What is the significance of this line spoken by Polonius in Act 2, scene 2, of Hamlet? "Though this be madness, yet there is method in it."

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The key to understanding the meaning of this line is the word "method."  In this case, it should be taken to mean "purpose" or "artfulness."  Read this way, what Polonius is saying is more like "although this is madness, there's a purpose (or a plan, or artfulness) in it.  So what he's suggesting is that Hamlet really isn't crazy but is only pretending.

As it turns out, we know he is right.  Hamlet is not truly crazy but is only acting like he is crazy so as to fool his mother and uncle.

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