12. To find the contents of a cube or parallelopipedon.

RULE. Multiply the length, height, and breadth, continually together, and the product is the contents.

13. To find the solidity of a prism.

RULE. Multiply the area of the base, or end, by the height.

14. To find the solidity of a cone or pyramid.

RULE. Multiply the area of the base by of its height. 15. To find the surface of a cone.

RULE. Multiply the circumference of the base by half its slant height.

16. To find the solidity of the frustum of a cone, or pyramid.

RULE. Multiply the diameters of the two bases together, and to the product add of the square of the difference of the diameters; then multiply this sum by .785398, and the product will be the mean area between the two bases; lastly, multiply the mean area by the length of the frustum, and the product will be the solid contents.

Or, find when it would terminate in a cone, and then find the contents of the part supposed to be added, and take it away from the whole.

17. To find the solidity of a sphere or globe.

RULE. Multiply the cube of the diameter by .5236. 18. To find the convex surface of a sphere or globe. RULE. Multiply its diameter by its circumference. 19. To find the contents of a spherical segment.

RULE. From three times the diameter of the sphere, take double the height of the segment; then multiply the remainder by the square of the height, and the product by the decimal .5236 for the contents; or to three times the square of the radius of the segment's base, add the square of its

N*

height; then multiply the sum by the height, and the product by .5236 for the contents.

20. 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 that product by 3, and extract the square root of the quotient for the answer.

21. To find the number of gallons, &c., in a square vessel.

RULE. Take the dimensions in inches; then multiply the length, breadth, and height together; divide the product by 282 for ale gallons, 231 for wine gallons, and 2150.42 for

bushels.

22. To find the contents of a cask.

RULE. Take the dimensions of the cask in inches ; viz. the diameter of the bung and head, and the length of the cask. Note the difference between the bung diameter and the head diameter. If the staves of the cask be much curved between the bung and the head, multiply the difference by .7; if not quite so much curved, by .65; if they bulge yet less, by 6; and, if they are almost straight, by .55; add the product to the head diameter; the sum will be a mean diameter by which the cask is reduced to a cylinder.

Square the mean diameter thus found, then multiply it by the length; divide the product by 359 for ale or beer gallons, and by 294 for wine gallons.

23. To find the contents of a round vessel, wider at one end than the other.

RULE. Multiply the greater diameter by the less; to this product, add of the square of their difference, then multiply by the height, and divide as in the last rule.

24. To measure round timber.

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

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

1. What are the contents of a board 25 feet long and 3 feet wide? Ans. 75 feet. 2. What is the difference between the contents of two floors; one is 37 feet long and 27 feet wide, and the other is 40 feet long and 20 feet wide? Ans. 199 feet. 3. The base of a rhombus is 15 feet, and its perpendicu lar height is 12 feet; what are its contents?

Ans. 180 feet. 4. What are the contents of a triangle, whose base is 24 feet, and whose perpendicular height is 18 feet?

Ans. 216 feet. 5. What are the contents of a triangular piece of land, whose sides are 50 rods, 60 rods, and 70 rods? Ans. 1469.69+ rods. 6. What is the circumference of a circle, whose diameter is 50 feet? Ans. 157.0796+ feet. 7. We have a round field 40 rods in diameter; what is the side of a square field, that will contain the same quantity? Ans. 35.44+ rods. 8. What is the side of an equilateral triangle, that may be inscribed in a circle 50 feet in diameter ?

Ans. 25.46

Ans. 35.35 feet. 9. If the diameter of a circle be 200 feet, what is the area? Ans. 31415.92+ feet. 10. What is the diameter of a circle, whose circumference is 80 miles ? miles. 11. I have a circular field 100 rods in circumference ; what must be the side of a square field, that shall contain the same area? 12. Required the side of a triangle, that may be inscribed in a circle, whose circumference is 1000 feet. Ans. 275.66+ feet. 13. How large a square field may be inscribed in a circle, whose circumference is 100 rods?

Ans. 28.2+ rods.

Ans. 22.5+ rods square. 14. How many cubic feet are there in a cube whose sides are 8 feet? Ans. 512 feet. 15. What is the difference between the number of cubic feet in a room 30 feet long, 20 feet wide, and 10 feet

high, and the number of square feet in the surface of the room? Ans. 6000 solid feet. 2200 square feet. 16. 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 feet. 17. 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. 18. 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. 19. Required the convex surface of a cone, whose side is 50 feet, and the circumference at its base 12 feet. Ans. 300 feet 20. Required the solid contents of Bunker Hill monument, whose height is 220 feet, and being 30 feet square at its base, and 15 feet square at its vertex.

Ans. 115500 cubic feet. 21. 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 ? 22. What is the solidity of a 20 inches?

Ans. 9.163 feet. sphere, whose diameter is Ans. 4188.8 inches.

23. What is the convex surface of a globe, whose diameter is 20 inches?

Ans. 1256.6 inches.

24. What are the contents of a spherical segment 3 feet in height, cut from a sphere 10 feet in diameter ?

25. What is the solidity of a height being 8 inches, and the inches?

Ans. 113.0976 feet.

segment of a sphere, its diameter of its base 20 Ans. 1224.7232 inches. inscribed in a sphere 10 Ans. 5.773 inches.

26. How large a cube may be inches in diameter ? 27. How many wine gallons will a cubical box contain, that is 8 feet long, 4 feet high, and 3 feet wide?

Ans. 718.1+ gallons. 28. How many bushels of grain will a box contain, that is 12 feet long, 5 feet wide, and 4 feet high?

Ans. 192.8+ bushels.

29. What are the contents of a cask, in wine gallons, whose bung diameter is 30 inches, head diameter 24 inches, and length 40 inches? Ans. 108.19+ gallons. 30. How many cubic feet in a stick of timber, which is 40 feet long, and whose girt is 60 inches?

Ans. 62 feet.

Section 51.

MISCELLANEOUS QUESTIONS.

1. What is the difference between 7 pence and 10 cents?

Ans. d. be added, the Ans. 78.

2. What number is that, to which, if sum will be 71 ? 3. What number is that, from which, if 34 be taken, the remainder will be 41 ? 4. What number is that, to which, if 33 be added, and the sum divided by 54, the quotient will be 5?

5. From of a mile take 7 of a furlong.

Ans. 7.

Ans. 234.

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 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. 1239 seconds. 9. If of a ton cost $49, what cost lcwt. ?

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.