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A silly fellow, or noodle: see NOODLE. Also a child’s penis. Doodle doo, or Cock a doodle doo; a childish appellation for a cock, in imitation of its note when crowing.
Definition taken from The 1811 Dictionary of the Vulgar Tongue, originally by Francis Grose.
Donkey * Doodle SackNathan Bailey's 1736 Dictionary of canting and thieving slang
John S. Farmer's collection of canting songs and slang rhymes
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Francis Grose was independently wealthy, having inherited money from his father, a jeweller. Finding himself overspending, he published a number of books; his Provincial Glossary seems to have been the starting-point for the Vulgar Tongue reproduced here.