BY THE SLIDING RULE.

Find a mean proportional (✓ (40 × 30) = 34·64,) between the length and breadth at the top, and a mean proportional (✓ (30 × 20) = 24·49,) between the length and breadth at the bottom; the sum of these is 59.13, twice a mean proportional between the length and breadth in the middle. Then.

2. Suppose the top and bottom of a vessel are parallelograms, the length of the top is 100 inches, and its breadth 70 inches; the length of the bottom 80, and its breadth 56, and the depth 42 inches; what is its content in imperial gallons? Ans. 862-59 imperial gallons.

THE GAUGING, OR DIAGONAL ROD.

The diagonal rod is a square rule, having four faces, and is generally 4 feet long. It folds together by joints. This instrument is employed 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 cask with the stave opposite the bung; being the longest line that can be drawn from the middle of the bung-hole to any part within the cask.

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, there are two scales of ale and wine gallons, expressing the contents of casks having the corresponding diagonals.

All the other lines on the instrument are similar to those on the sliding rule, and are used in the same manner.

Example. The diagonal, or distance between the middle of the bung-hole to the most distant part of the cask, as found by the diagonal rod, is 34-4 inches: what is the content in gallons?

To 344 inches correspond, on the rod, 90 ale gallons, or 111 wine gallons, 92 imperial gallons, the content required.

Note. The contents shown by the rod answer to the most common form of casks, and fall in between the 2nd and 3rd varieties following.

OF CASKS AS DIVIDED INTO VARIETIES.

Casks are usually divided into four varieties, which are easily distinguished by the curvature of their sides.

1. The middle frustum of a spheroid belongs to the first variety.

2. The middle frustum of a parabolic spindle belongs to the second variety.

3. The two equal frustums of a paraboloid belong to the third variety.

4. And the two equal frustums of a cone belong to the fourth variety.

If the content of any of these be found in inches by their proper rules, and this divided by 277-274, or 22182, the quotient will be the content in imperial gallons, or bushels, respectively.

PROBLEM VII.

To find the content of a vessel in the form of the frustum of a cone.

RULE. To three times the product of the two diameters, add the square of their difference; multiply the sum by of

the depth, and divide the product by 353-0362 for imperial gallons, and by 2824-289 for malt bushels.

1. What is the content of a cone's frustum, whose greater diameter is 20 inches, least diameter 15 inches, and depth 21 inches?

2. The greater diameter of a conical frustum is 38 inches, the less diameter 20-2, and depth 21 inches; what is the content in old ale gallons? Ans. 51-07 gallons.

To find the content of the frustum of a square pyramid.

RULE. To three times the product of the top and bottom sides, add the square of their difference, multiply their sum by of the depth, and divide the product by 282 and 231, for old ale and wine gallons, respectively; and by 277-274, for imperial gallons,

1. Suppose the greater base is 20 inches, the less base 15 inches, and depth 21 inches; required the content in old wine measure.

20 x 15 x 3 = 900

2015 5

Note.

Then, 5 x 5 = 25

925 × 7231 27.8 gallons.

The content of the frustum of a pyramid is found just like that of a cone, with the exception of the tabular divisor, or multiplier, the cone requiring the circular factor, and the pyramid the square one.

PROBLEM IX.

To find the content of a globe.

RULE. Multiply the diameter of the globe by its circumference, and the resulting product by of the diameter; then the last product multiplied or divided by the circular factor, will give the content in gallons.

1. Let the diameter be 34 inches, what is its content?
34 x 34 x 34 x 5236 20579.5744.

Then, 20579.5744 ÷ 282 = 72-9772 old ale gallons.
And, 20579.5744 ÷ 231 89′08 old wine gallons.

RULE II. Or cube the diameter of the globe, which multiply by 001888 ( of 002832) for the content in imperial gallons.

34339304; then 39304 x 001888 74-2 imperial gallons.

2. What is the content of a globe in old ale and wine measure, the diameter being 20 inches?

Ans.

(14-848 old ale gallons. 18.128 old wine gallons.

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3. Required the content of a globular vessel, whose diameter is 100 inches?

Ans. 1888 imperial gallons.

PROBLEM X.

To find the content of the segment of a sphere, as the rising crown of a copper still, &c.

RULE. Measure the diameter, or chord of the segment, and the altitude just in the middle. Multiply the square of half the diameter by 3; to the product add the square of the altitude; multiply this sum by the altitude, and the product again by 001856, or 002266, for old ale or wine measure, respectively, and by 001888 for imperial gallons.

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1. The diameter of the crown of a copper still is 27-6 its depth 9.2; required its content.

To gauge a copper having either a concave or convex bottom; or what is called a falling bottom, or rising crown.

RULE. If the side of the vessel be straight with a falling bottom, find the content of the segment Cy D, by Prob. X; find also the content of the upper part A B D C, by Prob. VII; the sum of both will give the content of the copper.

When the copper has a rising crown, find the content of A B CD, by Prob. VII, from which deduct the content of the segment C x D, and the remainder will be the content of the vessel A B D x C.