What clues does Holmes use to deduce his visitor is the King of Bohemia in "A Scandal in Bohemia"?

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Sherlock Holmes deduces that his mysterious visitor is the king of Bohemia from the latter's imperious manner and ostentatious dress.

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At the beginning of the story, Sherlock Holmes tells Watson that he has received a mysterious letter. In the letter, the writer reveals that he needs to consult Holmes "upon a matter of the very deepest moment."

The writer also says that he will call upon Holmes that very night at a quarter to eight. He ends the letter by telling Holmes to be available at that time and to look for a visitor wearing a mask. For his part, Holmes also notices that the letter-writer expresses himself like one of German descent. According to Holmes:

A Frenchman or Russian could not have written that. It is the German who is so uncourteous to his verbs.

Later, when the visitor arrives, Holmes notices that the latter is dressed ostentatiously and that he has an imperial manner. And, upon looking out the window, Sherlock sees that the visitor has arrived in a horse-drawn carriage. There are two horses, which Sherlock estimates at 150 guineas each. By modern calculations, the horses would be worth more than £30,000 (approximately $40,000) each.

Guineas were used in England until about 1816. Luxury items were often priced in guineas, and as we can see, Holmes frames the cost of the opulent horse-drawn carriage in guineas. This, along with his dress, shows that the visitor is extremely wealthy:

Heavy bands of astrakhan were slashed across the sleeves and fronts of his double-breasted coat, while the deep blue cloak which was thrown over his shoulders was lined with flame-coloured silk and secured at the neck with a brooch which consisted of a single flaming beryl. Boots which extended halfway up his calves, and which were trimmed at the tops with rich brown fur, completed the impression of barbaric opulence which was suggested by his whole appearance.

Although the visitor introduces himself as a Bohemian nobleman, his manner, dress, and mode of transportation speak to his royal heritage. Thus, Holmes deduces that his visitor is actually the king of Bohemia.

After Holmes unveils the truth, the visitor admits that he is, indeed, the king. The latter reveals that he must deal with the matter himself so as not to put himself in the power of another.

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