Guest post: Mathematical Allusions in Sherlock Holmes

Various versions of Holmes-from ‘Sherlock’ (Cumberbatch), ‘Mr Holmes’ (McKellen), released today, and ‘Sherlock Holmes’ (Rathbourne). Images from Wikimedia Commons, Youtube, and http://www.epicreads.com/ respectively

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Sherlock Holmes, the brainchild of Sir Arthur Conan Doyle (1859-1930), is arguably the most well-known fictional detective of all time. He is most famous for his outstanding skill of deduction. But the topic of this post is not about his deduction skill. Instead, I am going to outline some allusions to mathematics found scattered throughout the Holmes saga. References to logic are omitted, being too numerous to include.

Similar Triangles

In the short story The Musgrave Ritual, Holmes locates the would-be position, at a given time of day, of the tip of the shadow of a long-ago felled 64 foot elm tree. Setting up two lengths of a fishing rod, which came to just 6 feet, at the spot of the former elm tree, Holmes found the shadow cast by the rod to be 9 feet. Therefore a tree of 64 ft would throw a shadow of 96 feet, along the line of the fishing pole’s shadow.

Conclusions of a trained observer

In Chapter 2 of A Study in Scarlet, we read that Holmes once wrote an article entitled “The Book of Life”, in which he claimed that the conclusions of one trained to observation and analysis would be “as infallible as so many propositions of Euclid. So startling would his results appear to the uninitiated that until they learned the processes by which he had arrived at them they would well consider him a necromancer.”

An Elopement in Euclid’s Fifth Proposition

In Chapter 2 of The Sign of Four, acknowledging that he had perused the account of his solution of the Jefferson Hope case as narrated by Watson in A Study in Scarlet, Holmes deflates his friend by remarking, “I glanced over it. Honestly, I cannot congratulate you upon it. Detection is, or ought to be, an exact science and should be treated in the same cold and unemotional manner. You have attempted to tinge it with romanticism, which produces the same effect as if you worked a love-story or an elopement into the fifth proposition of Euclid.” [Note: The fifth proposition of Euclid, which proves that the base angles of an isosceles triangle are equal, became known as the ‘pons asinorum’, or ‘donkeys’ bridge’, a reference to the bridge-like appearance of the figure accompanying the proposition and the fact that many beginners experience difficulty in “getting over” it.]

The Rule of Three

There is another mathematical reference in The Sign of Four, this one in Chapter 6. A small barefoot Andaman Islander named Tonga inadvertently stepped into a puddle of creosote, thus rendering it an easy task to track him down. Holmes comments, “I know a dog that would follow that scent to the world’s end. If a pack can track a trailed herring across a shire, how far can a specially trained hound follow so pungent a smell as this? It sounds like a sum in the rule of three.” [Noted: The rule of three states the method of finding the fourth term, ‘x’, of a proportion a : b = c : x, where a, b, c are known.]

Professor Moriarty and the Binomial Theorem

In describing, in “The Final Problem,” his great arch enemy Professor James Moriarty, Holmes says, “His career has been an extraordinary one. He is a man of good birth and excellent education, endowed with a phenomenal mathematical faculty. At the age of twenty-one he wrote a treatise on the binomial theorem, which has had a European vogue. On the strength of it he won the mathematical chair at one of our small universities, and had, to all appearance, a most brilliant career before him.” [Note: Many eminent names in mathematics, Isaac Newton among them, have become associated with the binomial theorem.]