Ironically, in the attempts of Poe's narrator to explain his rationality in planning the murder of the old man, he actually points to his madness in his fixation on the man's "vulture eye" and his obsessive care that he takes in order to open the door so that the old man will not hear him while he spends hours watching for the eye to open.
Clearly, this typically unreliable narrator of Poe's demonstrates his mental instability in the third and fourth paragraphs of the story. His methodical movements are so incredibly slow that he even acknowledges that others would scoff at them,
I thrust in my head. Oh, you would have laughed to see how cunningly I thrust it in! I moved it slowly—very, very slowly, so that I might not disturb the old man's sleep. It took me an hour to place my whole head within the opening so far that I could see him...
Then, he repeats this procedure for seven days--the religious number, significantly--watching for what he calls the "Evil Eye" to open. Further, in the fourth paragraph, Poe's narrator now feels empowered,
Never before that night had I felt the extent of my own powers—of my sagacity. I could scarcely contain my feelings of triumph.
For seven more days, the narrator says, he "cautiously" shines a single ray of light on just the old man's eye.
And then, when my head was well in the room, I undid the lantern cautiously—oh, so cautiously—cautiously (for the hinges creaked)—I undid it just so much that a single thin ray fell upon the vulture eye.
All the time that he conducts himself so, the narrator does not become vexed with anyone. In Spartan fashion, he does not complain; instead, he goes to the old man and asks him congenially how he has slept. He has been automatic.