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The business of cask-guaging is commonly performed by two instruments, namely, the gauging or sliding rule, and the gauging or diagonal rule.
1. OF THE GAUGING RULE. This instrument serves to compute the contents of casks, &c. after the dimensions have been taken. It is a square rule, having various logarithmic lines on its four sides or faces, and three sliding pieces, running in grooves, in three of them.
Upon the first face are three lines, namely, two marked A, B, for multiplying and dividing; and the third, MD, for malt depth, because it serves to guage malt. The middle one B is on the silder, and is a kind of double line, being marked at both the edges of the slider, for applying it to both the lines A and MD. · These three lines are all
of the same radius or distance from 1 to 10, each containing twice the length of the radius. A and B are placed and numbered exactly alike, each beginning at 1, which may be either 1, or 10, or 100, &c. or .1, or .01, or .001, &c. but whatever it is, the middle division, 10, will be ten times as much, and the last division 100 times as much. But 1 on the line MD is opposite 215, or more exactly 2150.4. on the
other lines, which number 2150.4 denotes the cubic inches in a small bushel; and its divisions numbered retrograde to those of A and B. On these two lines are also several other marks and letters; thus, on the line A are MB, for malt bushel, at the number 2150.4; and A for ale, at 282, the cubic inches in an ale gallon; and on the lin B is W, for wine, at 231, the cubic inches in a wine gallon; also, si, for square inscribed, at .707, the side of a square inscribed in a circle whose diameter is 1; se, for square equal, at .886, the side of a square which is equal to the same circle; and c, for circumference, at 3.1416, the circumference of the same circle.
On the second face, or that opposite the first, are a slider and four lines, marked D, C, D, E, at one end, and root, square, root, cube, at the other; the lines C and D containing respectively the square and cubes of the opposite numbers on the lines D, D; the radius of D being double to that of A, B, C, and triple to that of E; so that whatever the first 1 on D denotes, the first on C is the square of it, and the first on E the cube of it; so if I begin with 1, C and E will begin with 1; but if D begin with 10, C will begin with 100, and E with 1000; and so on. On the line C are marked oc at .0796, for the area of the circle, whose circumference is 1; and od at .7854, for the area of the circle whose diameter is 1. Also on the line D, are WG, for wine guage, at 17.15; AG for ale guage, at 18.95; and MR for malt round, at 52.32; these three being the guage points for round and circular measure, and are found by dividing the square roots of 231, 282, and 2150.4 by the square root of .7854: also, MS, for malt square, are marked at 46.37, the malt guage point for square measure, being the square root of 2150.4.
On the third face are three lines, one on a slider marked N; and two on the stock, marked SS and SL, for segment standing and segment lying, which serve for ullaging standing and lying casks.
And on the fourth, or opposite face, are a scale of inches, and three other scales marked spheroid, or 1st variety, 2d variety, 3d variety; the scale for the fourth or copic variety,
being on the inside of the slider in the third face. The use of these lines is to find the mean diameters of casks.
Besides all those lines, there are two others on the insides of the first two sliders, being continued from the one slider to the other. The one of these is a scale of inches, from 121 to 36; and the other is a scale of ale gallons, bctween the corresponding numbers 435 and 3.61; which form a table to show, in ale gallons, the contents of all cylinders whose diameters are from 121 to 36 inches, their common altitude being 1 inch.
The use of the Gauging Rule.
PROBLEM I. To multiply two numbers, as 12 and 25. Set 1 on B, to either of the given numbers, as 12, on A, then against 25 on B, stands 300 on A; which is the product.
To divide one number by another, as 300 by 25. Set 1 on B, to 25 on A; then against 300 on A, stands 12 on B for the quotient.
PROBLEM III. To find a fourth proportional, as to 8, 24, and 96. Set A on B, to 24 on A; then against 96 on B, is 288 on A, the 4th proportional to 8, 24, 96 required.
To extract the square root, as of 225. The first one on C standing opposite the one on D, on the stock; then against 225 on C, stands its square root 15 on D.
The line D on the slide being set straight with E; opposite 3375 on E stands its cube root 15 on D.
PROBLEM VI. To find a mean proportional, as between 4 and 9. Set 4 on C, to the same 4 on D; then against 9 on C, stands the mean proportional 6 on D.
PROBLEM VII. To find numbers in duplicate proportion. As, to find a number which shall be to 120, as the square
of 3 to the square of 2. Set 2 on D, to 120 on C; then against 3 on D, stands 270 on C; for the answer.
PROBLEM VIII. To find numbers in subduplicate proportion. As, to find a number which shall be to 2 as the root of 270
to the root of 120. Set 2 on D, to 120 on C, then against 270 on C, stands 3 on D, for the answer.
PROBLEM IX. To find the numbers in triplicate proportion. As, to find a number which shall be to 100, as the cube of
36 is to the cube of 40. Set 40 on D, to 100 on E; then against 36 on D, stands 72.9 on E, for the answer.
PROBLEM X. To find numbers in subtriplicate proportion. As, to find a number which shall be to 40, as the cube root
of 72.9 is to the cube root of 100.
Set 40 on D, to 100 on E; then against 72.9 on E stands 36 on D, for the answer.
PROBLEM XI. To compute malt bushels by the line MD. As, to find the malt bushels in the couch, floor, or cistern,
whose length is 230, breadth 58.2, and depth 5.4 inches.
Set 230 on B, to 5.4 on MD; then against 58.2 on A stands 33.6 bushels on B, for the answer.
Note.—The uses of the other marks on the rule, will appear in the examples further on.
OF THE GAUGING OR DIAGONAL ROD.
The diagonal rod is a square rule, having four faces; being commonly four feet long, and folding together by joints. This instrument is used both for gauging and measuring casks, and computing their contents, and that from one dimension only, namely, the diagonal of the cask, or the length from the middle of the bung-hole to the meeting of the head of the cask with the stave opposite to the bung, being the longest straight line that can be drawn within the cask from the middle of the bung. And, accordingly, on one face of the rule is a scale of inches for measuring this diagonal; to which are placed the areas, in ale gallons, of circles to the corresponding diameters, in like manner as the lines on the under sides of the three slides in the sliding rule.
On the opposite face, are two scales of ale and wine gallons, expressing the contents of casks having the corresponding diagonals. And these are the lines which chiefly form the difference between this instrument and the sliding rule; for all their other lines are the same, and are to be used in the same manner.
EXAMPLE. The rod being applied within the cask at the bung-hole, the diagonal was found to be 34.4 inches; required the content in gallons.