Bugaev was born in Georgia, Russian Empire into a somewhat unstable family, and at the age of ten young Nikolai was sent to Moscow to find his own means of obtaining an education. He succeeded, graduating in 1859 from Moscow University, where he majored in mathematics and physics. He went on to study engineering, but in 1863 wrote a Master's thesis on the convergence of infinite series. This document was sufficiently impressive to win him a place studying under Karl Weierstrass and Ernst Kummer in Berlin. He also spent some time in Paris studying under Joseph Liouville. He earned his doctoral degree in 1866 and returned to Moscow, where he taught for the remainder of his career. Some of his most influential papers offered proofs of previously unproven assertions of Liouville, but his most original work centered around the development of formal analogies between arithmetic and analyticoperations. Bugaev was the active member of the Moscow Mathematical Society and the president of it. He also wrote influential philosophical essays in which he trumpeted the virtues of mathematical analysis and decried the influence of geometry and probability. Many feel he is largely responsible for the pronounced predilection towards "hard analysis" which is characteristic of so much of the bestRussian mathematics. Through Bugaev's star student, Dmitri Egorov, many famous Russian mathematicians, such as Andrei Kolmogorov and Nikolai Luzin, directly "descend" from Bugaev—and thus from the Prince of Mathematicians, Carl Friedrich Gauss. Nikolai Bugaev was also a talented chess player; some of his games are available at . Bugaev was a memorable "character" whose life was touched by scandal. He was not, it is said, much admired for his looks, but his wife was brilliant, beautiful, and rich, and the Bugaevs were socially prominent. Their mathematically, musically, and artistically talented son, Boris Nikolaevich Bugaev, went on to adopt the pseudonym Andrei Bely, under which name he helped found the Symbolist movement. Professor Korobkin, the main character of Bely's innovative novel Moscow, was inspired by Nikolay Bugayev. In view of his father's prejudices, Boris Bugaev was fascinated by probability and particularly by the notion of entropy, which is mentioned in several of his novels and poems.
