chapter 4 In the scarlet letter
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Chillingworth intends to have his revenge on the father of Hester's child (Dimmesdale). Gradesaver's analysis is as follows;
"It is difficult to establish what motivates Roger Chillingworth to remain and seek revenge. He is an educated man with superb skills in medicine and literature. Why then would he choose to remain in Boston and attempt to destroy Dimmesdale? There are few good explanations for Chillingworth's behavior and desire to not be known. The most likely reasons are revenge and the challenge of solving the mystery. The motive of revenge is clear enough from Hester’s infidelity and the damage that revealing himself would do to his reputation and future ability to marry. He also might seek vengeance on the true father for stealing his chance at a family. In that society, it would make sense to go after the father rather than Hester, and he admits in this chapter that he married Hester even though he knew she did not love him.
Even so, Chillingworth could have left town and tried to start a new family elsewhere. But there is still the mystery. Chillingworth's behavior is too sublimely cruel for that to be the only motivation, so it seems that he is motivated both by revenge and the mystery. A third possibility is that Chillingworth is also trying to remove the father from the scene in order to make a second attempt to win Hester’s heart. This idea seems unlikely, but it goes hand-in-hand with the acts of revenge Chillingworth carries out in his parasitic attack on Dimmesdale, sucking the virility out of the man."