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The purpose of the conversation near the water can be found in Pearl's reflection in the pool. It is in this chapter that we can find the true characters of our characters revealed.
Pearl reflects innocence;
"And Pearl, stepping in mid-leg deep, beheld her own white feet at the bottom; while, out of a still lower depth, came the gleam of a kind of fragmentary smile, floating to and fro in the agitated water."
Hester is finally able to discern the man that Chillingworth has become;
"All this while Hester had been looking steadily at the old man, and was shocked, as well as wonder-smitten, to discern what a change had been wrought upon him within the past seven years. It was not so much that he had grown older; for though the traces of advancing life were visible he bore his age well, and seemed to retain a wiry vigour and alertness. But the former aspect of an intellectual and studious man, calm and quiet, which was what she best remembered in him, had altogether vanished, and been succeeded by a eager, searching, almost fierce, yet carefully guarded look. It seemed to be his wish and purpose to mask this expression with a smile, but the latter played him false, and flickered over his visage so derisively that the spectator could see his blackness all the better for it."
The Scarlet Letter