When Hamlet says he'll show Gertrude the "inmost part" of herself, what is he suggesting about her current state of mind and what is he suggesting he'll "show" her?

In act 3, scene 4 of the play, Hamlet urges his mother to look at the “inmost part of herself” in a mirror. He is suggesting that she has been tricked by evil forces, and later attributes her obsession with Claudius to a devil having blinded her. Hamlet specifically wants his mother to stop having sexual relations with Claudius, and more broadly to believe that Claudius killed Hamlet, Sr.

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Act 3, scene 4, often referred to as the closet scene, takes place in Gertrude’s bed-chamber or closet. Through the course of the scene, Hamlet argues with his mother about the dangerous situation into which she has placed herself by marrying Claudius. Hamlet has become convinced that Claudius murdered Hamlet, Sr., but he desperately wants to believe Gertrude innocent of any involvement in the assassination. In imploring her to look at herself in a “glass,” or mirror, he employs a metaphor to mean engage in soul searching.

One central element of Hamlet’s concern is that Claudius is manipulating his mother through sexual desire. Part of him is appalled, as a child might think of their parents, that his mother has a sex life at all. Even more, he worries that obsessive desire is keeping her from realizing Claudius’s true villainous nature.

Early in the scene, Gertrude attempts to chastise him for offending his “father,” referring to Claudius. He counters that it is she who has offended his father, meaning Hamlet, Sr. Urging her to face the facts, he restrains her as she tries to leave.

Come, come, and sit you down; you shall not budge;
You go not till I set you up a glass
Where you may see the inmost part of you.

Soon after this he kills Polonius, mistaking him for Claudius, and she is justifiably upset. Hamlet still insists on speaking further with her about her relationship with Claudius, hoping his words will penetrate her heart.

Leave wringing of your hands: peace! sit you down,
And let me wring your heart; for so I shall,
If it be made of penetrable stuff….

He continues with a long speech, in which he speculates that a devil must have blinded her. He wonders why she is not ashamed of her behavior.

What devil was't
That thus hath cozen'd you at hoodman-blind?...
O shame! where is thy blush?

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