What clues does Holmes follow to explain who wrote the letter Sir Henry found at the hotel?

Holmes's treatment of clues is poignant in Chapter IV. His process here indicates his basic approach: he uses an understanding of types - a classification system - to explain the meaning of a particular clue. First, we see his familiarity with typescripts: he can identify to what medium a particular clipping belongs based off of its type. He then follows the trail to deduce the type of person who would read this kind of newspaper.

His next step is to ask Sir Henry if anything else strange has happened. Naturally, Sir Henry did not think enough of the boot to mention it on his own; such small events happen to us all the time. However, Holmes knows that the devil is in the details, that the answer is often in the places we otherwise think meaningless. Instead of judging the boot incident as trivial (as Dr. Mortimer does), Holmes wishes to methodically compare the ordinary to the unusual. In other words, the usual type (both boots remain) is compared to a specific instance (one boot is missing), to determine the reality of the situation.

Put another way, Holmes's method involves using general types to analyze specific incidents. This approach mirrors the scientific theories of Doyle's day, which classified various animal and plant species according to their types. In this way, understanding the class to which something belongs can help Holmes understand more about a particular entity. At the same time, understanding how something differs from its general class allows the observer to determine what specifics comprise clues worthy of consideration.