What did Hamlet mean when he said "there's a divinity that shapes our ends" (v.ii)?

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In stating that "there's a divinity that shapes our ends," Hamlet means that God is in control of what happens to us, guiding us even as we try to forge our own paths through life.

Hamlet makes this statement to Horatio, who has long been his trusted confidante. He goes on to explain what he means, telling Horatio that he happened to be bold enough while on board the ship to England to open the letters that Rosencrantz and Guildenstern were carrying. He learned that they held a command from Claudius that he be killed as soon as he set foot on shore.

Hamlet believes his discovery of this information was providential, the hand of God at work. He was, he tells Horatio, able to write a new letter to replace the one he found, with orders that Rosencrantz and Guildenstern be killed. When Horatio asks how he could possibly have affixed the royal seal, legitimizing the letter, Hamlet says that God provided the way for that too. He had a copy of the Danish seal on his father's ring, which he was carrying in his purse:

Why, even in that was heaven ordinant.
I had my father’s signet in my purse,
Which was the model of that Danish seal.

This series of events seems almost miraculous and has saved Hamlet's life, convincing him that there is a God who is in charge of our destinies, no matter what we might think.

This gives Hamlet a sense of peace that he has not formerly had. Earlier in the play, as he contemplated suicide, he wondered if God and the afterlife really existed. It was only fear that they might, not any sense of surety, that kept him from killing himself. Now, because of the way events have unfolded, Hamlet rests assured that God is in charge. He no longer is torn with anguish. He even decides he will be kind to Laertes.

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These words are found near the opening of the final scene in the play and foreshadow the resolution of conflict. The meaning is best examined in full context:

let us know
Our indiscretion sometimes serves us well
When our deep plots do pall, and that should teach us
There’s a divinity that shapes our ends,
Rough-hew them how we will— (V.ii.7-11)

Hamlet has certainly been involved in both "deep plots" and in "indiscretion" as the plot has unfolded. He mistakenly murdered the father of his former lover. He has used Ophelia to further his own plans. He has ordered the execution of Rosencrantz and Guildenstern, his former friends. And he has allowed Claudius to live until the final scene, second-guessing the ghost of his father for several scenes.

At this point, Hamlet sort of steps back from all of it, looking at how everything has brought him to this moment. He comes to believe that no matter what he has plotted, destiny has brought him to his point. However it turns out, God ("divinity") will create the "ending" that He desires.

Hamlet repeats this same sentiment later in the same scene when Horatio urges him not to fight if he has a bad feeling about how things will turn out:

Not a whit. We defy augury. There’s a special
providence in the fall of a sparrow. If it be now, ’tis
not to come. If it be not to come, it will be now. If it
be not now, yet it will come—the readiness is all.
Since no man of aught he leaves knows, what is ’t to
leave betimes? Let be. (V.ii.220-225)

Here, Hamlet is reiterating that God is ultimately in control of his life and therefore of the outcome of this fight. If God wishes him to die, he will die. And if God wishes him to die later, he will not die during this fight. His only job is to be ready to meet death when it comes.

In the end, the plotting of Hamlet, Claudius, and Laertes does not bring the victory they originally desired. Claudius and Gertrude die. Hamlet and Laertes die. In the midst of the carnage, Fortinbras swoops in to claim the Danish crown. Hamlet would therefore say that it was the destiny of Fortinbras and the favor of God that brought him to that victory.

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There's a divinity that shapes our ends,
Rough-hew them how we will.

This line talks about the intersection of individual free will with a specific kind of fate. We "rough hew" our ends (the end results of our actions and choices) in that we make choices about what to do and think and feel, but there is a "divinity" (a fate of a religious quality) that makes it so circumstances overpower our individual choices causing events to work out in unforeseen and perhaps negative ways.

This creates the paradox and conflict that is felt and observed between what is envisioned and what comes to be, between will and fate. This "divinity" gives unpredicted substance to the shapes our lives take.

Remember that Hamlet is about to explain to Horatio what happened at sea concerning Guildenstern and Rosencrantz. So when Hamlet speaks of "divinity that shapes our ends," he is speaking of discovering a plot against him; discovering the commission to have him assassinated in England; his forgery of a new letter to go to the King of England; and the requested assassination of Guildenstern and Rosencrantz once they arrive in England: "He should the bearers put to sudden death,..."

Remember also that very shortly Hamlet will tell Horatio that he regrets how he treated Laertes and will seek his pardon ("I'll court his favours. / ... his grief did put me / Into a tow'ring passion"). Remember also that, even while he speaks, Osric enters to request Hamlet choose a weapon while saying Laertes weapons are "Rapier and dagger." Remember then that both Laertes and Hamlet fall in death by the blades of the poisoned swords. 

Though Shakespeare has Hamlet speak of a "divinity that shapes" and a "heaven ordinant," the context of the statements--death and interrupted apologies made ineffectual and impending death--indicate that "divinity that shapes" is one of the situational ironies of the drama and that it is one of the underlying questions of the tragedy: Is the fate, divinity, ghostly importunacy, ordinate heaven that governs and "shapes our ends" benevolent or malevolent? This is a question that Hamlet never satisfactorily answers; he abdicates the question along with the throne of Denmark to Fortinbras,--who never questioned, as Hamlet did, the dominating power of the cultural norm of revenge killing--with his dying breath.

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