In Hamlet by Shakespeaere, how does Polonius try to prove his theory about the cause of Hamlet's madness in Act 2?
Polonius is perhaps kicking himself here. He totally misjudged the depth of the relationship between Ophelia and Hamlet. In Act 1 he forbids Ophelia to see Hamlet again, telling her that Hamlet, like all young men, is only after one thing--sex. He underestimates his daughter's intelligence and sensitivity, and does not believe that Hamlet is seriously considering marrying Ophelia. Being the dutiful daughter that she is, Ophelia breaks off her relationship with Hamlet.
After she does, Hamlet appears to her in in disarray and acting mad. When Polonius hears of Hamlet's actions, he thinks he has blown his chances of having his daughter marry a prince. So, he scrambles to position himself as the king's main advisor. To do so, he hastens to Claudius to tell him that he knows the secret to Hamlet's madness: heartbreak, and then he proceeds to read (and criticize) a love letter that Hamlet had sent to Ophelia.
The king is not totally convinced, so Polonius tries to pump Hamlet for more information. The quick-witted Hamlet only insults Polonius and gives him nothing. The next step is to go through with the plan conceived earlier: spy on Hamlet and Ophelia as they interact.
Polonius is indeed a "fishmonger," a term which many believe to mean a pimp. Hamlet calls Polonius this when Polonius is interrogating him. He also calls Polonius Jephthah--an Old Testament figure who sacrificed his daughter for a military victory. Polonius is using his daughter to ingratiate himself with the king.
He is an unscrupulous, conniving old fool, lacking Hamlet's wit and intelligence. He is no match for Hamlet, who sees through Polonius's "knee-crooking" ways.
Polonius loves his theory that Hamlet is mad because of his love for Ophelia; he loves it because he believes he will benefit several ways if it's true. His position at court would be even more secure.
Once he posits his theory to the King and Queen, Polonius proposes a plan to "loose" his daughter to Hamlet some afternoon when and where Hamlet typically spends his afternoons reading. That plan actually happens in Act III. For now, Polonius takes his leave of the royal couple and accosts Hamlet as he is walking, hoping to determine for himself if his theory is correct.
He talks with Hamlet, asking some innocuous (harmless, innocent) questions. Hamlet is teasing with him, calling him a fishmonger and shifting the course of their conversation at random moments. It seems as if Polonius has not accomplished his goal, however, because he can't make the determination. At one point in their conversation he said in an aside:
How say you by that? Still harping on my daughter. Yet he knows me not at first; he said I was a fishmonger. He is far gone.
Mere lines later, Polonius says this famous line:
Though this be madness, yet there is method in't.
Since he is unable to make a definitive judgment about Hamlet's madness--feigned or real--he will enact his original plan in the next act.
Polonius, with the king's approval, has his daughter Ophelia meet up with Hamlet in the courtyard while he and the king spy on the two.
Hamlet seemingly knows that this is a set up and a goes through his antics of denying his love for Ophelia, all the while knowing that he is being spied upon.