TANGENT (Lat. tangcns, pros. part. of tanycre, to touch ; connected with Goth. tekan, feel. taka„ AS. tacan, Eng. take). An unlimited straight line which meets a curve in but one point, without cutting it. The point is called the point of contact or point of tangency. A tangent may be thought of as the limiting posi tion of a secant, the point of tangency being the point in which the points of intersection of the secant with the curve coincide. In the circle there is but one distinct tangent at any point in the circumference, and this is perpendicular to the radius at that point. The segments of two tangents to a circle from an external point, lim ited by the point and the points of contact, are equal. If the point moves up to the circumfer ence, the segments become zero and the tangents coincide; if the point moves inside the circum ference, the tangents become imaginary. (See
CONTINUITY.) All conic sections being of the second class (see CunvE) admit of but two tan gents from any point. Curves belonging to high er classes admit of a greater number, depending upon the class. Tangents are of different orders of contact (see CONTACT) according to the num ber of coincident points at the point of tangency, and thus serve to distinguish various singulari ties of curves. (See CURVE.) In coOrdinate geometry the projection, upon the X-axis. of the segment of the tangent between the point of contact and the X-axis is called the subtang,ent. An important class of tangents are known as asymptotes (q.v.). In the geometry of surfaces the tangent plane corresponds to the tangent line of curves. The extensive list of properties involving tangents is best obtained from works on analytic geometry.