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mid-19c. in various senses, noun derivative of bounce (v.) in its original sense of "thump, hit." Earliest attested is "boaster, bully, braggart" (1833); also "large example of its kind" (1842); "enforcer of order in a bar or saloon" (1865, American English, originally colloquial).
"The Bouncer" is merely the English "chucker out". When liberty verges on license and gaiety on wanton delirium, the Bouncer selects the gayest of the gay, and -- bounces him! ["London Daily News," July 26, 1883]