Janos Bolyai

“Out of nothing I have created a strange new universe”

- Janos Bolyai, writing to his father upon discovering non-Euclidean Geometry

Janos Bolyai was born Dec 12, 1802 in Kolozsvar, Hungary to mother Zsuzanna Benko and father Farkas Bolyai. The elder Bolyai had studied at Gottingen with fellow classmate Carl Gauss, and the two became lifelong friends. In 1804, the Bolyai family moved to Maros-Vasarhely, where Farkas had accepted the post of professor of mathematics, which he would hold until 1851. Over those years, Farkas Bolyai formulated several attempts, all unsuccessful, to prove Euclid's parallel postulate.

As a child, Janos Bolyai displayed incredible competence in music, linguistics, astronomy and, of course, mathematics. Despite the prestigious post held by his father, the family struggled financially. After Bolyai had mastered calculus by the age of 13, his father asked Gauss if he would accept his son into his household as a student. Unfortunately Gauss rejected, so instead Bolyai began studying at the Royal Engineering College at Vienna in 1818. He completed the seven year program in just four years then began his service in the army engineering corps.

Around 1820 Bolyai began to follow in his father's footsteps in tackling Euclid's parallel postulate. Upon hearing this, Farkas Bolyai pleaded with his son to abandon what he considered a fruitless endeavor. Despite his father's admonitions, he persisted and through the method of indirect proof focused on the results which would follow a negation of the postulate. From these results, the younger Bolyai developed a general geometry (or “Absolute Geometry” in his words) which would contain Euclidean geometry as a special case. In November of 1823, he sent his father a letter outlining this geometric system.

Farkas Bolyai had himself been busy working on what would be the most important work of his lifetime, the Tentamen – his presentation of the elements of mathematics. The younger Bolyai's findings were included as an appendix to the Tentamen, published in 1832. Gauss was sent a copy of the publication by Farkas Bolyai and after reading the Appendix, wrote back, saying “To praise it would amount to praising myself. For the entire content of the work ... coincides almost exactly with my own meditations”. Gauss indeed had arrived at conclusions similar to those of Janos Bolyai, as demonstrated in written to F. A. Taurinus. Meanwhile, in Kazan, Nikolai Lobachevsky was composing manuscripts also dealing with a geometry more general than Euclidean.

Apparently, the fact that Gauss had presaged many of his concepts greatly upset Janos Bolyai and affected his health. His physical condition weakened to the point that he was forced to retire from the military in 1833. After living for a short time with his father, he lived at the family estate in Domald. He became involved with Rozália Kibédi Orbán and they lived together at Domáld from 1834 to1852. They did not marry due to financial reasons but had two children together.

For the remainder of his life, Bolyai continued to develop mathematical theories but was never published again. In response to a call for papers by the Jablonowsky Society in Leipzig, he developed a geometric concept of complex numbers as ordered pairs of real numbers. When he died of pneumonia at the age of 57, he left behind tens of thousands of pages of mathematical writings. These are now in the Bolyai-Teleki library in Tirgu-Mures. In 1945 a university in Cluj was named after him but it was closed down by Ceaucescu's government in 1959. A crater on the moon is also named after him.

The significance of the work by Janos Bolyai in the development of non-Euclidean geometry, as well as that of Gauss and Lobachevsky, was not fully realized and appreciated until decades later. One would hope that Bolyai at least savored the fact that he possessed the conviction to publish his ideas which Gauss lacked.

References

D.M.Y. Somerville, The Elements of Non-Euclidean Geometry, (Dover Publications 1958)

Biography in MacTutor

Biography in Encyclopaedia Britannica

Selected sites related to Janos Bolyai

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bolyai.html

http://www.britannica.com/eb/article-9080524

http://library.thinkquest.org/22584/temh3019.htm

http://scienceworld.wolfram.com/biography/BolyaiJanos.html

Quotes

“I have traversed this bottomless night, which extinguished all light and joy in my life.”

- Farkas Bolyai to his son regarding Euclid's parallel postulate

“All that I have sent you previously is like a house of cards in comparison with a tower.”

- Jonas Bolyai to his father regarding his discovery of a new system of geometry