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PROBLEM I. To find the area, or superficial content of a board or plank.
RULE. MULTIPLY the length by the breadth, and the product will be the content required.
Note.—When the board is tapering, add the breadths of the two ends together, and take half the sum for the mean breadth.
BY THE SLIDING RULE.
Set 12 on B to the breadth in inches on A, then against the length in feet on B is the content on A, in feet and fractional parts as required.
1. What is the value of a plank, whose length is 8 feet 6 inches, and breadth 1 foot 3 inches throughout, at 2 d. 10 7 6
10 7 6 the content
2d. is 11 8
1 is 5
2s. 64d. the Answer.
BY THE SLIDING RULE.
As 12 on B : 15 on A :: 81 on B : 104 on A. 2. What is the content of a board, whose length is 5 feet 7 inches, and breadth 1 foot 10 inches ?
Ans. 10 fe. 2 in. 10 pa. 3. At lqd. per foot, what is the value of a plank whose length is 12 feet 6 inches, and breadth 11 inches throughout?
Ans. ls. 5d. 4. Find the value of 5 oaken planks at 3d. per foot, each being 171 feet long, and their particular breadths as follows: viz. two of 131 inches in the middle, one of 141 inches in the middle, and the two remaining ones, each 18 inches at the broader end, and 11f at the narrower?
Ans. 11. 5s. 9 d.
To find the solidity of squared or four-sided timber.
Multiply the mean breadth by the mean thiekness, and this product again by the length, and it will give the solidity required.
* Note 1. If the stick be equally broad and thick throughout, the breadth and thickness, anywhere taken, will be the mean breadth and thickness.
BY THE SLIDING RULE.
As the length in feet on C : 22 on D :: quarter girth in inches on D : solidity on C.
1. The length of a piece of timber is 204 feet, the breadth at the greater end is 1 foot 9 inches, and the thickness 1 foot 3 inches; and at the less end the breadth is 1 foot 6 inches, and the thickness 1 foot; what is the solidity?
2. If the tree tapers regularly from one end to the other, the breadth and thickness, taken in the middle, will be the mean breadth and thickness.
3. If the stick does not taper regularly, but is thicker in some places than in others, let several different dimensions be taken, and their sum divided by the number of them will give the mean dimensions.
This method of finding the mean dimensions is mostly used in practice, but, in many cases, it is exceedingly erroneous.
The quarter girth, likewise, which is mentioned in the proportion by the sliding rule, is subject to error. It is not the fourth part of the circumference, but the ́square root of the product arising from multiplying the mean breadth by the mean thickness.
In order to show the fallacy of taking one-fourth of the girth for the side of a mean square, take the following example:
Suppose a piece of timber to be 24 feet long, and a foot square throughout, and let it be slit into two equal parts, from end to end.
Then the sum of the solidities of the two parts, by the quarter girth method, will be 27 feet, but the true solidity is 24 feet; and if the two pieces were very unequal, the difference would be still greater
1.125=mean thickness. Now 1.625 x 1.125 x20.5=37.4765625=content required.
A8 1 upon B : 1912 upon A :: 131 upon B : 2639 upon A, the mean square. As 16 upon C:4 upon D::1.8
C: 16.2.upon D, the side of the mean square. As 201 upon
D:37 iz upon C, the answer.
2. The length of a piece of timber is 24.5 feet, and its ends are equal squares, whose sides are each 1.04 feet : what is the solidity ?
Aps. 26 feet 6 inches. 3. The length of a piece of timber is 20.38 feet, and the ends are unequal squares, the sides of the greater being 19] inches, and that of the less 97 inches : what is the solidity ?
Ans. 29.756 feet. 4. The length of a piece of timber is 27.36 feet; at the greater end the breadth is 1.78 feet, and the thickness 1.23 feet; and at the less end the breadth is 1.04 feet, and the thickness .91 feet: what is the solidity ?
Ans. 41.278 feet.
To find the solidity of round or unsquared timber.
Multiply the square of the quarter girth (or one-fourth of the circumference) by the length, and the product will be the content, according to the common practice.
* Let c=girth or circumference, and l=length of the tree.
cal xl= = content of the tree according 4^
16 to the rule. c?
c7 And 4x 3.1416
= true content, accord
12.5664 ing to the rule for finding the content of a cylinder. co1
differs from 12.5664
16 by nearly one-fourth part of the whole, and therefore the rule is exceedingly erroneous.
When the tree is tapering, the mean girth is found in the same manner as in board measure. Or if the tree be very irregular, the best way is to divide it into a certain number of lengths, and find the content of each part separately.
When trees have their bark on, an allowance is generally made, by deducting so much from the girth as is judged sufficient to reduce it to such a circumference as it would have without its bark. In oak this allowance is about jo or in part of the girth ; but for elm, beach, ash, &c. whose bark is not so thick, the deduction ought to be less.