Answer:
One of the more predominant themes in Sir Arthur Conan Doyle's Sherlock Holmes stories is man versus man. 'The Red-Headed League' is no exception. In addition, one finds themes of reason and greed as Sherlock prevents another crime based on minuscule evidence.
Man Versus Man
Based on the reputation Sherlock Holmes has garnered through literature and the movies, it is hard to imagine anyone presenting him with a difficult challenge. He has shown an aptitude to solve crimes and mysteries before the clues have even been revealed. In the story 'The Red-Headed League,' Sherlock confronts an individual with whom he has 'had one or two little turns.' These previous encounters provide incentive for Sherlock to capture this villain. It also enhances the theme of man versus man.
Sherlock must prevent the antagonist, John Clay, from stealing gold bullion from 'one of the principal London banks.' With John Clay's plan already in motion, Sherlock looks for clues that will help him develop a counter move. When Sherlock finally confronts John Clay, he admits that his 'idea was very new and effective.' Clay, in turn, gives credit to Sherlock for having 'done the thing very completely. I must compliment you.' Although they go at each other as adversaries, trying to outwit the other, when their conflict is resolved, they acknowledge the other's talents. But what drives this conflict?
Greed
Greed comes in to play in the story on two different fronts. The first is with the pawnbroker, Jabez Wilson. He becomes Sherlock's client. He explains to Sherlock that he has 'a business already,' but was still enticed by earning four pounds a week to 'copy out the Encyclopedia Brittannica.' When this supposed job terminates, Vincent is upset at losing 'four pound a week.' Despite having his own business and employing an assistant, he bemoans the loss of this additional, yet minimal, income. The lure of money creates his current predicament.
John Clay, Sherlock's adversary, is motivated by illegally acquiring thousands in gold bullion. John Clay is a habitual criminal who is known for being a 'murderer, thief, smasher, and forger.' Knowing that all that gold sits unattended in the cellar of a bank is too big of a lure for a criminal such as Clay. It motivates him to contrive a plan to lure out the pawnbroker from his house so he can dig a tunnel from that basement to the bank. All in an effort to become rich. Greed motivates both characters to act in a manner that places them in some difficulty.
Man Versus Self
Sherlock is known for his adherence to, and belief in, logic. This practice is evident as he makes his observations while attempting to solve the mystery behind the red-headed league. Watson mentions to Sherlock at the conclusion of the story that he 'reasoned it out beautifully.' Watson also notes that Sherlock's 'exactness and astuteness represented. . . the reaction against the poetic and contemplative mood which occasionally predominated him.' With these words, Watson indicates that there is a conflict within Sherlock in which reason battles against his artistic nature.
In 'The Red-headed League' Sherlock attends a performance 'gently waving his long, thin fingers in time to the music.' He is enthralled by the music. It isn't until the 'lust of the chase would suddenly come upon him, and that his brilliant reasoning power would rise.' Sherlock needs cases such as this to engage his reasoning prowess. Perhaps he uses music as a means to balance the two sides of his personality, but when he goes without the mental challenge, it wears on him, and he finds life to be a bit dull and boring.
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