Hey diddle diddle, the cat and the fiddle shows up on the occasional pub sign even today. Why THAT incongruous pairing? (I think I may start a pub and call it "The Sloth and Stopwatch.") As it happens, back in 1829 somebody thought they knew why.
***The Spectator, also, if I remember right, declared the old sign of the Cat and the Fiddle to be quite beyond his comprehension. In truth, no two objects in the world have less to do with each other than a cat and a violin; and the only explanation ever given of this wonderful union, appears to be, that once upon a time a gentleman kept a public house with the sign of a Cat, and a lady one, with the sign of a Fiddle, or vice versa; that these two persons fell in love, married, and set up an Inn, which, to commemorate their early loves, they called the Cat and the Fiddle. Such reasoning is exceedingly poetical, and also (mind, also, not therefore) exceedingly nonsensical. No, Sir, the Cat and the Fiddle is of greater antiquity. Did you ever read the history of Rome? Thence comes the Cat and the Fiddle, in somewhat a roundabout way, perhaps, but so it is:
Vixtrix causa Diis placuit, sed victa Catoni. (Or, The victorious cause pleased the gods, but the conquered cause pleased Cato - curator)
. . . In the days of good Queen Bess, when those who had borne the iron yoke of Mary, ventured forth and gloried in that freedom of conscience which had lately been denied them, a jolly innkeeper having lately cast off the shackles of the old religion, likened himself to the old Roman (Cato, who wrote the sentence above- curator), and wrote over his door L'Hostelle du Caton fidele. The hostelle and its sign lasted longer than the worthy gentleman, and having gone shockingly to decay, was many years afterwards reestablished. But alas! the numerous French words once mixed with our language had vanished, become barbarized, and ground down into a heterogeneous mass of sounds; and le Caton fidele was no longer known to his best friends when resuscitated under the anomalous title of the Cat and the Fiddle!
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Sounds plausible? From The Atheneum, Or Spirit of the English Magazines (Vol. II, April - September 1829; Boston, John Cotton), p. 327.