tangent

American Heritage® Dictionary of the English Language, Fourth Edition

1. adj. Making contact at a single point or along a line; touching but not intersecting.
2. adj. Irrelevant.
3. n. A line, curve, or surface meeting another line, curve, or surface at a common point and sharing a common tangent line or tangent plane at that point.
4. n. Mathematics The trigonometric function of an acute angle in a right triangle that is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
5. n. A sudden digression or change of course: went off on a tangent during the courtroom argument.
6. n. Music An upright pin in a keyboard instrument, especially in a clavichord, that rises to sound a string when a key is depressed and stops the string at a preset length to set the pitch.

Century Dictionary and Cyclopedia

1. n. One of the keys or finger-levers of the hurdy-gurdy.
2. n. In railroading, a straight piece of track beginning and ending at a curve.
3. Touching; in geometry, touching at a single point: as, a tangent line; curves tangent to each other.
4. n. In geometry: A straight line through two consecutive points (which see, under consecutive) of a curve or surface. If we take the line through any two points of the locus, and then, while one of these points remains fixed, consider the other as brought by a continuous and not infinitely protracted motion along the locus into coincidence with the former, the line in its final position will be a tangent at that point. The idea of time which appears in this definition is only so far essential that some parameter must be used in order to define a tangent at a singular point, and this parameter must be such as to present no discontinuity or point-singularity at that point. A tangent at an ordinary point of a curve or surface may be defined, without the use of any parameter, simply as a line through two points infinitely close together; although, if the doctrine of limits is used to explain away the idea of infinity, a parameter will be used for that purpose. A curve has only one tangent at an ordinary point, or a mere line-singularity, or a cusp, but has two or more tangents at a node. A surface has a single infinity of tangents lying in one plane at an ordinary point; and two of these (real or imaginary), called the inflectional tangents, pass through three or more consecutive points of the surface. On the nodal curve of a surface the tangents lie in two or more tangent planes; at a conical point they are generators of a quadric cone. The tangents of a curve in space form two sets which are all generators of one developable. There are points upon some curves and surfaces at which, according to the doctrine of limits, there are no tangents. Such is the point in the second figure where the two multiple tangents intersect; for, as a second point on the curve moves toward this, the line through the two points will oscillate faster and faster, without tending toward any limit. In the same sense, a curve may have no tangent at any point; it may be an undulating line with small undulations on the large ones, and still smaller on these, and so on ad infinitum, the lengths and amplitudes of the undulations being duly proportioned. But an intelligence situated on such a curve might see that the tangent had a definite direction, for there is no logical absurdity in this. It is antagonistic to the principle of duality which rules modern geometry to define the tangent of a plane curve as the line through two consecutive points on the curve. On the contrary, the definition of a plane curve is a locus described by the parametric motion of a line with a point upon it, the point slipping along the line and the line turning about the point; and such a generating line is a tangent. In like manner, a surface is the locus formed by a plane with a point upon it, the position of the point in the surface and the aspect of the surface about the point varying, the one and the other, according to the variations of the same pair of independent parameters. Such a plane is a tangent plane, and a tangent may equally be conceived as the line through two consecutive ineunt-points, or as the line of intersection of two consecutive tangent planes. The tangent plane of a spacious curve is a line lying in a plane and having a point upon it, the plane turning continuously about the line, the point moving along the line, and the line turning in the plane around the point as a center. Euclid's definition of a tangent (“Elements,” bk. iii., def. 2) as a line meeting a circle and not crossing it when produced does not extend to curves having inflections. The definition of the tangent as the limiting case of a secant, which is due to Descartes (but was perfected by Isaac Barrow, 1674), may well be considered as the foundation of modern mathematics.
5. n. The length cut off upon the straight line touching a curve between the line of abscissas and the point of tangency.
6. n. In trigonometry, a function of an angle, being the ratio of the length of one leg of a right triangle to that of the other, the angle opposite the first leg being the angle of which the tangent is considered as the function. Formerly the tangent was regarded as a line dependent upon an arc—namely, as the line tangent to the arc at one extremity, and intercepted by the produced radius which cuts off the arc at the other extremity, Abbreviated tan.
7. n. In the clavichord, one of the thick pins of brass inserted in the back ends of the digitals so that the fingers should press them against the strings, and produce tones. Its action was not like that of the pianoforte-hammer, since it remained in contact with the string, and fixed the pitch of the tone by the place where it struck. If pressed too hard, it raised the pitch by increasing the string's tension. Accordingly the tone of the clavichord was necessarily weak.
8. n. Any method of drawing a tangent to a curve.
9. To bear or hold the relation of a tangent to.

Wiktionary

1. n. geometry A straight line touching a curve at a single point without crossing it there.
2. n. trigonometry In a right triangle, the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Symbols: tan, tg
3. n. A topic nearly unrelated to the main topic, but having a point in common with it.
4. n. A small metal blade by which a clavichord produces sound.
5. adj. geometry Touching a curve at a single point but not crossing it at that point.
6. adj. Of a topic, only loosely related to a main topic.

GNU Webster's 1913

1. n. (Geom.) A tangent line curve, or surface; specifically, that portion of the straight line tangent to a curve that is between the point of tangency and a given line, the given line being, for example, the axis of abscissas, or a radius of a circle produced. See Trigonometrical function, under function.
2. adj. (Geom.) Touching; touching at a single point. meeting a curve or surface at a point and having at that point the same direction as the curve or surface; -- said of a straight line, curve, or surface

WordNet 3.0

1. n. a straight line or plane that touches a curve or curved surface at a point but does not intersect it at that point
2. n. ratio of the opposite to the adjacent side of a right-angled triangle