[Height. Area Seg.'| Height. Area Seg. Height. Area Seg.

In the table, each number in the column of area seg. is the area of the circular segment, whose height, or the versed sine of its half arc, is the number immediately on the left of it, in the column of heights; the diameter of the circle being 1, and its whole area '785398.

See the

The use of this table is to find the area of a segment of any other circle, whatever be the diameter. rule, page 69.

In dividing the given height by the diameter, if the quotient do not terminate in three places of decimals without a fractional remainder, then for the area, answering to that fractional part, proportion ought to be made thus ;having found the tabular area answering to the first three decimals of the quotient, take the difference between it and the next following tabular area, which difference multiply by the fractional remaining part of the quotient, and the product will be the corresponding proportional part to be added to the first tabular area.

EXAMPLE.

If the height of the proposed segment be 3, and the diameter 50; required the area.

72

When the segment to be found is greater than a semicircle, subtract the quotient of its versed sine, divided by its diameter, from 1; subtract the tabular segment, which corresponds to the remainder, from 785398; and multiply the remainder by the square of the diameter.

GAUGING.

GAUGING signifies the art of measuring all kinds of

vessels, and determining their capacity, or the quantity of fluid or other matter they contain. The term is from a gauge, or rod; as the practitioners of the art perform the business, or make the calculations, commonly by means of instruments, called the gauging or diagonal rod, and the gauging or sliding rule. The vessels are principally pipes, tuns, barrels, rundlets, and other casks; also backs, coolèrs, vats, &c.

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 the proper rule, and this be divided by 282, Vol. II.

K

231, or 2150 ̊4, the quotient will be the content in ale gallons, wine gallons, or malt bushels, respectively. Be

Or, the particular rule for each will be as in the following PROBLEMS.

PROBLEM I.

To find the content of a cask of the first form.

RULE.*

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 00114, or divided by 882, for wine gallons.

EXAMPLES.

1. Required the content of a spheroidal cask, whose

* DEMONSTRATION. Let B = the bung diameter, H = the head diameter, L = the length. Then, by Problem xxIII, Case 1, Mens. of Solids,

2B2+H2 x L x 00113333; which is the Rule.