What does Chillingworth mean when he mutters "A strange sympathy betwixt soul and body! Were it only for the art's sake, I must search this matter...

Chillingworth, in saying these words, is basically stating
that he is shocked at his own obsession with Dimmesdale. He, a man who was once in
control of himself, his wife, his career, and his life, is now falling under his own
weaknesses and demons. Namely, these weaknesses and demons include jealousy, rancor, the
sadistic need to hurt others, a hunger for revenge, and
hatred.

It is apparent to the reader, and to Chillingworth
himself, that he has never felt these emotions before. Now, they are taking the best of
him. This is a huge event for him because, perhaps for the first time in his life, he is
allowing emotion to take over common sense. This is what he means when he says: "A
strange sympathy betwixt soul and body".

The second part of
the exclamation, "were it only for the art's sake..." is a slight allusion to his
natural curiosity, as well as his tendency for scientific inquiry: The two most
predominant factors in a physician's mind. Hence, this latter phrase is sort of an
excuse that Chillingworth tells himself to satisfy his morbid and desperate want of
information as to what is "eating" Dimmesdale inside. It is a way of saying: "I am
naturally curious so, even if it is for that matter, I must know what is going
on."

In Chillingworth, Hester, and Dimmesdale we find
nothing but broken people. Each of these characters allowed nature to take over their
senses and, for this same reason, they all failed in life. Chillingworth is nothing but
a sad example of a good man gone bad.