Most of the above definitions apply also to the hyperbola.
If the side, A B, Figure 3, of a right angle or square, A B c, be applied to the straight-edge, A D. of a ride, and a thread, equal in length to B c. be fiistened to the end. c. of the right angle, with the other end to the fixed point. F; and if any point, E, be taken in the line, then if the edge, A B, of the square be moved along the straight-edge, A 0, keeping the variable point, E, upon the side, B C, of the square, and the two portions e E and E F stretched, the point E will trace out a curve, which is a parabola.
The point F is called the focus.
The line A n is the directrix.
The line L K passing through the focus perpendicular to the direetrix, is Pie afix.
point 1, where the axis cuts the curve, is the vertex.
Any line parallel to the axis, terminated at one extremity by the curve, and on the concave side of it, is called, a diameter.
Any line parallel to a tangent at the limited end of a dia meter, is called a double ordinate to that diameter.
The limited part of a diameter, contained by the curve and a double ordinate, is called the abscissa of that double ordinate.
h'iyures 4, 5, 6. An abscissa, the ordinate, and the diameter being given, to describe the ellipsis or hyperbola.
Let A It be the diameter, A c the abscissa, and c n the ordi nate. Dra• A E parallel to C D, and D E parallel to C A. In D c take any number of points. le, 3.3, II, and divide D E in the
same proportion q, h. Draw It F 1, BG K, B11 L; likewise f I A, L A, 11.1111 the points 11, 1, E, 1,, A, draw a curve. In the sante manner may the curve for the opposite ordinate be drawn, \\lien the extremities of the diameter are on different sides of the ordinate, the curve is an ; but when the extremities of tire diameter are on the same side of the ordi nate, the curve is an When the diameter A 13, is i_ifinlinite length, the ordinate{, F c K, 11 L, will he parallel, then the curve is a parabola. Therefore, in Figure 4, the lines drawn from the points F, 0, 0, parallel to the transverse axis, or abscissa, A a, instead of being drawn to the point n, as in Figures I, 3, make the only difference.
It is hardly possible to conceive more convenient or easier !nodes of description than these ; their correctness may be proved by showing that their common properties are similar to those demonstrated of conic sections.
Figures7,8, 9.—Let A n be the diameter, c n the ordinate, and A c the abscissa, as before. hi C D take any point, a, and divide n E by y, in the same ratio as Incis by the point, G ; draw q K A, u C K, _Figu•e 7, and B E c, Figure S ; then, because of tire similar trianf_des, it N E and B C U. It N : B C : NK:CO; and also, because of the similar triangles A N K and