Post-crime narrative line is a story in which the crime -- in one sense the climax -- has already been committed and is part of the back story; the rest of the tale, as Miheala Prioteasa points out, is "entirely intellectual." In "Purloined," the devious Minister D has already stolen the letter when the prefect visits Dupin. The narrator describes them all as enjoying "the twofold luxury of meditation and a Meerschaum," a phrase that could describe Holmes at Baker Street. The exposition from the prefect tells Dupin, and the reader, what has already happened; the second climax of the story is the solving of the crime.
Ratiocination is the reasoning process that leads the detective to the solution. This thought chain is seldom obvious to the narrator, or the reader, as the story progresses, but is made clear at the end. In "Purloined," Dupin apparently lets a month go by without action; it is only after he produces the stolen letter, which he has in turn stolen from the evil Minister D, that the reader realizes he has been busy indeed. Sherlock Holmes, Hercule Poirot and a host of other literary detectives also show an apparent inactivity; Agatha Christie's elderly Miss Marple is completely sedentary, solving crimes while knitting in her chair. Nevertheless, all of them are mentally at fever pitch throughout.
The end piece of any detective story is the exposition; the sleuth tells all that the reader has missed. This section of the tale is usually lengthy, as it has to both recap the story and fill in missing information. Dupin takes more than half the text of "Purloined" to explain solving the riddle of the hidden letter, as he delves into both abstract philosophy and cognitive principles to aid in the reader's comprehension.
"Purloined Letter" is unique among Poe's tales; it is the only story in which Dupin is not near the scene of the crime, and the only tale of the three that hinges entirely on the final explanation of the case. Sir Arthur Conan Doyle himself satirized Poe's technique in the first Holmes story, "Study in Scarlet," by having Holmes speak scathingly of Dupin's exposition, a method Holmes proceeds to use himself. Even tongue-in-cheek, the masterly Conan Doyle recognized the debt all detective story writers owe to Poe and Dupin.