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In Act 2, Scene 2, lines 204-209 of Hamlet, what does Polonius say about madness and sanity?

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In Act 2, Scene 2, lines 204-209 of Hamlet, Polonius says that madness can bring about a happiness that rationality and sanity cannot achieve. He notes that Hamlet's seemingly mad replies are often profound, highlighting the fine line between sanity and madness. Polonius believes Hamlet's madness is due to lovesickness for Ophelia, echoing Shakespeare's exploration of love and madness in other works.

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Polonius indicates that there's a "method" or rationale to madness and, in these specific lines, notes that it achieves a happiness or joyfulness that sane, reasonable people often can't attain. (Polonius has no idea that Hamlet is playing with him or that Hamlet finds him a tiresome old fool.)

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a deeper level, Polonius's musings unconsciously interrogate the sharp distinction we make between the states of sanity and insanity. By noting the happiness Hamlet exhibits

A happiness that often madness hits on, which reason and sanity could not so prosperously be delivered of

Polonius calls into question the way we denigrate madness and automatically assume it is a less desirable state than sanity.

Polonius thinks Hamlet's madness may be a form of lovesickness for his daughter Ophelia. This brings to mind, too, the connection between love and madness that Shakespeare explores more thoroughly in A Midsummer's Night Dream.Hamlet, in a darker way, implicitly raises the question of whether being "mad" or behaving seemingly irrationally is worse than conforming to a cold rationality.

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Evidently you are asking about the conversation between Polonius and Hamlet in Act 2, Scene 2. This is a comical scene because Hamlet is only pretending to be mad, and doing such a good job of it that he has Polonius completely fooled. Polonius, in an aside, tells himself, quite correctly, "How pregnant sometimes his replies are! A happiness that often madness hits on, which reason and sanity could not so prosperously be delivered of." This is an astute observation. Insane people often say things that not only seem perfectly rational but seem strikingly profound and original. Hamlet apparently knows this and is deliberately replying to Polonius's questions with answers that are not to the point but are not totally inappropriate either. In this, Hamlet is using the same kind of literalness that is being employed by the gravedigger in Act 5, Scene 1. For example:

Hamlet    What man dost thou dig it for?

Clown      For no man, sir.

Hamlet     What woman, then?

Clown      For none, neither.

Hamlet     Who is to be buried in't?

Clown      One that was a woman, sir. But rest her soul, she's dead.

Unlike Polonius, Hamlet knows the gravedigger is only playing mind-games with him. Polonius is a complex character. He is both wise and foolish. He is old and no longer mentally agile or adaptable.

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Because the lines referred to in Act II, Scene 2, are said as an aside, they are private revelations of the thoughts of Polonius.  Here in Shakespeare's "Hamlet," Polonius, the corrupt sycophant of the Danish court, attempts to convince Claudius and Gertrude that Hamlet is mad.  When he talks with Hamlet their conversation has many non-sequiturs in it, such as the lines in which Polonius asks Hamlet what he reads and Hamlet replies, "Words, words, words" (II,ii,190) and punned insults in which Hamlet says that Polonius might grow old as Hamlet is if "like a crab, you could go backward."  So, when the conversation continues between Hamlet and Polonius and the courtier asks,  "Will you walk out of the air, my lord?"(II,ii,202), and Hamlet queries, "Into my grave?" (II,ii,203), then Polonius says in an aside, "How pregnant...." 

Polonius remarks are much like those of Emily Dickinson's in her poem "Much Madness is divinest Sense."  That is, the mind that loses its focus on reality is sometimes better able to discern other matters--perhaps, in much the way intuition works, by a sixth sense.  For, Hamlet, perhaps, intuitively senses that Polonius, too, wishes to do Hamlet ill, just as Rosencrantz and Guildenstern do.  "How pregnant," how full of meaning, Hamlet's remarks are, like the 'divinest sense" (divine as in to guess) of which Miss Dickinson writes.

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I'm not really sure which lines you're talking about because different versions seem to have different line numbers. I am assuming that you are thinking of these lines that Polonius says

How pregnant (220)
sometimes his replies are! a happiness that often madness
hits on, which reason and sanity could not so prosperously
be delivered of.

The basic idea here is that Polonius is saying that only a crazy person would be able to speak as well as Hamlet is speaking in this scene -- only a crazy person would be able to use words so well.

I think that what he means here is that sane people are too confined by what people expect and can't come up with really creative ways of expressing themselves the way a crazy person can.

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