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 Ans. 4712.38+ cubic feet.

30 and 10 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. ?

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. 784 cubic feet.

2. What are the contents of a stick, whose length is 30 feet, and girt 30 inches? Ans. 11.7+ solid feet.

§ XLVI.

MISCELLANEOUS QUESTIONS.

1. WHAT number is that, to which if be added, the sum will be 7? Ans. 78. 2. What number is that, from which if 3 be taken, the remainder will be 41 ? Ans. 78. 3. What number is that, to which if 34 be added, and the sum divided by 5%, the quotient will be 5? Ans. 239.

4. From of a mile take of a furlong.

Ans. 4fur. 12rd. 8ft. 8in. in § of an hour, but Thomas of an hour. Both started Boston, the distance being 12

5. John Swift can travel 7 miles Slow can travel only 5 miles in from Danvers at the same time for miles. How much sooner will Swift arrive in Boston than Slow? Ans. 123 seconds.

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

Ans. $3.92. 7. 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.

8. 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.

9. How many tiles, 8 inches square, will cover a floor 18 feet long, and 12 feet wide? Ans. 486 tiles. 10. If it cost $18.25 to carry 11cwt. 3qr. 191b. 46 miles, how much must be paid for carrying 83cwt. 2qr. 11lb. 96 miles? Ans. $266.702049. 11. A merchant sold a piece of cloth for $24, and thereby lost 25 per cent.; what would he have gained had he sold it for $34? Ans. 61 per cent.

12. Bought a hogshead of molasses, containing 120 gallons for $30; but 20 gallons having leaked out, for what must I sell the remainder per gallon to gain $10? Ans. $0.40. 13. Bought a quantity of goods for $128.25, and having kept them on hand 6 months, for what must I sell them to gain 6 per cent.? Ans. $140.02. 14. If a sportsman spends of his time in smoking, & in "gunning," 2 hours per day in loafing, and 6 hours in eating, drinking, and sleeping, how much remains for useful purposes? Ans. 2 hours.

15. If a lady spend of her time in sleep, in making calls, at her toilet, in reading novels, and 2 hours each day in receiving visits, how large a portion of her time will remain for improving her mind, and for domestic employments?

Ans. 327 hours per day.

16. If 5g ells English cost $15.16, what will 71 yards cost?

17. If a staff 4 feet long cast a shadow 5 height of a steeple whose shadow is 150 feet?

Ans. $155.39. feet, what is the

Ans. 107 feet. 18. Borrowed of James Day $150 for six months; afterwards I lent him $100; how long shall he keep it to compensate him for the sum he lent me ? Ans. 9 months.

19. A certain town is taxed $6045.50; the valuation of the town is $293275.00; there are 150 polls in the town, which are taxed $1.20 each. What is the tax on a dollar, and what does A pay, who has 4 polls, and whose property is valued at $3675? Ans. $0.02. A's tax $78.30.

20. D. Sanborn's garden is 234 rods long, and 134 rods wide, and is surrounded by a good fence 74 feet high. Now, if he shall make a walk around his garden within the fence, 75 feet wide, how much will remain for cultivation?

Ans. 1A. 3R. 7p. 851381ft.

12

21. J. Ladd's garden is 100 feet long and 80 feet wide; he wishes to enclose it with a ditch 4 feet wide; how deep must it be dug that the soil taken from it may raise the surface one foot? Ans. 51 feet.

22. How many yards of paper, that is 30 inches wide, will it require to cover the walls of a room that is 15 feet long, 114 feet wide, and 72 feet high? Ans. 5517 yards. 23. Charles Carleton has agreed to plaster the above room, at 10 cents per square yard; what will be his bill?

Ans. $6.544.