While the very formidable Carl Friedrich Gauss had explored the binomial theorem as a teenager and made a name for himself 
in the international astronomical community by his accurate predictions of the orbit of the newly-observed asteroid Ceres, there are
other indications that Arthur Conan Doyle might have had someone else in mind.  A possible candidate is Simon Newcomb (1835-1909)
The Binomial Theorem has a long history going back to Euclid in the 4th century BC and Pingala in India in the 3rd century BC.
In the 10th century Indian mathematician Halayudha wrote a commentary on Pingala and the 10th century Persian mathematician
Abu Bakr ibn Muhammad ibn al Husayn al-Karaji also wrote at length on what became known as Pascal's Triangle. From China there  
was similar work by Yang Hui (1238-1298) as well as his predecessor Jia Xian (1010-1070). Blaise Pascal (1623-1662), despite poor
health for many years, managed to collect and extend many results in 1658, but it was Newton's work in the 1660s that really made the
binomial theorem a powerful tool. Newcomb's own work on the Binomial Theorem was not notable, although he did make a great many
contributions to astronomy. Of interest in the context of BEISA is the use of older French observations of planetary positions by
Newcomb to correct  marine navigation tables (based on smaller samples and fewer years). It is not obvious why Doyle chose the
Binomial Theorem or asteroid orbital calculations as high water marks for Moriarty.
With considerable regret, we must go further back in time to 1810. Alas, not to watch the Feanorian mind of Gauss at work, but rather
to confront a mystery worthy of Sherlock Holmes.
© 2018 Peter F. Zoll. All rights reserved.
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