in which may be inscribed an undecagon (elevensided figure,) one side of which is 13 inches long?

Place 282 opposite fig. 1: then opposite 13 inches, found on the circular, is 46.1 inches, the answer.

To find the greatest diameter of each of three equal circles that can be inscribed within a circle

of a given diameter.

RULE.-Place, 464 opposite fig. 1: then opposite any diameter on the fixed part, is the diameter of one of the three inscribed circles.

Example. What is the greatest diameter of each of three circles, that can be inscribed within a circle 25 inches in diameter ?

Place 464 opposite fig. 1: then opposite 25 on the fixed part, is 11.6 inches, the answer.

To find the greatest diameter of four equal circles that can be inscribed within another circle of a given diameter.

RULE. Place 416 opposite fig. 1: then opposite any given diameter on the fixed part, is the diameter of each of the four inscribed circles.

Example.-What is the greatest diameter of each of four equal circles that can be inscribed in another circle 22 inches in diameter ?

Place 416 opposite fig. 1: then opposite 22, on the fixed part, is 9.15 (9) inches, the answer.

"o find the Solidity of a Cylinder, or to measure

Round Timber.

RULE. First find the area of the

base by the rule for finding the area

of a circle, place that area opposite 144, then opposite the length in feet, is the answer in feet and decimals of a foot.

NOTE.-If the diameter be given in feet, place the area opposite 1, instead of placing it opposite 144.

Example.-What are the solid contents of a cylinder 5 inches in diameter, and 13 feet long?

Place 25 the square of 5) opposite 1: then opposite A is 1.965. Now place 1.965 opposite 144. then opposite 13 (the length) is 1.77 feet, the

answer.

How many solid feet in a round log 15 inches in diameter, and 14 feet long?

Place 225 (the square of 15) opposite 1: then opposite A is 1.77 the area. Now place 1.77 oppo

site 144: then opposite 14 is

17-2 feet, the answer.

In a log 12 feet long, 14 inches diameter ?
Answer, 12.8 feet.

In a log 16 feet long, 11 inches in diameter ?
Answer, 10.5 feet.

In a log 7 inches diameter, 15 feet long?
Answer 48 feet.

NOTE.-If the diameter and length are both given in inches, place the square of the diameter opposite 1728: then opposite the inches in length, is the answer in feet.

NOTE.-A cylinder that is 12 inches in diameter and 12 inches long, and a globe that is 12 inches in diameter, and a cone that is 12 inches high and 12 inches diameter at its base, bear a proportion to each other as 3, 2 and 1. Therefore if you place the contents of any cylinder on the circular opposite to 3 on the fixed part, then opposite 2 on the fixed part is the contents of an inscribed globe, and opposite fig. 1 is the contents of an inscribed cone.

To find how many Solid Feet a Round Stick of Timber will contain, when hewn Square.

RULE.-Place double the square of half the diameter opposite 144: then opposite the length is the

answer.

Example. In a log 28 feet long, 22 inches diameter, half the diameter is 11, the square of which is 121. This doubled, is 242. Now place 242 opposite 144: then opposite 28 (the length) is 47+ the

answer.

To find how many feet of Boards can be sawn from a Log of given Diameter.

RULE -Find the solid contents of the log wher

made square, then place 12 opposite the thickness of the board (including the saw-calf:) then opposite the solid contents is the answer in feet.

To find the Area of a Globe or Ball.

RULE. Place the diameter opposite 1: then opposite the circumference is the answer. Example.-How many square inches of leather will cover a ball 3 inches in diameter? Place 3 opposite 1: then opposite D. is 11, the circumference. Opposite 11 is the area, 381⁄2 inches.

How

many square feet on the surface of a globe 4 feet in diameter ?

Place 4 opposite 1: then opposite D. is 12.55 feet, the circumference. Opposite 12.55 is 50-4, the

answer.

To find the Solid Contents of a Globe or Ball.

RULE. First find its area by the preceding rules: then multiply its area by of its diameter.

Example.-What are the solid contents of a ball 14 inches in diameter?

Place 14 opposite 1: then opposite D. is 44 inches, the circumference. Opposite 44 is 617, the area.

of the diameter, is 2-33. Place this opposite 1: then opposite 617 (the area) is 1437 inches, the solid contents.

What are the solid contents of a ball 5 inches in diameter ?

Place 5 opposite 1: then opposite D. is 15.7 inches, the circumference. Also, opposite 15.7 inches is 78-4 inches, the area. of 5 is 835. Place this opposite 1 then opposite 78-4 inches (the area) is 654 inches, the solid contents.

There is a ball 20 inches in circumference: what are its solid contents?

Place 20 opposite letter D. Opposite 20 is 127, the area. of the diameter is 1.06. Place this opposite 1: then opposite 127 is 1350 inches, the solid contents.

To find the Area of an Ellipse.

RULE. Place the product of the transverse diameter multiplied by the conjugate diameter opposite 1: then opposite letter A is the

answer.

Example. What is the area of an ellipse whose transverse diameter is 12 inches, and conjugate diameter 10 inches?

10 × 12=120. Place 120 opposite 1: then opposite letter A is 94.25, the area.