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János Bolyai wrote a treatise on a complete system of non-Euclidean geometry.Bolyai was born in Hungary (although the town Kolozsvár is now renamed Cluj and is in Romania). By the time Bolyai was 13, he had mastered the calculus and other forms of analytical mechanics, his father Farkas Bolyai giving him instruction. Bolyai also became an accomplished violinist and he performed in Vienna. He studied at the Royal Engineering College in Vienna from 1818 to 1822. Immediately after this he joined the army engineering corps in which he spent 11 years. He was the best swordsman and dancer in the Austro-Hungarian Imperial Army.
He neither smoked nor drank, not even coffee, and at the age of 23 he was reported to still retain the modesty of innocence . He was an accomplished linguist speaking nine foreign languages including Chinese and Tibetan.
Between 1820 and 1823 he prepared a treatise on a complete system of non-Euclidean geometry. Before the work was published, however, Bolyai discovered that Gauss had anticipated much of his work. Although Gauss had never published his work in this area, probably because he did not feel confident to publish, this was a severe blow to Bolyai. However Bolyai's work was published in 1832 as an Appendix to an essay by his father.
Gauss, on reading the Appendix, wrote to a friend saying
I regard this young geometer Bolyai as a genius of the first order .
To Bolyai's father he wrote
To praise it would amount to praising myself. For the entire content of the work ... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years .
In 1848 Bolyai discovered that Lobachevsky had published a similar piece of work in 1829.
In addition to his work in geometry, Bolyai developed a rigorous geometric concept of complex numbers as ordered pairs of real numbers.
Bolyai was plagued with a fever which frequently disabled him and in 1833 he was pensioned off from his army career. Although he never published more than the 24 pages of the Appendix he left more than 20000 pages of manuscript of mathematical work when he died. These are now in the Bolyai-Teleki library in Tirgu-Mures.
In 1945 a university in Cluj was named after him the name was later removed by the Roumanian authorities.
The picture above is taken from a stamp issued by the Hungarian Post Office to celebrate the centenary of his death.
References (21 books/articles)
References elsewhere in this archive:
Tell me about Bolyai's work on non-Euclidean geometry and in the development of group theory
Tell me about his part in the history of mathematics
There is a Crater Bolyai on the moon (named after this mathematician). You can see a list of lunar features named after mathematicians.