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Set 1 un B, to either of the given numbers, as 12, on A ; then against 25 on B, stands 300 on A; which is the product. Prob. 2. To divide one number by another, as 300 by 25.

Set 1 on B, to 25 ou A ; then against 300 on A, stauds 12 ou B, for the quotient.

Prob. 3. To find a fourth proportional, as to 8, 24, and 96.

Set 8 on-B, to 24 on A ; then against 96 on B, is 288 on A, the Ath proportional to 8, 24, 96, required.

Prob. 4. To extract the square root, as of 225.

The first I on C standing opposite the one on D, on the stock ; then against 225 on C, stands its square root 15 on D.

Prob. 5. To extract the cube root, as of 3375.

The line D on the slide being set straight with E; then opposite 3375 on E, stands its cube root 15 on D.

Prob. 6. 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.

Prob.7. 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.

Prob. 8. To find numbers in subduplicate proportion. As lo firid a Number which shall be to 2 us the Root of 270 to the

Root of 120. Set 2 on D, to 120 on C; then against 270 on C, stards 3 on D, for the answer.

Prob. 9. To find numbers in triplicate proportion.
As, to find a Number which shall be to 100, as the Cube of 36 is to

the Cule of 40. Set 40 on D, to 100 on E; then against 36 on D, stands 72.9 on E, for the answer.

Prob. 10. To find numbers in subtriplicate proportion. As, to find a Number which shall be to 40, us 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 or E, stands 36 on D, for the answer.

Prob. 11. To compute malt bushels by the line MD. As, to find the Malt Bushels in the Couch, Floor, or Cistern, whose

Length is 230, Breadih 582, und Depth 5:4 Inches. wet 230 on B, to 5:4 on MD; then against 582 on A stands 33:6 oushels op B, for ihe answer.

Note.—The uses of the other marks on the rule will appear in the examples farther 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 or 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 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.

Ex. The rod being applied within the cask at the bang-hole, the diagonal was found to be 34.4 inches; required the content in gallons.

Now to 34:4 inches correspond, on the rod, 90$ ale gallons, or all wine gallons, the content required.

Nole.—The contents exhibited by the rod, answer to the most common form of casks, and fall in between the ed and 3d varieties following.

of Casks as divided into Varieties. It is usual to divide casks into four cases or varieties, which are judged of from the greater or less apparent curvature of their sides ; namely,

1. The middle frustum of a spheroid,
2. The middle frustum of a parabolic spindle,
3. The two equal frustums of a paraboloid,
4. The two equal frustums of a cone.

And if the content of any of these be computed in inches, by their proper rules, and this be divided by 282, or 231, or 21504, the quotient will be the content in ale gallons, or wine gallons, or malt busbels, respectively. Because

282 cubic inches make 1 ale gallon.
231

I wine yalion.
215004

1 malt bushel. And the particular rule will be for each as in the following problem: Prob. 12. To find the content of a cask of the first form.

To the square of the head diameter, add double the square of the bung diameter; and multiply the sum by the length of the cask. Then let the product

be multiplied by •0009, or divided by 1077, for ale gallons : and multiplied by '0011, or divided by 882, for wine gallons.

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Ez. 1. Required the content of a spheroidal cask, whose length is 40, and bung and head diameters 32 and 24 inches.

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By the Gauging Rule.
Having set 40 on C, to the ale gauge 32.82 on D, against

24 on D, stands 21•3 on C
32 on D, stands 38.0 on C

the same 380

sum 973 ale gallons.

And having set 40 on C, to the wine gauge 29 7 on D, against

24 on D, stands 26.1 on C
32 on D, stands 46.5 on C

the same 465

sum 119'1 wine gallons.

Er. 2. Required the content of the spheroidal cask, whose length is 20, and diameters 12 and 16 inches.

Answer { 12:136 ale gallons,

14.869 wine gallons. Prob. 13. To find the content of a cask of the second form.

To the square of the head diameter, add double the square of the bung diameter, and from the sum take 2-5ths or 1-40th of the square of the difference of the diameters ; then multiply the remainder by

a

the length, and the product again by '000gł for ale gallons, or by *0011 for wine gallons.

Er. 1. The length being 40, and diameters 24 and 32, required the content.

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By the Gauging Rule. Having set 40 on C, to 32:82 on D, against 8 on D, stands 2:4 on C; the 4-10th of which is 0.96. This taken from the 97*3 in the last form, leaves 963 ale gallons.

And having set 40 on C, to 297 on D, against 8 on D, stands 2:9 on C; the 4-10th of which is 1'16. This taken from the 119 1 in the last form, leaves 1179 wine gallons.

Er. 2. Required the content when the length is 20, and the diameters 12 and 16

,

Answer { 12:028 gallons,

Prob. 14. To find the content of a cask of the third form.

To the square of the bung diameter, add the square of the head diameter ; multiply the sum by the lengtk, and the product again by the length, and the product again by .0014 for ale gallons, or by 0017 for wine gallons.

Ex. 1. Required the content of a cusk of the third form, when the length is 40, and the diameters 24 and 32.

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108.8 wine.

By the Gauging Rule.
Set 40 ou C, to 26 8 on D, then against

24 on D, stands 32 0 on C
32 on D, stands 57:3 on C

sum 89:3 ale gallons. And having set 40 on C, to 24-25 on D, then against

24 on D, stands 39*1 on C
32 un D, stands 69.8 on C

sum 108:9 wine gallons. Er. 2. Required the content when the length is 20, and the diameters 12 and 16.

S112 ale gallons. Answer

13.6 wine gallons. Prob. 15. To find the content of a cask of the fourth form.

Add the square of the difference of the diameters, to 3 times the square of their sum ; then multiply the sum by the length, and the product again by 00023} for ale gallons, or by 00028+ for wine gallons.

Ex. 1. Required the content, when the length is 40, and the diameters 24 and 32 inches.

8
56
8
378980

378880
336
64
*00023 5

*00028';
280 9408

1136640

303 1040
3136 9472 757760

757760
3
40
75776

126293

56

9408

378880 ale

87-90016 gallons 107.34933 wine.

By the Sliding Rule.
Set 40 on C, to 65.64 on D; then against

8 on D, stands 0.6 on C
56 on D, stands 29.1 on C

29.1
29:1

sum 879 ale gallons.
And set 40 on C, to 59:41 on D; then against

8 on D, stands 0.7
56 on D, stands 35.6

35.6 35.6

sum 107°3 wine gallons.

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