Chapter XIII of The Scarlet Letter, appropriately titled "Another View of Hester", is set seven years after the birth of Pearl. At this point, Pearl is an older child but still challenging to Hester, Reverend Dimmesdale is showing the signs of decrepitude that are finishing slowly with his life, and Chillingworth is at his worst trying to probe Dimmesdale's mind.
Yet, Hester's presence demonstrates a strength and calmness of character that, as the narrator states,
It is remarkable, that persons who speculate the most boldly often conform with the most perfect quietude to the external regulations of society...So it seemed to be with Hester.
With these words the reader gets a perception of Hester that surprises even Hester, herself. No longer is she the child of passion and rebellion that led her to the affair with Dimmesdale. The passing years have awarded her with enough time to analyze her own behavior in depth. Even after the scaffold, the imprisonment, the public scorn, and her general humilliation, Hester Prynne still looks back and sees that, perhaps, her actions were not made with the purpose of sinning; she simply is a woman ahead of her times. She thinks differently, and feels differently.
She sees how her own strength of character and temperance far surpasses Dimmesdale's and Chillingworth's, even though both men are educated and more powerful, in comparison to her. Hence, Hester is slowly realizing that, in the end, the same letter which was first used to insult her, deter her, and limit her, has actually empowered her, and taught her how to become independent, and in peace with herself.
She might have come down to us in history, hand in hand with Ann Hutchinson, as the foundress of a religious sect. She might, in one of her phases, have been a prophetess.
Therefore, the original purpose of the scarlet letter has never been met. Far from submitting Hester to the rules of the settlement, the letter effectively detached Hester from her insignificant and ignorant surroundings. The letter taught her how to appropriately measure the influence of other people's thoughts, beliefs, and opinions, on her life. Hester basically dominated and transformed the scarlet letter; it did not happen the other way around, as was expected.
[It] had not done its office.