Who makes the following statement and what is its significance?
"This man... pure as they deem him, --all spiritual as he seems, --hath inherited a strong animal nature from his father or his mother. Let us dig a little further in the direction of this vein."
A careful reading of the text will indicate that Chilllingworth is not necessarily trying to discover what ails Dimmesdale; rather, he knows exactly what the situation is -- stating that the minister has a "strong animal nature" and, as Hester's husband, is well aware, like the rest of the village, that Hester has borne a child without a named father, but unlike the villagers, he knows exactly who it is. What motivates Chillingworth is not any concern towards Dimmesdale; it is simply a cunning, lethal, and prolonged revenge. Chillingworth knows that Dimmesdale will never confess; to do so would mean ostracization from the community and possibly the incursion of the death penalty.
Recalling the Reverend Dimmesdale's having placed his hand upon his heart after saying "She will not speak" after Hester's refusal to name the father of her child while on the scaffold, in Chapter IV of "The Scarlet Letter," Roger Chillingworth tells the imprisoned Hester,
'Think not that I shall interfere with Heaven's own method of retribution, or, to my own loss, betray him in the gripe of human law. Neither do thou imagine that I shall contrive aught against his life; no, nor against his fame, if, as I judge, he be a man of fair repute. Let him live! Let him hide himself in outward honour, if he may! Not the less he shall be mine!'
Now, in Chapter X, Chillingworth commences his plan to make Dimmesdale "mine." After Chillingworth's analysis of Dimmesdale, Hawthorne writes,
The soil where this dark miner was working had perchance shown indications that encouraged him.
The metaphor of the miner searching for the "gold" of knowledge of the soul of Dimmesdale continues as Chillingworth delves into the consience of the minister in order to discover the sin of adultery upon the heart of his patient. But, Dimmesdale becomes
vaguely aware that something inimical to his peace had thrust itself into relation with him.
However, with stealth and cunning, the dissembling Chillingworth seeks to delve into the heart of Dimmesdale by telling the minister that the body cannot be healed unless the patient "first lay open to him the wound or trouble in your soul?"
Dimmesdale rebukes the "leech," rushing away from him. Yet, Chillingworth reinstablishes intimacy with the minister, and, one night, as the physician continues his search "to the bottom," he notices that Dimmesdale is in a profound sleep, unlike his usual fitful rests. It would seem that Chillingworth has administered something to the minister, for the physician advances towards Dimmesdale with no apprehension. He lays aside the "vestment that, hitherto, had always covered it [his chest] even from the professional eye."
Chillingworth turns away after looking with
a wild look of wonder, joy, and horror!...a ghastly rapture...making itself even riotously manifest....Had a man seen old Roger Chillingworth,...he would have had no need to ask how Satan comports himself, when a precious human soul is lost to heave, and won into his kingdom.
Chillingworth has fulfilled his vow of Chapter IV; he has made Dimmesdale his; he has violated the sacredness of the man's heart.
The line you cite is from Chapter 10 -- The Leech and His Patient. The line is spoken by Roger Chillingworth, but it is not spoken to anyone. He is talking to himself.
A little context. By the point of this chapter, Chillingworth is obsessed with Dimmesdale's secret. He really wants to know what is going on. In is in this context that he speaks the lines you cite.
What they mean is that he does not think Dimmesdale is pure -- he thinks Dimmesdale has some secret. It is saying that allthe people of the town think Dimmesdale is pure and spiritual. But in reality, he has a strong animal nature (meaning that he is not above lust and other things "of the flesh").
Of course, Chillingworth is right, because Dimmesdale is Pearl's father.
In chapter 10 Roger Chillingsworth suspects some interesting but uncertain things about the Reverend Dimmsdale. He thinks to himself:
"This man... pure as they deem him, --all spiritual as he seems, --hath inherited a strong animal nature from his father or his mother. Let us dig a little further in the direction of this vein."(112)
Roger is thinking that if he investigates and probes Reverend Dimmsdale a little bit more, he may be able to find out some secrets above the Reverend. Roger is aware that the congregation identifies the Reverend as being innocent and pious but he suspects that like man, Dimmsdale has human desire and the likelihood of sin hidden away as his secret. He wants to find out about it.