CONIC SECTIONS.
The figures known as the Conic Sections are the Ellipse, the Parabola, and the Hyperbola.
The Cone may have other sections in addition to these, such as the section through any point below the apex, on the axis, and taken parallel to the base; this would be a circle, and a section through the apex perpendicular to the base would be an isosceles triangle.
The Ellipse is the curve of the section made by a plane passing obliquely through a cone from side to side.
The Parabola is the curve of the section made by a plane passing through a cone parallel to one of its sides.
The Hyperbola is the curve of a section made by a plane passing through a cone parallel to its axis, or inclined at a greater angle to its base than its side, but not through its apex.
[Fig. 31] The elevation of a cone is shown at A B C. A section through point X at right angles to the axis of the cone is a Circle. A section passing through and across the cone from point X, but not at right angles to the axis, is an Ellipse, as at X 1. A section through X parallel to the opposite side A C is a Parabola, as at X 2. A section through X parallel to the axis, as at X 3, or a section through X at any other angle greater than the angle made by the side and base, as at X 4, is a Hyperbola.
Figs. [32], [33], and [34] show the actual shape of the sections X 1, X 2, and X 3 respectively.
[Fig. 32] In this figure the major or transverse axis of the Ellipse is equal to X 1. To find the minor or conjugate axis bisect X 1 ([Fig. 31]) in H, draw through it F G parallel to A B, drop a perpendicular from F to f, and describe the semicircle f h g. From H drop a perpendicular to A B, and produce it to h to meet the semicircle, k h is then half the length of the minor axis of the Ellipse, as C D. Divide A E into any number of equal parts, and A G into the same number. Draw from C lines through the divisions as 1, 2, 3 &c., and from D lines to 1´ 2´ 3´ &c. The curve of the required Ellipse will pass through the intersections of these lines, as at 1´´ 3´´ 5´´ &c.
[Fig. 33] In this figure, the Parabola, the line C D is equal to X 2 ([Fig. 31]), while A B is twice the length of D 2 ([Fig. 31]). Divide G B into any number of equal parts, and join the points of the divisions to C. Divide D B into the same number of equal parts, and draw lines from the points of division parallel to D C to meet the similar numbered lines drawn from B G; through these meeting points the curve of the Parabola will be drawn.