In chapter 10 of The Scarlet Letter, "The Leech and his Patient", Chillingworth tries everything he can to extract from Reverend Dimmesdale the truth about the ailment that has been consuming him alive. Since Dimmesdale refuses to give way, Chillingworth opts to hint at different things that "could" be wrong with the minister, namely, his spiritual health.
The men keep going back and forth, with Dimmesdale reminding Chillingworth that he (Dimmesale) is the one to deal with his own spiritual matters. Chillingworth lets Dimmesale go along with the certainty that he still will be needed.
At the end of this chapter, Reverend Dimmesdale falls asleep at around noon while reading. This is rare since Dimmesdale's physical state often rendered him sleepless and nervous. Chillingworth is aware of this and enters the room "without any extraordinary precaution". It is here where he moves the Reverend's robe aside and sees "something".
The reader does not get to know in this chapter exactly what it is that Roger actually sees, but we later find out that Arthur has carved a letter "A" on his own flesh. This means that he commiserates with Hester and her own scarlet letter. This is when Chillingworth makes the connection that Dimmesdale is the secret man that Hester refuses to identify as the father of her child born out of wedlock.
Chillingworth's reaction is quite surprising for an elder man. He feels a "ghastly rapture", and he essentially becomes extremely ecstatic. He is compared to Satan himself, and how the devil would behave itself in a situation where it had won someone's soul. However, Hawthorne clarifies:
But what distinguished the physician's ecstasy from Satan's was the trait of wonder in it!