Contributed by The Wendell Urth Society: I recommend very highly "Sherlock Holmes as Chemist" by Isaac Asimov from his book The Roving Mind published 1983,page 127. This is a brilliant essay on Arthur Conan Doyle's numerous bloopers in mathematics, astronomy, and chemistry in the Sherlock Holmes stories. Asimov gives devastating examples from nine Holmes stories: The Adventure of the Blue Carbuncle, The Adventure of the Copper Beeches, The Adventure of the Engineer's Thumb, The Adventure of Shoscombe Old Place, A Case of Identity, The Final Problem, The Sign of the Four, A Study in Scarlet, The Valley of Fear.
Asimov writes: In "The Final Problem," Holmes says of [the arch-villain James Moriarty] "At the age of twenty-one he wrote a treatise upon the Binomial Theorem, which has had a European vogue."
Moriarty was 21 years old in 1865 (it is estimated), but forty years earlier than that the Norwegian mathematician Niels Henrik Abel had fully worked out the last detail of the mathematical subject known as "the binomial theorem," leaving Moriarty nothing to do on the matter. It was completely solved and has not advanced beyond Abel to this day.