by bricklayers for setting out templets for wliat they describe as " elliptic " arches.
Draw the span or major axis A-A. Divide this into three equal parts, in points 1-2. With these points as centres, and 2-A as radius, describe circles intersecting in point 3. Draw lines trom the intersection through points 1 and 2 to cut the circles, and these give the radius for describing the central part of the curve, the two ends being formed by segments of the circles already drawn. It will be seen that only two shapes are required for the bricks in this arch.
tiib: cone and its properties
Definitions.—There are two kinds of cones—right and oblique. A right cone has its axis perpendicular to its base ; an oblique cone has its axis at some other angle than a right angle with its base (see sketches, Figs, x and 3, page 156). The axis of a solid is a straight line drawn, or imagined, connecting the centres of its opposite ends.
A right cone may be defined as a solid with a circular base, and sides tapering to a point. If may also be described as a circular pyramid. When the top portion of a cone (or other similar solid) is cut off parallel with its base, the remaining portion is called the Frustum (see Fig. 2). When the line of section is oblique, as in Fig. 4, the portion remaining is called Ungula. a Latin word signifying a hoof, which the solid somewhat resembles.
Projections of Cones, Figs. 5 to 8, are respectively elevations (or sections, as they are exactly alike) and plans of a right cone and its frustum; the projectors indicate how the plan is obtained from the elevation. Figs. 9 and 10 are the projections of an oblique cone, and as the method of obtaining the plan is similar to that for obtaining the section in the next case, its explanation will be dealt with there.
The Conic Sections are five in number, named respectively the_triangle, see Fig. 5, a section through the
axis and diameter of its basest he circle (see Fig. 6), a section at right angles to the axis ; the ellipse (see Fig. 11 a), a section through the two inclined sides at. an oblique angle to the axis; the parabola (see Fig. 14), a section through one side and the base, parallel with the other side; and the hyperbola, Fig. iG, a section made by a plane passing through the base and side of the cone, and making a greater angle with the base than the side of the cone makes.
It is not possible to cut a cone in any other direction than the above named, and, though the sizes of the sections will vary according to their positions on the cone, their shapes will be constant. It may be noted that all the five figures can be obtained by other methods than as sections of cones, but no other solid contains them all, hence the generic term of conic sections. Some geometricians hold that there are only three conic sections, because the triangle and circle are common to other solids, and they define these as particular cases of the parabola and the ellipse respectively.
The Ellipse, Fig. n.—To obtain this section, draw the elevation of the cone (and it may be here interpolated that it will be advisable for the student to draw these figures at least four times the size shown, and to take more points than are shown on the drawing, where they are limited to avoid a contusion of lines) and the line of section as A-B £ project the extremities to the base b b, and describe a semicircle cutting the po'nts as shown. Divide the semicircle into any number of equal parts as 1, 2, 3, 4, 5,6, and project these upwards to cut the line of section, then project ordin-ates from these points, at right angles to the section lines. Draw a line a-b across them, parallel with the line of section (this is merely done for convenience, the section could be drawn upon the line of section if desired). Make each of these ordinates equal ill length, on each side of the line a-b, to the corresponding ordinate in the semicircle, which is a plan of the section, and draw the outline of the section through the points so found. The more points used the truer will be the figure. It will be noticed that the semi-
15« PARABOLIC SECTION OF CONE
circle might equally well represent the half plan of a cylinder whose sides are cut by the line A-B, and obviously the same section would be obtained.
The Parabola, Fig. 14.—To produce this, draw the desired line of section A-B, Fig. 1 z, and divide it into a number of tqual parts, as 1, 2, 3, 4. Through these points draw lines parallel with the base, ami cutting the sides of the cone in points 1', 2', 3', 4'. These lines may be taken to represent a series of horizontal sections of the cone, and their planes will becirdes. To determine their size, drop projectors —dotted — into the plan, cutting the diameter in a, b, c, d; from the middle of the diameter as centre and these points as radii, describe concentric circles. Next drop projectors—full lines—from points t, 2, 3, 4, and the extremities in the line of section A-B, into the plan, cutting the appropriate circles in points B", 1", 2", 3", 4". a', and draw a curved line through these points on each side of the diameter as shown ; draw also the plan of the base of ¡Jane of section B' -B", thus obtaining the shape of the section in plan. To obtain its real shape, draw the line a -d parallel with A-B, and project ordinates across it, from the pomts in section line. Make these equal in length on each side of a-d, to the similar numbered ordinates in the plan, measured from the diameter b'~b", and draw the curve through the points so found.
The Hyperbola (see Fig. 16).--Two plans and elevations are given, the first pair to show the position of the line of section E-I) in elevation, and e-e' in plan. Obviously we are looking at the edge of the section, therefore cannot see its shape. To obtain this the plan must be revolved on its centre until the line of section is parallel to the vertical plane. This is shown in the right-hand plan, and a new elevation is projected from it. Divide the diameter into a number of equal parts, and from the centre describe circles through them as at a", b", c", c"; project these points into the elevation, cutting the side of the cone in a', b', c', c', and draw lines through them parallel to the base : these lines
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