32 multiply 20 equal 640, and 10 multiply 6 equal 60, and 640 multiply 60 equal 38400, the square root whereof is 195,959, the mean proportional: then add 640, the greater base, and 60, the lesser base, to the mean proportional, and the sum will be 895,959, which multiply by 6, a third of the length, and the product will be 5375,754, which divide by 144, and the quo tient is 37,33 feet, answer, the true way.

32 add 10 equal 42, sum, and 20 add 6 equal 26, sum: then half of 42 is 21, and half of 26 is 13: then 21 multiply 13 equal 273, area in the middle: then 273 multiply 18, the length, and the product is 4914, which divide by 144, and the quotient will be 34,12 feet, answer, the false way.

Of round timber, whose bases are equal.

96 divided by 4 equal 24, and 24 multiply 24 equal 576, area of the base: then 576 multiply 18, the length, and the product is 10368, which divide by 144, and the quotient will be 72 feet, answer, the false way.

96 multiply 96 equal 9216, multiply ,07958 equal 733,40928, the area: then 733,40928 multiply 18 equal 13201,36704, which divide by 144, and the quotient will be 91,67 feet, answer, the true way.

86 divided by 4 equal 21,5, and 21,5 multiply 21,5 equal 462,25, area: then 462,25 multiply 20 equal 9245,00, which divide by 144, and the quotient will be 64,2 feet, answer, the false way.

86 multiply 86 equal 7396, multiply,07958 equal 588,57368, area: then 588,57368 multiply 20 equal 11771,47360, which divide by 144, and the quo tient will be 81,74 feet, answer, the true way.

Of round timber, whose bases are unequal.

36 multiply 9 equal 324, the rect. then 36 less 9 equal 27, diff. then 27 mul. tiply 27 equal 729, which divided by 3 equal 243, one-third: then 324 add 243 equal 567, multiply ,7854 equal 445,3218, a mean area: then 445,3218 multiply 24 equal 10687,7232, which divide by 144, and the quotient will be 74,22 feet, the answer.

136 multiply 32 equal 4352, rect. then 136 less 32 equal 104, diff. then 104 multiply 104 equal 10816, which divided by 3 equal 3605,333, one-third: then 4352 add 3605,333 equal 7957,333, a mean cir. squared: then 7957,333 multiply,07958 equal 633,24456014, the mean area: then 633,24456014 multiply 21 equal 13298,13576294, which divide by 144, and the quotient will be 92,34 feet, the answer.

OF THE FIVE REGULAR BODIES.

1st. Of a tetraedron.

10,392 is the perpendicular composing the body; then 10,392 multiply 6, fhalf the side.) equal 62,352, the area of the triangle. One-third of 62,352 is 20,784; then 20,784 multiply 9,798, (the perpendicular height,) equal 203,641632 inches, the answer.

2nd. Of an octaedron.

5,6568, a third part of its length: 12 multiply 12 equal 144, the area of the equare base; then 5,6568 multiply 144 equal 814,5792, the solidity. Ans.

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Sd. Of a dodecaedron.

13,36219 is the altitude of one of the pentangled pyramids, and 8,258292, the perpendicular of the pentagon: then 8,258292 multiply 50, (half the sum of the sides,) equal 247,748760, the area of the pentagon: then 247,748760 multiply 13,362192 equal 1103,48783, content of one pyramid, which multiply by 12, and the product will be 13241,85392, the solidity. Ans.

4th. Of an icosaedron.

10,39224, the perpendicular of the triangle, multiplied by 6, half the side, is 62,35344, which multiplied by 20 equal 1247,06880: then 62,35344 multiply 3,0230456, (a third part of the altitude of the pyramid,) = 188,497292, multiply 20, (the number of triangles,) and the product will be 3769,945840, the solidity. Ans.

By the tabular numbers..

7,663119, tabular number, multiplied by 1728, the cube of the side, the product will be 13241,869632, the solid content, nearly as before. Ans.

20,645729, tabular number, multiplied by 144, the square of the side, the product will be 2972,984976, the superficial content. Ans.

,4714045, tabular number, multiplied by 8000, the cube of the side, the product will be 3771,2360000, the solid content. Ans.

3,464102, tabular number, multiplied by 400, the square of the side, the 4 product will be 1385,640800, the superficial content. Ans.

To measure an irregular body.

32 multiply 32 equal 1024, multiply ,7854 equal 804,2496, the area of the base: then 804,2496 multiply 14, (the fall of the water,) equal 11259,49, which divide by 1728, (the solid inches in a foot,) and the quotient will be 6,51 feet, the solidity. Ans.

PRACTICAL QUESTIONS.
Question 1.

47,75 multiply 18,5 equal 883,375, which divide by 9, and the quotient will be 98 yards 1 foot, the answer.

Question 2.

21,5 multiply 17,5 equal 376,25; then 1,5 multiply 1,5 equal 2,25, the area of one stone. 376,25 divided by 2,25, the quotient is 167 stones. Ans.

Question 3.

109,75 multiply 92,5 equal 1015,1875, the content of the room: then 5,75 multiply 6,5 equal 87,375, multiply 2 equal 74,750, the content of the windows; and 1015,1875 less 74,750 equal 940,4375: then 3,75 multiply 3 equal 1,25, (the square feet in a yard of tapestry,) which divided thus, 940,4375→→

Question 4.

5,1416 multiply 27,5 equal 86,394, the circumference; then 86,394 multiply 27,5, (the diameter,) equal 2375,8350, which divide by 144, and the quotient will be 16,49 feet, the superficial content. Ans.

Then 2375,8350, (inches in the superficial content,) divided by 6, the quotient is 395,9725, which multiplied by 27,5 equal 10889,24375, which divide by 1728, and the quotient will be 6,3 feet, the solid content. Ans.

Question 5.

24 multiply 24 equal 576, and 576 multiply,7854 equal 452,3904, and 400 multiply,7854 equal 314,1600, and 20 multiply 20 equal 400: then 314,1600 add 452,3904 equal 766,5504, the curve superficies, to which add 452,3904, the base, and the sum will be 1218,9408, the superficial content. Ans.

12 multiply 12 equal 144, multiply 3 equal 432, add 100, (the square of the altitude,) equal 532, which multiplied by 10, (the altitude,) equal 5320, multiply,5236 equal 2785,5520, the solidity in inches. Ans.

Question 6.

18,5 divided by 4 equal 4,625; then 4,625 multiply 4,625 multiply 24, and divide by 40, equal 12 tons 33 feet 4 inches. Ans.

3620 8 4324 and the quotient is 11 floors 56 feet, the answer.

Question 8.

48,75 multiply 35,5 equal 1780+, (the decimals of a foot not noticed:) then 1730 divided by 100, the quotient will be 17 squares 30 feet. Ans.

1764 multiply,7854 equal 1385,4456, multiply 94 equal 130231,8864÷3 and the quotient will be 43410,6288: then say, as 43410,6288: 830584 :: 3456 to 66124, the cube of the length; and the cube root of 66124 is 40,43 inches, the length upon the perpendicular. Ans.

If it had been 3 feet, the length would be 46,29 inches.

All the solid bodies are in triplicate reason of their homologous sides, by Euclid, 12, 12; 12, 18; and 11, 33.

Question 10.

21 less 3 equal 18, difference; then say, to find the length of the whole pyramid, as 18: 12: 21 to 14, the length of the whole pyramid 14 feet, or 168 inches: then the solid content of the whole pyramid is 24696 inches: then 1728 multiply 5 equal 8640 inc. the solid content of 5 feet: then 24696 less 8640 equal 16056 inches: then 168X168×168=4741632. Then say,

As 24696 4741632:: 16056 to 3082752, whose cube root is 145,54. Subtract this root from 168, the length, and there will remain 22,46 inches, which is the length of 5 solid feet, at the great end. Ans.

Question 11.

All circles are in duplicate reason of their diameters, by Euclid 12, 2.

20 multiply 20 equal 400; then say, as 20: 400 :: 9 to 180, equal to the square of the semidiameter of the circle belonging to the first man: then as 20: 400: 6 to 120, equal to the square of the semidiameter of the circle belonging to the second man. Again say, as 20: 400 :: 5 to 100, equal to the square of the semidiameter of the circle belonging to the third man: then 400 less 180 equal 220, whose square root is 14,83 inches, which subtract from 20 inches, (the semidiameter of the whole stone,) and there remains 5,17 inches, the breadth of the ring, or part of the stone, which must be ground down by the first man. Then 220 less 120 equal 100, whose square root is 10; then 14,83 less 10 equal 4,83 inches, the breadth or part of the stone to be ground down by the second man. Then 5,17 inches by the first man, and 4,83 inches by the second man, make 10 inches, which taken from 20 inches, (the semidiameter of the whole stone,) there remains 10 inches, the breadth or ring belonging to the third man.

Question 12.

12X12×2×20÷144-40 feet, the solid content. Then say, As 14: 40 :: 12 : 384 feet of boards, the answer Question 13.

11X11X2X20÷÷÷144-33,6+ the solidity, when hewn square. Ang Question 14.

All such questions as this admit of no work, but cut it at one-third of its altitude.

Question 15.

First suppose 3 numbers, according to the given proportion, and suppose the depth 7 feet; then the width will be 10 feet, and the length 42 feet: then 7X10X42=2940, and 567,5X1728-635040 inc. then 10X10X10, (the width,)=1000. Then say, as 2940: 1000 :: 635040 to 216000, the cube of the true width, whose cube root is 60, and seven-tenths thereof 42, the true depth; and 6 times 42 is 252 inches, the true length. Then 60 multiply 42 multiply 252 equal 635040÷2150,42, (the solid inches in a bushel,) and the quotient will be 295+ bushels, the answer.

Question 16.

18,5 multiply 18,5 equal 341,25, multiply,7854 equal 268,803150, multiply 8 equal 2150,425200, the solid inches in a bushel: then 2150,4252-7,5— 286,72336,7854-365,0666, whose square root is 19,107 inc. the diameter, when the depth is 7,5 inches, the answer.

Question 17.

660 feet, the length of an acre. 66 feet, the breadth of an acre.

43560 the square in an acre; then say, as 1 : 1,2732 :: 43560 to 55460,5920, whose square root is 235,5, (diameter of an acre, when in a circle,) ÷ 2 = 117,75, half the diameter: therefore the cord must be 117 feet 9 inches long, the answer.

Question 18.

16,5 multiply 16,5 equal 272,25, the area of the old crib: then 272,25 multiply 3 equal 816,75, whose square root is 28,57, or 28 feet 7 inches nearly,

the answer.

Question 19.

20 multiply 20 multiply 20 equal 8000; then say, (by Euclid, 11, 33,) as 217: 8000: 1000 to 36870, whose cube root is 33,28 inches, the length

7

Then say, as 20 : 33,28 :: 15 : 24,96 breadth.

4 multiply 4 multiply & equal 64, and 9 multiply 9 multiply 9 equal 729: then say, (by Euclid, 12, 18,) as 64: 9:: 729 to 102,515 lbs. or 102 lbs. 8 oz. 4 drs. the answer.

Question 21.

5 multiply 5 multiply 5 equal 125, and 30 multiply 30 multiply 30 equal 27000: then say, (by Euclid, 12, 12,) as 125 : 12,25::. 27.000 to 2646 lbs. the

answer.

Question 22.

6 multiply 6 multiply 6 equal 216; then say, as 11: 216 ::.500 to 981,1818, whose cube root is 21,4 inches, the diameter. Ans.

Question 23.

4,8284 the tabular number. 12 multiply 12, divide by 3, equal 48. 237 the square of a mean side. 21 multiply 9, add 48, equal 237:

1144,3308X15÷144119,2 solid feet, the answer..

Question 24.

First find the length of the whole cone, thus: 36 less 20 equal 16; then say, as 16 215: 36 to 483,75 inches, the whole length of the cone; and then find the content of the whole cone, thus: 36 multiply 36 multiply,7854 equal 1017,8784, area of the base: then 1017,8784 multiply 52,161 equal 164132,88-1728-94,98 feet, the length of the whole cone.

Then find the content of the top-part, that is wanting: ,7854, the area of unity: 20 multiply 20 equal 400, multiply,7854 equal 314,1600, (area of the lesser base,) 3 104,72, a third part: then 104,72×268,75, (altitude of the top-part,)÷1728-16,28 feet, content of the top-part.

Then from 94,98 feet, the content of the whole cone, take 16,28 feet, the content of the top-piece, and there remains 78,70 ft. content of the frustum.

Question 25.

200 multiply 200 multiply 200 equal 8000000; then 15961026171 185781, the sum: then say, as 26171: 8000000 :: 185781 to 56789881, whose cube root is 384,3 inches, the length of the whole cone.

Then, to find the diameter of the lesser base, 26171200X3=392,565, area of the lesser base: then by problem 10, (the proportion of a circle,) in mensuration, say, as 1: 1,2732 :: 392,565 to 499,8157580, whose square root is 22,35: then say, as 200: 22,35 :: 384,3 to 42,94, greater diameter, and the lesser diameter 22,35.

Question 26.

5 multiply 5 equal 25, and 5 multiply 2 equal 10, and 2 multiply 2 equal 4: then 25 add 10 add 4 equal 39, the sum. 1234, and 2045 feet, the triple of a mean area: then say, as 1 : 1,27324:: 5 to 6,3662, the triple square of a mean diameter. Then say, as 39: 6,3662:: 25 to 4,080897, the square of the greater diameter, whose square root is 2,020123 feet, which is 24,24147 inches, the greater diameter. Then, to find the lesser, say,

As 5 24,24147:: 2 to 9,69659 inches, the lesser diameter.

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