By the fliding rule.

Set the divifor upon B to the diameter on A, and against the diameter on B is the area on A.

SUPERFICIES having already been so largely treated of, it will perhaps be thought needless to give rules for each particular figure.

In general, the area of any back, tun, or cooler, or of any other vessel, may be obtained thus:-Find the area of its bottom or top * by the rule for its proper form, and divide this area (in square inches) by the divisors, or multiply by the multipliers in Table I. of right-lined furfaces, and the result will be the areas in ale, wine, corn gallons, or malt bushels.

But when the vessel is of a polygonous form, the following method is one of the most practical :-Divide it into triangles, by the help of a chalk'd line, such as carpenters use, by striking diagonals: And, having found the diagonals, the perpendiculars may be afcertained thus :- Fix one end of the chalk'd line in one of the angles; move it to and fro upon the stretch till you find the nearest distance from the angle to the fide which fubtends it, there strike a line. In like manner find the other perpendiculars; then, by a scale of inches and decimals of an inch, measure the perpendiculars, and the diagonals or fides on which they fall. Compute the area of each triangle feparately, and divide or multiply their fum as above, for gallons and bushels respectively.

EXAMPLES

• The vessel is here supposed to be of equal width from top to bottom.

EXAMPLES for practice.

EXAMPLE I.

How many ale, wine gallons, and malt bushels are in the area of a rhombus, whose side is 60 inches, and perpendicular breadth 50 inches ?

Ans. 10.63 ale gallons, 12.98 wine gallons, and 1.395

malt bushels.

Ex. 2. How many ale, wine gallons, and malt bushels are in the area of a rectangle, whose length is 96 inches, and breadth 50 inches ?

Anf. 17.02 ale gallons, 20.779 wine gallons, and 2.23 malt bushels.

Ex. 3. How many ale, wine gallons, and malt bushels are in the area of a rhombus, whose length is 120 inches, and perpendicular breadth 100 inches ?

Ans. 42.55 ale gallons, 51.948 wine gallons, and 5.58 malt bushels.

Ex. 4. How many ale, wine gallons, and malt bushels are in the area of a triangle, whose three fides are 80, 100, and 60 inches ?

Anf. 8.51 ale gallons, 10.389 wine gallons, and 1.116 malt bufhels.

Ex. 5. How many ale, wine gallons, and malt bushels are in the area of a triangle, whose base is 25 inches, and perpendicular 24 inches ?

Anf. 1.0638 ale gallons, 1.2987 wine gallons, and .1395 malt bushels.

Ex. 6. How many ale, wine gallons, and malt bushels are in the area of a trapezoid, whose parallel fides are 120 and 80 incho, and their perpendicular distance 50 inches ?

Ans. 17.73 ale gallons, 21.645 wine gallons, and 2.325 malt bushels.

Ex. 7. How many ale, wine gallons, and malt bushels are in the area of a trapezium, whose diagonal is 175 inches, and perpendiculars falling upon it, from the opposite angles, 80 and 120 inches?

Ans. 62.056 ale gallons, 75.757 wine gallons, and 8.13 malt bushels.

Ex. 8. How many ale, wine gallons, and malt bushels are in the area of a pentagon, whose side is 20 inches ?

Ans. 2.44 ale gallons, 2.978 wine gallons, and .32 malt bushels.

Ex. 9. How many ale, wine gallons, and malt bushels are in a hexagon, whose side is 20 inches ?

Anf. 3.686 ale gallons, 4.498 wine gallons, and .4833 malt bushels.

Ex. 1o. How many ale, wine gallons, and malt bushels are in a circle whose diameter is 40 inches ?

Anf. 4.456 ale gallons, 5.44 wine gallons, and .584 malt bushels.

Ex. 11. How many ale, wine gallons, and malt bushels are in a segment of a circle, whose diameter is 60 inches, and height of the segment 10 inches ?

Ans. 1.449 ale gallons, 1.769 wine gallons, and .19 malt bushels.

Ex. 12. How many ale, wine gallons, and malt bushels are in the sector of a circle, when the arch is 60 inches and radius 50 inches?

Anf. 5.319 ale gallons, 6.493 wine gallons, and .697 malt bushels.

Ex. 13. How many ale, wine gallons, and malt bushels are in an ellipfe, whose transverse and conjugate diameters are 40 and 30 inches ?

Anf. 3.341 ale gallons, 4.08 wine gallons, and .438 malt bushels.

Ex. 14. How many ale, wine gallons, and malt bushels are in an elliptic segment, cut off at the distance of 36 from the bentre, the axis being 120 and 40 inches ?

Anf. 1.903 ale gallons, 2.323 wine gallons, and .249 malt bushels.

These examples are so essentially necessary to the practice of gauging, that an officer in the excise ought not to be satisfied with less than the very principles on which they are performed, before he venture upon the following part of gauging.

PROBLEM IV.

To find the content of any cube, parallelopiped, prism, or of the cylinder, in ale, wine gallons, and malt bushels.

RULE I.

Find the area of the base in ale, wine gallons, or malt bushels, and multiply that area by the height, or depth; the product will give the content in ale gallons, &c.

RULE 2. Find the solid content, (in inches) as taught in Prob. II. IV. of folids, and this content, divided by the divisors, or multiplied by the multipliers in Table I. will give the confent in ale, wine gallons, &c.

EXAMPLE I.

How many ale, wine gallons, and malt bushels will a vefel, in the form of a parallelopipedon, contain, the length being 60 inches, breadth 50, and depth 36 inches?

282)108000(382.97 Content in ale gallons. 231)108000(467.54 Content in wine gallons. 2150)108000(50.23 Content in malt bushels.

And fo of the reft.

The small difference in the answers is not to be imputed to any defect in either of the preceding rules; for if the quots are extended to a few more decimal places, the answers will agree to the greatest nicety. In practice, however, it is customary to add to the integral part of the answer, when the decimal exceeds.5, and to neglect it when less. The fame is to be ob served of the following examples.

EXAMPLE II.

Required the content of a cubical vessel in ale, wine gallons, and malt bushels, the side being 20 inches.

Ans. 28 ale gallons, 35 wine gallons, and 4 malt bushels. Ex. 3. How many ale, wine gallons, will a cylinder contain whose diameter is 25 inches and depth 20 ?

Ans. 35 ale gallons, and 42 wine gallons.

Ex. 4. How many bushels malt will a vessel contain whose base is a rectangle of 50 inches by 404, and depth 40 inches? Anf.. 38.

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