Janos Bolyai

1802-1860

Hungarian Mathematician

Janos Bolyai was among the founders of non-Euclidean geometry. Non-Euclidean geometry concerns itself with internally consistent mathematical systems in which Euclid's parallel axiom does not apply. The parallel axiom states that only one line can be drawn parallel to a given line through a point not on the line. Non-Euclidean geometry later gained importance as the mathematical foundation of the general theory of relativity.

Bolyai was born in Kolozsvar, Hungary (now Cluj, Romania), on December 15, 1802. His father, Farkas, also called Wolfgang, was a mathematician and a lifelong friend of Carl Friedrich Gauss (1777-1855). By the time he was 13 years old, Janos Bolyai had been taught geometry and calculus by his father. He was also a gifted violinist, and later became an accomplished swordsman. Bolyai continued his education at the Royal Engineering College in Vienna from 1818 through 1822, and served as an officer in the engineering corps of the Austrian army.

Farkas Bolyai, inspired by his experience of tutoring his son, published his principal work, Tentamen Juventutem Studiosum in Elementa Matheseos Purae Introducendi ("An Attempt to Introduce Studious Youth to the Elements of Pure Mathematics") in 1832. This treatise on the foundations of mathematics included a rigorous presentation of geometry.

The elder Bolyai had long been almost obsessed with proving the parallel axiom. His son became absorbed in this quest as well, in spite of his father's warnings. "I entreat you, leave the science of parallels alone," wrote Farkas Bolyai in a letter to Janos. "I have traveled past all reefs of the infernal Dead Sea and have always come back with a broken mast and torn sail."

In 1820, Janos Bolyai convinced himself that a proof of the parallel axiom was impossible. This led him to try to construct a complete and consistent geometry that did not depend on it. In 1823, he drafted a presentation of such a system, and sent it to his father, exclaiming in his youthful exuberance: "I have created a new universe from nothing!" The 24-page work was published as an appendix to the Tentamen, entitled "Appendix Spatii Absolute Veram Exhibens," or "Appendix Explaining the Absolutely True Science of Space." This appendix was the only work of Janos Bolyai that was published during his lifetime, although he also applied himself to the field of complex variables.

When Farkas Bolyai sent the Tentamen to his friend Gauss, proudly pointing to the appendix written by his son, Gauss replied that he himself had had the same ideas decades earlier. In fact this assertion is confirmed in his private papers; he had even written of his thoughts to Farkas Bolyai. Gauss, fearing ridicule, had never published his revolutionary ideas on the subject, so he had no valid claim of precedence, nor had he systematized them to the extent that Janos Bolyai had.

Farkas Bolyai interpreted Gauss's remark that his son's ideas "coincide almost entirely with my meditations which occupied my mind for the last thirty or thirty-five years" as high praise. Yet Gauss's claim of precedence was a profound shock to Janos Bolyai, one he never truly overcame. In a further blow, he later discovered that the Russian mathematician Nikolai Lobachevsky (1792-1856) had also been working independently on non-Euclidean geometry, and had published a paper very similar to his three years earlier.

The appendix to the Tentamen gained little attention, and the unhappy Bolyai remained largely unknown during his lifetime. Later, however, George Bruce Halsted called his work "the most extraordinary two dozen pages in the whole history of thought."

Gauss was acquainted with Lobachevsky as well as the Bolyais, but never bothered to introduce the two younger mathematicians, and there is no evidence that Lobachevsky knew of Bolyai at all. Bolyai, although too retiring to make Lobachevsky aware of his own work, did read a German translation of one of Lobachevsky's books in 1848, and praised it warmly. Janos Bolyai died on January 27, 1860, in Marosvasarhely, Hungary (now Targu Mures, Romania).

SHERRI CHASIN CALVO