The murderer keeps insisting that he is sane. In fact, he seems to be telling, or writing, this story specifically in order to prove his sanity. In doing this, however, he gives more and more evidence that he is insane. Evidently he has been arrested and is being held in custody. He has apparently been examined and declared to be insane. He says, "...but why will you say that I am mad?" and "You fancy me mad." It would seem that his best course under the circumstances would be to let his keepers think he is mad. Otherwise he would be tried and found guilty of premeditated murder and executed. The fact that he is trying to prove he is sane seems to offer proof to the contrary, because he would only be hurting himself by succeeding in proving he is not mad. The strongest proof that he is insane is in his hearing the victim's heart beating after he has not only killed him but "...cut off the head and the arms and the legs." Poe seems to have described this dismemberment in order to make the reader feel perfectly certain that the heart could not possibly be really beating. The murderer/narrator had been mad all along but had succeeded in concealing it from everybody until the officers came and he began imagining he was hearing that heart beating louder and louder. It might have been his own heart he could hear beating. He begins his story by saying that he is "...very, very dreadfully nervous."
murderer is insane.