HYPERBOLA, in geometry, the sec tion, GEH, (Plate V11. Miscel.,fig. 5) of a cone, ABC, made by a plane, so that the axis, EF, of the section inclines to the op posite leg of the cone, BC, which, in the parabola, is parallel to it, and in the ellip sis intersects it. The axis of the hyper bolical section will meet also with the op posite side of the cone, when produced above the vertex at D.
Definitions. 1. If at the point E (fig. 6.) in any plane, the end of the rule EH be so fixed, that it may be freely carried round, as about a centre; and at the other end of the rule H there is fixed the end of a thread shOrter than the rule, and let the other end of the thread be fixed at the point F, in the same plane ; but the distance of the points EF must be greaterthan the excess of the rule above the length of the thread; then let the thread be applied to the side of the rule EH, by the help of a pin G, and be stretch.
ed along it ; afterwards let the rule be carried round, and in the mean time let the thread, kept stretched by the pin, be constantly applied to the rule: a certain line will be described by the motion of the pin, which is called the hyperbola. But if the extremity of the same rule, which was fixed in the point E, is fixed in the point F, and the end of the thread . is fixed in the point E, and the same things performed as before, there will be described another line opposite to the former, which is likewise called an hy perbola; and both together are called opposite hyperbolas. These lines may be extended to any greater distance from the points EF, viz. if a thread is taken of a length greater than that distance. 2. The points E and F are called the foci. 3. And the point C, which bisects the right line between the two focus's, is called the centre of the hyperbola, or of the opposite hyperbolas. 4. Any right line passing through the centre, and meeting the hyperbolas, is called a trans verse diameter ; and the points in which it meets them, their but the right line, which passes through the centre, and bisects any right line terminated by the opposite hyperbolas, but not pass ing through the centre, is called a right diameter. 5. The diameter which passes through the foci is called the transverse axis. 6.1f from A or a, the extremities of the transverse axis, there is put aright line AD, equal to the distance of the cen tre C from either focus, and with A, as a centre, and the distance AD, there is a circle described, meeting the right line which is drawn through the centre of the hyperbola, at right angles to the trans.
verse axis, in B b ; the line B b is called the second axis. 7. Two diameters, ei ther of which bisects all the right lines parallel to the other, and which are ter. minated both ways by the hyperbola, or opposite hyperbolas, are called conjugate diameters. 8. Any right line, not passing through the centre, but terminated both ways by the hyperbola, or opposite by. perbolas, and bisected by a diameter, is called an ordinate applied, or simply an ordinate to that diameter : the diameter, likewise, which is parallel to that other right line ordinately applied to the other diameter, is said to be ordinately applied to it. 9. The right line which meets the hyperbola in one point only, but produc ed both ways falls without the opposite hyperbolas, is said to touch it in that point, or is a tangent to it. 10. If through the vertex of the transverse axis a right line is drawn, equal and parallel to the se.
cond axis, and is bisected .by the trans verse axis, the right lines drawn through the centre, and the extremities of the pa rallel line, are called asymptotes. 11. The right line drawn through the centre of the hyperbola, parallel to the tangent, and equal to the segment of the tangent between the asymptotes, and which is bisected in the centre, is called the se cond diameter of that which is drawn through the point of contact. 12. A third proportional to two diameters, one of which is transverse, the other second to it, is called the lutes rectum, or parame ter of that diameter, which is the first of the three proportionals. And, 13. Lastly, fig. 9. If upon two right lines A a, 13 b, mutually bisecting each other at right an gles, the opposite hyperbolas AG, a g, are described; and if upon the same right lines there are described two other op posite hyperbolas, BK, b k, of which the transverse axis, B b, is the second axis of the two first and the second axis of the two last, A a, is the transverse axis of the two first ; these four are called con jugated hyperbolas, and their asymptotes shall be common.