In Scene 4, Cauchon agrees that if he cannot get Joan to repent, he will cast her out from the Church and hand her over to Warwick for punishment. Cauchon uses a simile to describe how the Church deals with heretics; Warwick seizes on the simile and turns it into a metaphor:
CAUCHON. [...] When The Church cuts off an obstinate heretic as a dead branch from the tree of life, the heretic is handed over to the secular arm. The Church has no part in what the secular arm may see fit to do.
WARWICK. Precisely. And I shall be the secular arm in this case. Well, my lord, hand over your dead branch; and I will see that the fire is ready for it.
Cauchon describes how the Church "cuts off an obstinate heretic as a dead branch from the tree of life." The tree of life appears in the Old Testament as one of the trees in the Garden of Eden. Adam and Eve are made mortal and cast out of the garden for eating fruit from the tree of the knowledge of good and evil, and they are barred from reentering the garden so that they cannot eat from the tree of life and restore their immortality. Cauchon compares an "obstinate heretic" (someone who goes against Church teachings and will not repent) to a dead branch on the tree of life. To protect the rest of the tree from the disease of this dead limb, the Church cuts it off. At that point, the "secular arm" of government can do whatever it wants with the dead limb.
For Cauchon, who still believes that he may be able to get Joan to repent, all of this is hypothetical. He uses a simile rather than a metaphor to avoid suggesting that Joan is ready to be cast out of the Church. Only if she fails to fall in line with institutional teachings will she turn into a "dead branch." Warwick, on the other hand, is all too eager to execute Joan. He assures Cauchon that he is ready to be the "secular arm" of which Cauchon speaks. He tells Cauchon to "hand over your dead branch," as though Joan's "obstinacy" as a heretic is a foregone conclusion. The metaphor dehumanizes her. By calling her a "dead branch" instead of a human woman, and by referring to her as "it," he makes it seem only logical that he is going to throw her on a fire. Burning a human might look closer to madness than sanity, but burning a dead branch makes sense. Warwick is careful to pick up on the language Cauchon is using so that Cauchon will be able to square the mistreatment of Joan with the institutional position of the Church.
The play does not paint either man in a flattering light. Warwick is bloodthirsty and manipulative, and Cauchon is so invested in speaking for the Church that he fails to think for himself and spot how Warwick is twisting his words. Neither man considers that Joan might be a live branch, thinking for herself and growing in a new direction.
