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14,6X 3=43,8-half sum of the sides; then 12,64, the perpendicular, nul.

tiplied by 43,8 equal 553,632 feet, the answer.

Of a dodecagon. By trigonometry, thus: 360=12=30 degrees, one of the angles, the half thereof i5 degrees, another angle, whose complement to 90 degrees is 75 degrees, another angle of the dodecagon. Then say, as 15 degrees is to ,5 balf the side, so is 75 deg. to the perpendicular, 1,866025. Then 1,866025, multiplied by ,5 half the side, gives the area, 9,330125. Ans.

Of a circle and its parts. 35,5, half the circumference, multiplied by 11,3, half the diameter, gives

401,15, the area, or answer.

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As 7 is to 22, so is the diameter to the circumference. Or, as 113 is to 355, 30 is the diameter to the circumference. Or, as 1 is to 3,141593, so is the diameter to the circumference. Thus,

As 7 : 22 :: 22,6 71,028 feet, the circumference.
As 13 355 :: 22,6 71,000 feet, circumference.

As 1 : 3,141593 :: 22,6 · 71,0000018 feet, the circumference. As 1 is to ,318309, so is the circumference to the diameter. Or, as 355 is to 113, so is the circumference to the diameter. Or, as 22 is to 7, so is the circumference to the diameter. Thus,

As 1 : ,318309 :: 71 : 22,599939 feet, the diameter.
As 355 113 71 : 22,6 feet, the diameter.
As 22 : 7 :: 71 : 22,59 + feet, the diameter.

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As 1, (the square of the diameter,) is to ,7854, so is 510,76, (the square of 22,6, the diameter of the given circle,) to 401,15 ft. the area, by Van Culen. Or, as 452 : 355 :: 510,76 to 401,15 feet, the area, as before, by Metius. Or, as 14 : 11 :: 510,76 to 401,31 it. the area, by Archimedes' proportion.

355 X4=1420; then say, as 1420 : 113 :: 5041, (the square of the circum

ference,) to 401,15 feet, the area.

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As 1 : ,8862 :: 22,6 20,02812 feet, the answer.

,8862 is equal to the side of a square, when the diameter is 1.

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As 1 : ,2821

:: 71 20,0291 feet, answer. If the circumference of any circle be 1, the side of the square equal will be ,2821.

As 1 : ,7071 :: 22,6 : 15,98046 feet, answer. If the diameter of a circle be 1, the side of the square inscribed will be ,7071.

As 1 : ,2251 :: 71 : 15,9821 feet, answer.
If the circumference be 1, the side of the square inscribed will be ,2251.

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As 1 : 1,2732 :: 401,15 510,744180, equal to the square of the diameter, whose square root is 22,599 feet, the answer. If the area of a circle be 1, the square of the diameter thereof will be 1,2732.

As 1 : 12,56637 : : 401,15 : 50$0,9993255, whose square root is 70,999 + feet, answer. If the area of a circle be 1, the square of the circumference will be 12,56637.

As 1 : 401,15 : : ,6366 255,372090, whose square root is 15,98 feet, answer.

If the area of a circle be 1, the area of a square inscribed within that circle will be ,6366.

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As 1 : 1,4142 :: 15,98 : 22,598916, the diameter, answer.

If the side of a square be 1, the diameter of a circle that will circumscribe that square, will be 1,4142.

As 1 : 1,128 :: 20,0291 : 22,5928248 feet, the answer. If the side of a square be 1, the diameter of a circle equal thereunto will be 1,128.

As 1 : 4,449 :: 15,98 : 70,99914 feet, the answer. If the side of a square be 1, the circumference of a circle that will encom pass that square, will be 4,443.

As 1 : 3,545 :: 20,0291 : 71,0031595 feet, answer. If the side of a square be 1, the circumference of a circle that will be equal thereunto, shall be 3,545.

Of a semicircle. 17,75, half the circumference, multiplied by 11,3, half the diameter, giver 200,575 feet, the area. Ans.

11,3 the semidiameter.
8,875 half the circumference.

100,2875 feet, the area. Ans.

24,5 semidiameter.
22,8 half the arch line.

358,6 feet, the area. Ans.

U

17,5 half the chord line.
17,5

306,25 -9,6=31,9 +9,6=41,5, the diameter of the wirele. 41,5--2=20,75, the semidiameter, less 9,6 equal 11,15, the perpendicular,

which multiplied by 17,5, half the chord lipe, equal 195,125, the area of the triangle. Then + 306,25 the square of 9,6=398,41, sum. 398,415 19,96X82159,68—35=124,68 3 =41,562520,78 X 20,75, semidiameter, equal 431,1850, area of the sector, from which take 195,125, area of the triangle, leaving 236,060, area of the segment. Ang

$1,6 the transverse diameter. 44,4 the conjugate diameter.

2735,04 the rectangle, multiplied by ,785432148,100416 feet, the area.

53,75 X 39,25=2109,6875 X 2=4219,3750+-31406,4583 feet, arexo

OF SOLIDS,
17,5 x 17,5X 17,5 equal 5359,375 inches; or 3 ft. 175 ine. Ans.

21 X 21 equal 441 X 15 ft. the length, and the product is 6615--144 equal 45,9 +- feet, the solidity. Ans.

13,51 the perpendicular. 7,8 half the base.

105,878 the area of the base, multiplied by 19,5 feet, the length, gives 2054,8710;-144514,27 feet, the solidity. Ang.

18,5X18,5 equal 342,25, area of the base, which multiplied by 5, one-third of the altitude, equal 1711,25 inches, which divide by 144, and you will have 11,88 feet, the solidity. Ans.

21,5 X 21,5 equal 462,25, the rectangle, which multiplied by ,7854, equal 363,05115, area of the base, which multiplied by 16 feet, the length, equal 5808,8164 inches, which divide by 144, and the quotient is 40,34 feet. Ans.

26,5 X 26,5 equal 702,25, which multiplied by ,7854, equal 551,54, the area of the base, which multiplied by 5,5, a third part of 16,5, the product is 3033,47--144, and the quotient is 21,06 feet. Ans.

18 X 18 equal 324

12X12 equal 144 18X12 equal 216; then 216 +324 +-144 equal 684, which multiply by 6, a third part of 18, and the product is 4104, which divide by 144, and the Quotient is 28,5 feet. Aps.

18X18 equal 324

9X9 equal 81 18X9 equal 162; then 324 +81 +162 equal 567, which multiply by ,7854, and the product is 445,3218=144 equal 3,09 feet, the triple area of a mean base, which multiplied by 4,75, a third of 14,25, the product is 14,6775 feets the solidity. Any

38 add 15 equal 53 multiply 16 equal 848 reserve. 30 add 19 equal 49 multiply 12 equal 588

1436 sum, which multiplied by 2, a Chird of 6, the product is 2872, which divide by 144, and the quotient will be 19,94 feet. Aos.

44 add 13 equal 57, sum, which multiplied by 14 equal 798, reserved. 28 add 22 equal 48, sum, which multiplied by 26 equal 1248, add 798 equal 2046, which multiplied by ,7854 equal 1606,9284, which multiplied by 3 equal 4820,7852, which divide by 144, and the quotient will be 33,47 feet, the answer.

20 multiply 20 multiply 20 equal 8000, multiply Il equal 88000, which divided by 21 equal 4190,47 inches, or 2,424 feet. Ans.

33 multiply 33 equal 1089, multiply 55 equal 59895, multiply ,5236 equal 31361,0220 inches, the solidity. Ans.

36 multiply 36 equal 1296, multiply ,7854 equal 1017,8784, which multiplied by 33 equal 33589,9872, divided by 2 equal 16794,9936, divided by 1728, the quotient is 9,71 feet, the answer.

36 multiply 36 equal 1296, multiply , 41888 equal 542,88848, multiply 99 equal 53743,97952, divide by 1728, and the quotient is 31,10184 feet. Ans.

Of carpenters' work. 47,75 maltiply 53,5 equal 2554,625 feet decimally, which divide by 10, and the quotient is 25 squares, and 54 feet, the answer.

12,25 multiply 82,5 equal 1010,625 feet decimally, which divided by 10% equal 10 squares and 10 feet, the answer.

44,5 multiply 18,25 equal 812,125 the flat.

406,62 half flat.

10)12,18 flat and half flat Ans. 12 squares 18 feets

Bricklayers' work. 72,5 multiply 19,25 egnal 1395,625 multiply 11, (the number of halses its 5} bricks,) equal 15351,875, which divided by 3, equal 5117,291, which divided by 272,25, equal 18 rods 3 quar. 12 feet. Ans.

245,75 multiply 16,5 equal 4054,875, multiply 5, (the number of halves in 24 bricks,) equal 20270, rejecting decimals: then 20270 divided by 3, equal 6756, which divided by 272, equal 24 rods 3 quar. 24 feet. Ans. To find how many bricks are necessary to build a house of

given dimensions: 8 multiply 4 multiply 2,5 equal 80 solid inches in a brick : then 1728 divided by 80, equal 21,6 bricks in a solid foot. 44 add 44 add 40 add 40 equal 168 less 4 equal 164, which multiplied by 20, (the height,) equal 3289), which multiplied by 21,6, (the bricks in a solid foot,) equal 70848 bricks 18,75 multiply 12,5 equal 284,375, the first story. 14,5 multiply 9 equal 130,5, the second story. 10,25 multiply 7 equal 71,75, the third story. 13,75 multiply 6,5 equal 89,375, the shaft. 1,25 multiply 12 equal 15, the fetter. These added all together, the sum is 541 feet, which doubled is 1082, and this divided by 272, equal 3rods 3 quar. 62 feet. Ans.

Plasterers' work. 59,75 multiply 24,5 equal 1463,875 feet, which divided by 9, the quotient is 182,65 yards, the answer.

141,5 multiply 11,25 equal 1591,875, which divided by 9, equal 176,87 yards, the answer.

Joiners' work. 126,25 multiply 15,75 equal 1988,4375, which divided' by 9, equal 220,8 yards, the answer.

Masons' work. 97,417 multiply 18,25 multiply 2,25 equal 4000 feet 2 inches. And

107,75 multiply 20,5 equal 2208,875 superficial feet, the answer.

Of board measure. 16 multiply 13 equal 208, which divided by 12, equal 17 ft. À inc. answer. Or, 156, (the inches in 18 feet,) multiplied by 16, equal 2496, which divide by 144, and the quotient is 17 feet 4 inches, the answer.

As 12 i 13 16 17 ft. 4 inc. Ans.

Of squared timber. 15 multiply 15 multiply 18 equal 4050, which divide by 144, and the quów tient is 28 feet 14 inches, the answer.

Or say,

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33 multiply 19 maltiply 16,75 equal 10502,25, which divided by 144, equal 72 feet 11 inches, answer, the true way.

33 add 19 equal 52, som, and half thereof 26, which multiplied by 26, equal 676: then 676 multiplied by 16,75, the length, is 11323, which divide by 144, and the quotient is 78,63 feet, answer, the false way.

The true content is 72,93 feet, and the false content 78,63 feet, too much by 5,70 feet.

18 maltiply 18'egual 324; then 1728 divided by 324, equal 54 inc: Ans. 22 multiply 15 equal 330; then 1788 divided by 330, equal 5,23 inc. Ans.

Of unequal squared timber. 25 add 9 equal 34, sum, and half thereof is 17, the side of the square in the middle: then 17 multiply 17 multiply 20 equal 5780, which divide by 144, and the quotient will be 40,13 feet, answer, the false way.

25 multiply 9 equal 225, and 25 less 9 equal 16, difference of the sides : then 16 multiply 16 equal 256, which divided by 3, equal 85,333: then 85,333 add 225 equal 310,333, which multiplied by 20, equal 6206,660, which divide by 144, and the quotient is 43,101 feet, the true way: so the difference is 3

feet nearly

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