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EXAMPLES. 1. What is the solid content of a round stick of timber, or a marble column, of equal bigness from end to end, whose diameter is 18 inches, and length 20 feet ?
18 inches=1,5ft. 1,5x1,5x,7854–1,76715 area of the base. 1,76715 x 20 length=35,343 solid feet. Ans. Or, 18x18x,7854254,4696 inches, area of the base. 254,4696 x 20
--=35,347 as before.
144 2. What is the solid content of a round stick of timber, of equal bigness from end to end, whose diameter is 35 inches, and length 35 feet? Ans. 233,842 feet. CASE 5.- To find how many solid feet a round stick of
timber, equally thick from end to end, will contain, when
Rule.-Multiply twice the square of its semidiameter, in inches, by the length in feet; then divide the product by 144, and the quotient will be the answer.
EXAMPLES. 1. If the diameter of a round stick of timber be 22 inches, and its length 20 feet, how many solid feet will it contain when hewn square ?
11x11x2x20 Half diameter=ll, and
144 solidity when hewn square, the answer. 2. If the diameter of a round stick of timber be 24 inches from end to end, and its length 20 feet, how many solid feet will it contain, when hewn square, and what will be the content of the slabs which reduce it to a square ?
40 feet solidity when hewn square, Ans.
and 22,832ft. the solidity of the slabs, Case 6.—To find how many feet of square edged boards,
of a given thickness, can be sawn from a log of a give en diameter.
RULE.—Find the solid content of the log, when made square, by the last Case ; then say, as the thickness of the board, including the saw calf, is to the solid feet, so are 12 inches to the number of feet of boards.
EXAMPLES. 1. How many feet of square edged boards, 14 inch thick, including the saw calf, can be sawn from a log 20 feet long, and 24 inches diameter ? 12x12x2x20
=40ft. solid content when hewn square. 144
As 11 : 40 : : 12 : 384 feet. Ans. 2. How many feet of square edged boards, 14 inch thick, including the saw gap, can be sawn from a log 12 feet long, and 18 inches diameter ? Ans. 108 feet.
Note.--A short rule for finding a number of feet of one inch boards that a log will make, is to deduct of its diameter in inches, and of its length in feet; then for each inch of diameter that remains, reckon 1 board of the same width as this reduced diameter, and of the same length as this reduced length of the log: thus a log 12 feet long, and 12 inches through, gives 9 boards, 9 feet long, 9 inches wide, or 60 feet-a log 16 feet long, and 16 inches through, gives 12 boards, 12 inches wide, 12 feet long, or 144 feet. CASE 2.-The length, breadth, and depth of any cubical
box being given, to find how many bushels it will contain.
RULE.—Multiply the length, breadth and depth together, in inches, and divide the last product by 2150,425, the solid inches in the statute bushel, and the quotient will be the answer.
EXAMPLE. There is a square or cubical box; the length of its bottom is 50 inches, breadth of ditto 40 inches, and its depth 60 inches; how many bushels of corn will it hold ? 50 x 40 x 60
-55,8+ or 55bush. 3 pecks. Ans. 2150,425
Case 8.-To find the solidity of a cone or pyramid,
whether round, square, or triangular. Definition.Solids which decrease gradually from the base till they come to a point, are generally called cones or pyramids, and are of various kinds, according to the
figure of their bases ; round, square, oblong, triangular, &c. ; the point at the top is called the vertex, and a line drawn from the vertex, perpendicular to the base, is called the height of the pyramid.
Rule. Find the area of the base, whether round square, oblong, or triangular, by some one of the foregoing rules, as the case may be; then multiply this area by one-third of the height, and the product will be the solid content of the pyramid.
EXAMPLES. 1. What is the solid content of a true-tapered round stick of timber, 24 feet perpendicular length, 15 inches diameter at one end, and a point at the other ? 15x15x,7854x8
-9,8175 solid feet. Ans.
144 2. What is the solid content of a square stick of timber of a true taper, 30 feet perpendicular length, 18 inches square at one end, and a point at the other? Ans. 224feet.
3. What is the solid content of a triangular tapering stick of timber, 21 feet long, 10 inches each side of the triangle, 8% inches the perpendicular of the triangle at the large end, and the other end a point ? Half perpendicular=4,33 and 4,33x10x7
=2,1ft. + Ans.
144 Note.--If a stick of timber be hewn three square, and be equal from end to end, you find the area of the base as in the last question, in inches, multiply that area by the whole length, and divide the product by 144, to obtain the solid content.
4. If a stick of timber be hewn three square, be 12 feet long, and each side of the base 10 inches, the perpendieular of the base being 83 inches, what is its solidity ?
Ans. 3,6+feet. CASE 9.-To find the solidity of the frustum of a cone
or pyramid. Definition.— The frustum of a cone is what remains after the top is cut off by a plane parallel to the base, and is in the form of a log greater at one end than the other, whether round, or hewn three or four square, &c.
Rule.--If it be the frustum of a square pyramid, multiply the side of the greater base by the side of the less; to this product add one third of the square of the difference of the sides, and the sum will be the mean area between the bases; then multiply this sum by the height, and it will give the content of the frustum. Or, if it be a tapering square stick of timber, take the girth of it in the middle ; square of the girth, (or multiply it by itself in inches ;) then say, as 144 inches to that product, so is the length, taken in feet, to the content in feet.
EXAMPLE. What is the content of a tapering square stick of timber, whose side of the largest end is 12 inches, of the least end, 8 inches, and whose length is thirty feet, calculating it by both rules ?
4x4 By the first Rule: 12x8=96. 12-8-4 =51
3 And 96 +54 x30
114 By the second Rule : 12+8
=10in. = of the girth in the middle.
2 Then 10x10=100=area in the middle of the stick.
And, as 144 : 100 : : 30ft. : 20, 83+feet. Ans. Rule.--If it be a triangle pyramid, or a tapering three square stick of timber, multiply the sum of the mean area, as found in the first rule, by ,433—and that product by the height or length. Or, multiply the area in the middle, as found in the second rule, by ,433—and then state the proportion as before.
EXAMPLE What is the content of a tapering three-square stick of timber, whose side of the largest end is 15 inches, of the least end, 6 inches, and whose length is 40 feet, calculating it by both rules ?
9x9 By the first Rule: 15x6=90. 15—6=9. =27.
3 And 90+27 X,433 x 40
E14,0725ft. Ans. 144
15+6 By the second Rule : -=10,5 in.=t of the girth
2 in the middle, if it were four-square. Then 10,5x10,5x,433=47,73825in.=area in middle. And, as 144: 47,73825 : : 40 feet : 13,260625ft. Ans.
RULE.-If it be a circular pyramid or cone, multiply the diameters of the two bases together,and to the product add one third of the square of the difference of the diameters; then multiply this sum by ,7851-and it will be the mean area between the two bases; multiply this area by the length of the frustum, and it will give the solid content.
Or multiply each diameter into itself; multiply one diameter by the other; multiply the sum of these products by the length; annex two ciphers to the product, and divide it by 382; the quotient will be the content, which divide by 144 for feet as in other cases.
What is the solid content of a tapering round stick of timber, whose greatest diameter is 13 inches, the least 64 inches, and whose length is 24 feet, calculating it by both rules ? 13x6,5=84,5 13–6,5=6,5 6,5x6,5
=12,904 +feet. Ans.
And 1858,115144=12,903+ft. Ans. * To find the content of timber in the tree, multiply the square of 1-5 of the circumference at the middle of a tree, in inches, by twice the length in feet, and the product divided by 144 will be the content, extremely near the truth. In oak an allowance of 1-10 or 1-12 must be made for the bark, if on the tree; in other wood less Trees of irregular growth must be measured in parts.