How does Conan Doyle present Holmes as a complex and unusual character in the first two chapters of The Sign of Four?

In The Sign of Four, through Watson's questioning Holmes about his use of cocaine, we learn that Holmes has a great deal of contempt for "the dull routine cases" that come Watson's way. He depends on cocaine to stimulate his senses and to clear away the fog which surrounds him in everyday life. The only real stimulation he finds comes from the tasks he takes on as a detective, and because these are so rare, he uses cocaine to relieve boredom and monotony. We learn that Holmes considers himself unique among humanity—the only person who is doing the kind of work he does. He feels superior to ordinary people because they do not have the mental capacities or energy to do what he does.

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As mentioned in the other answer, we find out that Sherlock Holmes uses cocaine. Watson, as a doctor, is alarmed by this habit:

[It] involves increased tissue-change and may at last leave a permanent weakness. You know, too, what a black reaction comes upon you.

As Watson questions Holmes about it, we gain insights into Holmes's character. These insights reveal to readers how complex and unusual Holmes is. We learn, as Holmes asks Watson to look out the window, that he, Holmes, sees the world as an unsatisfying place. Humdrum reality does not appeal to him—people are dull—and he describes the world in terms that will find their echoes later in T.S. Eliot's "The Love Song of J. Alfred Prufrock":

A dreary, dismal, unprofitable world? See how the yellow fog swirls down the street and drifts across the duncolored houses. What could be more hopelessly prosaic and material?

Holmes explains to Watson that he takes cocaine as a form of escape from the tedium of not having an intricate problem to work on and also as a way to enhance reality. He calls cocaine "so transcendently stimulating and clarifying."

We learn too that Holmes perceives himself as set apart from other humans, calling himself the only person in the world doing the work he is doing. He is the "only unofficial consulting detective":

I am the last and highest court of appeal in detection.

He emphasizes over and over that mental stimulation and the life of the mind are all that makes life worth living to him. He even tells Watson that the problem Watson proposes to lay out for him will keep him from a second dose of cocaine.

We learn too that Holmes looks down on Watson's tendency to add a romantic, emotional, human element to his crime stories, stating that the emphasis should be on the science and the facts. He has no interest in the kind of romantic love that Watson feels for Miss Morstan.

From Watson's probing into his friend's cocaine habit, we learn that Holmes is world-weary, sets himself apart from the average person, and finds the mental stimulation of detection, along with other mental activities, all that makes life vital for him. Doyle does his best to reveal that Holmes is not an ordinary human being.

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The Sign of Four is presented by Mary Morstan as a complex case. Holmes, being a complex man, is just the man to help solve it. In the first two chapters, Watson introduces various unusual character traits which will become familiar not just throughout the course of the book, but also in subsequent Sherlock Holmes adventures.

The story begins with Watson observing Holmes taking cocaine. He has been witnessing Holmes doing this three times a day for several months. Drug addiction was much more unusual in 19th-century England than it is today. The reason that Holmes gives for taking cocaine is also unusual; his mind "rebels at stagnation." Holmes needs to keep his mind active in order to solve such complex mysteries and regular injections of cocaine help him to do this. At the very least, he thinks it does.

Conan Doyle also allows us to marvel at Holmes's incredible powers of observation, as displayed in his brilliant deduction about the background history of the pocket watch belonging to Watson. Holmes correctly deduces that the watch once belonged to Watson's alcoholic brother from a simple examination of scratch marks and grooves.

It is here we see another character trait of Holmes that marks him out from the rest of humanity: a certain cold-bloodedness. When Holmes reveals the provenance of Watson's pocket watch, Watson is rather offended. Holmes means no offense; he is merely exercising his capacity for abstract logic to the utmost.

In chapter 2 we see this characteristic of Holmes when he is introduced to the comely Miss Morstan. After she presents Holmes with the facts of the case, Watson immediately observes how attractive the young lady is. Holmes, however, claims not to have noticed as he casually lights a pipe and sits back in his chair.

An exasperated Watson rebukes Holmes, describing him as an "automaton," and a "calculating machine"; indeed, there is something "positively inhuman" about him. Perhaps Watson is being a little harsh; however, we can still accept that Sherlock Holmes is no ordinary human being, as Conan Doyle makes perfectly clear in the first two chapters of The Sign of Four.

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