tangent

Etymology

It is borrowed from Latin tangēns (“touching”), the present participle of tangō (“to touch”),^{[2][3][4][5][6]} used in the phrase līnea tangēns (“touching line”).^[4][7][8] The Danish mathematician Thomas Fincke used the participle as a noun in his “Geometria Rotundi” (1583),^[2][3][9] and the German mathematician Bartholomäus Pitiscus took it up in 1595.^[9] Merriam-Webster and Barnhart treat the noun as New Latin.^[10][5] The Latin verb, with the perfect tetigī, is probably cognate with Ancient Greek τεταγών (tetagṓn, “having grasped”) and perhaps with Old English þaccian (“to pat, stroke”),^[5][9][10] from Proto-Indo-European *tag- (“to touch, handle”).^[3][4][5] The geometrical noun is tangente in French.^[2]

Earliest recorded meaning

In Blundeville's Exercises (1594), both as an adjective, of a line touching a curve, and as a noun, one of the three basic functions of trigonometry.^[2][5] Etymonline dates the noun to the 1580s.^[3]

Semantic evolution

The trigonometric tangent was at first a function of a circular arc, not of an angle,^[2] and Johnson (1755) still defined it as a straight line raised at right angles on the end of a radius, touching the circle without cutting it.^[11] The geometrical noun, a straight line touching a curve, is recorded from 1655.^[2][3] The image of a body flying off along the tangent when released from circular motion gave the figurative phrase fly off at a tangent, first in Smollett's Humphry Clinker (1771),^[2][3][12] and the adjective "divergent, erratic" in a letter of Burns (1787).^[2][5] The derivative tangential is recorded from 1630;^[5] Barnhart dates its sense "slightly connected with a subject" to 1825 and "digressive" to 1867,^[5] while Etymonline gives the 1825 sense under tangent itself.^[3] Technical senses followed in the 19th century: the striking pin of a clavichord key (1878), a bicycle wheel with spokes tangent to the hub (1886), and in American colloquial use a straight stretch of railway track (1895).^[2] A verb, to go off at a tangent, is recorded from 1920;^[2] the Shorter Oxford calls it literary.^[13]

Timeline

• 1594 – adjective, of a line touching a curve; noun, the trigonometric function.^[2][5]
• 1630 – the adjective tangential.^[5]
• 1655 – noun, a straight line touching a curve.^[2][3]
• 1771 – figurative, in fly off at a tangent.^[2][3]
• 1787 – adjective, divergent, erratic.^[2][5]
• 1802 – relating to the sense of touch.^[2]
• 1825 – slightly connected with a subject.^[3]
• 1878 – the striking pin of a clavichord key.^[2]
• 1886 – of a bicycle wheel with spokes tangent to the hub.^[2]
• 1895 – US colloquial, a straight stretch of railway track.^[2]
• 1920 – verb, to go off at a tangent.^[2][13]

Meanings

Noun

1. A straight line, curve or surface that touches a curve or surface at one point without crossing it there.^[7][10][2]
2. In trigonometry, the function of an angle that in a right-angled triangle is the ratio of the opposite side to the adjacent side, equal to the sine divided by the cosine; abbreviated tan.^[7][10][2]
3. A sudden change of course, or a digression from the subject, especially in go off at (or on) a tangent: Halfway through the meeting she went off on a tangent about parking.^[2][3]
4. In surveying, the straight part of a line between curves; in American colloquial use, a straight stretch of railway track.^[7][2]
5. In a clavichord, the small metal piece at the back of a key that strikes the string.^[7][2]

Adjective

1. Touching at a single point: a plane tangent to the sphere.^[10][2]
2. Of two curves or surfaces, having a common tangent line or plane at a point.^[10]
3. Diverging from an original purpose or course; only slightly connected; tangential.^[10][2]

Verb

1. (literary) To go off at a tangent; to diverge from one's course.^[2][13]

Word family

• arctangent
• cotangent
• subtangent
• tangential
• inverse tangent
• tangent plane

Related words

• tangency
• tact
• tactile
• contact
• tangible
• contingent
• intact
• attain
• integer
• task
• tax
• taste

References

1. ^ Wiktionary, the free dictionary, https://en.wiktionary.org
2. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z 27 Oxford English Dictionary Online (accessed March 2026), Oxford: Oxford University Press
3. ^ a b c d e f g h i j k Harper, Douglas. Online Etymology Dictionary, https://www.etymonline.com
4. ^ a b c The American Heritage Dictionary of the English Language (online 2024), 5th ed., Boston: Houghton Mifflin Harcourt
5. ^ a b c d e f g h i j k Barnhart, Robert K. (1988), The Barnhart Dictionary of Etymology
6. ^ Hoad, T. F., ed. (1986), The Concise Oxford Dictionary of English Etymology, Oxford: Clarendon Press
7. ^ a b c d e Collins English Dictionary (2023), Glasgow: HarperCollins
8. ^ Dictionary.com Unabridged (2016), based on the Random House Unabridged Dictionary, New York: Random House
9. ^ a b c Klein, Ernest. A Comprehensive Etymological Dictionary of the English Language. 2 vols. Amsterdam: Elsevier, 1966–1967.
10. ^ a b c d e f g Merriam-Webster Unabridged Dictionary (2024), online ed., Springfield, Mass.: Merriam-Webster
11. ^ Johnson, Samuel (1755), A Dictionary of the English Language, London; 4th ed. 1773; Johnson's Dictionary Online, ed. Beth Rapp Young et al.
12. ^ Weekley, Ernest (1921), An Etymological Dictionary of Modern English, London: John Murray
13. ^ a b c Shorter Oxford English Dictionary on Historical Principles (2002), 5th ed., ed. William R. Trumble and Angus Stevenson, Oxford: Oxford University Press