C

ART. 341. A CONE is a solid, having a circle for its base, and its top terminated in a point, called the

vertex.

The altitude of a pyramid and of a cone, is a line drawn from the vertex perpendicular to the plane of the base; as, BC.

The slant height of a cone is a line drawn from the vertex to the circumference of the base; as, A C.、

L

ART. 342. To find the surface of a pyramid and of a cone. RULE. Multiply the perimeter or the circumference of the base by half its slant height, and the product is the surface.

1. How many yards of cloth that is 27 inches wide, will it require to cover the sides of a pyramid whose slant height is 100 feet, and whose perimeter at the base is 54 feet?

Ans. 400 yards. 2. Required the convex surface of a cone, whose slant height is 50 feet, and the circumference at its base 12 feet? Ans. 300 square feet.

ART. 343. To find the solidity of a pyramid and of a cone. RULE. - Multiply the area of the base by one third of its altitude, and the product will be its solidity.

1. The largest of the Egyptian pyramids is square at its base, and measures 693 feet on a side. Its height is 500 feet. Now, supposing it to come to a point at its vertex, what are its solid contents, and how many miles in length of wall would it make, 4 feet in height and 2 feet thick?

Ans. 80,041,500 cubic feet; 1894.9 miles in length. 2. What are the solid contents of a cone, whose height is 30 feet and the diameter of its base 5 feet?

Ans. 196.3 feet.

ART. 344. A FRUSTUM OF A PYRAMID is the part next to the base, that remains after cutting off the top, by a plane parallel to the base.

QUESTIONS.-Art. 341. What is a cone? What is the altitude of a pyramid and of a cone? What is the slant height of a cone? -Art. 342. What is the rule for finding the surface of a pyramid and of a cone?- Art. 343. What is the rule for finding the solidity of a pyramid and of a cone? ---Art. 344. What is the frustum of a pyramid?

ART. 345. A frustum of a CONE is the part next to the base, that remains after cutting off the top, by a plane parallel to the base.

ART. 346. To find the surface of a frustum of a pyramid or of a cone.

RULE. - Add the perimeters or the circumferences of the two ends together, and multiply this sum by half the slant height. Then add the areas of the two ends to this product, and their sum will be the surface. 1. There is a square pyramid, whose top is broken off 20 feet slant height from the base. The length of each side at the base is 8 feet, and at the top 4 feet; what is its whole surface? Ans. 560 square feet. 2. There is a frustum of a cone whose slant height is 12 feet, the circumference of the base 18 feet, and that of the upper end 9 feet; what is its whole surface?

Ans. 194.22+ square feet.

ART. 347. To find the solidity of a frustum of a pyramid or of a cone.

RULE. Find the area of the two bases of the frustum; multiply these two areas together, and extract the square root of the product. To this root add the two areas, and multiply their sum by one third of the altitude of the frustum; the product will be the solidity.

1. What is the solidity of the frustum of a square pyramid, whose height is 30 feet, and whose side at the bottom is 20 feet, and at the top 10 feet? Ans. 7000 cubic feet. 2. What are the contents of a stick of timber 20 feet long, and the diameter at the larger end 12 inches, and at the smaller end 6 inches? Ans. 9.162+ feet.

THE SPHERE.

ART. 348. A SPHERE is a solid, bounded by one continued convex surface, every part of which is equally distant from a point within, called the

centre.

The axis or diameter of a sphere is a line passing through the centre, and terminated by the surface.

QUESTIONS. Art. 345. What is a frustum of a cone? - Art. 346. What is the rule for finding the surface of a frustum of a pyramid or of a cone? - Art. 347. What is the rule for finding the solidity of a frustum of a pyramid or of a cone?-Art. 348. What is a sphere? What is the diameter or axis of a sphere?

ART. 349. To find the surface of a sphere.

RULE. — Multiply the diameter by the circumference, and the product will be the surface.

1. What is the convex surface of a globe, whose diameter is 20 inches? Ans. 1256.6+ square inches.

2. If the diameter of the earth is 8000 miles, what is its convex surface? Ans. 201061888 square miles.

ART. 350. To find the solidity of a sphere.

RULE.- Multiply the cube of the diameter by .523598, and the product is the solidity.

1. What is the solidity of a sphere, whose diameter is 20 inches? Ans. 4188.7 inches.

2. If the diameter of a globe or sphere is 5 feet, how many Ans. 65.44+ cubic feet.

cubic feet does it contain?

ART. 351. To find how large a cube may be cut from any given sphere, or be inscribed in it.

RULE.

Square the diameter of the sphere, divide the product by 3. and extract the square root of the quotient for the answer.

1. How large a cube may be inscribed in a sphere 10 inches in diameter ? Ans. 5.773+ inches. 2. What is the side of a cube that may be cut from a sphere 30 inches in diameter ?

THE SPHEROID.

Ans. 17.32+ feet.

ART. 352. A SPHEROID is a solid, generated by the revolution of an ellipse about one of its diam

eters.

If the ellipse revolves about its longer or transverse diameter, the spheroid is prolate or oblong; if about its shorter or conjugate diameter, the spheroid is oblate or flattened.

ART. 353. To find the solidity of a' spheroid.

RULE.-1. Multiply the square of the shorter axis by the longer axis, and this product by .523598, if the spheroid is prolate, and the product will be its solidity.

QUESTIONS. Art. 349. What is the rule for finding the surface of a sphere? Art. 350. What is the rule for finding the solidity of a sphere ? — Art. 351. What is the rule for finding how large a cube can be cut from a given sphere?-Art. 352. What is a spheroid? What is a prolate spheroid? What an oblate spheroid?-Art. 353. What is the rule for finding the solidity of a spheroid?

2. If it is oblate, multiply the square of the longer axis by the shorter axis, and this product by .523598; the last product will be the solidity.

1. What is the solidity of a prolate spheroid, whose transverse axis is 30 feet, and the conjugate axis 20 feet?

Ans. 6283.17+ cubic feet.

2. What is the solidity of an oblate spheroid, whose axes are 30 and 10 feet? Ans. 4712.38+ cubic feet.

§ XLVI.

MENSURATION OF LUMBER AND

TIMBER.

ART. 354. All rectangular and square lumber and timber, as planks, joists, beams, &c., are usually surveyed by board measure, the board being considered to be one inch in thickness. Round timber is sometimes measured by the ton, and sometimes by board measure.

ART. 355. To find the contents of a board.

RULE.-Multiply the length of the board, taken in feet, by its breadth taken in inches, and divide this product by 12; the quotient is the cor.tents in square feet.

1. What are the contents of a board 18 inches wide and 16 feet long? Ans. 24 feet. 2. What are the contents of a board 24 feet long and 30 inches wide? Ans. 60 feet.

ART. 356. To find the contents of joists, beams, &c.

RULE.- Multiply the depth, taken in inches, by the thickness, and this product by the length, in feet; divide the last product by 12, and the quotient is the contents in feet.

1. What are the solid contents of a joist 4 inches wide, three inches thick, and 12 feet long? Ans. 12 feet. 2. What are the contents of a square stick of timber 25 feet long and ten inches thick? Ans. 2081 feet.

ART. 357. To find the contents of round timber.

RULE.-Multiply the length of the stick, taken in feet, by the square of one fourth the girt, taken in inches; divide this product by 144, and the quotient is the contents in cubic feet.

QUESTIONS. Art. 354. By what measure are planks, joists, &c., usually surveyed What is the usual thickness of a board? How is round timber measured?- Art. 355. What is the rule for finding the contents of a board? Art. 356. What is the rule for finding the contents of joists, &c.?

NOTE 1.

The girt is usually taken about one third the distance from the larger to the smaller end.

NOTE 2.-A ton of timber, estimated by this method, contains 50% cubic feet.

1. How many cubic feet of timber in a stick, whose length is 50 feet and whose girt is 60 inches? Ans. 78 cubic feet. 2. What are the contents of a stick whose length is 30 feet and girt 30 inches? Ans. 11.7 solid feet.

§ XLVII. MISCELLANEOUS QUESTIONS. 1. WHAT is the difference between 7 pence and 10 cents?

Ans. ¿d.

Ans. 7.

2. What number is that, to which if be added, the sum will be 7? Ans. 78. 3. What number is that, from which if 34 be taken, the remainder will be 41 ? 4. What number is that, to which if 3 be added, and the sum divided by 53, the quotient will be 5? 5. From of a mile take

of a furlong.

Ans. 239.

Ans. 4fur. 12rd. 8ft. 8in.

6. From 7 acres take of a rood.

Ans. 6A. 3R. 7p. 74ft. 36in.

7. John Swift can travel 7 miles in & of an hour, but Thomas Slow can travel only 5 miles in 7 of an hour. Both started

from Danvers at the same time for Boston, the distance being 12 miles. How much sooner will Swift arrive in Boston than Slow? Ans. 123 seconds.

8. If of a ton cost $ 49, what will lcwt. cost?

Ans. $3.92. 9. How many bricks, 8 inches long, 4 inches wide, and 2 inches thick, will it take to build a wall 40 feet long, 20 feet high, and 2 feet thick? Ans. 43200 bricks.

10. How many bricks will it take to build the walls of a house, which is 80 feet long, 40 feet wide, and 25 feet high the wall to be 12 inches thick; the brick being of the same dimensions as in the last question? Ans. 159300 bricks.

11. How many tiles, 8 inches square, will cover a floor 18 feet long, and 12 feet wide? Ans. 486 tiles.

12. If it cost $ 18.25 to carry 11cwt. 3qr. 191b. 46 miles,