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At first the Duke is supportive of Barbantio's complaint.
"Whoe'er he be that in this foul proceeding
Hath thus beguiled your daughter of herself
And you of her, the bloody book of law ....."
Indeed the Duke would have liked to have a say in such a high profile union like Othello and Desdemona. Othello's explanation, however, wins the Duke over. Othello says that he won Desdemona not by witchcraft but loved him "for the dangers I had pass'd, And I loved her that she did pity them." This is the same reason that the Duke and nobleman love Othello so much as well; he wins wars for them.
Othello's contribution to Cyprus much outweighs this indiscretion. In the end, the Duke agrees that what is done is done and let it pass. , "To mourn a mischief that is past and gone Is the next way to draw new mischief on. "
Othello Act 1 scene 3