Look you lay home to him.

How does Polonius have the authority to tell the Queen of Motherfucking Denmark how she should speak to her son?
She is the Queen. He is – NOT the Queen.
Not only is he not the Queen, he’s not even some other royalty of some kind. He’s, like, the royal what? The role is never explicitly stated. Nor is it clear how long he has held this position.
The Queen, however, has been the motherfucking Queen for some time. How is it possible that this guy a) has the authority to tell her what to do and b) has the audacity to? And why does she LISTEN? She does exactly as he’s instructed her – when several scenes before she wasn’t having any of his nonsense. She more mattered and less arted him not long before.
Does he have something over her?
On her?
Has he blackmailed her? Does he know what’s up and he’s threatened to share it with her son?
Is he being all Tulkinghorn to her Lady Deadlock?
I’d expect the Gertrude of earlier scenes to give him a whack and say, “Don’t you tell me how to talk to my son. I don’t care if he is crazy – you don’t give me instructions. I am the motherfucking Queen!”
But I guess the crazy is the wild card here. If Polonius has somehow set himself up on the authority of crazy, he’s cast himself a bit like a Doctor and anyone will listen to a doctor when their loved one is at stake. So maybe she tolerates this as a kind of prescription. It is instructions are followed because she thinks Polonius knows how to handle a crazy person. Maybe she thinks he knows what’s best for Hamlet. Which is bonkers. Because he doesn’t and nothing she’s seen would suggest Polonius does know anything about Hamlet or madness.

This physic but prolongs thy sickly days.

What is the physic? The prayers?
I’ve always thought of it as being the reprieve – as if Hamlet’s NOT KILLING him were the medicine. But it’s not terribly logical having the medicine be NOT doing something. I suppose, in modern medicine, you can be prescribed some Not Doing. Don’t Work or Don’t Have Sex or Don’t Drink Grapefruit Juice – or something.
But this is a physic – not a prescription. A temporary cure of some kind – some kind of palliative medicine.
I think it’s the prayers. I think physic is Claudius’ prayers.

Then trip him, that his heels may kick at heaven, And that his soul maybe as damned and black As hell, whereto it goes.

Hamlet does, in fact, manage to kill Claudius in the midst of an act that has no relish of salvation in’t. Claudius has just killed Gertrude (not on purpose, exactly, but he doesn’t stop her drinking that poison either) and meets his end while trying to scramble out of responsibility for any of it. As far as this worldview goes, he is pretty surely going to hell.
Hamlet, though, while, certainly he’s exchanged forgiveness with Laertes – so he’s probably in the clear there, murder-wise – I’m not sure so sure he’s in the clear for murdering Polonius, and (by proxy) Rosencrantz and Guildenstern. Claudius’s death – well, due to the confusing vengeance clause in religious ideals – Hamlet may be in the clear for that one – but I don’t know….murderer or not – Claudius is still murdered.
All in all, I’m a little concerned that Hamlet’s soul isn’t crystal clear at the end of this play. I mean, not really, it’s not my worldview – but – just because Horatio thinks he’s going to heaven doesn’t mean he really is. I guess we have to hope Hamlet got in a good confession before committing his final murders.

When he is drunk asleep, or in his rage, Or in th’incestous pleasure of his bed, At game, a-swearing, or about some act That has no relish of salvation in’t –

Maybe this is where that bonkers Freudian theory about this play began. Because one of those things nestled in this list of horrible things his uncle does is Hamlet’s mother. So he is, momentarily, proposing busting in on Claudius pleasuring his mother – which means he’d have to bust in on his mom having sex – and even though he means to do it so he can kill the guy – most children would go to some lengths to avoid witnessing the sexual congress of a parent.
Not Hamlet, though, not only is he proposing murdering Claudius while he’s fucking his mom, he describes it as “pleasure.” Which is funny. I mean it may be incestuous but it’s still pleasure. (And who knows, maybe the incestuousness makes it more pleasurable.) He’s choosing a word that suggests that he imagines the sexual relationship between Claudius and Gertrude as a mutual, pleasurable one. There are many many many words for sex – and he chooses pleasure. (Shakespeare himself used a LOT of them) and he chooses pleasure.
It’s funny.

Up, sword, and know thou a more horrid hent.

My students are often baffled by the notion of putting up one’s sword. They usually take it literally and take it mean that one should raise the sword, ready to fight. But it’s more like being asked to put up your toys when you’re done playing.

I was baffled by “hent” myself. I’d always assume it was a location for a sword – like a hole or target – like the thing you’d stick the blade in. But it’s mostly a verb – meaning to seize or catch hold of. Which is similar to what I thought.

As a noun, though, as it’s used here – it reportedly means “way” – which, while not as disgustingly visceral, does make a whole lot more sense. Because he is looking for a more horrid way to kill Claudius – not necessarily a different place to put the sword. He’ll kill him in the same place, regardless, one would think. This is why it pays to look up words even when you think you know.

No.

In my edition, this sentence gets its own line of verse. This leaves 9 beats of silence.
But it would seem like you’d want to use those 9 beats BEFORE the “No” and not after it. Like, to really feel the weight of this decision those 9 beats need to be full of struggle, to be thinking, really and truly – “Am I revenged? Should I do this now?”
If you say the “No” and THEN wait 9 beats. Well, I’m not sure what you do. But that too might be interesting. If he says, “No” and then doesn’t know what to do with himself for a little while. Like he spends 9 beats trying to work out where to put his sword and then determines that he should put it up, away from its target.
Of course, I imagine some editions would put this NO at the end of the previous line or, probably, more likely, the beginning of the next – both of those lines are somewhat irregular already – it might be just fine to, say, put three stressed words in a row to read. “No. Up, sword.”
But no – all on its own is definitely bold. And dramatic. But you’d have to really fill it.

And am I then revenged, To take him in the purging of his soul, When he is fit and seasoned for his passage?

I find I am confused by the way revenge fits into religion. Like, do you get a pass for murder if you’re doing it for revenge?
Because Hamlet here is very concerned for Claudius’ soul but has no real concern for his own. Like – wouldn’t it better to kill Claudius now if he’s going to do it? Won’t his own set of sins be slightly reduced by doing his murdering at this moment?
It’s like two competing codes of ethics here – the Honor code, which would seem to be in direct opposition to the Religious order.
And yet they are also intertwined.
They sit a little uncomfortably next to one another – Religion probably not going so far as to say a murder done for revenge isn’t a sin – but probably giving more of a pass in this area than would seem QUITE right in the Thou Shalt Not Kill department.
And then Revenge – which is not only concerned with a violent kind of redemption – but which is also seemingly concerned with the souls of the murdered.
The two ways are both ancient and have probably been doing this funny little dance for ages. Old primal urges competing and cooperating.

But In our circumstance and course of thought, ‘Tis heavy with him.

We’re still talking about Hamlet Sr, right? We haven’t switched to Claudius yet? Because him could be either him.
I’m not entirely clear what our circumstance and course of thought actually is. Who is “Our”?
I can try and make this mean something like “on further thought” or “now that I think about it, the weight on my Dad is definitely pretty heavy – now I’m remembering all the sulphorous and fiery and flames and what not.”

But I’m not entirely sure that this is what’s actually going on.
Is the air a royal air?
A reference to what’s just happened with the Mousetrap?
If so, it would seem to point to Claudius’ heaviness, not Hamlet Sr’s.
Things are heavy with both kings, it would seem. That’s probably part of the deal of being king – things are heavy with him always.

And how his audit stands, who knows save heaven?

I assume he means his father’s spiritual audit. And not his father’s financial audit. But he must have a SMALL sense of how his father’s audit stands because his father has told him that he’s enjoying some flames to purge away his sins – so it’s not a clean bill of soul health. If I were Hamlet, I’d be pretty curious about what my dad did that merited the tormenting flames treatment.

It’s funny to call it an audit. It makes me think of the soul audit as an excel document. St Peter – or some middle manager officer – comes along with his empty spreadsheet and gives the dead man a soul once over. He puts a mark in each column for each sin – and some columns are more heavily weighted than others. The auditor takes a look at the whole picture and then decides the final equation.
X number of sins = X number of tormenting flames.