"cop"

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does not stand for "constable on patrol" or "constabulary of police".
The noun "cop" (first attested meaning "policeman" in 1859) is short
for "copper" (first attested meaning "policeman" in 1846).  "Copper"
in this sense is unlikely to derive from copper buttons or shields
worn by early policemen.  Rather, dictionaries derive it from "to
cop" (first attested meaning "to grab" in 1704 and meaning "to
arrest" in 1844).  "To cop" may come Dutch kapen = "to steal"; or
it may come from Old French dialect caper = "to take", from Latin
capere.