In Edgar Allan Poe's classic short story "The Tell-Tale Heart," the narrator believes, and wants the reader to believe, that he's not mad because he so perfectly calculated and carried out every step of the murder of an old man, from the conception of the murder to the cover-up.

You should have seen how wisely I proceeded—with what caution—with what foresight—with what dissimulation I went to work!

As further proof that he's not mad, the narrator invites the reader to "Hearken! And observe how healthily—how calmly I can tell you the whole story."

To convince himself and the reader that he's not insane, the narrator explains everything he did in great detail and praises his own persistence—even his courage—in the way he sticks to his plan and never deviates from his goal of ridding himself of the old man and his "Evil Eye."

The eye of a vulture—a pale blue eye, with a film over it.

The narrator wants the reader to understand how fully and how deeply he understands the old man, how he understands the old man's fear, and how he even knows what the old man is thinking.

I knew what the old man felt ... I knew that he had been lying awake ever since the first slight noise, when he had turned in the bed. His fears had been ever since growing upon him. He had been trying to fancy them causeless, but could not. He had been saying to himself—“It is nothing but the wind in the chimney—it is only a mouse crossing the floor,” or “It is merely a cricket which has made a single chirp.” Yes, he had been trying to comfort himself with these suppositions.

The narrator commends himself with how he observes the old man's eye with "perfect distinctness" and how he directs the narrow ray of the lantern to the eye "as if by instinct, perfectly upon the damned spot."

[N]ow, I say, there came to my ears a low, dull, quick sound, such as a watch makes when enveloped in cotton. I knew that sound well, too. It was the beating of the old man's heart.

It's not madness, says the narrator, "but over-acuteness" of his sense of hearing caused by his nervousness, as he says in the opening paragraph of the story.

The disease had sharpened my senses—not destroyed—not dulled them. Above all was the sense of hearing acute. I heard all things in the heaven and in the earth. I heard many things in hell.

The narrator's obsession shifts from the old man's "Evil Eye" to the old man's loudly beating heart. When, finally, he commits the murder, the narrator is concerned much more about the old man's heart than he is about his pale blue, filmy, "vulture eye." In fact, by this point in the story, the old man's eye is little more than an afterthought.

I placed my hand upon the heart and held it there many minutes. There was no pulsation. He was stone dead. His eye would trouble me no more.

He never mentions the old man's eye again.

If still you think me mad, you will think so no longer when I describe the wise precautions I took for the concealment of the body.

Once again, the narrator praises his perfect execution of his plan.

I took for the concealment of the body. The night waned, and I worked hastily, but in silence. First of all I dismembered the corpse. I cut off the head and the arms and the legs.

I then took up three planks from the flooring of the chamber, and deposited all between the scantlings. I then replaced the boards so cleverly, so cunningly, that no human eye—not even his—could have detected any thing wrong.

When police officers arrive to investigate information from a neighbor who heard a shriek during the night, the narrator boldly takes them into the old man's room, the room where he committed the murder.

I led them, at length, to his chamber. I showed them his treasures, secure, undisturbed. In the enthusiasm of my confidence, I brought chairs into the room, and desired them here to rest from their fatigues, while I myself, in the wild audacity of my perfect triumph, placed my own seat upon the very spot beneath which reposed the corpse of the victim.

In time, the narrator's perfect plan is undone by his own nervousness and acute sense of hearing. As he sits in his chair directly above the old man's corpse, he hears the beating of the old man's heart. He first hears the heart beating as if from a distance; then, the sound grows increasingly louder until it seems to fill the room and inhabit the narrator's entire being, causing him unbearable agony.

By the end of the story, the beating heart drives the narrator into raving, raging madness—unless, of course, he was already mad when he started to tell the story.

The unreliable narrator of Poe's classic short story "The Tell-Tale Heart" insists that he is not a madman because of the way he cleverly manipulated the old man into feeling a sense of security before killing him. He uses his presumably calm disposition while narrating his actions as evidence of his mental stability. In the first paragraph of the story, the narrator tells the audience that he is not mad and claims that he is simply nervous. The narrator proceeds to blame a nondescript disease on his nervous condition and challenges the audience to observe how calmly he tells the story of his horrific crime. Ironically, the narrator does not calmly relate his story and proceeds to speak in staccato sentences: he uses an excessive amount of exclamation points and struggles to convey his genuine feelings. The fact that the narrator mentions that he has supernatural hearing abilities further undermines his argument that he is sane.

The narrator then describes his motivation for murdering the old man by elaborating on his "vulture" eye, which makes his blood run cold and fills him with rage. After explaining his motivation to kill the old man, the narrator suspects that the audience views him as insane and mentions:

Now this is the point. You fancy me mad. Madmen know nothing. But you should have seen me. You should have seen how wisely I proceeded—with what caution— with what foresight—with what dissimulation I went to work! I was never kinder to the old man than during the whole week before I killed him.

The narrator apparently believes that his cautious, calculating methods to deceive and harm the old man reflect his sanity. After describing how he stalked and murdered the old man, the narrator continues to prove his sanity by elaborating on the way he dismembered the body and hid it beneath the floorboards. The narrator says,

If still you think me mad, you will think so no longer when I describe the wise precautions I took for the concealment of the body. The night waned, and I worked hastily, but in silence.

Despite his attempts to prove his sanity, the unreliable narrator once again undermines his goal by describing how he heard the dead man's heart beating, misinterpreted the officers' behavior and attitude, and revealed the dismembered body. Overall, the narrator believes that he is not mad and is afflicted with a simple nervous condition, which gives him an acute sense of hearing. He also views his cunning actions as proof of his sanity and believes that he is calmly relating his story.

The narrator does not want his listeners to believe that he is mad because he wants what he has to say to be taken seriously and not written off as the ravings of a lunatic. Further, he truly believes that he is not mad, only "nervous." He feels that his nervousness has "sharpened [his] senses—not destroyed—not dulled them." He believes that he can hear "all things in the heaven and in the earth" as well as "many things in hell." Therefore, in his mind, his experiences differ from the average person's not because he is insane but because his senses are more acute, more powerful than ours. Further, he says, "observe how healthily—how calmly I can tell you the whole story." He believes that his ability to remain calm proves how healthy he is. In reality, he doesn't do a very good job of remaining calm at all (notice how many exclamation points crop up during his narration), and the events and feelings he goes on to describe most certainly oppose his claim to sanity.

The narrator in The Tell-Tale Heart claims that he is not mad because of his actions, which he claims were too well thought out to be done by a mad person. He starts off by saying that he has a disease (though he does not specify what the disease actually is), which heightened his senses and made him more acutely aware of things; it did not hinder him. Also, he claims that he could not possibly be mad because he can tell the story of what he did so calmly. He believes that because he was so systematic and methodical in his plans to murder the old man, he could not possibly be insane. After all, how could an insane man come up with such an elaborate plan and execute it so perfectly?

What evidence does the narrator give that he is not mad?

The narrator in "The Tell-Tale Heart" begins by announcing that he is not mad and arguing that nobody could possibly claim that he was. He justifies his argument by pointing to the fact that he can hear everything that is going on—not only in the real world in which he is living but also in heaven and hell. His senses, rather than being diminished by supposed madness or disease, have actually been sharpened. He notes that because he can hear everything that is happening in the earthly world, as well as in heaven and in hell, he cannot possibly be mad but rather is more perceptive than ever before.

The narrator, of course, is obviously mad from the beginning, as Poe's narration makes clear. However, he gives this evidence stating otherwise as a means of trying to convince the reader that what he is saying is true and that he is able to hear and perceive unexpected and surprising things, not because of any inherent madness but simply because his senses have been heightened as a result of his "disease." The narrator seems to recognize, however mad he may be, that he is likely to be disbelieved and dismissed. As such, he provides some reasoning as to why his acute perception of the world around him should be taken seriously.

What evidence does the narrator provide for his claim that he is not mad in "The Tell-Tale Heart"?

In "The Tell-Tale Heart" by Edgar Allan Poe, the narrator provides little real evidence to support his claim that he is not mad, which only serves to convince the reader that he is. In fact, Poe begins the story by having the narrator admit that he has been ill or at least extremely nervous.

True!—nervous—very, very dreadfully nervous I had been and am; but why will you say that I am mad?

The narrator attempts to prove that he is not mad by showing how “cunning” he was and “how wisely” he proceeded with his crime. After all, he reasons, “Madmen know nothing.” In contrast, he knew how to plan his crime and commit it in an ingenious (in his demented mind) way. He says he proceeded “with what caution— with what foresight—with what dissimulation ...” In other words, a madman would not have had the mental acuity to carry out the plan as well as he did.

Moreover, he was patient, another thing that perhaps a madman would not be. He takes “an hour to place [his] whole head within the opening so far that I could see him as he lay upon his bed.” The narrator then asks the reader, “would a madman have been so wise as this?”

He was also calm, according to his telling of the story. The implicit question he asks is whether a madman would have had the ability to commit the crime and maintain his presence of mind and calmness as he did. To explain why his actions might seem mad to some people, he says, “what you mistake for madness is but over-acuteness of the senses.”

He then describes how the “beating of the old man’s heart” was as loud to him as a drum beat because of this “over-acuteness of the senses.” Even though the heartbeat drove him to fury, he nevertheless “kept still.” This is another piece of evidence to show that he was in possession of his mental faculties and not mad.

Finally, after he has killed the old man, he has another point to prove that he is not mad. He says,

If still you think me mad, you will think so no longer when I describe the wise precautions I took for the concealment of the body.

What evidence does the narrator provide for his claim that he is not mad in "The Tell-Tale Heart"?

The narrator begins by insisting that whatever his "disease," it has

sharpened my senses—not destroyed—not dulled them. Above all was the sense of hearing acute. I heard all things in the heaven and in the earth. I heard many things in hell. How, then, am I mad? Hearken! and observe how healthily—how calmly I can tell you the whole story.

He argues, then, that because his senses seem sharper to him (rather than dulled), he cannot be mad. He thinks that if he were mad, his senses would be less acute or even destroyed, but since they have become more sensitive (or so he thinks), he interprets this as evidence of his health. The narrator obviously does not realize how crazy it sounds to say that he can hear everything in heaven and on earth as well as some things in hell. He also seems to equate calmness with health, as he believes that telling his story in a calm fashion will prove that he is not mad.

The narrator also says,

Madmen know nothing. But you should have seen me. You should have seen how wisely I proceeded—with what caution—with what foresight—with what dissimulation I went to work!

He believes that he has acted with a lot of intelligence and wisdom, something that a mad person would not be able to do. He thinks that by telling his audience (perhaps the police) how carefully and cautiously he sneaked into the old man's room each night, he will prove that he is not mad, when it very well may have the opposite effect.

What evidence does the narrator provide for his claim that he is not mad in "The Tell-Tale Heart"?

The evidence that the narrator provides for his claim that he is not mad can be found in the first two paragraph's of Poe's short story, "The Tell-Tale Heart."  First, the narrator claims that if he was mad, he would not be able to tell the story so calmly.  He also says that he was very, very nervous, but not mad.  This nervousness caused him to have sharpened senses, as opposed to dull ones, and that he has a heightened sense of hearing, enabling him to hear all things in Heaven and Hell.  In the second paragraph, the narrator says, "Madmen know nothing," but he calculated and executed the murder so well that he obviously knows quite a bit. 

In the story "The Tell-Tale Heart," what are the four examples in which the the narrator tries to convince the readers he is not mad?

From the outset, the narrator tells the audience that he is not crazy, yet in doing so he emphasizes the very thing he seeks to disavow. "But will you say that I am mad? The disease had sharpened my senses, not destroyed..." Later the narrator accuses the audiences, "You fancy me mad. Madmen know nothing," and the very assertion he seeks to disprove is reinforced. After setting his plan in motion, the narrator tries to convince the audience further. "And now have I not told you that what you mistake for madness is but over-acuteness of the senses?" The last example occurs just before the police show up when the narrator reassures the audience "If still you think me mad, you will think so no longer when I describe the wise precautions I took for the concealment of the body." At every turn the narrator reveals his madness  even as he attempts to deny it.

Enotes has a great resource for this at the following link.

http://www.enotes.com/tell-tale-heart