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A simpleton or fool. Also a kind of low cart, with a seat before it for the driver, used in and about Dublin, in the manner of a hackney coach: the fare is just half that of a coach, for the same distance; so that for sixpence one may have a set down, as it is called, of a mile and half, and frequently a tumble down into the bargain: it is called a noddy from the nutation of its head. Knave noddy; the old-fashioned name for the knave of trumps.
Definition taken from The 1811 Dictionary of the Vulgar Tongue, originally by Francis Grose.
See also the definition in Nathan Bailey's 1736 dictionary of canting and thieving slang.
Noddle * Noisy Dog RacketNathan 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.