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Polonius informs Claudius and Gertrude that Hamlet has been driven mad by love for Ophelia. To prove his point, he reads off some love letters that the Prince wrote, often discussing what a magnificent figure she has (seriously – an "excellent white bosom").
They decide to…spy on their children. They plan to set up a meeting between Hamlet and Ophelia, in the location where Hamlet has taken to pacing insanely, 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 thing's that seem to be irrelevant. But are they?
Polonius first asks if Hamlet recognizes him, and Hamlet replies he knows him as a "fishmonger" (that's a guy who sells fish but it's also code for "pimp"), which maybe means that Hamlet knows Polonius is using his daughter and her romantic relationship with Hamlet, for personal gain. (Spying for Claudius = getting in good with the king).
Hamlet also quips with Polonius that honest men are rare, and that the sun would breed maggots in a dead dog because that flesh is good enough to be kissed by the sun. He continues by asking if Polonius has a daughter. Polonius confirms this, and Hamlet replies that she should be kept out of the sun, as the sun would likely conceive with her too, thus likening our delicate Ophelia's womb to a dead dog rotting in the sun, which would then breed maggots.
Polonius, who continues to not get it, doesn't take any offense to this daughter/dead dog comparison, and instead 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, and thus misses the whole point of Hamlet's offensive speech about rotting and breeding.
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.
Polonius suggests Hamlet should take a walk out in the fresh air, and Hamlet takes the opportunity to say he'd like to walk out of the fresh air and into his grave.
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.
Polonius offers to take leave of Hamlet, and Hamlet replies no other loss would make him happier, except the losing of his life.
As Polonius is leaving, Hamlet dismisses him as a tedious old fool, which makes it clear that though Polonius thinks he's understood Hamlet, he's only fallen into Hamlet's trap of making everyone think that he's mad.