While Holmes does not fully explain how he solved the case in A Study in Scarlet , knowing more about his deduction style can help explain how he knew. As a "consulting detective," Holmes primarily uses the powers of deductive and inductive reasoning to solve crimes. Through deductive reasoning, Holmes...
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uses minute details from a crime scene to create a logical, evidence-based hypothesis. He then uses inductive reasoning by using criminal, scientific, or medical knowledge he already has to flesh out his deduction and solve the crime.
In A Study in Scarlet, the crime scene itself is primarily where Holmes gathers evidence to form his deduction. The bloodstains left in the room without any weapons or signs of a struggle indicates that the blood is from the murderer, not the deceased man. This could indicate that he knew it was Jefferson Hope because a red, flushed face is a symptom of those who get frequent nosebleeds. Hope's flushed face indicated to Sherlock that Hope had suffered a nosebleed on a crime scene, which explains the lack of a struggle or weapon. Hope's arrest and subsequent confession (and death) both solve the crime and justify Holmes's adherence to deductive and inductive reasoning.
What are the methods Holmes used to solve the case in A Study in Scarlet?
Holmes uses the process of deduction, which involves making careful observations of physical evidence and the use of reasoning to lead to larger conclusions. For example, when he arrives at the crime scene, he observes the imprint of a carriage in the driveway and concludes that the tracks are from a cab, not a private carriage, as it has a narrow wheel gauge. He also observes the footprints in the driveway and concludes that they were made by two men, one of whom was tall (based on the length between his strides) and one of whom was well-dressed (based on the elegant impression that his shoes left in the mud).
When Holmes finds the murder victim, who is fashionably dressed, he concludes that a tall man must have killed him. He then finds the smell of poison on the murdered man's lips, and, based on the look of horror on his face, Holmes reasons that the man was killed by someone or something that he had foreseen. As nothing was taken from the premises, Holmes believes that the crime was likely committed for personal or political reasons. As the murderer's prints were found all over the room, indicating that he had been present for a while, Holmes believes that the crime was committed for personal reasons—likely for a woman. This type of deductive reasoning involving the careful analysis of evidence helps Holmes solve the crime.