Where did the term copper for the police come from?

They had copper buttons on their uniforms.

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While it's entirely possible that policemen had copper buttons, or badges, the term may well come from the slang term to catch, or "to cop."

Added: Another theory is that the term derives from the duty rosters kept at the police station houses. When an officer went out on his "beat" the letters

"C.O.P." (meaning 'Constable On Patrol') was written next to their names.