393. A PRISM is a solid that has two similar, equal, parallel faces, called bases, and all its other faces parallelograms.

NOTE. A prism is triangular, quadrangular, pentagonal, etc., according as its bases are triangles, quadrangles, pentagons, etc.

A CYLINDER is a round body whose diameter is the same throughout its entire length, and whose ends or bases are equal, parallel circles.

394. PROBLEM 6. To find the surface of a prism or cylinder:

RULE. Multiply the perimeter or circumference of the base by the length of the solid, and to the product add the area of the two ends.

Ex. 1. What is the surface of a prism, whose length is 10 inches and base 4 inches square? Ans. 192sq. in.

2. What is the surface of a cylinder, whose length is 20 feet and diameter 4 feet?

395. PROBLEM 7. To find the solid contents of a prism or cylinder:

RULE. Multiply the area of the base by the altitude.

Ex. 1. What are the contents of a cylinder, whose length is 20 inches and whose diameter is 10 inches?

Ans. 1570.796c. in. 2. What are the contents of a quadrangular prism, whose length is 25 feet and whose base is 3 feet square?

396. A PYRAMID is a solid, having a polygonal face, called the base, and all its other faces are triangles which meet at a common point, called the vertex of the pyramid. The slant hight is the distance from the vertex to the middle of one side of the base.

393. What is a Prism? A Cylinder? 394. Rule for finding the surface of a prism or cylinder? 395. Rule for finding the contents of a prism or cylinder? 396. What is a Pyramid? Its vertex? Slant hight?

NOTE. A pyramid is triangular, quadrangular, etc., according as its base is a triangle, quadrangle, etc.

A CONE is a solid, like a pyramid, except that its base is a circle. The altitude of the pyramid or cone is its perpendicular hight.

397.

PROBLEM 8. To find the contents of a pyramid or of a cone:

RULE. Multiply the area of the base by one third of the altitude.

Ex. 1. What are the contents of a cone, whose base is 10 feet in diameter and whose altitude is 24 feet?

Ans. 628.3184 cu. ft. 2. What are the contents of a pyramid, whose altitude is 12 inches and whose base is a triangle, having its base 6 inches and its altitude 8 inches?

399. PROBLEM 9.

398. The FRUSTUM of a pyramid or cone is the part remaining after a portion next the vertex has been cut off by a plane parallel to the base. The two ends are called the upper and lower bases.

To find the contents of the frustum of a pyramid or cone:

RULE. Multiply the sum of the two bases, added to the mean proportional between the two bases, by one third of the altitude of the frustum.

Ex. 1. What are the contents of the frustum of a quadrangu

396. What is a Cone? Altitude of a pyramid or cone? 397. Rule for finding the solid contents? 398. What is the Frustum of a pyramid or cone? 899. Contents of a frustum, how found?

lar pyramid, whose altitude is 21 feet and whose bases are 5 feet and 3 feet square? Ans. 343cu. ft. 2. What are the contents of the frustum of a cone, whose hight is 12 feet and whose bases are 6 feet and 4 feet in diameter?

400. A SPHERE or GLOBE is a solid bounded by a curved surface, all parts of the surface being equally distant from a point within, called the center.

A diameter of the sphere is a line passing through the center, and limited in both directions by the surface.

401. PROBLEM 10. To find the surface of a sphere:

RULE. Multiply the circumference by the diameter.

Ex. 1. What is the surface of a sphere, whose diameter is 100 inches? Ans. 31415.92sq. in. 2. What is the surface of the earth, supposing it to be a sphere 8000 miles in diameter?

3. What is the surface of the sun, supposing it a sphere whose diameter is 885680 miles?

402. PROBLEM 11. To find the contents of a sphere: RULE 1. Multiply the surface of the sphere by one third of the radius.

RULE 2. Multiply the cube of the diameter by the decimal .523599; i. e. by of 3.141592.

Ex. 1. What are the contents of a sphere, whose diameter is 100 inches? Ans. 5235983c. in. 2. What is the volume or solidity of the earth, supposing it a sphere whose diameter is 8000 miles?

3. What is the volume or solidity of the sun, supposing it a sphere whose diameter is 885680 miles?

400. What is a sphere? Its diameter? 401. Rule for finding the surface of a sphere? 402. Rule for finding the volume or solid contents of a sphere? Second rule?

MISCELLANEOUS EXAMPLES.

1. What number increased by of itself gives 20?
2. What number diminished by 43 gives 21?

3. The sum of two numbers is 54 and one of the numbers is 8 times the other; what are the numbers?

4. Three roods and ten rods are what part of an acre?

5. The difference between two numbers is 37 and the smaller number is 12; what is the larger?

6. What number multiplied by 33 gives 1000 ?

7. What number divided by 37 gives 64?

8. What is the greatest common divisor of 84 and 144? 9. What is the least common multiple of 72 and 364? 10. What is the interest of $756.64 for 8m. 17d.?

11. The difference between two numbers is 25, and the smaller number is 10; what is the larger? What the sum of the two numbers?

12. The difference of two numbers is 563492, and the larger number is 3642538; what is the smaller? What the sum of the two numbers? 1st Ans. 3079046.

13. How many bricks 8 inches long, 4 inches wide, and 2 inches thick, will be required to build a wall 20 feet long, 16 feet high, and 2 feet thick?

14. How many bricks whose dimensions are 8', 4', and 2', will it take to build the walls of a house 40ft. long, 28ft. wide, and 22ft. high, the walls to be 1ft. 6′ thick, and no allowance made for doors and windows?

15. The salary of the President of the United States is $25000 per annum; what sum may he expend daily, and yet save $41560 in one term of office, viz. 4 years? Ans. $40. of itself, will produce 121?

of itself, will produce 27?

16. What number, multiplied by 17. What number, multiplied by 18. How many square feet of boards will it take to lay a floor 20ft. long and 16ft. wide?

19. How large a square floor can be laid with 676 square feet of boards?

20. The fore wheel of a carriage is 9 feet, and the hind wheel 10 feet in circumference; how many times will each turn round in running from Boston to Andover, 20 miles?

21. A rectangular piece of land, containing 60 acres, has its length to its breadth as 3 to 2, what are its length and breadth?

22. Bought a cask of molasses, containing 84 gallons, for $28; but 9 gallons having leaked out, at what price per gallon must I sell the remainder to gain $4.25? Ans. 43 cents.

23. If a pipe 6 inches in diameter will discharge a certain quantity of water in 4 hours, in what time will a 4-inch pipe discharge the same quantity?

Ans. 9 hours.

24. In 12gal. 3qt. 1pt. 2gi., how many gills?
25. In 1846542 seconds how many weeks, days, etc.?
26. Resolve 25740 into its prime factors.

Ans. 2, 2, 3, 3, 5, 11, 13.

27. Reduce, 12, 1, and to equivalent fractions having the least common denominator.

28. Reduce 3s. 4d. 2qr. to the fraction of a pound.

29. Reduce of a pound to shillings and pence. 30. Add lb. oz. dwt. gr. together.

31. From lb take 13.

32. A colonel, arranging his men in a square battalion, found that he had 31 men remaining; but, increasing the rank and file by 1 soldier, he wanted 20 men to make up the square. Of how many men did his regiment consist? Ans. 656.

33. How shall I mark gloves that cost me 80c. per pair so that I may discount 33 per cent. from the marked price and yet gain 25 per cent. on the cost? Ans. $1.50.

34. Suppose that in a shower the water falls to the depth of 2 inches, how many gallons will fall upon a township that is 6 miles square, each gallon containing 231 cubic inches?

35. How many bricks 8' long, 4′ wide, and 2′ thick, will be required to build a house 32ft. long, 243ft. wide, and 20ft. high, the walls being 1ft. 4' thick, the house having 2 doors, each 4ft. wide and 8ft. high, and 21 windows, each 3ft. wide and 6ft. high, no allowance being made for the space occupied by the mortar? 36. What is the square root of the square root of 16 times 81?