To cover with shade; to make shady or dark.
From 'be-' prefix + 'shade.' This follows the Old English pattern of creating verbs from nouns to mean 'cover or fill with' that thing. The word survives mostly in poetry.
Shakespeare and other Renaissance writers loved using 'be-' verbs—'beshade' sounds like something a poet would write to describe trees darkening the forest floor, which is exactly what it means.
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