Why do Polonius and Claudius spy on Hamlet with Ophelia?
The scene in which Claudius and Polonius spy on Hamlet and Ophelia is probably intended to be staged in an artificial way. Hamlet would be standing with his back turned to Claudius and Polonius and facing Ophelia, who would be standing at extreme stage right. Claudius and Polonius would be standing in plain sight at some small distance behind Hamlet. The two spies would be leaning sharply forward at close to a 45-degree angle, each man with a hand cupped around one ear to show they were listening intently. They would look slightly ridiculous just from their postures. They are listening for the signs of young love, but they have forgotten what young love is like. They wouldn't recognize it if they heard it. Hamlet would not see them but would know they are there because Ophelia could see them very clearly over Hamlet's shoulder, and her guilty knowledge would show on her innocent face. She is like a mirror in which he can see behind him. The scene would be funnier if it were played in an...
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In the middle of 2.2, after the ambassadors have departed, Polonius goes off on his loquacious ramblings about Hamlet being mad. Polonius has a theory that Hamlet has gone mad over the loss of Ophelia's love. Both Gertrude and Claudius are sceptical. Claudius says, "How may we try it further?" Polonius responds, "If circumstances lead me I will find where truth is hid, though it were hid indeed within the center."
Now, go back quickly to 2.1 where Polonius is talking to Reynaldo about spying on Laertes. He explains to Reynaldo the aim of his tactic: "Your bait of falsehood takes this carp of truth." Polonius now extends this idea to catch the truth behind Hamlet's madness using Ophelia as the bait of falsehood. Really, he is looking for confirmation of his own belief; but he must convince Claudius and Gertrude. So in 2.2 he continues as he explains how Hamlet walks the halls at times. He will then "loose" his daughter to Hamlet. The King in hiding can then judge for himself.
In 3.1 Polonius baits his hook with poor Ophelia. "Ophelia walk you here...Read on this book, that show of such an exercise may color your loneliness." Polonius figures that if Hamlet is mad for Ophelia's love then dangling her in front of Hamlet should generate a reaction. The falsehood of this bait is making it appear that Ophelia hasn't been planted. The appearance of loneliness serves to draw Hamlet out.