Janos Bolyai is most noted for having authored a complete treatise on a complete system on non-Euclidean geometry.
Born on December 15, 1802 in present-day Romania (then Hungary). By the tender age of 13, Bolyai had developed a mastership of calculus and other sorts of analytical mathematics and mechanics. Mainly, Janos was taught by his father, Farkas Bolyai, who gave his son unparalleled instruction. Meanwhile, Janos became an equally talented violinist who performed in Vienna. Thus, Janos embarked on a study of engineering at the Royal Engineering College in Vienna from 1818 to 1822. As soon as Janos completed his studies, he joined the then Austro-Hungarian Imperial Army as an engineer and he was known as the best swordsman and dancer in the army. Notably, Janos adhered to a puritanical style of life and he did not smoke and did not drink. His genius lead him to learn nine languages including Tibetan.
Janos Bolyai's treatise was an immense milestone in non-Euclidean development. From 1820 to 1823, Janos authored a treatise on a full non-Euclidean system. Sadly, Janos discovered that his ideas had been anticipated by Gauss. Discouraged and dismayed, Bolyai did not publish his work. His father later published it as an appendix to one of his works. Gauss, however, regarded Janos with the greatest respect calling him a "genius of the first order".
Later, in 1848, Janos discovered that Lobachevsky had published an identical work in 1829. No matter, Janos Bolyai developed an intricate geometric concept of "complex numbers as ordered pairs of real numbers".
Tormented by a horendous fever, Bolyai was pensioned from the army in 1833 and died on January 27, 1860. In his honor exists the Bolyai Crater and the Bolyai-Teleki library in Tirgu-Mures.
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Picture of Bolyai from http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Bolyai.html, a great site.