"His ignorance was almost as remarkable as his brilliance.”
This quote describes Holmes. Holmes possesses brilliant powers of deduction. However, there are many things that he knows nothing of. In fact, Holmes was unaware that the planets orbited around the sun. Watson was astounded that Holmes could identify cigars by their ashes, but was oblivious about something as commonplace as planetary theory.
“It is a capital mistake to theorize before you have all the information. It biases the judgement.”
This quote describes the foundation of Holmes’ deductions. He relies upon having all of the possible information about something before making conjectures about it. By knowing all the details, he can make extended, accurate judgement about things, proving his powers of deduction.
“You see but you do not observe.”
Here, Holmes explains more of how he comes to his deductions. He is not just passively seeing things, instead he is actively examining and observing his surroundings. This helps him gather evidence to deduce conclusions from.
“To Sherlock Holmes she is always the woman. I have seldom heard him mention her under any other name.”
This quote is about Irene Adler. Irene Adler is arguably the most famous female character from any of Doyle’s works, even though she only appeared in one story ("A Scandal in Bohemia"). She is one of few people who has outwitted Holmes, and for this she has earned his complete admiration. She is referenced in other stories, including “His Last Bow” and “The Five Orange Pips.”
“You know my method. It is founded on the observation of trifles.”
Here is another example of Holmes explaining his deductions. He notices everything around him, and believes that even the smallest clues are still clues that can help make a correct conclusion, solving the mystery. This quote is similar to one from “The Man with the Twisted Lip": “It is of course, a trifle, but there is nothing so important as trifles.”
“It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth.”
Here, Holmes describes how he comes to his final conclusions. After he has observed all he can, gathered all the evidence he can, thought about the puzzle all he can, he can make several theories. Next, by eliminating the ones that cannot be possible, he comes to the correct conclusion.
"'The law cannot, as you say, touch you,' said Holmes… 'yet there was never a man who deserved punishment more. If the young lady has a brother or a friend, he ought to lay a whip across your shoulders… it is not part of my duties to my client, but here's a hunting crop handy, and I think I shall just treat myself to—"
Holmes threatens to whip James Windibank, stepfather of Mary Sutherland, for his impersonating her lover in order to prevent her from marrying. This is significant as one of the few scenes in which Holmes takes a dramatic moral stance and even goes so far as to personally threaten the villain of the story with violence.
"Your experience has been a most entertaining one."
Holmes often says such things to clients in the process of listening to their stories; for, as he explains to Watson on more than one occasion, he wouldn't take cases unless they offered him some sort of intriguing puzzle to solve. In this case, he is speaking to Jabez Wilson.
"My dear fellow, is it possible you do not see how strongly it bears upon the case?"
Holmes frequently directs such quasi-comical outbursts of disbelief at Watson for not picking up on the significance of some clue both of them are aware of; naturally, he always continues by explaining.
"I would have spoken now had it not been for my dear girl. It would break her heart—it will break her heart when she hears that I am arrested."
John Turner, when confronted by Holmes about the Charles McCarthy, pleads for Holmes' sympathy by invoking his care for his daughter.
