# tangent

## Definitions

• WordNet 3.6
• n tangent ratio of the opposite to the adjacent side of a right-angled triangle
• n tangent a straight line or plane that touches a curve or curved surface at a point but does not intersect it at that point
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Webster's Revised Unabridged Dictionary
• n Tangent (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.
• a Tangent (Geom) Touching; touching at a single point
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Century Dictionary and Cyclopedia
• n tangent One of the keys or finger-levers of the hurdy-gurdy.
• n tangent In railroading, a straight piece of track beginning and ending at a curve.
• tangent Touching; in geometry, touching at a single point: as, a tangent line; curves tangent to each other.
• n tangent 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.
• n tangent The length cut off upon the straight line touching a curve between the line of abscissas and the point of tangency.
• n tangent 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.
• n tangent 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.
• n tangent Any method of drawing a tangent to a curve.
• tangent To bear or hold the relation of a tangent to.
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Chambers's Twentieth Century Dictionary
• n Tangent tan′jent a line which touches a curve, and which when produced does not cut it
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## Idioms

Go off on a tangent - If someone goes off on a tangent, they change the subject completely in the middle of a conversation or talk.
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## Etymology

Webster's Revised Unabridged Dictionary
L. tangens, -entis, p. pr. of tangere, to touch; akin to Gr. having seized: cf. F. tangente,. Cf. Attain Contaminate Contingent Entire Tact Taste Tax (v. t.)
Chambers's Twentieth Century Dictionary
L. tangens, -entis, pr.p. of tangĕre, to touch.

## Usage

### In literature:

The only thing he got out of all his fussing was the one new thought that seemed to fly out at a tangent and mock him.
"A Man to His Mate" by J. Allan Dunn
The wind sang by them as they tangented off from the road.
"Rimrock Trail" by J. Allan Dunn
But Thornton's unaccountable whims had flown at another tangent.
"The Roof Tree" by Charles Neville Buck
This is anything but a theological tangent.
"Holbein" by Beatrice Fortescue
They rode at a tangent for the dark shadow of the trees.
"The Coyote" by James Roberts
It swung off at a tangent.
"Astounding Stories of Super-Science, March 1930" by Various
Margaret Gannion was tangent to many lives.
"The Dominant Strain" by Anna Chapin Ray
His horse, already unnerved by the wild dash down the hill, now saw the fire and started to bolt off at a tangent.
"The Shepherd of the North" by Richard Aumerle Maher
Then he dwelt upon the dead man, looking nothing else as he now remembered him, and that sent him off at a final tangent.
"" by E.W. Hornung
I could not but think how good was the end of that long travel; and with that, my mind swung at a tangent to my lord.
"The Works of Robert Louis Stevenson - Swanston Edition, Vol. XII (of 25)" by Robert Louis Stevenson
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### In poetry:

It is a mist
now tangent to another
quiet. Here the leaves
come, there
is the rock in evidence
"The Innocence" by Robert Creeley

### In news:

Today we launch The Tangent , amNewYork's new entertainment blog, where we'll cover movies, television, gossip and anything else of a pop cultural bent.
The bottle is customized so it bears Tangent 's logo.
Reid laughed when he told the story about the time a bartender at Tangent 's Dixie Creek Saloon refused to serve Bass -- a man who liked his drink -- another round.
Tangent Hosts Aerial Firefighting Show.
Tangent 's DataCove DT Eases E-Mail Archiving.
Fortunately, there is help for performing these tasks, and Tangent 's DataCove DT appliance is one such solution.
TANGENT 'S RUGGED MINI is a fanless PC that can be attached to a cart or wall.
In Tangent Railroad Knocked doen telephone Poles.
The railroad knocked down telephone poles here in Tangent causing this damage.
Front man for Yalobushwhackers and Tangents dies.
Digital and direct marketing specialist Tangent Communications has appointed Andy Wheatley as its director of strategy for local marketing.
Tangent hires Wheatley to head local strategy.
Of course, we got off on our many tangents, because that is what talking about food does for anyone.
I have a friend who stays on a tangent from one issue to the next.
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### In science:

T∆,0 is the tangent space of ∆ at the origin and TS is the tangent bundle of S .
A Simple Proof that Rational Curves on K3 are Nodal
For a group G acting on the J 8 denote by XG the intersection TxX ∩ Tx (Gx), where TxX is the tangent space to X and Tx (Gx) is the tangent space to the G-orbit of the point x.
Simple singularities of multigerms of curves
There exist a tangent space T ˆX L(V ), tangent bundle TL(V ), ˆX L(V ), and cotangent bundle T⋆L(V ).
Quantum fluctuations and random matrix theory
The exist a tangent space T ˆX L(V ), tangent bundle TL(V ), ˆX L(V ), and cotangent bundle T⋆L(V ).
Applications of methods of random differential geometry to quantum statistical systems
In particular, the tangent vectors to the coordinate curves (the curves obtained by keeping all the coordinates ﬁxed except one) through P are called the natural basis for the tangent space.
Random matrix theory and symmetric spaces
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