Both Uses

muse
in
Persuasion, by Jane Austen
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She occupied her mind as much as possible in such like musings and quotations; but it was not possible, that when within reach of Captain Wentworth's conversation with either of the Miss Musgroves, she should not try to hear it; yet she caught little very remarkable.†
Chpt 10 *
Prettier musings of high-wrought love and eternal constancy, could never have passed along the streets of Bath, than Anne was sporting with from Camden Place to Westgate Buildings.†
Chpt 21

Definitions:

(1) (muse as in: her musings)
reflect (think) deeply on a subject -- perhaps aloud
(2) (muse as in: She was his muse)
the source of an artist's inspiration - especially a person or mythological goddess

The 9 muses were the daughters of the Greek gods Zeus & Mnemosyne. Each of the muses was thought to inspire and preside over one of the arts.
(3) (meaning too rare to warrant focus)
More rarely, muse is used as a synonym for "poet". In Shakespeare's time, muse could be used in the sense of being astonished or puzzled by something strange or unexpected.