The Duke actually begins to express his arrogance, though subtly, in the first few lines of the monologue:
. . . for never read
Strangers like you that pictured countenance,
The depth and passion of its earnest glance,
But to myself they turned . . .
And seemed as they would ask me if they durst,
How such a glance came there.
At least after we have read further, in looking back on these lines we can see he's already implying that there was something wrong with that "earnest glance." He then tells his auditor that
She had a hear—-how shall I say—
Too soon made glad, too easily impressed
At this point, his judgmental nature emerges without any subtlety, the complaint being, if we may paraphrase him, that she seemed to like everyone, not just the duke. She was not sufficiently awed by his "gift of a 900-year old name." Now the floodgates of his criticism are open, and he begins to elaborate. She smiled at others just as she did to him; he did not like this, and he states (almost with pride) that...
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