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1. What is the area of an 2. What is the area of an oval whose longest diameter oval whose longest diameter is 5 feet, and shortest 4 feet? | is 21, and shortest 17 ?. 5x4x.7854–15.708ft. Ans. |

Ans. 280.3878.

313. To find the area 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 ' 3. What is the area of the the surface of a globe whose surface of a cannon shot, diameter is 14 inches and cir- whose diameter is one inch? . cumference 44?

Aos. 3.1416 inches. 44x14=616 Ans. 4. How many square inches 2. How many square miles in the surface of an 18 inch in the earth's surface, its cir- artificial globe ? cumference being 25000, and

Ans. 1017.8784. its diameter 7957} miles ?

· Ans. 198943750.

2. Mensuration of Solids.

314. Mensuration of Solids teaches to determine the spaces includes: 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 nuinber of cubes, whose sides are inches, or feet, or yards, &c. contained in the body.

315. To find the solidity of a cube. RULE.-Cube one of its sides, that is, multiply the side by itsell, and that product by the side again, and the last product will be the answer.

1. If the length of the side! 2. How many cubic inches of a cube be 22 feet, what is in a cube whose side is 24 inits solidity ?

ches ?

Ans. 13824. 22x22x22=10648 Ans.

316. To find the solidity of a parallelopipedon. 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 2. How many feet in a stick parallelopipedon whose length of hewn timber 30 feet long, is 6 feet, its breadth 23 feet, i 9 inches broad, and 6 inches and its depth 19 feet ?

thick? 6 x 2.5 X 1.75=26.25, or

Ans. 114 feet. 264 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



1. The diameter of a round | 2. The diameter at the log at its smallest end is 16 smallest end being 24 inches, inches'; what will be the side | how large square will the of the largest squared stick of stick of timber hew? timber that can be hewed from

ADS. 16.97 in. it? V8 X 8 X2–11.31 in. Ans.

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

1. What is the content of a 2. What number of cobic triaogular prism, the area of feet in a round stick of timber wbose end is 2.7 feet, and whose diameter is 18 inches, whose length is 12 feet ?

and length 20 feet? 2.7*12=32.4 ft. Ans.

Id Ans. 35.343.

319. To find the solidity of a pyramid or cone. Role. --Multiply the area of the base by the height, and one third of the product will be the content.

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1. What is the content 2. What is the content of a cone whose height is | of a triangular pyramid, its 124 feet and the diameter | height being 141 feet, and of the base 24 feet? | the sides of its base being 5,

27x212X=*=61, 6 and 7 feet ? and 648.7854 X 121-23

Ans. 71.0357 20.453125, Ans.


320. To find the solidity of a sphere.* RULE.—Multiply the cube of the diameter by .5236, or mul. tiply the square of the diameter by one 6th of the circumference.

1. What is the content of a í 2. What is the solid consphere whose diameter is 12 tent of the earth, its circuminches ? 12x12x12x2.5236 | ference being 25000 miles ? =194.7808, Ans.

Ans. 26385814912 miles.



V, 3. 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. - 1. What is the content of a l 2. What is the content of a cask whose length is 40 inch. I cask whose length is 20 inches, and its diameters 24 and 32 | es, and diameters 12 and 16? inches ?

S 11.2 a. gal. 32x32.+24x24x40=64000A 64000>.0014=89.6 A. 64000><.0017=108.8 A.

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Ans. 3 13.6 w. gal,


1. Of the fall of Heavy Bodies.

322. Heavy Bodies near the surface of the carth, fall one foot ibe first quarter of a second, three feet the second quarter, five feel the third quarter, and seven feet the fourth quarter, equal to 16 feet the first secund. 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 10 64 ; and the third (3X3=9) nine times 16, equal to 144 feet, and so

* The surface of a sphere is found by multiplying its diameter by its circumference.

on. Or if 16 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 | 2. From what height a body fall to acquire the must a body fall to acquire velocity of a cannon ball, | a velocity of 1200 feet per which is about 660 feet per i seconii ? second ?

Ans. 22500 feet. 660-8=82.5 and 82.5 X 682.5=806.25ft. -177 iniles, Ans.

324. The time given to find the space fallen through. RULE.-Multiply the time in seconds by 4, and tbe square of the product will be the space fallen througb in the given time.

1. How mapy feet will al 3. Ascending bodies are re. body fall in 5 seconds ? | tarded in the same ratio that

5x4=20, and 20x20=400 descending bodies are accelefeet, Abs. .

rated; therefore, if a ball, fir

ed upwards return to tbe eartb 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, its as3*4=12, and 12% 12=144

cent; therefore, 8c4=32, seet, Aps.

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 | 2. How long must a body be falling to acquire a veloci- be falling to acquire a velocity of 160 feet per second ? Its of 400 feet per second ?

160:8=20, and 20:45 Ans. 124 seconds.
5 seconds, Ans.


326. The space given to find the time. the body has been falling.

Rule.--Divide the square root of the space fallen through by 4, and the quotient will be the time.

1. In how many seconds ! 2. In how many seconds will a body fall 400 feet ? | will a bullet fall through a

7400=20, and 20:4= space of 11025 feet ? 5 seconds, Ans.

Ans. 264 seconds. 327. To find the velocity per second, with which a body will begin to descend at any distance frow the earth's surface.

RULE.--As the square of the earth's semi-diameter is to 16 feet, so is the square of any other distance from the earth's centre, inversely, to the velocity with which it begins to desceòd per second.

1. Admitting the semi- } 2. How high above the rliameter of the earth to be | earth's surface must a ball 4000 miles, with what velo- be raised to begin to descend city per second will a body | with a velocity of 4 feet begin to descend, if raised per second ? 4000 miles above the earth's

Ans. 4000 miles. surface ? As 4000 X 0400 : 16:: 8000 X 8000 : 4 feet,

Ans. 328. To find the velocity acquired by a falling body, per second, at the end of any given period of time.

RULE.--Multiply the perpendicular space, fallen through by 64, and the square root of the product is the velocity required:

1. What velocity per 1 2. If a ball fall 484 feet in second does a ball acquire 5 seconds, with what velociby falling 2225 feet , i ty will it strike? 225 X 64=14400, and

Ans. 176. p 14400=120, Ans.

329. The velocity wilh which a body strikes, given to find the space fallen through.

Rule.--Divide the square of the velocity by 61, and the quotient will be the space required.

1. If a ball strike the ground I 2. If a stream move with a with a velocity of 56 feet per velocity of 12.649 feet per secsecond, from what height did ond, what is its perpendicular it fall? 56 56-64=49 feet, Ans.

Ans. 24 feet

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