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By having the diameter to find the circumference. RULE-Multiply the diameter by 3.1416, and you get the circumference.

If the diameter of a circle be 24, what is the circumference?

Ans. 75.3984.


By having the circumference of a cube to find the diameter.

RULE.—Multiply the circumference by .31831 and the product is the diameter.

PROBLEM 6. I have a circular field 50 rods in diameter, what is the side of a square field, that shall contain the same area?

Ans. 44.31.

Are figures having length breadth and thickness.

To find the content of a cube or paral- c

D lelopipedon, whose side is 18 inches.

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


B 1. How many cubic feet in a cube whose side is 18 inches?

Ans. 33 feet. 2. What is the content of a parallelopipedon whose length is 6 feet, height 2 feet, breadth 13 feet? Ans 18ft.

3. A cellar is 50 feet long, 38 feet wide and 12 feet deep, how many cubic yards of earth has been taken out in digging, and what was the expense of digging it at 10 «cts. per cubic yard?

Ans. 844.44 + cubic yds.; expense $84.45 nearly.

PROBLEM 8. To find the solidity of a Prism. A B RULE.—Multiply the area of a base or end by the height.

4. Required the solidity of a triangular prism whose length is 10 feet, and the three sides of its triangular base are 5, 4 and 3 ft. Ans. 60ft. C

D PROBLEM 9. To find the solidity of a Cylinder. RULE.-Multiply the area of the base by the length.

5. The diameter of the base of a cylinder is 10 inches and its length 24 feet, required the solidity?

Ans. 13.09 feet.

PROBLEM 10. To find the solidity of a cone or pyramid. RULE.-Multiply the area of the base by 1 of its height.

6. What is the solid content of a cone whose heighth is 12 feet, and the diameter of the base 24 feet?

Ans. 20.45 + feet. PROBLEM 11.


To find the superfices of a Cone. RULE.—Multiply the circumference of the base by half its slant height.

7. What is the convex surface of a cone, whose slant height is is 20 feet, and the circumference of its base 9 feet?

Ans. 90 feet. A


PROBLEM 12. 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 1 of the square of the difference of the diameters; then multiply this sum by .7854 and

the product will be the mean area between the two bases;

Frustum of a cone. lastly, multiply the mean area by the length of the frustum, and the product will be the solid content.

8. What is the content of a stick of timber whose length is 40 feet, the diameter of the larger end 24 inches, and the smaller end 12 inches? Ans. 733 ft. nearly.

PROBLEM 13. To find the solidity of a sphere or globe. Rule.—Multiply the cube of the diameter by .5236.

9. What is the solidity of a sphere or globe, whose axis or diameter is 12 inches? Ans. 904.78 + inches.



To find the convex surface of a sphere or

globe. Rule.—Multiply the diameter by the circumference.

10. Required the superficial content of a globe whose diameter is 24 inches?

Ans. 1809.55 + inches. 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.

1. How large a cube may be inscribed in a sphere 40 inches in diameter?

Ans. 23.09 + inches.

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To find the solid content of Casks. RULE.Find

TABLE. the mean diameter of the cask, by taking the mean of the bilge and head diameter, then multiply the mean diameter in inches by the length of the cask in inches. And, again, multiply this product by the mean diameter; deduct one-fifth of the sum so found for the roundness of the cask, and reduce the remainder to feet and inches, by the rule for measurement, which is correct if the content be required in solid feet, or, divide by 232 for ale gallons.

Example 1.-If a cask be in length 3 feet 9 inches, its head diameter 2 feet 6 inches, and its bung diameter 2 feet 10 inches, what are its solid contents in ale gallons? Length 3 ft. 9 in. Multiply

45 length. 12


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90 Head diameter, Bung diameter. 135

2 ft. 6 in. 2 ft. 10 in.


32 Again to be
30 Bung diam. 34 inches



4320 1)64

46080 Mean diam. 32 in. Deduct } 9216

gals. qts. 282)36864(130 3

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2. Each side of the square base of a vessel is 40 inches, and its depth 10 inches. Required its contents in ale gallons?

Ans. 56.7 gallons.

8. The diameter of a cylindrical vessel is 32 inches, and its internal depth 45.5 inches. Required its content in ale gallons?

Ans. 129.78 gals. 4. How many bushels of grain will a box contain that is 15 feet long, 5 feet wide, and 7 feet high?

Ans. 421.8 bushels.


OF MATTER. Matter, is possessed of the following properties, viz: solidity, extension, divisibility, mobility, inertia, attraction and repulsion.

1. Solidity is that property by which two bodies cannot occupy the same place at the same time. It is sometimes called the impenetrability of matter.

2. EXTENSION, like the solidity of matter, is proved by the impossibility of two bodies co-existing in the same place.

3. DIVISIBILITY, is that property by which bodies are capable of being divided into parts removable from each other.

4. MOBILITY expresses the capacity of matter to be moved from one position or part of space to another.

5. INERTIA, designates the passiveness of matter, which, if at rest, will forever remain in that state until compelled by some cause to move; and, on the contrary, if in motion, that motion will not cease, or abate, or change its direction unless the body be resisted.

1. SPACE is either absolute or relative.

2. ABSOLUTE SPACE, is merely extension, immoveable, illimitable and without parts, yet it is usually spoken of as if it had parts. Hence the expression:

Relative Space, signifies that part of absolute space occupied by any body, as compared with any part occupied by another body..

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