An archaic or dialectal form of 'besiege'; to surround and attack a place, or to crowd around someone persistently.
Variant or alternative spelling of 'besiege,' from Old French 'assieger' and Latin 'obsidere.' The 'be-' prefix is sometimes used in English as an alternative to 'siege,' though 'besiege' became the standard spelling.
English orthography (spelling) wasn't standardized until recently, so 'beseige' is a witness to spelling variation—what we now spell 'besiege.' Medieval scribes would write words different ways, and modern editors had to choose which versions became official.
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