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 (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 (source of Italian cacciare, Catalan casar, Spanish cazar, Portuguese caçar "to chase, hunt;" see catch (v.)).

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.
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)).