311. The area of a circle given to find the diameter and circumference.

RULE 1.-Divide the area by .7854, and the square root of the quotient will be the diameter.

2. Divide the area by .07958, and the square root of the quotient will be the circumference.

312. To find the area of an oval, or ellipsis. RULE.-Multiply the longest and shortest diameters together, and the product by .7854; the last product will be the area

313. To find the area, or surface, of a globe or sphere. RULE. Multiply the circumference by the diameter, and the product will be the area.

1. How many square feet in the surface of a globe whose diameter is 14 inches, and circumference 44?

44×14 616, Ans. 2. How many square miles in the earth's surface, its circumference being 25000, and its diameter 79573 miles?

Ans. 198943750.

3. What is the area of the surface of a cannon shot, whose diameter is 1 inch?

Ans. 3.1416 inches. 4. How many square inches in the surface of an 18 inch artificial globe?

Ans. 1017.8784.

2. Mensuration of Solids.

814. Mensuration of Solids teaches to determine the spaces included by contiguous surfaces, and the sum of the measures of these including surfaces is the whole surface of the body. The measure of a solid is called its solidity, capacity, content, or volume. The content is estimated by the number of cubes, whose sides are inches, or feet, or yards, &c. contained in the body.

315. To find the solidity of a cube. (254)

RULE. Cube one of its sides, that is, multiply the side by itself, and that product by the side again, and the last product will be the answer.

1. If the length of the side of a cube be 22 feet, what is its solidity?

22X22X22-10648, Ans.

2. How many cubic inches in a cube whose side is 24 inches?

Ans. 13824.

316. To find the solidity of a parallelopipedon. (69) RULE.--Multiply the length by the breadth, and that product

by the depth; the last product will be the answer.

1. What is the content of a parallelopipedon whose length is 6 feet, its breadth 23 feet, and its depth 13 feet? 6×2.5×1.75 26.25, or 261

feet.

2. How many feet in a stick of hewn timber 30 feet long, 9 inches broad, and 6 inches thick? Ans. 11 feet.

317. To find the side of the largest stick of timber that can be hewn from a round log.

RULE.-Extract the square root of twice the square of the semidiameter at the smallest end for the side of the stick when squared.

318. To find the solidity of a prism, or cylinder. RULE.-Multiply the area of the end by the length of the prism, for the content.

1. What is the content of a riangular prism, the area of hose end is 2.7 feet, and hose length is 12 feet?

2.7×12=32.4 ft. Ans.

2. What number of cubic feet in a round stick of timber whose diameter is 18 inches, and length 20 feet?

Ans. 35.343.

319. To find the solidity of a pyramid, or cone.

RULE.-Multiply the area of the base by the height, and one

third of the product will be the content.

1. What is the content of

a cone whose height is 12

2. What is the content of a triangular pyramid, its height

feet, and the diameter of the being 14 feet, and the sides basc 2 feet? of its base being 5, 6 and 7 Ans. 71.035+

2123=23=22=61, | feet?

and 6.7854× 12÷3

20.453125, Ans.

12*

138

OF THE FALL OF HEAVY BODIES.

320 322

320. To find the solidity of a sphere.*

RULE.-Multiply the cube of the diameter by .5236, or multiply the square of the diameter by one 6th of the circumference.

1. What is the content of a sphere whose diameter is 12 inches? 12X12X12X.5236 904.7808, Ans.

2. What is the solid content of the earth, its circumference being 25000 miles?

Ans. 26385814912 miles.

Guaging.

321. Guaging teaches to measure all kinds of vessels, as pipes, hogsheads, barrels, &c.

RULE. To the square of the bung diameter add the square of the head diameter; multiply the sum by the length, and the product by .0014 for ale gallons, or by .0017 for wine gallons.

PHILOSOPHICAL MATTERS.

1. Of the Fall of Heavy Bodies.

322. Heavy Bodies near the surface of the carth, fall one foot the first quarter of a second, three feet the second quarter, five feet the third quar ter, and seven feet the fourth quarter, equal to 16 feet the first second. The velocities acquired by falling bodies, are in proportion to the squares of the times in which they fall; that is, if 3 bullets be dropped at the same time, and the first be stopped at the end of the first second, the second at. the end of the second, and the third at the end of the third, the first will have fallen 16 feet, the second (2X2-4) four times 16, equal to 64; and the third (3x3=9) nine times 16, equal to 144 feet, and so on. Or, if 16

The surface of a sohero is found by multiplying its diameter by its circunfercuce

feet be multiplied by so many of the odd numbers, beginning at 1, as there are seconds in the given time, these several products will be the spaces passed through in each of the several seconds, and their sum will be the whole distance fallen.

323. The velocity given to find the space fallen through.

RULE.-Divide the velocity in feet by 8, and the square of the quotient will be the space fallen through to acquire that velocity.

1. From what height must a body fall to acquire the velocity of a cannon ball, which is about 660 feet per second? 660-8-82.5, and 82.5X 82.5-806.25ft 13 miles,

Ans.

2. From what height must a body fall to acquire a velocity of 1200 feet per second?

Ans 22500 feet.

324. The time given to find the space fallen through.

RULE.-Multiply the time in seconds by 4, and the square of the product will be the space fallen through in the given time.

1. How many feet will a body fall in five seconds? 5×4=20, and 20×20=400 feet, Ans.

|

3. Ascending bodies are retarded in the same ratio that descending bodies are accelerated; therefore, if a ball, fired upwards, return to the earth 2. A stone, dropped into a in 16 seconds, how high did it well, reached the bottom in 3 ascend? The ball being half seconds; what was its depth? | the time, or 8 seconds, in its 3×4-12, and 12×12=144 | ascent: therefore 8X4-32, feet, Ans. and 32x32 1024ft., Ans.

325. The velocity per second given to find the time.

RULE.-Divide the given velocity by 8, and one fourth part of the quotient will be the answer.

1. How long must a body be falling to acquire a velocity of 160 feet per second?

160 820, and 20-45 seconds, Ans.

2. How long must a body be falling to acquire a velocity of 400 feet per second?

Ans. 12 seconds.