Generally speaking there are three sorts of casks, which are judged of according to the curvature of their sides, viz. 1. If the sides of the cast are very curving from end to end, it may be said to represent the middle frustum of a spheroid, 'see the figure E F G H, case 13, solids.) 2. If the sides of the cask are more straight it more properly represents the middle frustum of an elliptic spindle, (see the figure, case 14, solids.) 3. If the cask is straight from the end to the bung, and of a true taper, (such a cask is rarely found) it represents two frustums of a round cone; the two bases form the middle diameter. (Şee the frustum of a round cone case 5, solids, figure first,) which represents half of such a cask. GENERAL RULE, Find the solidity of either of the above figures, (or any other figure) in solid inches, and divide the solidity by 231 for wine gallons ; by 282 for beer gallon; and by 268.8 for gallons of grain or corn: and by 2150•4 for bushels of corn; and the quotients will be the answers respectively. CASE I. To guage a spheroidical cask. Rule 1.--To twice the square of the bung diameter, add the square of the head diameter, and multiply the sum by the length of the cask, divide the product by 882 for wine, by 1077 for ale, or beer, and by 1026.3 for gallons of corn or grain ; and the quotients will be the answers respectively. Rule 2.–Find the solidity of the cask by the following rule, viz. To the square of the diam. at the end, add twice the square of the middle, or búng diameter; multiply the sum by the length of the cask, and again by 2618, the last pro. is the solidity; which divide by 231 for wine ; 282 for ale, or beer; and by 268.8 for gallons of corsi, or grain : 'the quotients will be the answers res; pectively. Exainples. 1. What is the content of a spheroidical cask, in wine, ale or beer, and corn gallons; whose middle or bung diameter B D (see figure, case 13, solids) is equal to 20 inches, anil the end diameter E G or FH is equal to 13 inches, and the length of the cask 22 inches? Solved by rule first. Bung diam. B D 20 x 20 x 2800 twice tlie square. *Head diam. E G or F H 13 X13—169 added. şum 969 X 22=21318 Pro. Pro. 21318+882—241.gal. of wine, Ans. 21318--1077–198677 gal. of ale or beer, Ans. 21318–1026:3=20102 gal. of corn, &c. Ans. The same question by rule second. Diam. E G or F H 13 x 13=169 square of E G or FH. Bung diam. BD 20 X 20 X2=800 twice square BD add. 969 sum. Sum 969 X 22 length=21318X•2618=5581.0524 sol. Solidity 5581•0524-231=24.167 gal. of wine, Ans. 558 1•0524-=-282=19•79+ gal. of ale, &c. Ans. 5581.0524--268.8=20.767 gal. of corn, &c. Ans. CASE IT. To guage a cask representing the middle frustum of an Note.---Dimensions are always taken in inches ; due allowance must be made for the thickness of the heads, &c. 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 bụng 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 figure E F, OP, CD, 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 G H 16; and a diameter at OP 20.5 inches ? Solved by rule first. Diam. C D 22x22=484X2=968 Diam, E F or G H 16X16-256 reserved sum. 1224 Diam. 22-16—6X636;144 sub. rem. remainder. 1209.6 1209:6X38=45964.8 product: 459648-882 52-1f gal, wine, Ans. 45964:8;-1077 =42:67 gal. ale, &c. Ans. 45964-87-1026-344.7+ gal. corn, &c. Ans. The same question by rule second, Diam. at 0 P 20:5X2=41X41=1681 square of twice O P. Diam. CD 22 x 22 484 square of CD. Diam. E For G H 16X16=256 square of E For GH, 2421 sum. 12042-5382--282=42:7+ gal. of ale, &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 1.—Add the two diameters together, square the sum and multiply the square by 3, and reserve the product ; find the difference of the diameters ; square the difference, and add the square of the difference to the reserved product, and multiply the sum by the length, and the product again by •00023; for ale, &c. by .00028} for wine. RULL 11.-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 į the length, and į 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 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. CD & A B 44-31-13x13169 add. Sum 17044 Sum 17044 X 55 X 000231=217.48144 ale gal. Ans. 17044X55X•00028}=265.60233 wine gal. Ans. Diam. 44 x by cir. 138:3+1=15213 area largest end. Dam. 31x by cir. 97.4x=754.87 area of small. end. Areas. 2276-1 reserved sum. 1521:3x754.8= 1148277.24=1071:5 root added. 2 3347.6 sum. Sum 3347.6 X half length 2751=30686.33. solidity of half the cask. 61372.66-231=265.67 wine gal. Ans. GUAGING, SECOND METHOD. Sometimes the following method is rised in guaging. Rule.-Measure the diameter of the cask at botlı ends, and take half the sum for the head diameter ; measure the diameter at the bung: or in the middle of |