What are the steps taken by Polonius to prove that Hamlet is mad?
Well, few people question Hamlet's madness. The real question is what causes the madness. After speaking with Ophelia about Hamlet's odd behavior, Polonius is convinced that Hamlet has gone made because he is in love with Ophelia and has been denied access to her. He intends to prove that Hamlet is mad for love and requests the king help him. Below is an excerpt from Act II Scene two wherein he and Claudius discuss a plot to determine the root of Hamlet's madness:
How can we find out more?
At such a time I'll loose my daughter to him:
Be you and I behind an arras then;
Mark the encounter: if he love her not,
And he not from his reason fall'n thereon
Let me be no assistant for a state,
But keep a farm and carters.
Polonius here says that he will get his daughter to speak to Hamlet and while he and Claudius hide and spy on the two young lovers. Polonius is convinced that the encounter will cause Hamlet to react in such a way that will prove his love for Ophelia and prove that his madness is a result of being denied that love.
When this meeting doesn't convince Claudius, Polonius asks for a second opportunity to spy on Hamlet. He will hide in Gertrude's chamber and eavesdrop on a conversation between Hamlet and his mother. He is still convinced that his spying will prove that Hamlet is indeed mad for love.