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the cask ; (dimensions must be taken upon the inside, or else allowance must be made for the thickness of the staves, Sc.); 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 diameter, 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 29 *•12 for wine ; by 359.05 for alc, &c.; by 342•25 for corn gallons; and by 2738 for bushels of
NOTE.-These divisors arise by dividing 231, 282 and 2688 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 2688 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 tha curvature of the cask; if the cask is very curving like figure, case 13th, solids, it wonld 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.
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= dif. X70=490+13=1:79 me ui diam.
7049-02-:-359:05=194 gal. ale, &c. Ans.
If the cask is more straight, the number •67 ought to be used.
2. What is the content of a cask representing figure EFCDG H, case 14th solids; diam. EF or G H equal 16, diam. C D 22 ; length 38 inches?
Diam. 22–16—6 dif. X.67–4:02+16=20:02 mean diam.
pro. Mean diam. 20.02 X 20-02-400:8004 X 38 len. =15230-4152 'Pro. 15230:4152--29412 517 gal. wine, Ans.
15230.4152---359.05—427 gal. ale, &c. Ans.
3. What is the content of a cask, formed by joining two frustums of a round cone together, (the frustuin A B C D figure first, case 5th, in solids, represents half of such a cask) diam. 44 and 31, and length 55 inches?
Nore.--A cask of this form requires that the difference between the diagneters be multiplied by .515, for ·52 produces too much.
Diam. 44—31513 dif. X.515—66957-31 diam. =37.695- mean
[diam. Mean diam. 37.695X37.695X55=78150-216375 product. Pro. 78150-216375--294:12—265.77 gal. wine, Ans.
78150.216375:-359.05=217.67 gal. ale, &c. Ans.
Nore. It is evident from the three last exainples, that the nulpber 62 would uot exactly apply in either case : the di. mensions 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 sanie answers as were produced by the first method. If the cask is a little straighter than the elliptic spindle, in case 14 solids, the pumber •62 will exactly apply ; this number should always be varied according to the shape of the cask, and at the judgment of
4. What is the content of a spheroidical cask whose length is 20; diameters 16 and 12 inches?
Ans. 14•87 wine, 12:27 ale, &c.
5. What is the content of a cask, representing an
Ans. 117.27 wine, 96.0f ale, &c.
To guage with calipers, or Gunter's scale.
RULE.For a cask representing the middle frustum of a spheroid, (see figure, case 13, solids) extend from 1 to •7; and with the same extent, set one foot of the dividers in the number expressing the difference between the bung and head diameter, and they will reach to a number, towards the left hand, which, when added to the head diameter will give the mean diameter of the cask; then set one foot of the dividers in the guage point, and extend to the mean diameter; and the same way set off twice that extent from the length of the cask, and they will reach to a number expressing the content of the cask.
NOTE 1. The same operation may be performed on the line of numbers, on Gunter's scale.
Note 2. If wine gallons are required use the guage point 17 -15; if ale gal. are reqnired nse the guiage point 18-95; and if córn gal. are required you may use the point 18•5 respectively.
Note 3. If the cask represents the middle frustum of an elliptic spindle (see figure, case 14, solids) extend from 1 to :67, or if a little more straight, extend from 1 to .62 ; and for 2 cask formed by two frustumus of a cone, extend from 1 to -51,
* These numbers are the square roots of the numbers 291:12 wd 359.05 respectively.
1. What is the content of a cask of rum, or wine whose length is 59.3 in.; and diameters 34.5, and - 3007; difference of diameters 3.8 ?
Extend from 1 to 62 ; and that extent reaches from 3.8, (the difference of the diameters) towards the left hand to 2-4 nearly, which add to the head diameter: 30•7+2:4=33:1 mean diam. ; then extend from the guage point 17:15 to 33•1 (the mean diameter), and set off twice that extent from 59.3 (the length of the cask), and they will reach to 220•8 nearly, which is the content of the cask.
2. What is the content of a cask of wine whose length is 38; diam. 22, and 16 in. : difference of diameters 6?
Ans. 52+ wine, 427 ale, &c.
Extend from 1 to .67 ; and that extent will reach from 6, (the difference) to 3:7 nearly : which being added to the head diameter 16, produces 19.7 for the mean diameter: then extend from the wine guage point to 19.7, (the mean diameter) and set off twice that extent from 38, the length; and the dividers will rest on 52 the answer.
Note.-Guagers make use of the guaging rod, to take the dimensions of the cask; and there are also, tables on it by means of which the outs are taken; after the preceding rules are well committed to memory, the young practitioner would soon obtain a complete knowledge of guaging, by a little practice with the instruments
Or to find whether water can be made to run from one place to another, or to find how much higher
one place is than another.
F is the fountain, and E the place assigned to carry the water to
Note.-To perform this business you must be provided with two staves divided into inches and tenths, which may be of any convenient length perhaps, 4 or 6 feet; and also a water level which may be made easily at any Joiners shop; the level may be of any convenient length dug out bollowing, so as to hold water; with two sights at the top to make the observations through : having provided the staves and water level, go to the fountain with two assistants and observe the following rule.
RULE.Order the first assistant to the fountain with one staff placed perpendicularly : and the second assistant to any convenient place as at A, with his staff per. pendicularly; then place the water level in the middle between them at w, and looking through the sights or. der the first assistant to move a piece of white paper up and down the staff till you can see it through the sights : then order him to notice the distance it rests from the ground; and going to the other end of the water level, order the second assistant to do likewise ; and after he has marked down the distance the papers rest from the ground, order the first assistant to take the place of the second, and the second to take a new stand, as at B; place the water level between them, make observations as before, and order the assistants to mark down the distances the paper rests from the ground, and proceed to take another station and make new obscrvations till you have arrived to the place assigned;
and order the assistants to cast up their notes, and as much as the seč