Bolyai was born in Bolya, a village near Hermannstadt, Grand Principality of Transylvania (now Buia, Sibiu County, Romania).

This meant that Bolyai was now treated as a member of one of the leading families in the country, and he became not only a tutor but a real friend to the count's son.

Bolyai, however, decided to go abroad with Simon Kemény on an educational trip in 1796 and began to study mathematics systematically at German universities first at Jena and then at Göttingen.

Soon thereafter he accepted a teaching position for mathematics and sciences at the Calvinist College in Marosvásárhely (today Târgu-Mureş), where he spent the rest of his life.

His main work, Tentamen juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi (An Attempt to Introduce Studious Youths to the Elements of Pure Mathematics; 1832),[2] was an attempt at a rigorous and systematic foundation of geometry, arithmetic, algebra and analysis.