Shakespeare enjoyed irony. In Act1 Scene 2,Polonius says, "Since brevity is the soul of wit...I will be brief." How is this ironic? 

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In Shakespeare's Hamlet, Polonius says in Act 2.2.91-96 (rather than 1.2): 

Therefore, since brevity is the soul of wit

And tediousness the limbs and outward flourishes,

I will be brief:  your noble son is mad.

Mad call I it, for, to define true madness,

What is 't but to be nothing else but mad?

But let that go. 

It is ironic when Polonius says he will be brief, because he is anything but brief.  This passage is just the beginning of Polonius' speeches in which he tries to convince Gertrude and Claudius that Hamlet is mad, but even in just these few lines we see how full of "art" he is, and how empty of substance.  He's talking in circles, defining and redefining.  He uses the word mad to define mad

Gertrude recognizes this immediately, telling him:

More matter, with less art.

Polonius is a nonsensical windbag, who talks much while saying little.  And he is anything but brief.

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