chase (n.1) Look up chase at Dictionary.com
mid-13c., chace, "a hunt," from Old French chace "a hunt, a chase; hunting ground" (12c.), from chacier (see chase (v.)). Meaning "a pursuit" (of an enemy, etc.) is early 14c.
chase (n.2) Look up chase at Dictionary.com
"bore of a gun barrel," 1640s, from French chas "eye of a needle; enclosure," from Vulgar Latin *capsum, variant of Latin capsa "box" (see case (n.2)).
chase (v.) Look up chase at Dictionary.com
c. 1300, chacen "to hunt; to cause to go away; put to flight," from Old French chacier "to hunt, ride swiftly, strive for" (12c., Modern French chasser), from Vulgar Latin *captiare "try to seize, chase" (source of Italian cacciare, Catalan casar, Spanish cazar, Portuguese caçar "to chase, hunt"), from Latin captare "to take, hold," frequentative of capere "to take, hold," from PIE root *kap- "to grasp."

Meaning "run after" developed mid-14c. Related: Chased; chasing. Older European words for "pursue" often also cover "persecute" (Greek dioko, Old English ehtan); modern ones often derive from words used primarily for the hunting of animals.