1. What are the contents of a triangular prism, whose length is 20 feet, and the three sides of its triangular end or base 5, 4, and 3 feet? Ans. 120 cubic feet.

2. How many cubic feet are there in a cube, whose sides are 8 feet? Ans. 512 cubic feet. 3. What is the number of cubic feet in a room 30 feet long, 20 feet wide, and 10 feet high? Ans. 6000 cubic feet.

THE CYLINDER.

ART. 351. A CYLINDER is a round solid, of uniform diameter, with circular ends.

The axis of a cylinder is a straight line drawn through it, from the centre of one end to the centre of the other.

ART. 352. To find the surface of a cylinder.

RULE.

Multiply the circumference of the base by the altitude, and to the product add the areas of the two ends; the sum will be the whole surface.

1. What is the surface of a cylinder, whose length is 4 feet, and the circumference 3 feet? Ans. 13.43+ square feet.

2. John Snow has a roller 12 feet long and 2 feet in diameter; what is its convex surface? Ans. 75.39+ square feet.

ART. 353. To find the solidity of a cylinder.

RULE.

Multiply the area of the base by the altitude, and the product will be the solidity.

1. What is the solidity of a cylinder 8 feet in length and 2 feet in diameter ?

Ans. 25.13+ cubic feet.

2. What is the solidity of a cylinder, whose diameter is 5 feet, and its altitude 20 feet? Ans. 392.69+ cubic feet.

THE PYRAMID AND CONE.

ART. 354. A PYRAMIL is a solid, standing on a triangular, square, or polygonal base, with its sides tapering uniformly to a point at the top, called the

vertex.

The slant height of a pyramid is a line drawn from the vertex to the middle of one of the sides of the base.

QUESTIONS. Art. 351. What is a cylinder? What is the axis of a cylinder?-Art. 352. What is the rule for finding the surface of a cylinder? Art. 353. What is the rule for finding the solidity of a cylinder?- Art. 354. What is a pyramid ? What is the slant height of a pyramid ?

ART. 355. A CONE is a solid, having a circle for its base, and tapering uniformly to 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 B C.

A

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

ART. 356. 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 convex 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. 357. 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. 358. A FRUSTUM OF A PYRAMID is the part that remains after cutting off the top, by a plane parallel to the base.

QUESTIONS. Art. 355. What is a cone? What is the altitude of a pyramid and of a cone? What is the slant height of a cone?- Art. 356. What is the rule for finding the surface of a pyramid and of a cone? - Art. 357. What is the rule for finding the solidity of a pyramid and of a cone? — Art. 358. What is the frustum of a pyramid?

ART. 359. A FRUSTUM OF A CONE is the part that remains after cutting off the top, by a plane parallel to the base.

ART. 360. 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. 361. To find the solidity of a frustum of a pyramid or of a cone.

RULE. Find the area of the two ends 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?

2. What are the contents of a and the diameter at the larger end end 6 inches?

Ans. 7000 cubic feet. stick of timber 20 feet long, 12 inches, and at the smaller Ans. 9.162 feet.

THE SPHERE.

ART. 362. 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. 359. What is the frustum of a cone? - Art 360. What is the rule for finding the surface of a frustum of a pyramid or of a cone? Art. 361. What is the rule for finding the solidity of a frustum of a pyramid or of a cone?-Art. 362. What is a sphere ? What is the diameter or axis of a sphere?

ART. 363. 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. 364. 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 ?

2. If the diameter of a globe or

cubic feet does it contain?

Ans. 4188.7 inches. sphere is 5 feet, how many Ans. 65.44+ cubic feet.

ART. 365. 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 ? Ans. 17.32+feet.

THE SPHEROID.

ART. 366. 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. 367. 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. 363. What is the rule for finding the surface of a sphere? Art. 364. What is the rule for finding the solidity of a sphere ?— Art. 365. What is the rule for finding how large a cube can be cut from a given sphere?-Art. 366. What is a spheroid? What is a prolate spheroid? What an oblate spheroid? - Art. 367. What is the rule for finding the solidity of a spheroid?

RULE 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.

§ XLV.

MENSURATION OF LUMBER AND

TIMBER.

ART. 368. ALL rectangular and square lumber and timber, as planks, joists, beams, &c., are usually surveyed by board measure, the board being considered to be 1 inch in thickness. Round timber is sometimes measured by the ton, and sometimes by board

measure.

ART. 369. 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 contents 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. 370. 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, 3 inches thick, and 12 feet long? Ans. 12 feet. 2. What are the contents of a square stick of timber 25 feet long and 10 inches thick?

Ans. 208 feet.

ART. 371. 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. 368. By what measure are planks, joists, &c., usually surveyed? What is the usual thickness of a board? How is round timber measured? Art. 369. What is the rule for finding the contents of a board? Art. 370. What is the rule for finding the contents of joists, &c. ?