a. and sb. [ad. L. tangens, tangent-em, pr. pple. of tang-ĕre to touch; used by Th. Fincke, 1583, as sb. in sense = L. līnea tangens tangent or touching line. tangent, -e adj., tangente sb. (Geom.), Ger. tangente sb.] A. adj.
1. Geom. Of a line or surface in relation to another (curved) line or surface: Touching, i.e., meeting at a point and (ordinarily) not intersecting; in contact.
A surface may also be tangent to another surface along a line (e.g., a plane in contact with a cylinder). In quot. 1869, Taking place along a tangent. Cf. B. 1 b.
1594. Blundevil, Exerc., II. (1597), 48 b. Our moderne Geometricians haue of late inuented two other right lines belonging to a Circle, called lines Tangent, and lines Secant.
1644. Digby, Nat. Bodies, xiii. § 8. 114. The reflexion must follow the nature of tangent surfaces.
1713. Berkeley, Guard., No. 126, ¶ 2. Hence … the earth,… without flying off in a tangent line, constantly rolls about the sun.
1866. Proctor, Handbk. Stars, 33. The cone, instead of being a tangent-cone, is supposed to be a secant-cone, intersecting the sphere.
1869. Tyndall, in Fortn. Rev., 1 Feb., 245. All the vibrations tangent to the little circle … are reflected perfectly polarized.
1876. Catal. Sci. App. S. Kens. Mus., § 102. Model exhibiting the simultaneous transformation … of the tangent paraboloid of the conoïd into the tangent plane of the cylinder.
† b. Cryst. Applied to a plane replacing an edge or solid angle of a crystal (which is more properly a secant plane). Obs.
1823. H. J. Brooke, Introd. Crystallogr., 109. Edges replaced by tangent planes.
1851. Richardson, Geol., v. 88. Crystals often present the appearance of having lost their edges and solid angles, which are then said to be replaced by tangent planes.
c. transf. Said of the wheel of a bicycle or tricycle having the spokes tangent to the hub.
1886. Bicycling News, 6 Aug., 664/1. Laced tangent wheels, hollow rims, Hancock’s tyres.
2. fig. ‘Flying off at a tangent’ (see B. 1 c); divergent, erratic.
1787. Burns, Lett. to Moore, 23 April. If once this tangent flight of mine were over, and I were returned to my wonted leisurely motion in my old circle.
1799. E. Du Bois, Piece Family Biog., I. 152. The voluble loquacity and tangent style of reasoning of their new companion.
3. In general sense. a. Touching, contiguous.
1846. Ellis, Elgin Marb., I. 107. Beaten together till the tangent surfaces were fitted to each other.
b. Of or pertaining to touch; tangent sense, sense of touch. nonce-use.
1802. E. Darwin, Orig. Soc., III. 424. Say, did these fine volitions first commence From clear ideas of the tangent sense?
B. sb.
1. Math. (ellipt. for tangent line.) [= Fr., Ger. tangente.] a. Trigonometry. One of the three fundamental trigonometrical functions (cf. SECANT, SINE), originally considered as functions of a circular arc, now usually of an angle (viz. that subtended by such arc at its center): orig. The length of a straight line perpendicular to the radius touching one end of the arc and terminated by the secant drawn from the center through the other end; in mod. use, the ratio of this line to the radius, or (equivalently, as a function of the angle) the ratio of the side of a right-angled triangle opposite the given angle (if acute) to that of the side opposite the other acute angle (the tangent of an obtuse angle being numerically equal to that of its supplement, but of opposite sign). Abbrev. tan.
Tables of tangents and cotangents were constructed and used by the Arab mathematicians of the 9th and 10th c. (see Nallino Al Battani, Opus astronomicum, Milan, 1903, I. 182); but began to be constructed in Christendom late in the 15th c. The names tangens and secans, introduced by Thos. Fincke (Finkius) in 1583, had no connection with the names used by the Arabs.
[1583. Fincke, Geometriæ Rotundi, v. 64. De semicirculi sinibus, tangentibus, secantibus. Ibid., 73. Recta sinibus connexa est tangens peripheriæ aut eam secans.]
1594. Blundevil, Exerc., II. (1597), 57 b. Of which Arke the line A D is the Tangent, and the line C D is the Secant thereof.
1635. [see COTANGENT].
1658. Phillips, Tangent,… a Mathematical Term used chiefly in Astronomy,… signifies, a right line perpendicular to the Diameter drawn by the one extream of the given Arch, and terminated by the Secant.
1690. Leybourn, Curs. Math., 397. Which Scales of Tangents … let be extended to 75 deg. at least.
1728. Pemberton, Newton’s Philos., 366. The refracting powers … will be in the duplicate proportion of the tangents of the least angles, which the refracted light can make with the surfaces of the refracting bodies.
1828. Hutton, Course Math., II. 3. As the arc increases from 0, the sines, tangents, and secants, all proceed increasing, till the arc becomes a whole quadrant…, and then the sine is the greatest it can be…; and both the tangent and secant are infinite.
b. Geom. A straight line which touches a curve (or curved surface), i.e., meets it at a point and being produced does not (ordinarily) intersect it at that point.
In Higher Geometry a tangent is regarded as the limiting position of a line intersecting a curve when the two (or more) points of intersection coincide, and is hence defined as a straight line passing through two (or more) consecutive points of the curve. If the curve be conceived as traced by a moving particle, the tangent at any point of it represents the direction of motion at that point; hence a body moving in a curve, when the restraining force is withdrawn, flies off at a tangent, i.e., along the tangent (cf. the fig. use in c). At a point of inflexion, where the curvature (i.e., deviation from the straight line) changes its direction, the tangent intersects as well as touches the curve.
1655. T. Gibson, Syntaxis Math., xiii. 142. To draw a tangent [cf. 1551 Recorde, Pathway, touche line] to any point assigned in any section, or from any point without the section.
1704. J. Harris, Lex. Techn., I. Tangent, of a Parabola, (or other Conick Section, or Geometrical Curve) is a Right Line Drawn, cutting the Ax Produced, and touching the Section in one point without cutting it.
1706. W. Jones, Syn. Palmar. Mathescos, 221. A Tangent to any point of the Circumference [of a circle] is Perpendicular to the Radius drawn to that Point.
1832. Nat. Philos., II. Introd. Mech., p. xvi. (Usef. Knowl. Soc.). If a stone, whirled round in a sling, gets loose at the point A…, it flies off in the direction AB: this line is called a tangent.
c. In general use, chiefly fig. from b, esp. in phrases (off) at, in, upon a tangent, i.e., off or away with sudden divergence, from the course or direction previously followed; abruptly from one course of action, subject, thought, etc., to another.
1771. Smollett, Humph. Cl. (1815), 219. After having twelve times described this circle, he lately flew off at a tangent to visit some trees at his country house in England.
1815. Paris Chit-Chat (1816), II. 92. The passengers on the roof, being at the highest point of projection flew off in a tangent, and were precipitated … into a field of new-mown hay.
1825. Bentham, Ration. Reward, 393. That manner which they have … of flying off in tangents when they are pressed.
1850. J. F. Cooper, Ways of the Hour, v. Here you are flying off at a tangent from all your engagements and professions, to fall at the feet of an unknown girl of twenty.
1865. Lecky, Ration. (1878), I. 284, note. Flying off at a tangent from his main subject.
1875. Whitney, Life Lang., viii. 150. To abandon the established habits of speech and go off upon a tangent.
1879. Miss Braddon, Clov. Foot, x. Smoking his cigar, and letting his thoughts wander away at a tangent every now and then.
2. The upright pin or wedge fixed at the back of each of the keys of a clavichord, which on the depression of the key pressed up against the string and caused it to sound, acting also as a bridge to determine the pitch of the note. [= Ger. tangent.]
[1614. Prætorius, Syntagma Musicum, III. 68. Es hat aber ein solch Geigenwerk an statt der Tangenten [etc.].]
1878. A. J. Hipkins, in Grove, Dict. Mus., I. 367. The tangents … not only produced the tones but served … to measure of the vibrating lengths required for the pitch of the notes.
1896. C. W. Naylor, Shaks. & Music, 63, note. The German clavichord had ‘tangents’ of brass at the ends of the key levers.
3. Short for tangent scale, tangent galvanometer: See C.
1861. W. H. Russell, in Times, 14 May. His guns were without screws, scales, or tangents.
1905. Preece & Sivewright, Telegraphy, 404. Perhaps the most useful galvanometer for general testing purposes is the Tangent.
4. A straight section of railway track. U.S. colloq.
1895. in Funk’s Stand. Dict.
C. Combinations and special collocations.
(Some of these are examples of the adj. qualifying a sb.)
Tangent backsight, = tangent scale (a); tangent balance, a balance in which the weight is shown on a graduated arc by a pointer attached to the beam; the bent-lever balance, common as a letter-balance; tangent compass = next; tangent galvanometer, a galvanometer in which the tangent of the angle of deflection of the needle is proportional to the strength of the current passing through the coil; tangent scale, (a) in Gunnery, a kind of breech-sight in which the heights of the steps or notches correspond to the tangents of the angle of elevation; (b) a graduated scale indicating the tangents of angles (see quot. 1902); tangent screw, a screw working tangentially upon a toothed circle or arc so as to give it a slow motion for delicate measurements or adjustments; tangent sight, = tangent scale (a).
For tangent cone, line, plane, surface, etc., see A. 1.
1862. Catal. Internat. Exhib., II. XI. 23. The *Tangent back-sight is elevated by a rack and pinion, the latter having a micrometer wheel for finer readings than the divisions on the tangent stem allow.
1873. Maxwell, Electr. & Magn. (1881), II. 325. The current is … proportional to the tangent of the deviation, and the instrument is therefore called a *Tangent Galvanometer.
1876. Preece & Sivewright, Telegraphy, 267. The insulation resistance is the only test which is taken by means of the tangent-galvanometer.
1859. F. A. Griffiths, Artil. Man. (1862), 51. A *Tangent scale is affixed to the breech of Guns, and Howitzers, by means of which the requisite elevation may be given.
1902. Sloane, Stand. Electr. Dict., Tangent scale, an arc of a circle in which the number of graduations in any arc starting from zero are proportional to the tangent of the angle subtended by such arc. The system is for use with tangent galvanometers.
1862. Catal. Internat. Exhib., II. XIII. 5. Circumferenter or miner’s dial, with *tangent screw adjustment.
1877. Knight, Dict. Mech., Tangent-screw, an endless screw tangentially attached to the index-arm of an instrument of precision, enabling a delicate motion to be given to the arm after it has been clamped to the limb, and permitting angular measurements to be made with greater exactness than could be done were the movement entirely effected by hand.
1908. Treat. Serv. Ordn. Roy. Artill., 513. The *tangent sights consist of triangular nickel-plated steel bars graduated on the rear face.
