**János Bolyai** was a pioneer of non-Euclidean geometry.

- One might suppose that this would mean that János's education was put first in the Bolyai household, but this was not so for Farkas believed that a sound mind could only achieve great things if it was in a sound healthy body, so in his early years most attention was paid to János's physical development.
- It is important to understand that although Farkas had a lecturing post he was not well paid and even although he earned extra money from a variety of different sources, János was still brought up in poor financial circumstances.
- In 1816 Farkas wrote to his friend Gauss asking him if he would let János live with him and take him on as a pupil so that he might receive the best possible mathematical education.
- Certainly it would have been a wonderful education for János and it is interesting to speculate what benefits might have come to the world of mathematics if he had accepted the plan.
- When János graduated from Marosvásárhely College on 30 June 1817 it was not clear how he might obtain a good mathematical education.
- The decision that János would study military engineering at the Academy of Engineering at Vienna was not taken without a lot of heartache and soul-searching but in the end this route was chosen as the least bad of the options.
- János remained for one further year at Marosvásárhely College attempting to gain entry to the Academy at Vienna at the highest possible level which he achieved.
- Of course he had received military training during his time in Vienna, for the summer months were devoted to this, but Bolyai's nature did not allow him to accept easily the strict military discipline.
- However Farkas Bolyai did not react enthusiastically which clearly disappointed János.
- Bolyai was posted to Arad in 1826 and there he found that Captain Wolther von Eckwehr, one of his old teachers of mathematics from the Academy in Vienna, was also stationed.
- Bolyai gave him a draft of the materials which he was writing on the theory of geometry, probably because he hoped for some constructive comments from him.
- However it would appear from Bolyai's later writings that he got nothing from von Eckwehr, in particular he never received the manuscript back.
- In 1830 Bolyai learnt that he was to be sent to a posting in Lemberg.
- Bolyai did not remain long in Lemberg for in 1832 he was posted to Olmütz where he was now a captain.
- The discovery that Gauss had anticipated much of his work, however, greatly upset Bolyai who took it as a severe blow.
- They did not marry, however, since the law insisted on money being deposited before a marriage could take place and Bolyai could not afford the money.
- Certainly Bolyai continued to develop mathematical theories while he lived at Domáld, but being isolated from the rest of the world of mathematics much of what he attempted was of little value.
- In addition to his work on geometry, Bolyai developed a rigorous geometric concept of complex numbers as ordered pairs of real numbers.
- János's paper was called Responsio and it was written to answer the question of whether the imaginary quantities used in geometry could be constructed.
- Bolyai chose to argue that the question was wrongly formulated, not perhaps the best way to find favour with judges.
- In 1846 Bolyai moved from the estate at Domáld to Marosvásárhely.
- In 1848 Bolyai discovered that Lobachevsky had published a similar piece of work in 1829.
- They express the thoughts and anxieties of János provoked by the perusal of the book.
- In spite of his mental agitation amidst which János put observations to paper, he preserved enough objectivity to highly appreciate the work of his rival.
- These are now in the Bolyai-Teleki library in Târgu-Mureș.
- The two universities combined to become the Babeș-Bolyai University in 1959.

Born 15 December 1802, Kolozsvár, Hungary (now Cluj, Romania). Died 27 January 1860, Marosvásárhely, Hungary (now Târgu-Mureş, Romania).

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Algebra, Geometry, Group Theory, Origin Romania

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive