One of my favorite things that we’ve been able to learn about this semester has been the different mathematicians that we’ve studied. It fascinates me to hear of their interaction with each other and how they affected one another’s work. As much as I love the math, the historical aspect is something that I had never heard and something that I love learning about. One of those mathematicians that I wanted to know more about was János Bolyai. Of him Gauss would say, “*I regard this young geometer Bolyai as a genius of the first order.*” Coming from someone like Gauss, that is quite the compliment. I wanted to know what made Gauss say that about such a young mathematician.

János Bolyai was born in Kolozsvár (which is now the city of Cluj in Romania) to Zsuzsanna Benkö and Farkas Bolyai, who was also a great mathematician, physicist and chemist at the Calvinist College. Like many fathers, Farkas wanted his son to follow in his footsteps and perhaps to achieve more than even Farkas himself had achieved in the field. So, he raised János with that goal in mind. However, Farkas was a firm believer that a strong body would lead to a strong mind, so in János’s younger years, most of the attention was spent developing his physical body. (O’Connor & Robertson, 2004)

János quickly became a child prodigy. According to Barna Szénássy in History of Mathematics in Hungary until the 20^{th} Century, “… *when he was four he could distinguish certain geometrical figures, knew about the sine function, and could identify the best known constellations. By the time he was five [he] had learnt, practically by himself, to read. He was well above the average at learning languages and music. At the age of seven he took up playing the violin and made such good progress that he was soon playing difficult concert pieces.” (Szenassy, 1992)*. Bolyai’s childhood and adolescence were fascinating. His father wanted to send him to live with Gauss as a student in order to accelerate his mathematical education, but Gauss would not agree to it. Because the Bolyai family didn’t have the financial assets to send János to an expensive university, they made the decision to send him to the Royal Academy of Engineering at Vienna to study military engineering. He truly was a “jack of all trades.” He finished the seven year engineering program in just four years, became an excellent sportsman and even performed as a violinist in Vienna. He was in the military for eleven years, where he became known as the greatest swordsman and dancer in the Austro-Hungarian Imperial Army. (O’Connor & Robertson, 2004) It wasn’t until 1820 that he began intense study on Euclid’s parallel postulate and the development of hyperbolic geometry. One of János Bolyai’s most recognized quotations comes from a letter that he wrote to his father when he said that he had, “*created a new, another world out of nothing*.”

The story is well-known of the publication of Bolyai’s work on hyperbolic geometry. During János’s military service, his father read the mathematical work that his son had sent him previously and then went to where János was stationed. Farkas then encouraged his son to publish his work. János later said, “*Had my father not happened to urge or even force me at Marosvásárhely, on my way to duty in Lemberg, to immediately put things to paper, possibly the contents of the Appendix would never have seen the light of day.” *When Farkas sent a copy of his son’s work to his old friend, Gauss, Gauss responded by saying, “*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.”*

Bolyai’s work on the parallel axiom led to the development of what would be known as a “pseudosphere,” which is an object that extends infinitely, but has a finite volume. This object was created by Beltrami many years later, but now is seen as an embodiment of hyperbolic geometry.

The story of János Bolyai ends as a sad one. He did not manage his money very well, gave very little care or attention to the family estate he had inherited, and finally left his wife and children. Years after his work on hyperbolic geometry, he found the works of another geometer named Lobachevsky, who he thought was fictional; a cover that Gauss had created in order to steal his work on hyperbolic geometry. He quit working on mathematics entirely and focused on “a theory of all knowledge.” (O’Connor & Robertson, 2004) Although he may not have felt like he received the credit that he deserved for his work, János Bolyai was indeed, as Gauss called him, “*a genius of the first order*.” He gave the world of mathematics a new way of understanding the concept of parallelism and the way in which mathematics relates to our natural world.

**Editor’s note: The portrait here, which also appears on postage stamps honoring János Bolyai, has long been associated with the mathematician but is not authentic. For more information, see “The Real Face of János Bolyai” by Tamás Dénes.*

# Works Cited

O’Connor, J., & Robertson, E. (2004, March). *János Bolyai.* Retrieved from MacTutor History of Mathematics: http://www-history.mcs.st-and.ac.uk/Biographies/Bolyai.html

Szenassy, B. (1992). *History of Mathematics in Hungary until the 20th Century.* New York: Springer-Verlag Berlin Heidelberg.