See ZN2571 in the broadside ballad index on my website for "A Choice of Inventions", 1632. www.erols.com. The ballad is reproduced in facsimile in the Opie's 'Oxford Dictionary of Nursery Rhymes', Plate 22 (2nd edition).

The 'wise men of Gotham' was a contradiction in terms because Gothamites were held to be incredibly stupid. There was an older chapbook with about a dozen tales of the stupidity of Gothamites.

The only tale I can recall off-hand is "The Penning of the Cuckoo", where in order to catch a cuckoo, the Gothamites built a pen around it. When the pen was done the cuckoo just flew away (no top on pen).