In Twelfth Night, why did not Olivia respond to the Duke's proposal?
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Olivia refuses the Duke's proposal because she is has unwittingly fallen in love with Cesario, who is really Viola in disguise. Olivia is a romantic who believes in love at first sight, and she waxes on about her sudden infatuation with "Cesario" in Act 1: Methinks I feel this youth's perfection with an invisible and subtle stealth / To creep in at mine eyes" (1.5.287-88)
Later, Cesario will try to woo Olivia on the Duke's behalf, but will not be successful. Fortunately, the Duke has fallen in love with Viola (Cesario) now, and everything works out happily for all involved in the end.