Hamlet and Polonius
At this point Hamlet knows his father was killed, he also knows the name of the murderer, the new kink and the queen, his mother. He also knows that they were lovers. But that is all based on a ghost's tale, so it must be proved.
Hamlet's mind is shaken by the events and he goes or pretend to go insane.
In fact on verse 1 his mother calls Hamlet 'the poor wretch'. Eventually Hamlet has a long talk with Polonius, in which he shows his strange state of mind (verso 8: you are a fishmonger) alternating contradictory considerations. Some of his stataments seem to come out of the blue sky: verso 15/16 'for... carrion' which is followed by 'have you a daughter?'. So speaking about dishonesty and comparing dishonest men to 'a dead dog' that gives life to worms (see how trivial the words he uses are...) he ends up asking Polonius if he has a daughter. Of course Hamlet knows Polonius has one: Ophelia. So he is pretending here. V18: 'let... sun.. conception.. conceive': that sounds strange again. But if 'to concieve' means 'to think', it also means 'to give birth'. Hamlet does not end his sentence, he lets Polonius draw a conclusion.
Verso 20: Polonius, talking to the audience, (nds, possibile collegamento con il metateatro di Plauto) understands Hamlet's words that way: 'Still sharpening...'. He also lets the audience know that at this point he does not stand Hamlet any longer. When asked what he is reading Hamlet replies 'words, words, words', but then he hits Polonius straight by attacking him as an aged man. He mentions all the negativity which is tied to that period of life. And he says it is all true, though he had first called what he was reading 'slanders' and the writer 'satirical rogue'. In this sense Hamlet is very cruel. Hamlet compares Polonius to a worm, that moves crawling. Again Polonius tells his doubts to the audience. Sometimes madness is brighter than insanity. See that short exchange: vv. 35-38, which sounds ambiguous, but expresses Hamlet's irony once more. Hamlet asks Polonius: 'Do you want me to die?'
On v. 43 Hamlet plays with Polonius's words: I will take leave of you' and tells him he's glad not to have him around. But, knowing how things are at that point, he adds: 'you won't take my life', you won't kill me, in other words. He ends saying 'this tedious old fools.'