An archaic or old-fashioned form of 'would,' used in Early Modern English (like in Shakespeare) when addressing one person directly, as in 'thou wouldst' (you would).
From Old English and Early Modern English second-person singular conditional form of 'will.' The -st ending was the archaic marker for 'you' (thou wouldst = you would), abandoned as English simplified its grammar after the 1600s.
When Shakespeare wrote 'Wouldst thou...?' he was using the 'thou/thee' form that showed familiarity or informality, like the French 'tu'—you'd only use it with close friends, family, or people of lower status. This grammatical feature reveals social relationships in every sentence of old plays.
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