Exam. 4. Required the content of the piece of timber, whose length is 27.36 feet; at the greater end the breadth is 1078, and thickness 1•23; and at the less end the breadth is 1.0+, and thickness 0.91 feet ? Ans. 41.278 feet. PROBLEM III. To find the Solidity of Round or Unsquared Timber. MULTIPLY the square of the quarter girt, or of of the mean circumference, by the length, for the content. By the Sliding Rule. 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 al, lowance for the squaring of the tree. EXAMPLES. ExAM. 1. A piece of round timber being 9 feet 6 inches long, and its mean quarter girt 42 inches; what is the content? 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? 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 174 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? Ans. 42:519525, CONIC 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. 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 7. The section is an hyperbola, when the cutting plane makes a greater angle with the base than the side of the cone makes. B 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 dbe. And further, if there be four cones N Q CMN, COP, CMP, CNo, having all the R same vertex c, and all their axes in the same plane, and their sides touching or L 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 GQR, FST, VKL, wil, of which each two opposites are equal, and 1 the other two are conjugates to them; as here in the annexed figure, and the same as represented in the two following pages. 9. The Vertices of any section, are the points where the cutting plane meets the opposite sides of the coné, 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 parabola 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. 11. The Centre c is the middle of the axis. Hence the centre of a parabola is infinitely distant from the vertex. And of an ellipse, the axis and centre lie within the curve; but of an hyperbola, without. 12. A Diameter is any right line, as AB or de, drawn through the centre, and terminated on each side by the curve; and the extremities of the diameter, or its intersections with the curve, are its vertices. Hence all the diameters of a parabola are parallel to the axis, and infinite in length. And hence also every diameter of the ellipse and hyperbola have two vertices; but of the parabola, only one; unless we consider the other as at an infinite distance. 13. The Conjugate to any diameter, is the line drawn through the centre, and parallel to the tangent of the curve. at the vertex of the diameter. So, FG, parallel to the tangent at D, is -the conjugate to do; and hi, parallel to the tangent at A, is the conjugate to AB. Hence the conjugate Hi, of the axis AB, is perpendicular to it. 14. An Ordinate to any diameter, is a line parallel to its conjugate, or to the tangent at its vertex, and terminated by the diameter and curve. So DK, EL, are ordinates to the axis AB; and MN, NO, ordinates to the diameter de. Hence the ordinates of the axis are perpendicular to it. 15. An Absciss is a part of any diameter contained between its vertex and an ordinate to it; as AK or BK, or DN or en. Hence, in the ellipse and hyperbola, every ordinate has two determinate abscisses; but in the parabola, only one; the other vertex of the diameter being infinitely distant. 16. The Parameter of any diameter, is a third proportional to that diameter and its conjugate. 17. The 17. The Focus is the point in the axis where the ordinate is equal to half the parameter. As K and L, where dk or EL is equal to the semi-parameter. The name focus being given to this point from the peculiar property of it mentioned in the corol. to theòr. 9 in the Ellipse and Hyperbola following, and to theor. 6 in the Parabola. Hence, the ellipse and hyperbola have each two foci; but the parabola only one. 18. If DAE, FBG, be two opposite hyperbolas, having AB for their first or transverse axis, and ab for their second or conjugate axis. And if dae, fbg, be two other opposite hys perbolas having the same axes, but in the contrary order, namely, ab their first axis, and as their second; then these two latter curves dae, fbg, are called the conjugate hyperbolas to the two former DAE, FBG, and each pair of opposite curves mutually conjugate to the other; being all cut by one plane, from four conjugate cones, as in page 94, def. 8. 19. And if tangents be drawn to the four vertices of the curves, or extremities of the axes, forming the inscribed rectangle HikL; the diagonals HCK, ICL, of this rectangle, are called the asymptotes of the curves. And if these asymptotes intersect at right angles, or the inscribed rectangle bé a square, or the two axes AB and ab be equal, then the hyperbolas are said to be right-angled, or equilateral. SCHOLIUM. The rectangle inscribed between the four conjugate hy perbolas, is similar to a rectangle circumscribed about an ellipse, by drawing tangents, in like manner, to the four exa tremities of the two axes; and the asymptotes or diagonals in the hyperbola, are analogous to those in the ellipse, cutting this curve in similar points, and making that pair of conjugate diameters which are equal to each other. Also, the whole figure formed by the four hyperbolas, is, as it were, an ellipse turned inside out, cut open at the extremities D, E, F, G, of the said equal conjugate diameters, and those four points drawn out to an infinite distance; the curvature being turned the contrary way, but the axes, and the rectangle passing through their extremities, continuing fixed. OF |