1590s, "meeting at a point without intersecting," from Latin tangentem (nominative tangens), present participle of tangere "to touch," from PIE root *tag- "to touch, to handle; border on; taste, partake of; strike, hit;" figuratively "affect, impress; trick, cheat; mention, speak of" (cognates: Latin tactus "touch;" Greek tassein "to arrange," tetagon "having seized;" Old English þaccian "stroke, strike gently"). First used by Danish mathematician Thomas Fincke in "Geomietria Rotundi" (1583). Extended sense of "slightly connected with a subject" is first recorded 1825. Related: Tangence; tangency.

1590s as a geometric function, from tangent, adj.. From 1650s as "a tangent line." Figurative use of off on a tangent is from 1771.