CASE II. To guage a cask representing the middle frustum of an elliptic spindle. RULE. 1-To twice the square of the bung diameter, add the square of the head diameter and reserve the sum; find the difference between the head and bung diameter, square the difference, and subtract of the square of the difference from the reserved sum; multiply the remainder by the length of the cask; divide the product by 882 for wine; by 1077 for ale or beer; and by 1026-3 for corn, &c.; the quotient will be the answers in gallons respectively. RULE 2. Find the solidity in solid inches by the following rule, viz. To the square of twice a diameter taken half way between the head and bung, (as at o P) add the squares of the head and bung diameters; multiply the sum by the length of the cask, and again by 1309 for the solidity; divide the solidity by 231 for wine; and 282 for ale, or beer; and by 268.8 for corn, &c.; the quotients will be the answers in gallons respectively. Examples. 1. What is the content of a cask representing the middle frustum of an elliptic spindle, (see figare E F, O P, C D, and G H, Case 14, solids) in wine, ale, and corn gallons; whose length is 38; bung diameter C D 22; and end diameter, E F, or GH 16; and a diameter at O P 20.5 inches? Solved by rule first. Diam. C D 22×22=484×2=968 reserved sum. 1224 Diam. 22-166×636÷144 sub. rem. remainder. 1209.6 1209 6X38-45964-8 product. 45964-888252.1† gal, wine, Ans. The same question by rule second. Diam. at O P 20·5×2=41×41-1681 square of twice O P. the sum. 2421x38x1309=12042-5382 solidity. Solidity 12042-5382÷÷231-521 gal. wine, Ans. 12042-5382-282-427+ gal. of ale, &c. Ans. 12042-5382-268-8=44.8+ gal. of corn, &c. Ans. CASE III. To guage a cask representing two frustums of a round cone, the bases being joined together to form the bung diameter, (See figure 1, Case 5, Solids,) RULE I.-Add the two diameters together, square the sum and multiply the square by 3, and reserve the product; find the differenee of the diameters; square the difference, and add the square of the difference to the reserved product, and mul tiply the sum by the length, and the product again by 00023 for ale, &c. by •000281 for wine. RULL II. Find the solidity of half the cask by the following rule, (half of such a cask would represent a churn, or mash tub, or basket.) viz. find the area of the two ends, supposing the cask to be cut in two, add the areas together and reserve the sum; multiply the areas together extract the square root of the product, and add the root to the reserved sum; multiply the sum by the length, and the product is the solidity of half the cask, which must be doubled to make the solidity; divide the solidity by 231 for wine; 282 for ale; by 268.8 for corn gals. &c. NOTE. To find the area of the ends, multiply the diameter by the circumference, and 4 of the product is the area. Examples. 1. What is the content of a cask representing two frustums of a cone, (see frustum A B C D, figure first, case 5, solids) diameter AB 31, and diameter C D 44; length 55? Solved by rule first. res'd. prc. Diam. AB & CD 31 +4475X755625X3=16875 Diam. C D & A B 44-31=13×13169 add. Sum 17044 Sum 17044X55X 00023217-4814 ale gal. Ans. 17044X55X00028265.60233 wine gal. Ans. Diam. 44xby cir. 1383÷1521.3 area largest end. Dam. 31xby cir. 97-4x=754-8† area of small. end. Areas. 2 2276 1 reserved sum. 1521-3×754-8=1148277·24=1071·5 root added. 3347-6 sum. Sum 3347-6xhalf length 27.530686-33 solidity of half the cask. 30686 33×261372-66 solidity of the cask. solidity 61372-66282217·6† ale gal, &c. Ans, 61372·66÷÷231—265.6† wine gal. Ans. GUAGING, SECOND METHOD. Sometimes the following method is used in guaging. RULE, Measure the diameter of the cask at both ends, and take half the sum for the head diameter; measure the diameter at the bung, or in the middle of the cask; (dimensions must be taken upon the inside, or else allowance must be made for the thickness of the staves, &c.); measure also the length of the cask; find the difference between the head and bung diameter; multiply it by 62, and add the product to the head di ameter, the sum is the mean diameter of the cask ;) and reduces it to a cylinder). Square the mean diameter. and multiply the square by the length of the cask, and divide by 294.12 for wine; by 359.05 for ale, &c.; by 342-25 for corn gallons; and by 2738 for bushels of corn. NOTE.-These divisors arise by dividing 231, 282 and 268.8 by 7854; if the square of the mean diameter were multiplied by the length and again by 7854, the last product would be the solidity of the cask; (see Cylinder, case 4th, solids)the solidity being divided by 231 for wine; 282 for ale; by 268-8 for gallons of corn; &c. the quotients would be the answer in gallons, respectively; although the number 62 is generally used by guagers, yet it ought to be varied according to the curvature of the cask; if the cask is very curving like figure, case 13th, solids, it would require that the difference between the diameters be multiplied by '69 or 7; If the cask is more straight the number used should be less. Examples. 1. What is the content of a cask in form of the figure in case 13th, solids, whose diameter E G or FH equal 13, and diameter B D 20; and length 22 inches? Diam. 20-13-7 dif. x70490+13=179 me diam. Product 7049-02-294·12—24† gal. wine, Ans. 7049 02--359-05-19† gal. ale, &c. Ans. 7049 02-342-25-20 gal. of corn, Aus. If the cask is more straight, the number '67 ought to be used. 2. What is the content of a cask representing figure E F C D.G H, case 14th solids; diam. E F or G H equal 16, diam. C D 22; length 38 inches? Diam. 22-166 dif. ×·674·02+16—20·02 mean diam. pro. Mean diam. 20·02×2002400-8004×38 len. =15230-4152 15230-4152-359-05—42† gal. ale, &c. Ans. 3. What is the content of a cask, formed by joining two frustums of a round cone together, (the frustum A B C D figure first, case 5th, in solids, represents half of such a cask) diam. 44 and 31, and length 55 inches? NOTE.--A cask of this form requires that the difference between the diameters be multiplied by 515, for 52 produces too much. Diam. 44-3113 dif. ×·515—6-695+31 diam. =37·695_mean [diam. Mean diam. 37-695×37·695×55-78150-216375 product. Pro. 78150 216375-294-12-265·7† gal. wine, Ans. 78150 216375-359-05-217-6† gal. ale, &c. Ans. NOTE. It is evident from the three last examples, that the number 62 would not exactly apply in either case: the dimensions of the three last examples are the same as the dimensions of the three first examples, by the first method; and the numbers which I have made use of produce the same answers as were produced by the first method. If the cask is a little straighter than the elliptic spindle, in case 14 solids, the number 62 will exactly apply; this number should always be varied according to the shape of the cask, and at the judgment of the guager. 4. What is the content of a spheroidical cask whose length is 20; diameters 16 and 12 inches? Ans. 14.8 wine, 12.2† ale, &c. In this example the number 7 is used. |