Do the verb "muse" and the noun "Muse" have a common etymology?
Etymonline says this about that:
muse "to be absorbed in thought," mid-14c., from O.Fr. muser (12c.) "to ponder, loiter, waste time," lit. "to stand with one's nose in the air" (or, possibly, "to sniff about" like a dog who has lost the scent), from muse "muzzle," from Gallo-Romance *musa "snout," of unknown origin. Probably influenced in sense by muse (n.). Related: Mused; musing.
[Middle English, from Old French, from Latin Mūsa, from Greek Mousa; see men-1 in the Appendix of Indo-European roots.]
Word History: Ever since Chaucer first mentions the Muses in a work from around 1390, English poets have invoked these goddesses like so many other versifiers since the days of Homer, who begins both The Iliad and The Odyssey with an invocation of his Muse. The word Muse comes from Latin Mūsa, which in turn is from Greek Mousa. In Greek dialects, this word is found in the variant forms mōsa and moisa, and together these indicate that the Greek word comes from an original *montwa. As to the further origins of this form, a clue is provided by the name of Mnemosyne, the goddess of memory and mother of the Muses. Her name is simply the Greek noun mnēmosunē, "memory"—the faculty of memory was indeed the mother of invention for the ancient Greek professional poets and bards whose job it was to compose new poems in traditional styles on festive occasions, to recite the verses of Homer, and to improvise material whenever they had a memory lapse. Greek mnēmosunē is derived from the root *mnā-, an extended form of the Greek and Indo-European root *men-, "to think." This is the root from which English also gets the words amnesia (from Greek), mental (from Latin), and mind (from Germanic). The reconstructed form *montwa, the ancestor of Greek Mousa, also comes from this root and probably originally referred to "mental power" that enables poets to craft verses—the Muses were the Greek poets' divinized conceptions of the faculties that help them to create and recite poetry.
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