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4. A curve meets the base at one extremity; the base is 2364; the perpendicular, at the other extremity, 758, and the others are 642, 587, 524, 432, 417, and 335 links. What is the area? Ans. 11198604 links=11 acres, 31 poles, 23.5 yards.

MENSURATION OF SOLIDS.

Art. 283.-The MENSURATION OF SOLIDS includes the mensuration of all bodies which have length, breadth, and thick

ness.

DEFINITIONS.

1. Solids are figures, having length, breadth, and thickness.

2. A prism is a solid, whose ends are any plane figures, which are equal and similar, and its sides are parallelograms.

OBS.-A prism is called a triangular prism, when its ends are triangles; a square prism, when its ends are squares; a pentagonal prism, when its ends are pentagons; and so on.

3. A cube is a square prism, having six sides, which are all squares. 4. A parallelopiped is a solid, having six rectangular sides, every opposite pair of which are equal, and parallel.

5. A cylinder is a round prism, having circles for its ends.

6. A pyramid is a solid, having any plane figure for a base, and its sides are triangles, whose vertices meet in a point at the top, called the vertex of the pyramid.

7. A cone is a round pyramid, having a circular base.

8. A sphere is a solid, bounded by one continued convex surface, every point of which is equally distant from a point within, called the centre. The sphere may be conceived to be formed by the revolution of a semicircle about its diameter, which remains fixed.

A hemisphere is half a sphere.

9. The segment of a pyramid, sphere, or any other solid, is a part cut off the top by a plane, parallel to the base of that figure.

10. A frustum is the part that remains at the bottom after the segment is cut off.

11. The sector of a sphere is composed of a segment less than a hemisphere, and of a cone, having the same base with the segment, and its vertex in the centre of the sphere.

12. The axis of a solid is a line drawn from the middle of one end to the middle of the opposite end; as between the opposite ends of a prism. The axis of a sphere is the same as a diameter, or a line passing through the centre, and terminating at the surface on both sides.

13. The height, or altitude of a solid, is a line drawn from its vertex, or top, perpendicular to its base.

Art. 284. To find the solidity of a cube.

RULE.

Multiply the length, breadth, and thickness together, and the product will be the area.

1. If the length of one side of a cubical block be 14 inches, what is its solidity? 14 × 14 × 14=2744 inches, Ans.

2. How many cubical feet in a mound, each side of which is 25.5 feet? Ans. 16581.375 feet.

Art. 285.-To find the solidity of a prism, or cylinder.

RULE.

Find the area of the end, and multiply it by the length. The product will be the area.

What is the solidity of a prism, the area of whose end is 2.6 feet, and whose length is 16 feet?

2.6 × 16 41.6 feet, Ans.

Art. 286.-To find the side of the largest stick of timber

that can be hewn from a round log

The circle, PEON, represents A the end of a round stick of timber; ABCD, a circumscribed square, and PEON, an inscribed square. It will be perceived that the square ABCD is double the square PEON. But the square ABCD is equal to the square of PO, the diameter of the circle; but PO is equal to Pa+a0=Pa+a E. Now Pa+aE=PE2, the side of D the largest inscribed square. Hence the

RULE.

B

Extract the square root of double the square of half the diameter at the smallest end of the stick, for the side of the stick when squared.

1. What will be the side of the largest stick of square timber which can be hewn from a round log, 18 inches in diameter at the smallest end?

√9x9x2=12.727+ inches, Ans.

2. The diameter of a log at the smallest end is 24 inches. What will be the side of the largest stick of timber that can be hewn from it? Ans. 16.97 inches

Art. 287.—To find the solidity of a pyramid, or cone.

RULE.

Multiply the area of the base by one third of the height, and the product will be the area.

1. What is the contents of a cone, whose height is 21 feet, and the diameter of the base 9.5 feet?

9.5 X 9.5.7854×21÷3=496.176 feet, Ans. 2. How many solid feet in a cone, whose height is 48 feet, and whose diameter at the base is 13 feet?

Ans. 2123.7216 feet. Art. 288.-To find the solid contents of a globe, or sphere.

RULE.

Multiply the cube of the diameter by .5236, or multiply the square of the diameter by one sixth of the circumference.

1. What is the solidity of a ball, 9 inches in diameter ? 9X9X9X.5236=381.7044, Ans. 2. What is the solidity of a globe, whose diameter is 13 inches? Ans. 1150.3492 inches. Art. 289.-To find the solid contents of the segment of a sphere, the height and base of the segment being given.

RULE.

To three times the square of the radius of the base of the segment, add the square of the height, and multiply this sum by the height of the segment, and this product by .5236.

How many cubic feet are there in a coal-pit, the diameter of whose base is 103 feet, and whose height is 9 feet? Ans. 37877.0931.

GAUGING.

Art. 290.-GAUGING is the art of measuring all kinds of vessels, such as pipes, hogsheads, barrels, etc.

RULE.

Add the square of the head diameter to the square of the bung diameter; multiply the sum by the length, and the product by .0014 for ale gallons, or by .0017 for wine gallons.

1. What is the contents of a cask, whose diameters are 18 and 26 inches, and its length 38 inches?

26 × 26+18 × 18×38=38000; then 38000 ×.0017=64.6 wine measure. 38000.0014=53.2 gallons, beer measure. 2. How many wine gallons will fill a cask 50 inches in length, bung diameter 38, head diameter 30 inches?

Ans. 199.24 gallons.

MEASURING GRAIN, ETC.

Art. 291.—WHEN the grain is heaped in the form of a cone.

RULE.

Measure the perpendicular height of the heap, and also the slanting height, from the top to the floor, in inches; then multiply the difference of the squares of those two heights by the perpendicular height, and this product by .0005. The last product will be the contents in bushels.

1. How many bushels in a parcel of wheat heaped in the form of a cone; the perpendicular height being 40 inches, and the slanting height 90 inches?

[blocks in formation]

2. What number of bushels in a conical heap of rye, the perpendicular height being 35 inches and the slanting height 65 inches? Ans. 52.5 bushels. Art. 292.-When grain is heaped against the side of the

barn.

RULE.

Multiply the difference of the squares of the heights by one half of the perpendicular height, and this product by .0005. The

result will be the contents in bushels.

1. How many bushels of oats are in a heap, the perpendicular height being 30 inches, and the slanting height 60 inches? Ans. 20.25 bushels.

2. How many bushels of beans are in a heap, the perpendicular height being 25 inches, and the slanting height 50 inches? Art. 293.-When grain is heaped in the corner of the barn.

RULE.

Multiply the difference of the squares of the heights by one fourth of the perpendicular height, and this product by .0005. The result will be the contents in bushels.

1. Required the number of bushels of grain heaped in the corner of the barn; the perpendicular height being 40 inches, and the slanting height 70 inches? Ans. 16.5.

2. How many bushels of barley in the corner of a box, the perpendicular height being 24 inches, and the slanting height 36 inches? Ans. 2.16 bushels.

TONNAGE OF VESSELS.

CARPENTERS' RULE.

Art. 294.-For single-decked vessels, multiply the length and breadth at the main beam, and depth in the hold, together, and divide the product by 95, and the quotient is the tons. But for a double-decked vessel, take half of the breadth of the main beam for the depth of the hold, and proceed as before.

1. What is the tonnage of a single-decked vessel, whose length is 67 feet, breadth 24 feet, and depth 12 feet? Ans. 2031 tons. 2. What is the tonnage of a double-decked vessel, whose length is 80 feet, and breadth 30 feet? Ans. 37818 tons.

GOVERNMENT RULE.

"If the vessel be double decked, take the length thereof from

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