Holmes compares Watson's accounts of their adventures as like working a love story into the Fifth Proposition of Euclid. This is the theorem that the angles opposite the equal sides of an isosceles triangle are themselves equal.
Small identifies the penal colony as being on "Blair Island." In fact the location was in Port Blair, on Chatham Island. The British operated a naval base and penal colony there from 1789-91. The colony was moved to Port Cornwallis on Great Andaman, and abandoned in 1796. The colony was then re-established at Port Blair in 1858, to handle prisoners convicted of offenses during the Indian Mutiny of 1857. Port Blair is now the capital of the Andaman and Nicobar Islands, a "union territory" of India.