How we cite our quotes:
O momentary grace of mortal men,
Which we more hunt for than the grace of God!
Who builds his hope in air of your good looks
Lives like a drunken sailor on a mast,
Ready with every nod to tumble down
Into the fatal bowels of the deep. (3.4.96)
Hastings realizes that he has been used by Richard, and he has no one to blame but himself. He curried favor with men rather than embracing the graciousness demanded of a Christian. (Think of how he delighted in the deaths of his enemies at Pomfret.) Here Hastings realizes that he was walking a fatal line with Richard, and he's just stumbled to his death.
We live to tell it-that the subtle traitor
This day had plotted, in the council-house,
To murder me and my good Lord of Gloucester.
Had he done so?
What! think you we are Turks or Infidels?
Or that we would, against the form of law,
Proceed thus rashly in the villain's death
But that the extreme peril of the case,
The peace of England and our persons' safety,
Enforc'd us to this execution?
Now, fair befall you! He deserv'd his death;
And your good Graces both have well proceeded
To warn false traitors from the like attempts.
I never look'd for better at his hands
After he once fell in with Mistress Shore. (3.5.6)
The mayor of London is all that stands in the way of the common people coming to realize that Richard is tyrannically killing people. So the mayor's sanction on Richard and Buckingham's hasty execution of Hastings is important. Buckingham outright lies to the mayor, and the mayor is somewhat skeptical.
What changes the mayor's tune is Richard's manipulation. Here Richard doesn't insist that Buckingham is right and bolster their story. Instead, he asks in feigned outrage whether the mayor is calling him a liar, suggesting that the mayor's skepticism is preposterous. Again, it's that tactic of asserting one's power over the situation and making the accuser answer for himself, thereby deflecting the question from the punk who's actually guilty.
The mayor quickly corrects his position, insists that he never trusted Hastings once the guy took up with a hooker, and seems satisfied to fall into Richard's manipulation. It's almost like Richard is the leader of a club (called the royalty) and if you want to stay cool, you better believe what he says, no matter how ridiculous it seems.
Yet know, whe'er you accept our suit or no,
Your brother's son shall never reign our king;
But we will plant some other in the throne
To the disgrace and downfall of your house;
And in this resolution here we leave you.
Come, citizens. Zounds, I'll entreat no more.
GLOUCESTER. O, do not swear, my lord of Buckingham.
Exeunt BUCKINGHAM, MAYOR, and citizens
CATESBY. Call him again, sweet Prince, accept their suit.
If you deny them, all the land will rue it.
GLOUCESTER. Will you enforce me to a world of cares?
Call them again. I am not made of stones,
But penetrable to your kind entreaties,
Albeit against my conscience and my soul. (3.7.12)
Buckingham manipulates the people by making it seem like he has to twist Richard's arm to take the crown. Richard can't seem too eager, or the people will suspect him of ambition. So he has Buckingham put on a really big show, and the people are made to feel like Richard's done them a favor by usurping the throne. (On the other hand, Richard may be less powerful than he thinks, as this little scheme is pretty transparent – to us, at least.)