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PROBLEM III.

To find divisors, multipliers, and gauge-poiuts for circular areas.

RULE.

Divide 282, 231, 268.8, and 2150.42 *, by :7854, (the area of a circle whose diameter is 1) and the quots will be a set of divisors; and divide .7854 by the same numbers, the quot will give a set of multipliers as exhibited in the following table. Also the square roots of these divisors will give their respective gauge-points t.

Table II. 'For circular areas.

Divisors. Multipliers. Gauge-points. 359 05 A. G. 1.00278 A. .G 18.95 A. G. 294.12 · W. G..0034 W. G.

.00292

C. G. 58.5 C. G. 2737

M. B.

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17.15 W.G.

342.24 C. G.

.000305 M. B.

22.32 M. B.

PROBLEM IV.

To find the area of a circle in ale, wine, corn gallons, and malt

bufbels.

RULE.

In practice, the decimal part is neglected, 2150 being reckoned sufficiently accurate : We shall therefore only use the integer for the future. The same is to be observed of the divisors in the preceding table.

+ The gauge-point for circular areas is the diameter of a circle whose area, at I inch deep, is 5 gallon, i bushel, &c.

RULE.

Divide the square of the diameter by the divisors, or multiply the same square by the multipliers, the result will be the area in ale, wine, corn gallons, or malt bushels.

EXAMPLE I.

Required the area of a circle, whose diameter is 80 inches, in ale, wine gallons, and malt bushels,

80 X 80 = 6400.

First by division, 359)6400(17.827 A. G. 294)6400(21.768 W. G. 2737)6400( 2.338 M. B.

then by multiplication: 6400 x .00278=17.792 A. G. 6400 x .0034521.76 W. G. 6400 X.000365=2.336 M. B.

Ex. 2. Required the area of an ellipse, whose diameters are 1734 and 90, in ale, wine gallons, and malt bushels.

1731 X90 = 15600, This may be reckoned as the square of the diameter in circular surfaces.

First by division,

then by multiplication. 359) 15600(43.4 A. G. 15600 X 00278 =43.4 A. G. 294), 5600(53.06 W.G. 15600 K0034 =53.0 W. G. 2737) 15600(5.7 M. G. 15600 X 000365=5.7

M. B.

Required the area of a circular back, whose diameter is 50 inches, in ale, wire gallons, and malt bushels. Ans. 0.95 ale gallons, 8.5 wine gallons, and .91 malt

bufhels.

By By the fliding rule.

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Set the divisor upon B to the diameter on A, and against the dianieter 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 surfaces, 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 ascertained 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 side which subtends it, tiere 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 sides on which they fall. Compute the area of each triangle separately, and divide or multiply their sum as above, for gallons and bushels respectively.

EXAMPLES

* The vesiel 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 fide is 60 inches, and perpendicular breadth 50 inches ? Anf. 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 y6 inches, and breadth

50 inches?

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Anfo 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? Anf. 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 sides are 80, 100, and 60 inches? Anf. 8.51 ale gallons, 10.389 wine gallons, and 1.116

malt bushels. 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 ? Ans: 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 sides are 120 and 80 incher, and their perpendicular distance 50 inches ? Anf. 17.73 ale gallons, 21.645 wine gallons, and 2.325 malt bushels.

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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, 8o and 120 inches ? Anf. 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 fide is 20 inches ? Anf. 2.44 ale gallons, 2.978 wine gallons, and .32 malt

bufhels. Ex. 9. How many ale, wine gallons, and malt bushels are in a hexagon, whose fide is 20 inches? Anj. 3.686 ale gallons, 4.498 wine gallons, and .4833

malt bushels. Ex. 10. 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 ? Anf. 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

? 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 ellipse, whose transverse and conjugate diameters are 40 and 30 inches ? Anf. 3.341 ale gallons, 4.08 wine gallons, and .438 malt bushels.

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