At the very end of Chapter 10, "The Leech and His Patient," Dimmesdale falls into a deep slumber. It is midday, and he has been reading. The sleep is suspicious, which Hawthorne suggests through his description of it. First, it is early in the day and Dimmesdale was studying. Second, it is a very deep sleep. Dimmesdale is not usually a heavy sleeper, "he was one of those persons whose sleep, ordinarily, is as light, as fitful, and as easily scared away, as a small bird hopping on a twig." Here, he is out cold.
Another reason that the sleep appears suspicious is the way in which Chillingworth enters the room. Rather than walk quietly, so as not to disturb Dimmesdale, he enters "without any extraordinary precaution." He then "advanced directly in front of his patient, laid his hand upon [Dimmesdale's] bosom, and thrust aside the vestment, that, hitherto, had always covered it even from the professional eye." Essentially, he walks in and opens the shirt of a typically light sleeper, and Dimmesdale only "shuddered, and slightly stirred."
Finally, to make the scene even more disturbing, Chillingworth reacts to what he sees
But with what a wild look of wonder, joy, and horror! With what a ghastly rapture, as it were, too mighty to be expressed only by the eye and features, and therefore bursting forth through the whole ugliness of his figure, and making itself even riotously manifest by the extravagant gestures with which he threw up his arms towards the ceiling, and stamped his foot upon the floor! Had a man seen old Roger Chillingworth, at that moment of his ecstasy, he would have had no need to ask how Satan comports himself, when a precious human soul is lost to heaven, and won into his kingdom.
Chillingworth sees something on Dimmesdale's chest, and it appears to convince him that the priest is Hester's "fellow sinner" that the old man has been searching for. He is astounded, so much that rather than allow the reader to also see, Hawthorne focuses his attention on the gestures of the old man, once again drawing a parallel between Chillingworth and Satan in the process. Chillingworth is ecstatic, in awe, and both happy and horrified by the sight. It makes him feel as if he has finally accomplished his goal.
Even without the connection to the devil, Chillingworth's actions are symbolic of his true nature. He is now nothing more than a sinister creature, living for little more than revenge. Whatever he perceives on Dimmesdale's chest has led him to be more animated than we have seen him before. He is truly the leech, living only to suck the life from his host, and he believes he has nearly succeeded.