found on D will be found the contents on C. Thus, in the above example, the mean diameter is found to be 29.6 inches; then having placed the length, 40 inches, over 17.15 on D, over 29.6 on D will be found 118.95 wine gallons; and placing the length over 18.95, over 29.6 you will find 97.44 ale gallons; and placing the length over 18.79, over 29.6 you will find 99.1 gallons imperial measure. 2. What is the content of a cask of the first variety, whose bung and head diameters are 24 and 20 inches, and length 30 inches, in wine, ale, and imperial gallons? Ans. 52.77; 43.22; and 43.96. 3. Find the content of a cask of the second variety, in wine, ale, and imperial gallons, the dimensions being the same as in example 1. Ans. 117.78; 96.49; and 98.1. To find the mean diameter of a cask of the second variety, add .68 of the difference between the diameters to the head diameter. 4. Find the content of a cask of the second variety, in wine, ale, and imperial gallons, its bung and end diameters being 48 and 36 inches, and length 60 inches. Ans. 398; 325.8; and 331.2. 5. Find the content of a cask of the second variety, in wine, ale, and imperial gallons, the bung and head diameters being 36 and 20, and its length 40 inches. Ans. 131.03; 107.34; and 109.12. When the difference between the diameters is great in proportion to the length of the cask, as in the last example, the mean diameter will be found more nearly by adding .69 of the difference between the diameters to the head diameter. The mean diameter of any cask of the first and second varieties may be found very nearly by the following rule:-Divide the square of the difference between the head and bung diameters by the sum of the head and twice the bung diameter: for casks of the first variety, add one-third of the quotient; and for casks of the second variety, four-thirtieths of the quotient to two-thirds of the difference between the head and bung diameters, and the sum will be the MEAN diameter. 6. Find the content of a cask of the third variety, in wine, ale, and imperial gallons, the bung and head diameters being 30 and 24 inches, and its length 36 inches. Ans. 90.33; 74; and 75.25. The factor for reducing casks of the third variety to a cylinder is .54; therefore, add .54 of the difference between the head and bung diameters to the head diameter, and the sum will be the mean diameter; then proceed as directed under example 1. Or, to find the mean diameter of any cask which is but little curved :-To the head diameter add half the difference of the diameters; and to this sum add the quotient obtained by dividing one-fourth part of the square of the difference of the diameters, by their sum. 7. What is the mean diameter of a cask of the third variety, and what is its content in wine, ale, and imperial gallons, its diameters being 29 and 15 inches, and its length 24 inches? Ans. 23.11; 43.4; 35.65; and 36.2. 8. Find the content of a cask of the fourth variety, in wine, ale, and imperial gallons, the bung diameter being 32, the end diameter 18, and the length 38 inches. Ans. 83.34; 68.8; and 69.4. : To find the mean diameter of a cask of the fourth variety :To the head diameter add half the difference of the diameters; and to this sum add one-twelfth of the quotient obtained by dividing the square of said difference by the sum of the bung and head diameters. 9. Find the mean diameter and content of a cask of the fourth variety in wine, ale, and imperial gallons, its bung and |