How does Sherlock Holmes use methodical ways to solve the crime in "The Red-headed League?"
Sherlock Holmes is methodical because he uses evidence he collects to make deductions about the case. He solves the case by observing, and determining the answer from what he observes. Watson is quite impressed.
“You reasoned it out beautifully,” I exclaimed in unfeigned admiration. “It is so long a chain, and yet every link rings true.” (p. 19)
Holmes begins by listening carefully to what Mr. Wilson tells him. From this he is able to determine that something funny is happening, and agrees to take the case. His suspicions are aroused by the absurdity of the league and the sudden appearance of Mr. Wilson’s new employee, who also brought the advertisement to his attention.
Holmes tested his deductions. He looks for new evidence. He goes to the shop and pretends to ask for directions so he can see the assistant. He recognizes him immediately.
He is, in my judgment, the fourth smartest man in London, and for daring I am not sure that he has not a claim to be third. I have known something of him before. (p. 13)
Holmes notices that he has dirt on his trouser knees, so he assumes that he has been kneeling. There are few reasons for a grown man to kneel, so Holmes taps his stick to see if the ground is hollow. He then uses this information to deduce that the men plan to rob the bank by digging a tunnel under it from the shop. He then calls the police, they wait, and the men are caught when they do exactly what Holmes said they would.