80 APPENDIX.-PART II. GAUGING OF CASKS. ART. 119. GAUGING is a practical art, which does not admit of being treated in a very scientific manner. Casks are not commonly constructed in exact conformity with any regular mathematical figure. By most writers on the subject, however, they are considered as nearly coinciding with one of the following forms; { The middle frustum of a parabolic spindle. The equal frustums of a paraboloid, of a cone. The second of these varieties agrees more nearly than any of the others, with the forms of casks, as they are commonly made. The first is too much curved, the third too little, and the fourth not at all, from the head to the bung. 120. Rules have already been given, for finding the capacity of each of the four varieties of casks. (Arts. 68, 110, 112, 118.) As the dimensions are taken in inches, these rules will give the contents in cubic inches. To abridge the computation, and adapt it to the particular measures used in gauging, the factor .7854 is divided by 282 or 231; and the quotient is used instead of .7854, for finding the capacity in ale gallons or wine gallons. If then .0028 and 0034 be substituted for .7854, in the rules referred to above; the contents of the cask will be given in ale gallons and wine gallons. These numbers are to each other nearly as 9 to 11. PROBLEM I. To calculate the contents of a cask, in the form of the middle frustum of a SPHEROID. 121. Add together the square of the head diameter, and twice the square of the bung diameter; multiply the sum by of the length, and the product by .0028 for ale gallons, or by .0034 for wine gallons. If D and d the two diameters, and l—the length; The capacity in inches (2D2+d2)×}l×.7854. (Art. 110.) And by substituting .0028 or .0034 for .7854, we have the capacity in ale gallons or wine gallons. Ex. What is the capacity of a cask of the first form, whose length is 30 inches, its head diameter 18, and its bung diameter 24? Ans. 41.3 ale gallons, or 50.2 wine gallons. PROBLEM II. To calculate the contents of a cask, in the form of the middle frustum of a PARABOLIC SPINdle. 122. Add together the square of the head diameter, and twice the square of the bung diameter, and from the sum subtract of the square of the difference of the diameters; multiply the remainder by of the length, and the product by .0028 for ale gallons, or .0034 for wine gallons. The capacity in inches .7854. (Art. 118.) (2D2+d2 —} (D—d)2)×}/× Ex. What is the capacity of a cask of the second form, whose length is 30 inches, its head diameter 18, and its bung diameter 24? Ans. 40.9 ale gallons, or 49.7 wine gallons. 11 To calculate the contents of a cask, in the form of two equal frustums of a PARABOLOID. 123. Add together the square of the head diameter, and the square of the bung diameter; multiply the sum by half the length, and the product by .0028 for ale gallons, or .0034 for wine gallons. The capacity in inches=(D2+d2)××.7854. (Art. 112. Cor.) Ex. What is the capacity of a cask of the third form, whose dimensions are, as before, 30, 18, and 24? Ans. 37.8 ale gallons, or 45.9 wine gallons. PROBLEM IV. To calculate the contents of a cask, in the form of two equal frustums of a CONE. 124. Add together the square of the head diameter, the square of the bung diameter, and the product of the two diameters; multiply the sum by of the length, and the product by .0028 for ale gallons, or .0034 for wine gallons. The capacity in inches (D2+d2+Dd)×11×.7854. (Art. 68.) = Ex. What is the capacity of a cask of the fourth form, whose length is 30, and its diameters 18 and 24? Ans. 37.3 ale gallons, or 45.3 wine gallons. 125. The preceding rules, though correct in theory, are not very well adapted to practice, as they suppose the form of the cask to be known. The two following rules, taken from Hutton's Mensuration, may be used for casks of the usual forms. For the first, three dimensions are required, the length, the head diameter, and the bung diameter. It is evident that no allowance is made by this, for different degrees of curvature from the head to the bung. If the cask is more or less curved than usual, the following rule is to be preferred, for which four dimensions are required, the hear and bung diameters, and a third diameter taken in the middle between the bung and the head. For the demonstration of these rules, see Hutton's Mensuration, Part v. Sec. 2. Ch. 5. and 7. PROBLEM V. To calculate the contents of any common cask from THREE dimensions. 126. Add together 25 times the square of the head diameter, 39 times the square of the bung diameter, and Multiply the sum by the length, divide the product by 90, and multiply the quotient by .0028 for ale gallons, or .0034 for wine gallons. The capacity in inches (39 D2+25d2+26 Dd)×; X.7854. 90 Ex. What is the capacity of a cask whose length is 30 inches, the head diameter 18, and the bung diameter 24? Ans. 39 ale gallons, or 47 wine gallons. PROBLEM VI. To calculate the contents of a cask from FOUR dimensions, the length, the head and bung diameters, and a diameter taken in the middle between the head and the bung. 127. Add together the squares of the head diameter, of the bung diameter, and of double the middle diameter; multiply the sum by of the length, and the product by .0028 for ale gallons, or .0034 for wine gallons. If D=the bung diameter, d=the head diameter, m=the middle diameter, and 7-the length; The capacity in inches (D2+d2+2m2)×÷l×.7854. Ex. What is the capacity of a cask, whose length is 30 inches, the head diameter 18, the bung diameter 24, and the middle diameter 22? Ans. 41 ale gallons, or 49 wine gallons. 128. In making the calculations in gauging, according to the preceding rules, the multiplications and divisions are frequently performed by means of a Sliding Rule, on which are placed a number of logarithmic lines, similar to those on Gunter's Scale. See Trigonometry, Sec. vi. and Note G. p. 141. Another instrument commonly used in gauging is the Diagonal Rod. By this, the capacity of a cask is very expeditiously found, from a single dimension, the distance from the bung to the intersection of the opposite stave with the head. The measure is taken by extending the rod through the cask, from the bung to the most distant part of the head. The number of gallons corresponding to the length of the line thus found, is marked on the rod. The logarithmic lines on the gauging rod are to be used in the same manner, as on the sliding rule. ULLAGE OF CASKS. 129. When a cask is partly filled, the whole capacity is divided, by the surface of the liquor, into two portions; the least of which, whether full or empty, is called the ullage. In finding the ullage, the cask is supposed to be in one of two positions; either standing, with its axis perpendicular to the horizon; or lying, with its axis parallel to the horizon. The rules for ullage which are exact, particularly those for lying casks, are too complicated for common use. The following are considered as sufficiently near approximations. See Hutton's Mensuration. PROBLEM VII. To calculate the ullage of a STANDING cask. 130. Add together the squares of the diameter at the surface of the liquor, of the diameter of the nearest end, and of double the diameter in the middle between the other two; multiply the sum by of the distance between the surface and the nearest end, and the product by .0028 for ale gallons or 0034 for wine gallons. |