In act 4, scene 5 of Shakespeare's Hamlet, what does Claudius mean when he says, “And where the offence is let the great axe fall”? What is ironic about this?

In act 4, scene 5 of Shakespeare's Hamlet, Claudius utters the line, "And where the offence is let the great axe fall," by which Claudius means to say that the "great axe" of Laertes's revenge for Polonius's death will fall on Hamlet. The irony of the line is that it foreshadows Claudius's own demise as the person who set in motion all of the evil that exists in the play because of his murder of Hamlet's father.

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In act 4, scene 5 of Shakespeare's Hamlet, Laertes has returned from France after the death of his father, Polonius. Laertes forces his way into a room in Elsinore Castle where he angrily confronts King Claudius and demands to know how Polonius died.

LAERTES. How came he dead? I'll not be juggled with:
To hell, allegiance! Vows, to the blackest devil!
Conscience and grace, to the profoundest pit!
I dare damnation. To this point I stand,
That both the worlds, I give to negligence,
Let come what comes; only I'll be revenged
Most throughly for my father. (4.5.140–146)

Although Claudius knows quite well that Hamlet killed Polonius while Polonius was eavesdropping on Hamlet's conversation with his mother, Gertrude, Claudius doesn't immediately respond to Laertes's question.

Instead, Claudius sees an opportunity to ingratiate himself to Laertes, and perhaps to persuade Laertes to join him in a plot to kill Hamlet which would satisfy Laertes's need to avenge his father's death and at the same time rid Claudius of Hamlet's potential threat against Claudius's throne and his life.

However, before Claudius can discuss his plan with Laertes, Ophelia interrupts their conversation. Rather than interfering with Claudius's plan, however, the sight of Ophelia's madness serves to strengthen Laertes's resolve to avenge his father's death.

LAERTES. Let this be so.
His [Polonius's] means of death, his obscure burial—
No trophy, sword, nor hatchment o'er his bones,
No noble rite nor formal ostentation,
Cry to be heard, as 'twere from heaven to earth,
That I must call't in question. (4.5.224–229)

"So you shall," says Claudius, and then he utters this fateful, ironic, and prophetic line:

CLAUDIUS. And where the offence is let the great axe fall. (4.5.231)

By that line Claudius means to say that the offender is Hamlet, that the "great axe" will fall on Hamlet, and Laertes's desire to avenge his father's death will be fulfilled.

The irony of Claudius's line is that, as Laertes says as he lies dying near the end of the play, "The King, the King's to blame" (5.2.328).

Claudius isn't to blame for Polonius's death or Ophelia's madness and death, which prompted Laertes to take revenge against Hamlet, but Claudius is at least partially to blame for every other death in the play: Hamlet, who is killed through Claudius's plot with Laertes; Gertrude, who is poisoned with wine intended for Hamlet; Laertes, who is a victim of the same plot that kills Hamlet; and Rosencrantz and Guildenstern, who are killed as a result Claudius's plot to have the King of England execute Hamlet.

Above all, Claudius is responsible for the death of Hamlet's father, whose murder sets in motion the events that lead to the deaths of all of these characters in the play, and which leads to the death of Claudius himself.

HAMLET. Here, thou incestuous, murderous, damned Dane,
Drink off this potion! Is thy union here?
Follow my mother.

[King dies.]

LAERTES. He is justly served. (5.2.332–335)

Indeed, he is.

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