
The first part of this scene, between Polonius and Reynaldo, is not necessary to the plot of the play, and is often cut from performances. That's a shame, because it's lots of fun. Polonius thinks he is very wise, and Reynaldo, a sort of superior butler, knows otherwise. 
Polonius is sending money and notes (maybe more advice) to his son, Laertes, but he's suspicious about what Laertes might be up to. He seems to assume that if Laertes is fooling around, he will lie about it, so Polonius has a plan. His fetch of wit (2.1.38) is for Reynaldo to find some friends of Laertes, and say certain things about him, such as he's very wild, and see what the friends say. (Later, Hamlet, in trying to discover if the King is really guilty of murder, will use a similar device to discover the truth: He'll watch the King's reactions while something like the murder of King Hamlet is shown in a play.) 
Polonius is quite proud of his plan, because, as he says, Thus do we of wisdom and of reach, / With windlasses and with assays of bias, / By indirections find directions out (2.1.61-63). Of course, when Polonius mentions we of wisdom and of reach, he's thinking of himself, despite the fact that he takes too long to say all of this, forgets what he's saying, and contradicts himself. Reynaldo politely tolerates all of this, and goes about his business. 
Exit Reynaldo. Enter Ophelia:
Just as Reynaldo leaves, Ophelia comes rushing in, badly frightened. Without warning, Hamlet has come into her closet (i.e., her study or sewing-room), seized her wrist, stared at her, sighed, and gone back out, all without saying a word. His clothes were unlaced and unbuttoned, and he had a look so piteous in purport / As if he had been loosed out of hell / To speak of horrors (2.1.79-81). It's interesting and puzzling that she should describe him very much as the Ghost might be described. However, Polonius isn't puzzled. He immediately jumps to the conclusion that This is the very ecstasy of love (2.1.99). He says that he is sorry he misjudged Hamlet, but he is most interested in rushing off to tell the King. 
Polonius' exit lines, This must be known; which, being kept close, might move / More grief to hide than hate to utter love (2.1.115-116), are not the clearest that Shakespeare ever wrote, but they need to be considered carefully. He apparently means that if he doesn't tell the King that Hamlet is crazy because Ophelia dumped him, there will be more trouble than if he does tell. This implies or assumes a couple of things. First, the King is very interested in finding out what is wrong with Hamlet. And, since this is the first time we've seen any sign of the antic disposition that Hamlet said he might put on, we may assume that the King hasn't seen any sign of it, either. Therefore, the antic disposition is probably not the reason for the King's interest in Hamlet's state of mind. We can guess that Claudius sees Hamlet as a potential political rival, and that Claudius senses danger in Hamlet's continued mourning for his father. Second, the phrase hate to utter love means that Claudius will hate to hear that the daughter of his close advisor has a relationship with Hamlet. This might lead us to guess that Polonius' real--though unstated--reason for putting a stop to the relationship was to make sure that he was on the right side: the King's. 
Summary of Act 2, Scene 2: The King tells Rosencrantz and Guildenstern to investigate Hamlet's madness . . . Polonius's theory of Hamlet's madness . . . Polonius examines Hamlet . . . Rosencrantz and Guildenstern examine Hamlet . . . The players arrive . . . Hamlet's second soliloquy. 
· Enter King and Queen, Rosencrantz and Guildenstern. 
--King: Welcome, dear Rosencrantz and Guildenstern (2.2.1). 
· Exeunt Rosencrantz and Guildenstern. Enter Polonius.
--Polonius: I have found / The very cause of Hamlet's lunacy (2.2.48-49). 
· Enter Ambassadors [Voltemand and Cornelius].
--King: Well, we shall sift him (2.2.58). 
· Exeunt Ambassadors [Voltemand and Cornelius].
--Polonius: My liege, and madam, to expostulate (2.2.86). 
· Enter Hamlet. Exeunt King and Queen.
--Queen Gertrude: But, look, where sadly the poor wretch comes reading (2.2.168). 
· Exit Polonius. Enter Rosencrantz and Guildenstern.
--Guildenstern: My honor'd lord! (2.2.222). 
· A flourish trumpets for the players. Enter Polonius.
--Hamlet: I am but mad north-north-west (2.2.378). 
· Enter Players.
--Hamlet: Come give us a taste of your quality, come, a passionate speech (2.2.431-432). 
· Exit all but Hamlet.
--Hamlet: O, what a rogue and peasant slave am I! (2.2.550).
Enter King and Queen, Rosencrantz and Guildenstern:
The King welcomes dear Rosencrantz and Guildenstern (2.2.1), and immediately gets down to business. They, friends of Hamlet, are supposed to hang out with him, so that they can find out what's wrong with him. The King says that he cannot dream of what might be wrong with Hamlet, other than his father's death. Of course, we've already learned that the King killed Hamlet's father, so we may suspect that what the King really wants to know is what Hamlet knows or suspects, and what Hamlet might do. 
The Queen seconds the King's request by telling them how much Hamlet likes them, and by suggesting that there might be some money in it for them, or--as she puts it--such thanks / As fits a king's remembrance (2.2.25-26). Rosencrantz and Guildenstern not only agree to do what they're asked, they suck up. They know, and say, that the King could simply command, rather than ask, and so they're glad he asked. 
Exeunt Rosencrantz and Guildenstern. Enter Polonius:
As soon as Rosencrantz and Guildenstern leave to seek out Hamlet, Polonius comes bustling in with two pieces of news: The ambassadors to Norway have had success, and he has discovered the very cause of Hamlet's lunacy (2.2.49). The King wants to hear what Polonius has to say about Hamlet, but Polonius insists on bringing in the ambassadors, and saving the news about Hamlet as the fruit [i.e., the dessert] to that great feast. Polonius steps out to fetch the ambassadors, and the King and Queen are alone for a moment. The King wonders aloud if Polonius really has found the source and head of Hamlet's distemper. The Queen replies with a bit of common sense: I doubt [suspect] it is no other but the main; / His father's death, and our o'erhasty marriage (2.2.56-57). 
Enter Ambassadors [Voltemand and Cornelius]:
The King and Queen's moment alone is soon over. In comes Polonius, with Voltemand and Cornelius in tow. Voltemand tells the King that the King of Norway, who was sick, thought that Fortinbras was raising his army to fight the Poles, but when he received the letter from the King, he called Fortinbras in, learned the truth, and gave him a rebuke. Now Fortinbras has promised never to direct any hostilities toward Denmark, and to use the army only to attack Poland. The King of Norway is happy with this, and wants the King's permission for Fortinbras to pass through Denmark on his way to Poland. All this may sound fishy, but the King seems satisfied and says he'll think about it. Later in the play, he has given the requested permission, because Fortinbras briefly appears, leading his army across the stage toward Poland. The King thanks Voltemand and Cornelius, and they exit, never to be seen again. 
Exeunt Ambassadors [Voltemand and Cornelius]:
As soon as the ambassadors are gone, Polonius, saying he will not expostulate on the obvious, expostulates. And after he says that brevity is the soul of wit / And tediousness the limbs and outward flourishes (2.2.90-91), he proceeds to be very tedious as he explains his theory of Hamlet's madness. He makes his case by reading a love-letter written by Hamlet to Ophelia, and then explaining how he, faithful and honorable, got Ophelia to lock herself away from Hamlet. He concludes in his windy way: 
And he, repulsed--a short tale to make--
Fell into a sadness, then into a fast,
Thence to a watch, thence into a weakness,
Thence to a lightness, and, by this declension,
Into the madness wherein now he raves,
And all we mourn for. (2.2.146-151)
The King and Queen are almost persuaded, but still doubtful, and so Polonius boasts that I will find / Where truth is hid, though it were hid indeed / Within the centre. The King asks how his theory may be tested, and Polonius offers to loose Ophelia to Hamlet while he and the King hide behind a curtain to overhear their conversation. 
Enter Hamlet. Exeunt King and Queen:
The King agrees to Polonius' plan for spying on Hamlet, but just then Hamlet himself comes wandering into the room, reading a book.Polonius is eager to examine Hamlet for himself, and he shoos away the King and Queen, so that he can board Hamlet. He starts right in, saying Do you know me, my lord? as though Hamlet is so far gone that he can't recognize Polonius. Hamlet replies, Excellent well; you are a fishmonger (2.2.174). And so goes the rest of the encounter, with Polonius asking more dumb questions and Hamlet replying with insults which Polonius doesn't understand because he thinks they only show just how crazy the prince is. In the course of the conversation Hamlet mocks Polonius' attitude towards Ophelia, telling him that conception is a blessing: but not as your daughter may conceive (2.2.184-185). And Hamlet also mocks Polonius' appearance and lack of self-knowledge by pretending to read a passage from his book that describes old men as having wrinkled faces and a plentiful lack of wit; of course, he is really describing Polonius. Polonius sort of gets the idea that something is going on, but all he can figure out is that Though this be madness, yet there is method in't (2.2.205-206). 
Exit Polonius. Enter Rosencrantz and Guildenstern:
Baffled, Polonius takes his leave of Hamlet, and just as he does, Rosencrantz and Guildenstern show up, so that they, too, can take a shot at finding out what's wrong with Hamlet. 
Hamlet greets his old friends heartily, and asks how they're doing, which leads to a good-old-boy off-color joke about the secret parts of Fortune. Then Hamlet asks, What news? He means what we mean when we say What's up? Rosencrantz and Guildenstern don't have a good answer to that question. They didn't come just to hang out with Hamlet, and they didn't just happen to run into him while they were doing something else. They came to find out what his problem is, but they're not supposed to tell him that. So Rosencrantz answers Hamlet's What news? with None, my lord, which is a little white lie. 
Hamlet then invites Rosencrantz and Guildenstern to be on his side. He asks, What have you, my good friends, deserved at the hands of Fortune, that she sends you to prison hither? (2.2.239-241). But Rosencrantz and Guildenstern are employed by the King of Denmark, so they can't jump in and agree that Denmark is a prison. When Hamlet insists that To me it is a prison, Rosencrantz takes that as an opportunity to divert the conversation to an interesting topic: Hamlet's ambition. If Rosencrantz and Guildenstern could report back to the King that Hamlet's problem is that he wants to be king, that would be news indeed. Hamlet denies that he is ambitious, saying, O God, I could be bounded in a nutshell and count myself a king of infinite space--were it not that I have bad dreams (2.2.254-256). 
However, Rosencrantz and Guildenstern don't give up easily, and spar a little over the meaning of ambition, until Hamlet gets tired of the whole thing and suggests that they go to th' court. Rosencrantz and Guildenstern say We'll wait upon you, as though they have nothing better to do than just tag around with him. This apparently reminds Hamlet that they never really answered his question, so he asks it again: But, in the beaten way of friendship, what make you at Elsinore? Rosencrantz replies with a half-truth: To visit you, my lord, no other occasion. Hamlet suddenly inuits the truth and asks Were you not sent for? (2.2.274). Rosencrantz and Guildenstern are surprised, and a little sarcastic brow-beating from Hamlet gets them to confess that they were indeed sent for. 
The discovery that his supposed friends are really the king's spies sends Hamlet into a kind of philosophical orbit. He tells Rosencrantz and Guildenstern that he knows that they were sent for because he has lost all of his mirth. Not only that, but to him the earth is nothing but a sterile promontory within a foul and pestilent congregation of vapors. He goes on, in a passage that is often quoted as an example of the Renaissance belief in the dignity of man: 
What a piece of work is a man! how noble in reason! how infinite in faculty! in form and moving how express and admirable! in action how like an angel! in apprehension how like a god! the beauty of the world! the paragon of animals! (2.2.303-307)
This kind of idea about man was one of the inspirations of famous Renaissance artists (think of Michelangelo's statue of David), but Hamlet's conclusion is a question, and yet to me what is this quintessence of dust? 
At this point Rosencrantz and Guildenstern are lost, or distracted, or feeling smug that Hamlet is talking crazy. Whatever, they are smiling, and Hamlet accuses them of having their minds in the gutter. He says, man delights not me: no, nor woman neither, though by your smiling you seem to say so (2.2.309-310). Rosencrantz and Guildenstern cover themselves by saying they were only thinking about how disappointed the players (a company of actors) are going to be when they show up. Rosencrantz and Guildenstern met them on the road and were told that they were coming to entertain Hamlet, but Hamlet doesn't seem to be in the mood. 
The explanation Rosencrantz and Guildenstern give for their smiling seems rather lame, but Hamlet is more interested in the players than in his two friends. He asks all about them, and finds--in a passage that is often cut from performances--that these tragedians of the city are on the road because boy actors have become more popular. This information prompts Hamlet to think about the current situation in Denmark. He reflects that it's not so strange that the public has suddenly taken a liking to the boy actors, because now people buy pictures of Claudius, despite the fact that before he was king, they made faces at him behind his back. But then again, it is too strange, because there is something in this more than natural, if philosophy could find it out (2.2.367-368). 
A flourish trumpets for the players. Enter Polonius:
As Hamlet is meditating on the strange shifts of popular opinion, we hear the sound of the players' trumpets. Now Hamlet decides that it's time to let Rosencrantz and Guildenstern know that he understands them better than they understand him. He tells them they are welcome to Elsinore, and shakes hands with them, then adds a kind of back-handed insult by saying that he is welcoming them because he's going to welcome the players more warmly than he has them, and he doesn't want them to feel too bad. Then he throws in the information that his uncle-father and aunt-mother are deceived if they think he mad, because he is but mad north-north-west(2.2.378). Hamlet thus presents Rosencrantz and Guildenstern with a kind of reverse catch-22, because if he is only pretending to be mad, why would he say that he is only pretending? 
Now Polonius comes bustling in. Hamlet tells Rosencrantz and Guildenstern that Polonius is a great baby, and tells them that he can tell, just from looking at Polonius, that he's come to announce the arrival of the players. Sure enough that's what Polonius has on his mind, and Hamlet mocks him, although Polonius doesn't seem to notice until Hamlet suddenly calls him Jephthah, judge of Israel (2.2.404). The story of Jephthah is a cruel tale that can be found in Judges 11. In short, Jephthah, who has only one daughter, promises God that if he is given victory in battle he will sacrifice whatsoever cometh forth of the doors of my house when he returns. He does win the battle. His daughter hears of his victory and comes out to meet him with timbrels and with dances. He keeps his promise to God. Hamlet's implication seems to be that Polonius, like Jephthah, has one daughter whom he claims to love passing well, and that Polonius, again like Jephthah, sacrificies her for his own advantage. 
Enter Players:
All of these implications probably pass right over Polonius' head, and besides, Hamlet is interrupted by the entrance of the players, whom he greets gladly. He knows these players well, and jokes about how one has a new beard, and how another has grown. 
Then Hamlet asks for a passionate speech. In fact, he has one in mind, Aeneas' tale to Dido, . . . especially when he speaks of Priam's slaughter, and recites the first thirteen lines. Then the First Player takes over, telling a story that we could expect Hamlet to be interested in, since it climaxes in a description of a woman grieving for her husband. Hecuba, the dead man's wife, made a clamor that would have made the very stars weep, would have made milch [milk] the burning eyes of heaven (2.2.517). By this time the player is himself weeping, and Polonius says Prithee, no more. 
Hamlet agrees to let the rest of the speech wait until later, and he asks Polonius to see to it that the players are well bestow'd. Polonius apparently thinks it's beneath him to be real nice to a bunch of traveling players, and answers that he will use them according to their desert. Hamlet gives him a tongue lashing, saying God's bodykins, man, much better: use every man after his desert, and who should 'scape whipping? (2.2.529-530). Polonius has nothing to say to that, and tells the players to follow him. As everyone is leaving Hamlet announces that we'll hear a play tomorrow, and then detains First Player to make the arrangements. Hamlet wants a particular play, The Murder of Gonzago, and he asks the player to memorize an extra speech which Hamlet would write and put into the play. Later, we find that this particular play interests Hamlet because it tells of a king who was, like King Hamlet, poisoned in his garden by his wife's lover. First Player says that it will be no problem to put Hamlet's speech into the play, but we never hear of it again. Rosencrantz and Guildenstern are still hanging around, but Hamlet coldly dismisses them, and he is alone with his thoughts. 
Exit all but Hamlet:
Hamlet's second soliloquy begins, O, what a rogue and peasant slave am I! (2.2.550). Hamlet is disgusted with himself because First Player could weep for Hecuba, but Hamlet can say nothing; no, not for a king, / Upon whose property and most dear life / A damn'd defeat was made (2.2.569-571). 
Remember that Laertes has gone to France and Polonius has sent Reynaldo after to spy on him, so some time has passed since Hamlet has promised the Ghost that he would sweep to revenge. Yet Hamlet is not blaming himself because he hasn't killed Claudius, but because he hasn't said anything. Half-mockingly, he says that if the player had the same motive and cue for passion he would drown the stage with tears. And then he turns the mockery back upon himself, saying that he is a dull . . . rascal. The Ghost told him that if he didn't take revenge he would be duller than the fat weed / That roots itself on Lethe wharf. So Hamlet seems to be accusing himself of not having the player's passion, of not hating Claudius strongly enough, of not loving his father strongly enough. 
Next, Hamlet asks, Am I a coward? But it's not a really a question. He's trying to work himself into a state of passion. He imagines someone insulting him in the most outrageous way, pulling his nose, calling him a liar, and says that he should take it because if he weren't pigeon-liver'd he would have killed Claudius, gutted him, and fed the guts to the hawks. He flies into a rage at the very thought of Claudius, calling him bloody, bawdy villain! / Remorseless, treacherous, lecherous, kindless villain! (2.2.580-581). 
As the words come rushing out, Hamlet hears himself, and is even more disgusted. Instead of doing anything, he Must unpack my heart with words, / And fall a-cursing like a very drab [a whore], / A stallion [a male whore]. Now he is cursing because he doing nothing but cursing, and he realizes it, saying About, my brains! About is an order, as you might give to a horse, when you want him to turn around. Hamlet is telling himself to stop, take a breath, and try to take a new look at the situation. 
He has heard that guilty people at a play have been struck so to the soul that they have betrayed their guilt. That's what he will try on Claudius. He'll keep a careful watch on Claudius during The Murder of Gonzago. Perhaps he suspects himself of just finding another reason to not do anything, because he justifies himself by reasoning that The spirit that I have seen / May be the devil (2.2.598-599), and may be misleading him to damn him. He would be damned if he killed an innocent man, but this is the first time that he has shown any doubt that the Ghost is anything but his father's spirit, or any doubt that Claudius is guilty of murder. Nevertheless, he declares that the play's the thing / Wherein I'll catch the conscience of the king. 
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