Editing János Bolyai (section)

==Career==
Bolyai became so obsessed with [[Euclid]]'s [[parallel postulate]] that his father, who had pursued the same subject for many years, wrote to him in 1820: "You must not attempt this approach to parallels. I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone...Learn from my example."<ref name=Ellenberg>{{cite book |last=Ellenberg |first=Jordan| author-link=Jordan Ellenberg |date=May 2014 |title=[[How Not to Be Wrong]] |location=[[New York City|New York, NY]] |publisher=[[Penguin Group]] |page=365 |isbn=978-0-14-312-753-6}}</ref>

János, however, persisted in his quest and eventually came to the conclusion that the postulate is independent of the other axioms of geometry and that different consistent geometries can be constructed on its negation. In 1823, he wrote to his father: "I have discovered such wonderful things that I was amazed...out of nothing I have created a strange new universe."<ref name="Ellenberg"/><ref name="Tóth_1965"/> Between 1820 and 1823 he had prepared a treatise on [[parallel lines]] that he called [[absolute geometry]]. Bolyai's work was published in 1832 as an appendix to a mathematics textbook by his father.

[[Carl Friedrich Gauss]], on reading the Appendix, wrote in a letter, "I regard this young [[geometer]] Bolyai as a genius of the first order."<ref name="St Andrews">{{MacTutor Biography|id=Bolyai}}</ref> To his old friend Farkas Bolyai, however, Gauss 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."<ref>[https://archive.org/details/briefwechselzwi00gausgoog/page/n146/mode/2up Letter from Gauss to Farkas Bolyai from 6 March 1832]</ref><ref name="Ellenberg"/><ref name="St Andrews"/><ref name="Tóth_1965"/> Indeed, Gauss had disclosed his discovery of a consistent non-Euclidean geometry in a letter in 1827, and in 1829 wrote that he feared backlash if he published about it.<ref>Greenberg 2008, p. 243-244</ref> János suspected that Gauss had been secretly informed about his discoveries by his father, causing a rift between him and his father.<ref>{{Cite journal |last=Osserman |first=Robert |date=2005 |title=Book Review: Non-Euclidean Geometry, and the Nature of Space. |url=https://www.ams.org/journals/notices/200509/rev-osserman.pdf |journal=Notices of the AMS |volume=52 |issue=9}}</ref> He later bitterly complained about Gauss's attitude.<ref>Greenberg 2008, p. 242</ref> 

In 1848 Bolyai learned that [[Nikolai Ivanovich Lobachevsky]] had published a piece of work similar to his appendix in 1829. Though Lobachevsky published his work a few years earlier than Bolyai, it contained only [[hyperbolic geometry]]. Working independently, Bolyai and Lobachevsky pioneered the investigation of [[non-Euclidean geometry]].

In addition to his work in geometry, Bolyai developed a rigorous geometric concept of [[complex numbers]] as ordered pairs of [[real numbers]]. Although he never published more than the 24 pages of the Appendix, he left more than 20,000 pages of mathematical manuscripts when he died. These can now be found in the [[Teleki Library|Teleki-Bolyai Library]] in [[Târgu Mureș]], where Bolyai died. His grave lies in the Lutheran Cemetery in Târgu Mureș.<ref>{{MacTutor|class=Extras|id=Bolyai_house_grave|title=Janos Bolyai's birth house and original grave}}</ref>