1590s, "meeting at a point without intersecting," from Latin tangentem (nominative tangens), present participle of tangere "to touch," from PIE root *tag- "to touch, handle." 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.