In Chapter 1, the reader meets the detective Sherlock Holmes and the unconventional—and highly principled—methods that he employs in his detective work. Holmes explains his philosophy of "detection" through simile and allusion:
'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 much the same effect as if you worked a love-story or an elopement into the fifth proposition of Euclid.'
The "fifth proposition of Euclid" is an allusion to the Euclidean geometry theory of the Ancient Greek mathematician Euclid. This particular proposition concerns how to determine the relative angles of intersecting lines. By asserting that detective work should be "cold and unemotional," Holmes compares the consideration of emotion in the act of detection to the romanticization of geometry—both, Holmes argues, would be equally nonsensical.
This is an early passage that speaks to the constant tension between the rational and the emotional throughout The Sign of the Four. While Holmes is dedicated to rationalism to the point of austerity, Watson is a highly emotional character who often lets himself get carried away by fancy. The two characters thereby function as foils to each other, drawing out the other's inherent qualities in greater intensity through relief. In that respect, Watson's role as narrator of the novel is chiefly responsible for highlighting Holmes' dogged commitment to reason above all else.
Doyle wrote The Sign of the Four at the height of the Victorian period, at the end of the 19th century, in the midst of a scientific revolution that presented rational analysis and scientific reasoning as possible solutions for life's biggest problems. Holmes is a sort of avatar for this attitude and a spokesperson for the beauty of an "exact science."
