All day I turned these facts over in my mind, endeavouring to hit some theory which could reconcile them all, and to find that line of least resistance which my poor friend had declared to be the starting-point of every investigation. (Empty House.1.7)
Watson describes Holmes's scientific methods here as he tries to follow his friends reasoning to solve a puzzle. Holmes seems to be a fan of Occam's Razor here, or the idea that the simplest solution is usually the right one.
"I don't think I could have done it in cold blood. It was a hundred times more difficult than getting up. But I had no time to think of the danger [....]" (Empty House.1.33)
This is one of the very few instances where Holmes doesn't actually praise cold detachment and logic. Instead, Holmes notes that his adrenaline and his panic, an emotional response, saved his life in a threatening situation.
"I have a theory that the individual represents in his development the whole procession of his ancestors, and that such a sudden turn to good or evil stands for some strong influence which came into the line of his pedigree. The person becomes, as it were, the epitome of the history of his own family."
"It is surely rather fanciful."
"Well, I don't insist upon it." (Empty House.2.19-21)
Holmes's fascinating theory of heritage and genetics is in line with a lot of the scientific theories emerging in the 1890s regarding genetics, evolution, family traits, and race. This idea of someone being the culmination of their family's history also implies that criminals are always born in the past rather than made in the present, which seems to not jive with Holmes's views on talent and choice. How do you reconcile these ideas?
"Ah! my dear Watson, there we come into those realms of conjecture, where the most logical mind may be at fault. Each may form his own hypothesis upon the present evidence, and yours is as likely to be correct as mine." (Empty House.2.23)
Even when Holmes is departing from the "realm of logic" he still uses and relies on the scientific method, with his reference to a hypothesis. It's notable that Holmes is talking about determining human motives here. Human behavior often departs from logic.
"That it was written in a train. The good writing represents stations, the bad writing movement, and the very bad writing passing over points. A scientific expert would pronounce at once that this was drawn up on a suburban line [....]" (Norwood Builder.61)
Holmes's use of handwriting analysis points to how forensic science was evolving and becoming more widespread and accepted in this period.
"I am a practical man, Mr. Holmes, and when I have got my evidence I come to my conclusions." (Norwood Builder.143)
Though it seems like Holmes would be down with Lestrade's statement here, Holmes is actually a big fan of theory and of never taking evidence at face value. Evidence has to be picked apart and turned around before it can produce any sort of conclusion. And Holmes sometimes works backwards too, theorizing first and then proving or disproving the theory based on evidence.
We continued our systematic survey of the edge of the sodden portion of the moor, and soon our perseverance was gloriously rewarded. (Priory School.152)
Watson's words here, like systematic and perseverance, highlight the scientific process of Holmes's criminal investigations.
"Have you tried to draw a harpoon through a body? No? Tut, tut, my dear sir, you must really pay attention to these details. My friend Watson could tell you that I spent a whole morning in that exercise. It is no easy matter, and requires a strong and practiced arm." (Black Peter.75)
Holmes's experiment with harpooning a dead pig emphasizes his very hands on approach to crime-solving. As a scientist, Holmes must test and question and prove or disprove everything.
"This is a first-class, up-to-date burgling kit, with nickle-plated jemmy, diamond-tipped glass-cutter, adaptable keys, and every modern improvement which the march of civilization demands." (Milverton.1.78)
Holmes's even uses science and gadgets to commit a burglary. His reference to the "march of civilization" also gives us some insight into the important role that science, and scientific progress, played in late Victorian society.
"Your fatal habit of looking at everything from the point of view of a story instead of as a scientific exercise has ruined what might have been an instructive and even classical series of demonstrations." (Abbey Grange.6)
This passage really highlights the personality differences and differing world views of Holmes the scientist and Watson the writer.