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6 inches long, and 3 feet 3 inches broad, at 8:lb. to the square foot ” Ans. 10911?".

ExAM. 2. What cost the covering and guttering a roof with lead, at 18s. the cwt ; the length of the roof being 43 feet, and breadth or girt over it 32 feet; the guttering 57 feet long, and 2 feet wide; the former 9.831 lb. and the latter 7:373 lb. to the square foot ” Ans. 115l. 9s. 1; d.

XII. TIMBER MEASURING.
PROBLEM I.

To find the Area, or Superficial Content, of a Board or Plank,

MULTIPLY the length by the mean breadth.

Note. When the board is tapering, add the breadths at the two ends together, and take half the sum for the mean breadth. Or else take the mean breadth in the middle.

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.

EXAMPLEs.

ExAM. 1. What is the value of a plank, at 1:d. per foot, whose length is 12 feet 6 inches, and mean breadth 11 inches 2 Ans. 1.f. 5d. ExAM. 2. Required the content of a board, whose length

is 11 feet 2 inches, and breadth 1 foot 10 inches 2 Ans. 20 feet 5 inches 8".

ExAM. 3. What is the value of a piank, which is 12 feet 9 inches long, and 1 foot 3 inches broad, at 2; d. a foot. Ans. 3s. 33d. ExAM. 4. Required the value of 5 oaken planks at 3d. per foot, each of them being 17; feet long; and their several breadths as follows, namely, two of 13+ inches in the middle, one of 14; inches in the middle, and the two remaining

ones, each 18 inches at the broader end, and 11+ at the narrower - Ans. 11. 5. 9; d.

PROBLISM

PROBLEM II,

To find the Solid Content of Squared or Four-sided Timler.

MULTIPLY the mean breadth by the mean thickness, and the product again by the length, for the content nearly.

By the Sliding Rule.

C D D C
As length ; 12 or 10 :: quarter girt : solidity.

That is, as the length in feet on c, is to 12 on D, when the quarter girt is in inches, or to 10 on D, when it is in tenths of feet; so is the quarter girt on D, to the content On C.

Note 1. If the tree taper regularly from the one end to the other; either take the mean breadth and thickness in the middle, or take the dimensions at the two ends, and half their sum will be the mean dimensions: which multiplied as above, will give the content nearly. 2. If the piece do not taper regularly, but be unequally thick in some parts and small in others; take several different dimensions, add them all together, and divide their sum by the number of them, for the mean dimensions.

ExAMPLEs.

ExAM. 1. The length of a piece of timber is 18 feet 6 inches, the breadths at the greater and less end 1 foot 6 inches and 1 foot 3 inches, and the thickness at the greater and less end 1 foot 3 inches and 1 foot; required the solid content 2 Ans. 28 feet 7 inches.

ExAM. 2 What is the content of the piece of timber, whose length is 244 feet, and the mean breadth and thickness each 104 feet Ans. 26; feet.

ExAM. 3. Required the content of a piece of timber, whose length is 20:38 feet, and its ends unequal squares, the sides of the greater being 19% inches, and the side of the less 9% inches • Ans. 29.7562 feet.

ExAM.

ExAM. 4. Required the content of the piece of timber, whose length is 27-36 feet; at the greater end the breadth is 1-78, and thickness 1:23; al.d at the less end the breadth is 1:04, and thickness 0-91 feet 2 , Ans. 41.278 feet.

PROBLEM III.
To find the Solidity of Round or Uniquared Timber.

MULTIPLY the square of the quarter girt, or of ; of the mean circumference, by the length, for the content.

By the Sliding Rule.

As the length upon c : 12 or 10 upon D ::
quarter girt, in 12ths or 10ths, on D : content on c.

Note 1. When the tree is tapering, take the mean dimensions as in the former problems, either by girting it in the middle, for the mean girt, or at the two ends, and taking half the sum of the two ; or by girting it in several places, then adding all the girts together, and dividing the sum by the number of them, for the mean girt. But when the tree is very irregular, divide it into several lengths, and find the content of each part separately.

2. This rule, which is commonly used, gives the answer about # less than the true quantity in the tree, or nearly what the quantity would be, after the tree is hewed square in the usual way: so that it seems intended to make an allowance for the squaring of the tree.

EXAMPLES.

ExAM. I. A piece of round timber being 9 feet 6 inches long, and its mean quarter girt 42 inches ; what is the content 2 Ans. 116; feet,

ExAM. 2. The length of a tree is 24 feet, its girt at the thicker end 14 feet, and at the smaller end 2 feet; required the content 2 - Ans. 96 feet.

ExAM. 3. What is the content of a tree, whose mean girt is 3:15 feet, and length 14 feet 6 inches? Ans. 8-9922 feet.

ExAM. 4. Required the content of a tree; whose length is 17; feet, which girts in five different places as follows, namely, in the first place 9:43 feet, in the second 7.92, in the

third 6' 15, in the fourth 4:74, and in the fifth 3:16 ° f Ans. 42'519.525,

C O N IC SECTIONS.

DEFINITIONS,

1. Conic Sections are the figures made by a plane cutting a cone.

2. According to the different positions of the cutting plane, there arise five different figures or sections, namely, a triangle, a circle, an ellipsis, an hyperbola, and a parabola: the three last of which only are peculiarly called Conic Sections,

3. If the cutting plane pass through the vertex of the cone, and any part of the base, the section will evidently be a triangle; as vab.

4. If the plane cut the cone parallel to the base, or make no angle with it, the section will be a circle; as ABD.

5. The section DAB is an ellipse when the cone is cut obliquely through both sides, or when the plane is inclined to the base in a less angle than the side of the cone is.

6. The section is a parabola, when the cone is cut by a plane parallel to the side, or when the cutting plane and the side of the cone make equal angles

with the base. - 7. The

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7. The section is an hyperbola, when the cutting plane makes a greater angle with the base than the side of the cone makes.

8. And if all the sides of the cone be continued through the vertex, forming an opposite equal cone, and the plane be also continued to cut the opposite cone, this latter section will be the opposite hyperbola to the former; as dee.

And further, if there be four comes cMN, cop, CMP, CNo, having all the same vertex c, and all their axes in the same plane, and their sides touching or coinciding in the common intersecting lines Mco, NCP ; then if these four cones be all cut by one plane, parallel to the common plane of their axes, there will be formed the four hyperbolas Gar, FST, vKL, whi, of which each two opposites are equal, and the other two are conjugates to them ; as here in the annexed figure, and the same as represented in the two follow

ing pages. 9. The Vertices of any section, are the points where the

cutting plane meets the opposite sides of the cone, or the sides of the vertical triangular section; as A and B.

Hence the ellipse and the opposite hyperbolas, have each two vertices; but the paraboia only one; unless we consider the other as at an infinite distance.

10. The Axis, or Transverse Diameter, of a conic section, is the line or distance AB between the vertices.

Hence the axis of a parabola is infinite in length, Ab being only a part of it.

Ellipse.

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