János Bolyai is remembered as the Hungarian mathematician whose work on non-Euclidean geometry was eclipsed by the work of Nikolai Lobachevsky in Russia and Karl Gauss in Germany. The son of Farkas (Wolfgang) and Susanna Arkos Bolyai, Bolyai was born in Kolosvar, Transylvania, Hungary, on December 15, 1802. Bolyai was education in Marosvasarhely at the Evangelical-Reformed College, where his father was a professor of mathematics, physics, and chemistry.

From childhood, Bolyai showed an aptitude for both mathematics and music; he was an accomplished violinist at an early age. His father--a notable mathematician in his own right who devoted considerable effort to trying to prove the Euclidean theory of parallels--wanted Bolyai to study mathematics in Germany with Gauss; instead, the young man enrolled in the imperial engineering academy in Vienna in 1818 and pursued a military education. However, he still shared his father's passion for proving the Euclidean...