Collier's New Encyclopedia (1921)/Tangent
TANGENT, in geometry, a straight line which meets or touches a circle or curve in one point, and which, being produced, will not cut it. In Euclid (III. 16, Cor.) it is proved that any line drawn at right angles to the diameter of a circle at its extremity is a tangent to the circle.
In trigonometry, the tangent of an arc or angle is a straight line, touching the circle of which the arc is a part at one extremity of the arc, and meeting the diameter passing through the other extremity; or it is that portion of a tangent drawn at the first extremity of an arc, and limited by a secant drawn through the second extremity. The tangent is always drawn through the initial extremity of the arc, and is reckoned positive upward, and consequently, negative downward. The tangent of an arc or angle is also the tangent of its supplement. The arc and its tangent have always a certain relation to each other, and when the one is given in parts of the radius, the other can always be computed by means of an infinite series. Tables of tangents for every arc from 0° to 99°, as well as of sines, cosines, etc., are computed and formed into tables for trigonometrical purposes. Two curves are tangent to each other at a common point, when they have a common rectilinear tangent at this point. A tangent plane to a curved surface is the limit of all secant planes to the surface through the point. The point is called the point of contact. Two surfaces are tangent to each other when they have, at least, one point in common; through which, if any number of planes be passed, the sections cut out by each plane will be tangent to each other at the point. This point is called the point of contact. Another definition is this: Two surfaces are tangent to each other when they have a common tangent plane at a common point. This point is the point of contact.
Artificial tangents, tangents expressed by logarithms. Methods of tangents, the name given to the calculus in its early period. When the equation of a curve is given, and it is required to determine the tangent at any point, this is called the direct method of tangents, and when the subtangent to a curve at any point is given, and it is required to determine the equation of the curve, this is termed the inverse method of tangents. These terms are synonymous with the differential and integral calculus. Natural tangents, tangents expressed by natural numbers. To go (or fly) off at a tangent, to break off suddenly from one course of action, line of thought, or the like, and go on to something else.