Jefferson Hope's aneurism burst that evening; he was found in his cell, dead, with a placid expression upon his face. After a few pensive moments following the news, Holmes brightened and commenced explaining to Watson how he had put the pieces of the case together and found the murderer within three days.
His skill centered upon being able to reason backwards rather than look at a chain of events and predict what they would lead to. The latter was a common skill but reasoning backwards was very rare and quite useful indeed. In this case Holmes put this strategy to effect.
When Holmes first arrived at the murder scene he noted the marks of a cab and deduced that it was there during the night. The footsteps in the yard revealed two men –one that was very tall (because of the long stride) and one that was fashionably dressed (because of the neat little boots). Inside, the well-dressed one was found dead. Holmes smelled poison upon him and saw that his contorted face was a result of the poison.
This crime was not committed for robbery, so it must have been political or for a woman. Political motivation was soon dropped, for "political assassins are glad to do their work and then fly." This must have been a private matter, not a public one. The word in blood on the wall was no doubt a blind. The ring of course answered the question, especially when Holmes learned the man came back for it.
Looking around the room at the clues Holmes learned more about the murderer. After he left he telephoned Cleveland and asked about a marriage of Enoch Drebber. He learned that Drebber had once invoked the protection of the law against a Jefferson Hope, a "rival in love" who was now in Europe as well. The man in the cab was no doubt Jefferson Hope; being a cab driver was an excellent way to follow someone in London.
Hope would not want to draw attention to himself by leaving his profession right away, so he would continue to drive the cab for a few days. Holmes sent his street urchin gang to every cab proprietor in London until he found the one Hope worked for. The murder of Stangerson was unforeseen, but could hardly have been prevented. Through that he came into contact with the pills.
Watson warmly lauded Holmes for his detective prowess and encouraged him to publish an account of the case. Holmes told Watson he could do as he pleased and handed him the paper. In it was a paragraph about the case they had just solved.
The paper lamented that the true facts of the case may never be known because of Hope's death, but that "we are informed upon good authority that the crime was the result of an old-standing and romantic feud, in which love and Mormonism bore a part." It said Lestrade and Gregson were responsible for the capture, and that an amateur named Sherlock Holmes helped and might someday "attain to some degree of their skill."
Holmes laughed at that, and reminded Watson of when they started this whole affair he had warned him of this –"that's the result of all our Study in Scarlet; to get them a testimonial!" Watson told him that he had all of the facts in his journal and would make them known to the public. In the meantime he should remember the Latin words of the Roman miser that said "the public hiss at me, but I cheer myself when in my own house I contemplate the coins in my strong-box."
A Study in Scarlet concludes with the death of Jefferson Hope, Holmes’ more thorough explanation of how he solved the case, and the newspaper's accolades for Lestrade and Gregson. Holmes learned about the two men who entered the house and deduced that the tall one was the murderer. He also surmised that the man who drove the cab was the murderer because a murder could not have been done before the eyes of a third party. He mused over what the possible motivations for such a crime could be as there was no hint of robbery, and settled on either political reasons or matters of the heart.
The latter speculation of it being a private, not a public matter, was confirmed when he received a wire back from Cleveland identifying a Jefferson Hope as a rival in love for Drebber. The ring was also key to supporting this theory. All that was left to do was employ the street gang in finding out which cab company Hope worked for, and then commissioning his services since it was only logical that Hope would not immediately cease driving after the crimes because it would raise suspicion.
Holmes further explains to Watson his method of working in this concluding chapter: "I have already explained to you that what is out of the common is usually a guide rather than a hindrance. In solving a problem of this sort, the grand thing is to be able to reason backwards. That is a very useful accomplishment, and a very easy one, but people do not practise [sic] it much" (123). He believes that there are far more people that can reason analytically but very few that can reason synthetically. Doyle scholar and editor Owen Dudley Edwards notes that this line is delivered with some self-mockery, for "of course the author works synthetically, from causes to effects, where the rare master of the analytical goes from results to their origins."
Although it is clear to the reader that Holmes was truly responsible for solving the case, the congratulations in the press go to Lestrade and Gregson, and Holmes is "an amateur, [who has] shown some talent in the detective line, and who, with such instructors, may hope in time to obtain some degree of their skill" (127). Holmes is not particularly upset; it seems as if he knows that this will be the result of the affair because he is not the official law enforcement and his abilities are so far beyond the scope of the average person's capacity to perform and comprehend that they are not ready to publicly celebrate him.
After Holmes laughs off the results of the case Watson comforts him with the Latin words of a Roman miser, Quintus Horatius Flaccus, from the First Satire, which translate to: "The public hiss at me, but I cheer myself when in my own house I contemplate the coins in my strong-box." Watson clearly assumes that Holmes has no problem with translating Latin or recognizing a quote from Horace. These last words reinforce the reality that while Holmes might not be publicly lauded for his work, he has the personal satisfaction of recognizing his talents and observing to what use they were put to.