Method to my madness
I'm Polonius. I'm a Danish Lord who's pretty self-absorbed and really impressed with the sound of my own voice. I love giving advice even if people aren't after it. And you know what I think?
Though this be madness, yet there is method
in 't. Will you walk out of the air, my lord? (2.2.205-206)
Who Said It and Where
When Hamlet finds out from a ghost that his uncle murdered his father, he has to get to the bottom of it. Complicating matters? Hamlet's mom (Gertrude) married his uncle (Claudius) right after his dad knocked off. So Hamlet wants to know if (1) Claudius is actually guilty of the crime and (2) his mom knows about it.
Naturally, he decides to go mad to get the goss. Wait, what? That's right, Hamlet thinks no one will pay any attention to him if he's mad—and that'll make it easier to eavesdrop on the shenanigans at court. Too bad Claudius is on edge (probably because of that guilty conscience and all). He's all too suspicious of Hamlet.
Polonius thinks he's found the source of Hamlet's madness. Duh, the guy's in love. To prove his point, Polonius reads off some love letters to Claudius that the Prince wrote about how sexy his daughter, Ophelia is (seriously— she's got an "excellent white bosom").
To get to the bottom of things, Polonius and Claudius decide to… spy on their children. (Parents, do not try this at home.) Specifically, they plan to set up a meeting between Hamlet and Ophelia and watch what happens. The King and Queen give the plan the go-ahead and, seeing Hamlet coming, they quickly rush out of the hall.
Hamlet comes in, deliberately misunderstands Polonius's questions, and says things that seem to be irrelevant and totally nuts. He insults Polonius a bunch of times, but Polonius doesn't get it, and writes it off as lovesick madness. Instead, he thinks that Hamlet mentioning his daughter is a good sign that he's got her on the mind. He declares in an aside that he, too, was crazy for love when he was young. If there's one thing Polonius is good at, it's tragically missing the point.
Polonius asks what Hamlet is reading, and Hamlet gives a nonsensical explanation about foolish and disgusting old men (now making fun of Polonius for being old). He quips that Polonius would be as old as Hamlet, if time and Polonius went backwards. Polonius, because he's not too clever, thinks there is some very sensible stuff here, only further proving Hamlet's point that Polonius is indeed a foolish man.
Then, Polonius suggests Hamlet should take a walk out in the fresh air, which Hamlet obviously uses as an opportunity to say he'd like to walk out of the fresh air and into his grave. He's always talking about death. Polonius yet again interprets Hamlet's deliberately foolish speech to be very sensible. He says men speak more honestly in madness than when they are sane.
We might be crazy, but we're going to call that one for Hamlet.