Couldn't Sherlock just verify his choice of the bottle by the chemical analysis (Study in Pink)?
In "Study in Pink", the killer wants Sherlock to play the game with the pills, betting on the fact that for Sherlock being right is even more valuable that being alive. The killer uses this incentive to force Sherlock to swallow the pills from the bottle Sherlock has chosen. Although Sherlock may indeed value his life less than his rightfulness, I don't think he really wants to bet his life in case there are other ways to check whether he was right or not.
For example, he could call the police to catch the killer, collect both bottles remembering his choice and check the pills say with chemical analysis or mice in the worst case. Unless I'm missing something, by no means this is something impossible. Instead, Sherlock chooses to risk his life. The only reason for the latter that comes to my mind is: Sherlock shows off that he is so sure that he doesn't think he risks anything. However, this seems to be a pretty weak reason - especially taking into account that the killer says that there is somebody beyond him (Moriarty), and hence why would Sherlock show off himself in front of this killer, knowing that the latter is just a pawn of Moriarty.
Best Answer
Sherlock wanted to play the game and win the game. He wanted to show the cabbie he could beat him and was prepared to bet his life he was the smarter of the two.
Refusing to drink from the bottle would result him having to tolerate the cabbie smirking and saying "But you weren't sure, were you?" that would have haunted Holmes.
How many times does Holmes put his life in danger trusting that his superior intellect will get him out?! At times his life seems to almost be a quest to prove his superiority over everyone.
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Answer 2
Sherlock with his curious mind would never have found about how he lured his victims and how he made me them take the pill because the cabbie remarks to him at the door that if he calls the cops he will not run instead surrender but will not reveal to Sherlock how he made them take the pill.
At that point it was far fetched for Sherlock to assume that the cabbie would've had the pills on him when he came to pick him, also you're completely underestimating the analytical powers of the cabbie he is just as intelligent as Sherlock is and is fairly good with deductions himself I don't think he would've allowed for the scenario to let sherlock or the cops to have the bottles for them to analyze even. Quite simply he would have confessed and let them scratch their heads out as to how he made his victims take the pills.
Finally Sherlock only knows about Moriarity only when the cabbie dies and he had no idea at that time as to if that Moriarity person held an entire criminal network web at his sleeve. He was only following on his hunch and curiosity to find the drive and motivation behind the cabbie's mind to drive him to commit all those murders.
Answer 3
No. The contents of each bottle were identical in every way possible. There is a theory going around that each of the victims had trouble swallowing the pills and therefore asked for a glass of water. The cabby must've actually put the poison in the water. And he must've just swallowed the pill straight down. The pills apparently weren't the poison. The water was.
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